### The Mizar article:

### Complex Numbers --- Basic Theorems

**by****Library Committee**

- Received April 10, 2003
Copyright (c) 2003 Association of Mizar Users

- MML identifier: XCMPLX_1
- [ MML identifier index ]

environ vocabulary ARYTM, ARYTM_1, RELAT_1, ARYTM_3, XCMPLX_0, COMPLEX1, OPPCAT_1; notation SUBSET_1, ORDINAL1, NUMBERS, ARYTM_0, XCMPLX_0; constructors ARYTM_0, XREAL_0, XCMPLX_0, ARYTM_3, XBOOLE_0; clusters NUMBERS, XREAL_0, XCMPLX_0, ARYTM_3, ZFMISC_1, XBOOLE_0; requirements SUBSET, BOOLE, NUMERALS, ARITHM; theorems XCMPLX_0, ARYTM_0; begin reserve a, b, c, d, e for complex number; :: '+' operation only theorem Th1: :: AXIOMS'13 a + (b + c) = (a + b) + c proof consider x1,x2,y1,y2 being Element of REAL such that A1: a = [*x1,x2*] and A2: b = [*y1,y2*] and A3: a+b = [*+(x1,y1),+(x2,y2)*] by XCMPLX_0:def 4; consider y3,y4,z1,z2 being Element of REAL such that A4: b = [*y3,y4*] and A5: c = [*z1,z2*] and A6: b+c = [*+(y3,z1),+(y4,z2)*] by XCMPLX_0:def 4; A7: y1 = y3 & y2 = y4 by A2,A4,ARYTM_0:12; consider x3,x4,yz1,yz2 being Element of REAL such that A8: a = [*x3,x4*] and A9: b+c = [*yz1,yz2*] and A10: a+(b+c) = [*+(x3,yz1),+(x4,yz2)*] by XCMPLX_0:def 4; A11: x1 = x3 & x2 = x4 by A1,A8,ARYTM_0:12; consider xy1,xy2,z3,z4 being Element of REAL such that A12: a+b = [*xy1,xy2*] and A13: c = [*z3,z4*] and A14: (a+b)+c = [*+(xy1,z3),+(xy2,z4)*] by XCMPLX_0:def 4; A15: z1 = z3 & z2 = z4 by A5,A13,ARYTM_0:12; A16: xy1 = +(x1,y1) & xy2 = +(x2,y2) by A3,A12,ARYTM_0:12; A17: yz1 = +(y1,z1) & yz2 = +(y2,z2) by A6,A7,A9,ARYTM_0:12; then +(x3,yz1) = +(xy1,z3) by A11,A15,A16,ARYTM_0:25; hence thesis by A10,A11,A14,A15,A16,A17,ARYTM_0:25; end; theorem Th2: :: REAL_1'10 a + c = b + c implies a = b proof assume a + c = b + c; then a + (c + -c) = b + c + -c by Th1; then a + 0 = b + c + -c by XCMPLX_0:def 6; then a = b + (c + -c) by Th1; then a = b + 0 by XCMPLX_0:def 6; hence thesis; end; :: '*' operation only Lm1: a-a=0 proof thus a-a = a+ -a by XCMPLX_0:def 8 .= 0 by XCMPLX_0:def 6; end; Lm2: a + b - c = a - c + b proof thus a + b - c = a + b + - c by XCMPLX_0:def 8 .= a + - c + b by Th1 .= a - c + b by XCMPLX_0:def 8; end; theorem :: INT_1'24 a = a + b implies b = 0 proof assume a = a + b; then 0 = a + b - a by Lm1 .= a - a + b by Lm2; then 0 = 0 + b by Lm1; hence thesis; end; :: using operation '*' Lm3: for x,y being Element of REAL holds opp +(x,y) = +(opp x, opp y) proof let x,y be Element of REAL; +(+(x,y),+(opp x, opp y)) = +(x,+(y,+(opp x, opp y))) by ARYTM_0:25 .= +(x,+(opp x,+(y, opp y))) by ARYTM_0:25 .= +(x,+(opp x,0)) by ARYTM_0:def 4 .= +(x,opp x) by ARYTM_0:13 .= 0 by ARYTM_0:def 4; hence thesis by ARYTM_0:def 4; end; theorem Th4: :: AXIOMS'16 a * (b * c) = (a * b) * c proof consider x1,x2,y1,y2 being Element of REAL such that A1: a = [*x1,x2*] and A2: b = [*y1,y2*] and A3: a*b = [* +(*(x1,y1),opp*(x2,y2)), +(*(x1,y2),*(x2,y1)) *] by XCMPLX_0:def 5; consider y3,y4,z1,z2 being Element of REAL such that A4: b = [*y3,y4*] and A5: c = [*z1,z2*] and A6: b*c = [* +(*(y3,z1),opp*(y4,z2)), +(*(y3,z2),*(y4,z1)) *] by XCMPLX_0:def 5; A7: y1 = y3 & y2 = y4 by A2,A4,ARYTM_0:12; consider x3,x4,yz1,yz2 being Element of REAL such that A8: a = [*x3,x4*] and A9: b*c = [*yz1,yz2*] and A10: a*(b*c) = [* +(*(x3,yz1),opp*(x4,yz2)), +(*(x3,yz2),*(x4,yz1)) *] by XCMPLX_0:def 5; A11: x1 = x3 & x2 = x4 by A1,A8,ARYTM_0:12; consider xy1,xy2,z3,z4 being Element of REAL such that A12: a*b = [*xy1,xy2*] and A13: c = [*z3,z4*] and A14: (a*b)*c = [* +(*(xy1,z3),opp*(xy2,z4)), +(*(xy1,z4),*(xy2,z3)) *] by XCMPLX_0:def 5; A15: z1 = z3 & z2 = z4 by A5,A13,ARYTM_0:12; A16: xy1 = +(*(x1,y1),opp*(x2,y2)) & xy2 = +(*(x1,y2),*(x2,y1)) by A3,A12,ARYTM_0:12; A17: yz1 = +(*(y3,z1),opp*(y4,z2)) & yz2 = +(*(y3,z2),*(y4,z1)) by A6,A9,ARYTM_0:12; +(*(opp x4,*(y3,z2)),*(opp x4,*(y4,z1))) = +(*(opp x4,*(y4,z1)),*(*(opp x2,y1),z4)) by A7,A11,A15,ARYTM_0:15 .= +(*(*(opp x2,y2),z3),*(*(opp x2,y1),z4)) by A7,A11,A15,ARYTM_0:15; then A18: +(*(x3,*(opp y4,z2)),+(*(opp x4,*(y3,z2)),*(opp x4,*(y4,z1)) )) = +(*(*(x1,opp y2),z4), +(*(*(opp x2,y2),z3),*(*(opp x2,y1),z4))) by A7,A11,A15,ARYTM_0:15 .= +(*(opp *(x1,y2),z4), +(*(*(opp x2,y2),z3),*(*(opp x2,y1),z4))) by ARYTM_0:17 .= +(*(*(opp x1,y2),z4), +(*(*(opp x2,y2),z3),*(*(opp x2,y1),z4))) by ARYTM_0:17 .= +(*(*(opp x2,y2),z3), +(*(*(opp x1,y2),z4),*(*(opp x2,y1),z4))) by ARYTM_0:25; A19: +(*(x3,yz1),opp*(x4,yz2)) = +(*(x3,yz1),*(opp x4,yz2)) by ARYTM_0:17 .= +(*(x3,+(*(y3,z1),*(opp y4,z2))),*(opp x4,yz2)) by A17,ARYTM_0:17 .= +(+(*(x3,*(y3,z1)),*(x3,*(opp y4,z2))), *(opp x4,+(*(y3,z2),*(y4,z1)))) by A17,ARYTM_0:16 .= +(+(*(x3,*(y3,z1)),*(x3,*(opp y4,z2))), +(*(opp x4,*(y3,z2)),*(opp x4,*(y4,z1)))) by ARYTM_0:16 .= +(*(x3,*(y3,z1)),+(*(*(opp x2,y2),z3), +(*(*(opp x1,y2),z4),*(*(opp x2,y1),z4)))) by A18,ARYTM_0:25 .= +(+(*(x3,*(y3,z1)),*(*(opp x2,y2),z3)), +(*(*(opp x1,y2),z4),*(*(opp x2,y1),z4))) by ARYTM_0:25 .= +(+(*(*(x1,y1),z3),*(*(opp x2,y2),z3)), +(*(*(opp x1,y2),z4),*(*(opp x2,y1),z4))) by A7,A11,A15,ARYTM_0:15 .= +(*(+(*(x1,y1),*(opp x2,y2)),z3), +(*(*(opp x1,y2),z4),*(*(opp x2,y1),z4))) by ARYTM_0:16 .= +(*(+(*(x1,y1),*(opp x2,y2)),z3), +(*(*(opp x1,y2),z4),*(opp *(x2,y1),z4))) by ARYTM_0:17 .= +(*(+(*(x1,y1),*(opp x2,y2)),z3), +(*(*(opp x1,y2),z4),opp *(*(x2,y1),z4))) by ARYTM_0:17 .= +(*(+(*(x1,y1),*(opp x2,y2)),z3), +(*(opp *(x1,y2),z4),opp *(*(x2,y1),z4))) by ARYTM_0:17 .= +(*(+(*(x1,y1),*(opp x2,y2)),z3), +(opp *(*(x1,y2),z4),opp *(*(x2,y1),z4))) by ARYTM_0:17 .= +(*(+(*(x1,y1),*(opp x2,y2)),z3), opp +(*(*(x1,y2),z4),*(*(x2,y1),z4))) by Lm3 .= +(*(+(*(x1,y1),*(opp x2,y2)),z3), opp*( +(*(x1,y2),*(x2,y1)),z4)) by ARYTM_0:16 .= +(*(+(*(x1,y1),*(opp x2,y2)),z3),*(opp xy2,z4)) by A16,ARYTM_0:17 .= +(*(xy1,z3),*(opp xy2,z4)) by A16,ARYTM_0:17 .= +(*(xy1,z3),opp*(xy2,z4)) by ARYTM_0:17; A20: +(*(opp*(x2,y2),z4),*(*(x2,y1),z3)) = +(opp*(*(x2,y2),z4),*(*(x2,y1),z3)) by ARYTM_0:17 .= +(*(x4,*(y3,z1)),opp*(*(x2,y2),z4)) by A7,A11,A15,ARYTM_0:15 .= +(*(x4,*(y3,z1)),opp*(x4,*(y4,z2))) by A7,A11,A15,ARYTM_0:15 .= +(*(x4,*(y3,z1)),*(x4,opp*(y4,z2))) by ARYTM_0:17; A21: +(*(opp*(x2,y2),z4),*(xy2,z3)) = +(*(opp*(x2,y2),z4),+(*(*(x1,y2),z3),*(*(x2,y1),z3))) by A16,ARYTM_0:16 .= +(*(*(x1,y2),z3),+(*(opp*(x2,y2),z4),*(*(x2,y1),z3))) by ARYTM_0:25 .= +(*(x3,*(y4,z1)),+(*(x4,*(y3,z1)),*(x4,opp*(y4,z2)))) by A7,A11,A15,A20,ARYTM_0:15 .= +(*(x3,*(y4,z1)),*(x4,yz1)) by A17,ARYTM_0:16; +(*(xy1,z4),*(xy2,z3)) = +(+(*(*(x1,y1),z4),*(opp*(x2,y2),z4)),*(xy2,z3)) by A16,ARYTM_0:16 .= +(*(*(x1,y1),z4),+(*(opp*(x2,y2),z4),*(xy2,z3))) by ARYTM_0:25 .= +(*(x3,*(y3,z2)),+(*(x3,*(y4,z1)),*(x4,yz1))) by A7,A11,A15,A21,ARYTM_0:15 .= +(+(*(x3,*(y3,z2)),*(x3,*(y4,z1))),*(x4,yz1)) by ARYTM_0:25 .= +(*(x3,yz2),*(x4,yz1)) by A17,ARYTM_0:16; hence thesis by A10,A14,A19; end; Lm4: a-(b+c)=a-b-c proof a-(b+c) =a+ -(b+c) + 0 by XCMPLX_0:def 8 .=a+ -(b+c) + (b+ -b) by XCMPLX_0:def 6 .=a+ -(b+c) + b+ -b + 0 by Th1 .=a+ -(b+c) + b+ -b + (c + -c) by XCMPLX_0:def 6 .=a+ -(b+c) + b+ -b + c + -c by Th1 .=a+(-(b+c) + b)+ -b + c + -c by Th1 .=a+ -b + (-(b+c) + b) + c + -c by Th1 .=a+ -b + (-(b+c) + b + c) + -c by Th1 .=a+ -b + -c +( -(b+c) + b + c) by Th1 .=a+ -b + -c +( -(b+c)+ (b + c)) by Th1 .=a+ -b + -c + 0 by XCMPLX_0:def 6 .=a-b + -c by XCMPLX_0:def 8 .=a-b-c by XCMPLX_0:def 8; hence thesis; end; Lm5: a-(b-c)=a-b+c proof a-(b-c)=a-(b+ -c) by XCMPLX_0:def 8 .=a-b-(-c) by Lm4 .=a-b+ -(-c) by XCMPLX_0:def 8 .=a-b+c; hence thesis; end; theorem :: REAL_1'9 c <> 0 & a * c = b * c implies a = b proof assume A1: c<>0; assume a * c = b * c; then a * (c * c") = b * c * c" by Th4; then a * 1 = b * c * c" by A1,XCMPLX_0:def 7; then a = b * (c * c") by Th4; then a = b * 1 by A1,XCMPLX_0:def 7; hence thesis; end; theorem Th6: :: REAL_1'23 :: right to left - requirements REAL a*b=0 implies a=0 or b=0 proof assume A1:a*b=0; assume A2:a<>0; a"*a*b=a"*0 by A1,Th4; then 1*b=0 by A2,XCMPLX_0:def 7; hence thesis; end; theorem Th7: :: REAL_2'37 b <> 0 & a * b = b implies a = 1 proof assume A1:b<>0 & a*b=b; then a*b*b"=1 by XCMPLX_0:def 7; then a*1=1 by A1,Th4; hence a=1; end; :: operations '+' and '*' only theorem Th8: :: AXIOMS'18 a * (b + c) = a * b + a * c proof consider y3,y4,z1,z2 being Element of REAL such that A1: b = [*y3,y4*] and A2: c = [*z1,z2*] and A3: b+c = [*+(y3,z1),+(y4,z2)*] by XCMPLX_0:def 4; consider x3,x4,yz1,yz2 being Element of REAL such that A4: a = [*x3,x4*] and A5: b+c = [*yz1,yz2*] and A6: a*(b+c) = [* +(*(x3,yz1),opp*(x4,yz2)), +(*(x3,yz2),*(x4,yz1)) *] by XCMPLX_0:def 5; consider x1,x2,y1,y2 being Element of REAL such that A7: a = [*x1,x2*] and A8: b = [*y1,y2*] and A9: a*b = [* +(*(x1,y1),opp*(x2,y2)), +(*(x1,y2),*(x2,y1)) *] by XCMPLX_0:def 5; A10: y1 = y3 & y2 = y4 by A1,A8,ARYTM_0:12; consider x5,x6,z3,z4 being Element of REAL such that A11: a = [*x5,x6*] and A12: c = [*z3,z4*] and A13: a*c = [* +(*(x5,z3),opp*(x6,z4)), +(*(x5,z4),*(x6,z3)) *] by XCMPLX_0:def 5; A14: x1 = x3 & x2 = x4 & x1 = x5 & x2 = x6 by A4,A7,A11,ARYTM_0:12; A15: z1 = z3 & z2 = z4 by A2,A12,ARYTM_0:12; consider xy3,xy4,xz1,xz2 being Element of REAL such that A16: a*b = [*xy3,xy4*] and A17: a*c = [*xz1,xz2*] and A18: a*b+a*c = [*+(xy3,xz1),+(xy4,xz2)*] by XCMPLX_0:def 4; A19: yz1 = +(y3,z1) & yz2 = +(y4,z2) by A3,A5,ARYTM_0:12; A20: xy3 = +(*(x1,y1),opp*(x2,y2)) & xy4 = +(*(x1,y2),*(x2,y1)) by A9,A16,ARYTM_0:12; A21: xz1 = +(*(x5,z3),opp*(x6,z4)) & xz2 = +(*(x5,z4),*(x6,z3)) by A13,A17,ARYTM_0:12; *(x4,yz2) = +(*(x2,y2),*(x6,z4)) by A10,A14,A15,A19,ARYTM_0:16; then opp*(x4,yz2) = +(opp*(x2,y2),opp*(x6,z4)) by Lm3; then A22: +(*(x3,z1),opp*(x4,yz2)) = +(opp*(x2,y2),xz1) by A14,A15,A21,ARYTM_0:25; A23: +(*(x3,yz1),opp*(x4,yz2)) = +(+(*(x3,y3),*(x3,z1)),opp*(x4,yz2)) by A19,ARYTM_0:16 .= +(*(x1,y1),+(opp*(x2,y2),xz1)) by A10,A14,A22,ARYTM_0:25 .= +(xy3,xz1) by A20,ARYTM_0:25; *(x4,yz1) = +(*(x2,y1),*(x6,z3)) by A10,A14,A15,A19,ARYTM_0:16; then A24: +(*(x3,z2),*(x4,yz1)) = +(*(x2,y1),xz2) by A14,A15,A21,ARYTM_0:25; +(*(x3,yz2),*(x4,yz1)) = +(+(*(x3,y4),*(x3,z2)),*(x4,yz1)) by A19,ARYTM_0:16 .= +(*(x1,y2),+(*(x2,y1),xz2)) by A10,A14,A24,ARYTM_0:25 .= +(xy4,xz2) by A20,ARYTM_0:25; hence thesis by A6,A18,A23; end; theorem :: REAL_2'99_1 (a + b + c) * d = a * d + b * d + c * d proof thus (a+b+c)*d=(a+b)*d+c*d by Th8 .=a*d+b*d+c*d by Th8; end; theorem :: REAL_2'101_1 (a + b) * (c + d) = a * c + a * d + b * c + b * d proof thus (a+b)*(c+d)=a*(c+d)+b*(c+d) by Th8 .=a*c+a*d+b*(c+d) by Th8 .=a*c+a*d+(b*c+b*d) by Th8 .=a*c+a*d+b*c+b*d by Th1; end; theorem Th11: :: SQUARE_1'5 2 * a = a + a proof A1: 1*a = a; thus 2*a = (1 + 1)*a .= a + a by A1,Th8; end; theorem Th12: :: REAL_2'88_1 3 * a = a + a + a proof A1:2*a = a+a by Th11; thus 3*a=(2+1)*a .=a+a+1*a by A1,Th8 .=a+a+a; end; theorem Th13: :: REAL_2'88_2 4 * a = a + a + a + a proof A1:2*a = a+a by Th11; A2:3*a=(2+1)*a .=a+a+1*a by A1,Th8 .=a+a+a; thus 4*a=(3+1)*a .=3*a+1*a by Th8 .=a+a+a+a by A2; end; :: using operation '-' theorem :: REAL_1'36 a - a = 0 by Lm1; theorem Th15: :: SQUARE_1'8 a - b = 0 implies a = b proof assume a - b = 0; then a + -b = 0 by XCMPLX_0:def 8; then -b = -a & --b = b & --a = a by XCMPLX_0:def 6; hence thesis; end; theorem :: REAL_2'1 b - a = b implies a = 0 proof assume b-a=b; then -b+(b-a)=0 by XCMPLX_0:def 6; then -b+(b+-a)=0 by XCMPLX_0:def 8; then -b+b+-a=0 by Th1; then 0+-a=--0 by XCMPLX_0:def 6; hence a=0; end; :: 2 times '-' theorem :: REAL_2'17_2 a = a - (b - b) proof a=a-0; hence thesis by Lm1; end; theorem :: SEQ_4'3 a - (a - b) = b proof thus a-(a-b)=a-a+b by Lm5 .=0+b by Lm1 .=b; end; theorem :: REAL_2'2_3 a - c = b - c implies a = b proof assume a-c =b-c; then a-c =b+-c by XCMPLX_0:def 8; then a+-c =b+-c by XCMPLX_0:def 8; hence a=b by Th2; end; theorem :: REAL_2'2_5 c - a = c - b implies a = b proof assume c-a=c-b; then c-a=c+-b by XCMPLX_0:def 8; then c+-a=c+-b by XCMPLX_0:def 8; then -a=-b by Th2; then --a=b;hence a=b; end; :: 4 times '-' Lm6: a + b - c = a - c + b proof a+b-c =a+b+-c by XCMPLX_0:def 8; hence a+b-c =a+-c+b by Th1 .=a-c+b by XCMPLX_0:def 8; end; theorem Th21: :: REAL_2'24_1 a - b - c = a - c - b proof a-b-c =a+-b-c by XCMPLX_0:def 8; hence a-b-c =a-c+-b by Lm6 .=a-c-b by XCMPLX_0:def 8; end; Lm7: a = a - b + b proof thus a = a - 0 .= a - (b - b) by Lm1 .= (a-b)+b by Lm5; end; theorem Th22: :: REAL_2'29_1 a - c = (a - b) - (c - b) proof thus a-c =a-c-b+b by Lm7 .=a-b-c+b by Th21 .=a-b-(c-b) by Lm5; end; Lm8: a = a + b - b proof a+(b+ -b)=a+b+ -b by Th1; then a+0=a+b+ -b by XCMPLX_0:def 6; hence thesis by XCMPLX_0:def 8; end; theorem Th23: :: JGRAPH_6'1_1 (c - a) - (c - b) = b - a proof thus (c-a)-(c-b)=c-a-c+b by Lm5 .=c+-a-c+b by XCMPLX_0:def 8 .=-a+b by Lm8 .=b-a by XCMPLX_0:def 8; end; Lm9: a=a+b+-b proof a+(b+-b)=a+b+-b by Th1; then a+0=a+b+-b by XCMPLX_0:def 6; hence thesis; end; Lm10: a+b=c+d implies a-c =d-b proof assume a+b=c+d; then a=c+d+-b by Lm9; then a=c+(d+-b) by Th1; then a-c =d+-b by Lm8;hence thesis by XCMPLX_0:def 8; end; Lm11: a+b=a--b proof thus a--b=a+--b by XCMPLX_0:def 8 .=a+b; end; Lm12: a-c =d-b implies a+b=c+d proof assume a-c =d-b; then a+-c =d-b by XCMPLX_0:def 8; then a+-c =d+-b by XCMPLX_0:def 8; then a--b=d--c by Lm10; then a+b=d--c by Lm11; hence thesis by Lm11; end; theorem :: REAL_2'15 a - b = c - d implies a - c = b - d proof assume a-b=c-d; then a+d=c+b by Lm12; hence thesis by Lm10; end; :: using '-' and '+' Lm13: a"*b"=(a*b)" proof per cases; suppose A1: a = 0 or b = 0; then a" = 0 or b" = 0 by XCMPLX_0:def 7; hence a"*b"= (a*b)" by A1,XCMPLX_0:def 7; suppose that A2: a<>0 and A3: b<>0; A4: a*b<>0 by A2,A3,Th6; thus a"*b"=a"*b"*1 .=a"*b"*((a*b)*(a*b)") by A4,XCMPLX_0:def 7 .=a"*b"*(a*b)*(a*b)" by Th4 .=a"*b"*a*b*(a*b)" by Th4 .=a"*a*b"*b*(a*b)" by Th4 .=a"*a*(b"*b)*(a*b)" by Th4 .=1*(b"*b)*(a*b)" by A2,XCMPLX_0:def 7 .=1*(a*b)" by A3,XCMPLX_0:def 7 .=(a*b)"; end; Lm14: a/(b/c)=(a*c)/b proof thus a/(b/c)=a/(b*c") by XCMPLX_0:def 9 .=a*(b*c")" by XCMPLX_0:def 9 .=a*(b"*c"") by Lm13 .=a*c*b" by Th4 .=(a*c)/b by XCMPLX_0:def 9; end; Lm15: b<>0 implies a/b*b=a proof assume A1:b<>0; thus a/b*b=a*b"*b by XCMPLX_0:def 9 .=a*(b"*b) by Th4 .=a*1 by A1,XCMPLX_0:def 7 .=a; end; Lm16: 1/a=a" proof thus 1/a= 1 * a" by XCMPLX_0:def 9 .=a"; end; Lm17: a<>0 implies a/a = 1 proof assume A1: a<>0; thus a/a=a*a" by XCMPLX_0:def 9 .=1 by A1,XCMPLX_0:def 7; end; :: using operations '-' and '+' theorem Th25: :: REAL_2'17_1 a = a + (b - b) proof a=a+0; hence a=a+(b-b) by Lm1; end; theorem :: REAL_1'30 a = a + b - b by Lm8; theorem :: SQUARE_1'6 a = a - b + b by Lm7; theorem :: REAL_2'28_1 a + c = a + b + (c - b) proof thus a+c =a+c+b-b by Lm8 .=a+b+c-b by Th1 .=a+b+(c-b) by Lm2; end; :: 2 times '-' theorem :: REAL_2'22_1, INT_1'1, REAL_1'17 a + b - c = a - c + b by Lm2; theorem :: REAL_2'23_1 a - b + c = c - b + a proof a-b+c =a+-b+c by XCMPLX_0:def 8; hence a-b+c =c+-b+a by Th1 .=c-b+a by XCMPLX_0:def 8; end; Lm18: -(a-b)=b-a proof (a-b) + (b-a) = a-b+b-a by Lm2 .=a-(b-b)-a by Lm5 .=a-0-a by Lm1 .=0 by Lm1; hence thesis by XCMPLX_0:def 6; end; Lm19: a-(b-c)=a+(c-b) proof thus a-(b-c)=a+-(b-c) by XCMPLX_0:def 8 .=a+(c-b) by Lm18; end; theorem :: REAL_2'28_2 a + c = a + b - (b - c) proof a+c =a+c+b-b by Lm8 .=a+b+c-b by Th1 .=a+b+(c-b) by Lm2; hence thesis by Lm19; end; theorem :: REAL_2'29_3 a - c = a + b - (c + b) proof thus a-c =a-c+b-b by Lm8 .=a+b-c-b by Lm2 .=a+b-(c+b) by Lm4; end; theorem :: REAL_2'13 a + b = c + d implies a - c = d - b by Lm10; theorem :: REAL_2'14 a - c = d - b implies a + b = c + d by Lm12; theorem :: REAL_2'16 a + b = c - d implies a + d = c - b proof assume a+b=c-d; then a+b=c+-d by XCMPLX_0:def 8; then a--d=c-b by Lm10; hence thesis by Lm11; end; :: 3 times '-' theorem :: REAL_1'27 a - (b + c) = a - b - c by Lm4; theorem :: REAL_1'28 a - (b - c) = a - b + c by Lm5; theorem :: REAL_2'18 a - (b - c) = a + (c - b) by Lm19; theorem :: REAL_2'29_2 a - c = (a - b) + (b - c) proof a-c = a-c-b+b by Lm7 .=a-b-c+b by Th21 .=a-b-(c-b) by Lm5; hence thesis by Lm19; end; :: using operations '-' and '*' Lm20: (-a)*b = -(a*b) proof thus (-a)*b = (-a)*b + 0 .= (-a)*b + (a*b + -(a*b)) by XCMPLX_0:def 6 .= (-a)*b + a*b + -(a*b) by Th1 .= (-a + a)*b + -(a*b) by Th8 .= 0*b+ -(a*b) by XCMPLX_0:def 6 .= -(a*b); end; theorem Th40: :: REAL_1'29 a * (b - c) = a * b - a * c proof thus a*(b-c)=a*(b+ -c) by XCMPLX_0:def 8 .=a*b + a*(-c) by Th8 .=a*b + -(a*c) by Lm20 .=a*b - a*c by XCMPLX_0:def 8; end; Lm21: (-a)*b=a*(-b) proof a*(-b)=-(a*b) by Lm20; hence thesis by Lm20; end; theorem :: REAL_2'98 (a - b) * (c - d) = (b - a) * (d - c) proof thus (a-b)*(c-d)=(-(b-a))*(c-d) by Lm18 .=(b-a)*(-(c-d)) by Lm21 .=(b-a)*(d-c) by Lm18; end; theorem :: REAL_2'99_4 (a - b - c) * d = a * d - b * d - c * d proof thus (a-b-c)*d=(a-b)*d-c*d by Th40 .=a*d-b*d-c*d by Th40; end; :: using operations '-' and '*', '+' theorem :: REAL_2'99_2 (a + b - c) * d = a * d + b * d - c * d proof thus (a+b-c)*d=(a+b)*d-c*d by Th40 .=a*d+b*d-c*d by Th8; end; theorem :: REAL_2'99_3 (a - b + c) * d = a * d - b * d + c * d proof thus (a-b+c)*d=(a-b)*d+c*d by Th8 .=a*d-b*d+c*d by Th40; end; theorem :: REAL_2'101_2 (a + b) * (c - d) = a * c - a * d + b * c - b * d proof thus (a+b)*(c-d)=a*(c-d)+b*(c-d) by Th8 .=a*c-a*d+b*(c-d) by Th40 .=a*c-a*d+(b*c-b*d) by Th40 .=a*c-a*d+b*c-b*d by Lm2; end; theorem :: REAL_2'101_3 (a - b) * (c + d) = a * c + a * d - b * c - b * d proof thus (a-b)*(c+d)=(a-b)*c+(a-b)*d by Th8 .=a*c-b*c+(a-b)*d by Th40 .=a*c-b*c+(a*d-b*d) by Th40 .=a*c-b*c+a*d-b*d by Lm2 .=a*c+a*d-b*c-b*d by Lm2; end; theorem :: REAL_2'101_4 (a - b) * (e - d) = a * e - a * d - b * e + b * d proof thus (a-b)*(e-d)=a*(e-d)-b*(e-d) by Th40 .=a*e-a*d-b*(e-d) by Th40 .=a*e-a*d-(b*e-b*d) by Th40 .=a*e-a*d-b*e+b*d by Lm5; end; :: using operation '/' theorem :: REAL_2'67_1 a / b / c = a / c / b proof thus a/b/c =a*b"/c by XCMPLX_0:def 9 .=a*b"*c" by XCMPLX_0:def 9 .=a*c"*b" by Th4 .=a/c*b" by XCMPLX_0:def 9 .=a/c/b by XCMPLX_0:def 9; end; :: 0 theorem Th49: :: REAL_2'19 a / 0 = 0 proof thus a/0 = a*0" by XCMPLX_0:def 9 .= 0; end; Lm22: a<>0 implies a"<>0 proof assume A1:a<>0; assume a"=0; then a*a"=0; hence contradiction by A1,XCMPLX_0:def 7; end; theorem Th50: :: REAL_2'42_2 a <> 0 & b <> 0 implies a / b <> 0 proof assume A1:a<>0 & b<>0; then a"<>0 & b"<>0 by Lm22; then b"*a<>0 by A1,Th6; hence thesis by XCMPLX_0:def 9; end; :: 2 times '/' theorem :: REAL_2'62_4 b <> 0 implies a = a / (b / b) proof assume A1:b<>0; a=a/1; hence a=a/(b/b) by A1,Lm17; end; Lm23: (a/b) * (c/d) = (a*c)/(b*d) proof thus (a/b) * (c/d) =(a*b")*(c/d) by XCMPLX_0:def 9 .=(a*b")*(c*d") by XCMPLX_0:def 9 .=a*b"*c*d" by Th4 .=a*c*b"*d" by Th4 .=(a*c)*(b"*d") by Th4 .=(a*c)*(b*d)" by Lm13 .=(a*c)/(b*d) by XCMPLX_0:def 9; end; Lm24: (a/b)"=b/a proof per cases; suppose A1: a = 0; hence (a/b)" =b*0" .=b/a by A1,XCMPLX_0:def 9; suppose A2: b = 0; hence (a/b)" =(a*0")" by XCMPLX_0:def 9 .=b/a by A2; suppose A3: a<>0 & b<>0; A4: (a/b) * (b/a) =(a*b)/(a*b) by Lm23; a*b<>0 by A3,Th6; then A5: (a/b)*(b/a) = 1 by A4,Lm17; A6: a/b=a*b" by XCMPLX_0:def 9; b"<>0 by A3,Lm22; then a/b<>0 by A3,A6,Th6; hence thesis by A5,XCMPLX_0:def 7; end; Lm25: a*(b/c) = (a*b)/c proof thus a*(b/c) = (a/1)*(b/c) .= (a*b)/(1*c) by Lm23 .= (a*b)/c; end; theorem :: TOPREAL6'5 a <> 0 implies a / (a / b) = b proof assume A1:a <> 0; thus a/(a/b) = a * (a/b)" by XCMPLX_0:def 9 .= a * (b/a) by Lm24 .= a*b/a by Lm25 .= a/a*b by Lm25 .= 1 * b by A1,Lm17 .= b; end; theorem :: REAL_2'31 c <> 0 & a / c = b / c implies a = b proof assume A1:c<>0 & a/c =b/c; then a=b/c*c by Lm15; hence thesis by A1,Lm15; end; :: 3 times '/' Lm26: b<>0 implies a=a*b/b proof assume A1:b<>0; a=a*1; then a=a*(b/b) by A1,Lm17; then a=a*(b*b") by XCMPLX_0:def 9; then a=a*b*b" by Th4; hence thesis by XCMPLX_0:def 9; end; theorem :: REAL_2'74 a / b <> 0 implies b = a / (a / b) proof assume A1: a/b<>0; then b<>0 by Th49; then a/b*b = a by Lm15; hence thesis by A1,Lm26; end; :: 4 times '/' Lm27: c<>0 implies a/b=(a*c)/(b*c) proof assume A1: c<>0; thus a/b =a*b"*1 by XCMPLX_0:def 9 .=a*b"*(c*c") by A1,XCMPLX_0:def 7 .=a*b"*c*c" by Th4 .=a*c*b"*c" by Th4 .=(a*c)*(b"*c") by Th4 .=(a*c)*(b*c)" by Lm13 .=(a*c)/(b*c) by XCMPLX_0:def 9; end; theorem :: REAL_2'55_1 c <> 0 implies a / b = (a / c) / (b / c) proof assume c<>0; then c"<>0 by Lm22; hence a/b=(a*c")/(b*c") by Lm27 .=(a/c)/(b*c") by XCMPLX_0:def 9 .=(a/c)/(b/c) by XCMPLX_0:def 9; end; :: 1 theorem :: SQUARE_1'16 1 / (1 / a) = a proof thus 1/(1/a) = (1*a) /1 by Lm14 .= a; end; Lm28: (a*b")"=a"*b proof thus (a*b")"=a"*b"" by Lm13 .=a"*b; end; theorem :: REAL_2'48_1 1 / (a / b) = b / a proof thus 1/(a/b)=1/(a*b") by XCMPLX_0:def 9 .=(a*b")" by Lm16 .=b*a" by Lm28 .=b/a by XCMPLX_0:def 9; end; theorem Th58: :: REAL_2'30_1 a / b = 1 implies a = b proof assume A1:a/b = 1; then b <> 0 by Th49; then a=1*b by A1,Lm15; hence thesis; end; Lm29: a"=b" implies a=b proof assume a"=b"; then a=b""; hence thesis; end; theorem Th59: :: REAL_2'33_2 1 / a = 1 / b implies a = b proof assume 1/a=1/b; then a"=1/b by Lm16; then a" = b" by Lm16; hence thesis by Lm29; end; :: 0 and 1 theorem :: REAL_1'37 a <> 0 implies a / a = 1 by Lm17; theorem :: REAL_2'39 b <> 0 & b / a = b implies a = 1 proof assume A1: b<>0 & b/a=b; then a <> 0 by Th49; then b=b*a by A1,Lm15; hence a=1 by A1,Th7; end; theorem :: REAL_2'41 a <> 0 implies 1 / a <> 0 proof assume a<>0; then a"<>0 by Lm22; hence thesis by Lm16; end; :: using operations '/' and '+' theorem Th63: :: REAL_1'40_1 a / c + b / c = (a + b) / c proof thus a/c + b/c =a*c" + b/c by XCMPLX_0:def 9 .= a*c" + b*c" by XCMPLX_0:def 9 .= (a+b)*c" by Th8 .= (a+b)/c by XCMPLX_0:def 9; end; theorem :: REAL_2'100_1 (a + b + e) / d = a / d + b / d + e / d proof thus (a+b+e)/d=(a+b)/d+e/d by Th63 .=a/d+b/d+e/d by Th63; end; Lm30: a/2 + a/2=a proof thus a/2 +a/2 = 2*(a/2) by Th11 .= a by Lm15; end; :: 2 theorem :: SQUARE_1'15 (a + a) / 2 = a proof thus (a + a) /2 = a/2 + a/2 by Th63 .= a by Lm30; end; theorem :: SEQ_2'2_1 a/2 + a/2 = a by Lm30; theorem :: TOPREAL3'4 a = (a + b) / 2 implies a = b proof assume a = (a+b)/2; then a + a = a + b by Lm30; hence thesis by Th2; end; :: 3 theorem Th68: :: REAL_2'89_1 (a + a + a)/3 = a proof thus (a+a+a)/3=3*a/3 by Th12 .=a by Lm26; end; theorem :: SEQ_4'5 a/3 + a/3 + a/3 = a proof thus a/3+a/3+a/3 = (a+a)/3 +a/3 by Th63 .= (a+a+a)/3 by Th63 .= a by Th68; end; :: 4 theorem Th70: :: REAL_2'89_2 (a + a + a + a) / 4 = a proof thus (a+a+a+a)/4=a*4/4 by Th13 .=a by Lm26; end; theorem :: REAL_2'90 a/4 + a/4 + a/4 + a/4 = a proof thus a/4+a/4+a/4+a/4=(a+a)/4+a/4+a/4 by Th63 .=(a+a+a)/4+a/4 by Th63 .=(a+a+a+a)/4 by Th63 .=a by Th70; end; theorem :: SEQ_2'2_2 a / 4 + a / 4 = a / 2 proof thus a/4 +a/4=(1*a+1*a)/4 by Th63 .=((1+1)*a)/4 by Th8 .=a*2*(2*2)" by XCMPLX_0:def 9 .=a*(2*(2"*2")) by Th4 .=a/2 by XCMPLX_0:def 9; end; theorem :: REAL_2'89_3 (a + a) / 4 = a / 2 proof thus (a+a)/4=2*a/(2*2) by Th11 .=a/2 by Lm27; end; :: using operations '/' and '*' theorem :: REAL_2'35_1 a * b = 1 implies a = 1 / b proof assume A1: a*b=1; then b<>0; then a*1=1*b" by A1,XCMPLX_0:def 7; hence a=1/b by Lm16; end; theorem :: SQUARE_1'18 a * (b / c) = (a * b) / c by Lm25; theorem :: REAL_2'80_1 a / b * e = e / b * a proof thus a/b*e=a*e/b by Lm25 .=e/b*a by Lm25; end; :: 3 times '/' theorem :: REAL_1'35 (a / b) * (c / d) = (a * c) / (b * d) by Lm23; theorem :: REAL_1'42 a / (b / c) = (a * c) / b by Lm14; Lm31: (a/b)/(c/d)=(a*d)/(b*c) proof thus (a/b)/(c/d) = (a/b) * (c/d)" by XCMPLX_0:def 9 .=(a/b) * (d/c) by Lm24 .=(a*d)/(b*c) by Lm23; end; theorem Th79: :: SQUARE_1'17 a / (b * c) = a / b / c proof thus a/(b*c) = a*1/(b*c) .= a/b/(c/1) by Lm31 .= a/b/c; end; theorem :: REAL_2'61_1 a / (b / c) = a * (c / b) proof thus a/(b/c)=(a*c)/b by Lm14 .=a*c*b" by XCMPLX_0:def 9 .=a*(c*b") by Th4 .=a*(c/b) by XCMPLX_0:def 9; end; theorem :: REAL_2'61_2 a / (b / c) = c / b * a proof a/(b/c)=(a*c)/b by Lm14 .=a*c*b" by XCMPLX_0:def 9 .=a*(c*b") by Th4 .=a*(c/b) by XCMPLX_0:def 9; hence thesis; end; theorem Th82: :: REAL_2'61_3 a / (b / e) = e * (a / b) proof thus a/(b/e)=(a*e)/b by Lm14 .=e*a*b" by XCMPLX_0:def 9 .=e*(a*b") by Th4 .=e*(a/b) by XCMPLX_0:def 9; end; theorem :: REAL_2'61_4 a / (b / c) = a / b * c proof a/(b/c)=(a*c)/b by Lm14 .=c*a*b" by XCMPLX_0:def 9 .=c*(a*b") by Th4 .=c*(a/b) by XCMPLX_0:def 9; hence thesis; end; Lm32: a*(1/b)=a/b proof thus a/b=a*b" by XCMPLX_0:def 9 .=a*(1/b) by Lm16; end; Lm33: 1/c*(a/b)=a/(b*c) proof a/b/c =c"*(a/b) by XCMPLX_0:def 9 .=1/c*(a/b) by Lm16; hence thesis by Th79; end; theorem :: REAL_2'70 (a * b) / (c * d) = (a / c * b) / d proof thus a*b/(c*d)=1/c*(a*b/d) by Lm33 .=1/c*(a*b)/d by Lm25 .=1/c*a*b/d by Th4 .=a/c*b/d by Lm32; end; :: 4 times '/' theorem :: REAL_1'82 (a / b) / (c / d) = (a * d) / (b * c) by Lm31; theorem :: REAL_2'53 (a / c) * (b / d) = (a / d) * (b / c) proof thus a/c*(b/d)=a*b/(d*c) by Lm23 .=a/d*(b/c) by Lm23; end; theorem :: IRRAT_1'5 a / (b * c * (d / e)) = (e / c) * (a / (b * d)) proof thus a/(b*c*(d/e)) = a/(b*c*(d*e")) by XCMPLX_0:def 9 .= a/(c*(b*(d*e"))) by Th4 .= a/(c*(b*d*e")) by Th4 .= a/(c*((b*d)/e)) by XCMPLX_0:def 9 .= a/((b*d)/(e/c)) by Th82 .= (e/c)*(a/(b*d)) by Th82; end; :: 0 theorem :: REAL_1'43 b <> 0 implies a / b * b = a by Lm15; theorem :: REAL_2'62_1 b <> 0 implies a = a * (b / b) proof assume A1:b<>0; a=a*1; hence thesis by A1,Lm17; end; theorem :: REAL_2'62_2 b <> 0 implies a = a * b / b by Lm26; theorem :: REAL_2'78 b <> 0 implies a * c = a * b * (c / b) proof assume A1:b<>0; thus a*c =a*1*c .=a*(b*b")*c by A1,XCMPLX_0:def 7 .=a*b*b"*c by Th4 .=a*b*(b"*c) by Th4 .=a*b*(c/b) by XCMPLX_0:def 9; end; :: 2 times '/' theorem :: REAL_1'38 c <> 0 implies a / b = (a * c) / (b * c) by Lm27; theorem :: REAL_2'55_2 c <> 0 implies a / b = a / (b * c) * c proof assume A1: c<>0; c*(a/(b*c))=c*((a*1)/(b*c)) .=c*(1/c*(a/b)) by Lm23 .=1/c*c*(a/b) by Th4 .=1*(a/b) by A1,Lm15 .=a/b; hence thesis; end; theorem :: REAL_2'79 b <> 0 implies a * c = a * b / (b / c) proof assume A1:b<>0; thus a*c =a*1*c .=a*(b*b")*c by A1,XCMPLX_0:def 7 .=a*b*b"*c by Th4 .=a*b*(b"*c) by Th4 .=a*b*(b*c")" by Lm28 .=a*b/(b*c") by XCMPLX_0:def 9 .=a*b/(b/c) by XCMPLX_0:def 9; end; theorem Th95: :: REAL_2'75 c <> 0 & d <> 0 & a * c = b * d implies a / d = b / c proof assume A1:c<>0 & d<>0; assume a*c = b*d; then a=b*d/c by A1,Lm26; then a=d*(b/c) by Lm25; hence thesis by A1,Lm26; end; theorem Th96: :: REAL_2'76 c <> 0 & d<>0 & a/d=b/c implies a*c = b*d proof assume A1:c<>0 & d<>0 & a/d=b/c; then c*(a/d)=b by Lm15; then a*c/d=b by Lm25; hence thesis by A1,Lm15; end; theorem :: REAL_2'77 c <> 0 & d <> 0 & a * c = b / d implies a * d = b / c proof assume A1:c<>0 & d<>0; assume a*c =b/d; then a*c*d=b by A1,Lm15; then a*d*c =b by Th4; hence thesis by A1,Lm26; end; :: 3 times '/' theorem :: REAL_2'55_3 c <> 0 implies a / b = c * (a / c / b) proof assume A1: c<>0; thus a/b=a*b" by XCMPLX_0:def 9 .=c*(a/c)*b" by A1,Lm15 .=c*(a/c*b") by Th4 .=c*(a/c/b) by XCMPLX_0:def 9; end; theorem :: REAL_2'55 c <> 0 implies a / b = a / c * (c / b) proof assume A1: c<>0; thus a/b=a*b" by XCMPLX_0:def 9 .=a/c*c*b" by A1,Lm15 .=a/c*(c*b") by Th4 .=a/c*(c/b) by XCMPLX_0:def 9; end; :: 1 theorem :: REAL_2'56: a * (1 / b) = a / b by Lm32; Lm34: 1/a"=a proof 1/a"=a"" by Lm16 .=a; hence thesis; end; theorem :: REAL_2'57 a / (1 / b) = a * b proof thus a/(1/b)=a/b" by Lm16 .=a*(1/b") by Lm32 .=a*b by Lm34; end; theorem :: REAL_2'80_3 a / b * c = 1 / b * c * a proof a/b*c = 1/b*a*c by Lm32; hence thesis by Th4; end; :: 3 times '/' theorem :: REAL_2'51 (1 / a) * (1 / b) = 1 / (a * b) proof thus (1/a)*(1/b)=a"*(1/b) by Lm16 .=a"*b" by Lm16 .=(a*b)" by Lm13 .=1/(a*b) by Lm16; end; theorem :: REAL_2'67_4 1 / c * (a / b) = a / (b * c) by Lm33; :: 4 times '/' theorem :: REAL_2'67_2 a / b / c = 1 / b * (a / c) proof a/b/c =a*b"/c by XCMPLX_0:def 9 .=a*b"*c" by XCMPLX_0:def 9 .=a*c"*b" by Th4 .=a/c*b" by XCMPLX_0:def 9 .=a/c/b by XCMPLX_0:def 9; hence a/b/c =b"*(a/c) by XCMPLX_0:def 9 .=1/b*(a/c) by Lm16; end; theorem :: REAL_2'67_3 a / b / c = 1 / c * (a / b) proof thus a/b/c =c"*(a/b) by XCMPLX_0:def 9 .=1/c*(a/b) by Lm16; end; :: 1 and 0 theorem Th107: :: REAL_1'34 a <> 0 implies a * (1 / a) = 1 proof assume A1:a<>0; thus a*(1/a)=a*a" by Lm16 .=1 by A1,XCMPLX_0:def 7; end; theorem :: REAL_2'62_3 b <> 0 implies a = a * b * (1 / b) proof assume A1:b<>0; a=a*1; then a=a*(b/b) by A1,Lm17; then a=a*(b*b") by XCMPLX_0:def 9 .=a*(b*(1/b)) by Lm16; hence thesis by Th4; end; theorem :: REAL_2'62_6 b <> 0 implies a = a * (1 / b * b) proof assume A1:b<>0; thus a=a*1 .=a*(1/b*b) by A1,Lm15; end; theorem :: REAL_2'62_7 b <> 0 implies a = a * (1 / b) * b proof assume A1:b<>0; a=a*1 .=a*(1/b*b) by A1,Lm15; hence thesis by Th4; end; theorem :: REAL_2'62_5 b <> 0 implies a = a / (b * (1 / b)) proof assume A1:b<>0; thus a=a/1 .=a/(b*(1/b)) by A1,Th107; end; theorem :: REAL_2'42_4 a <> 0 & b <> 0 implies 1 / (a * b) <> 0 proof assume a<>0 & b<>0; then a"<>0 & b"<>0 by Lm22; then a"*b"<>0 by Th6; then (a*b)"<>0 by Lm13; hence thesis by Lm16; end; theorem :: JGRAPH_2'1 a <> 0 & b <> 0 implies (a / b) * (b / a) = 1 proof assume A1: a<>0 & b<>0; A2:(b/a)=(a/b)" by Lm24; a/b<>0 by A1,Th50; hence thesis by A2,XCMPLX_0:def 7; end; :: using operations '*', '+' and '/' theorem Th114: :: REAL_2'65 b <> 0 implies a / b + c = (a + b * c) / b proof assume A1:b<>0; a/b+c =a/b+1*c .=a/b+b*b"*c by A1,XCMPLX_0:def 7 .=a/b+b*c*b" by Th4 .=a/b+c*b/b by XCMPLX_0:def 9 .=(a+c*b)/b by Th63; hence thesis; end; theorem Th115: :: REAL_2'92 c <> 0 implies a + b = c * (a / c + b / c) proof assume A1:c<>0; hence a+b=c*(a/c)+b by Lm15 .=c*(a/c)+c*(b/c) by A1,Lm15 .=c*(a/c+b/c) by Th8; end; theorem Th116: :: REAL_2'94 c <> 0 implies a + b = (a * c + b * c) / c proof assume A1:c<>0; hence a+b=a*c/c+b by Lm26 .=a*c/c+b*c/c by A1,Lm26 .=(a*c+b*c)/c by Th63; end; theorem Th117: :: REAL_1'41_1 b <> 0 & d <> 0 implies a / b + c / d =(a * d + c * b) / (b * d) proof assume A1: b<>0; assume d<>0; hence a/b + c/d=(a*d)/(b*d) + c/d by Lm27 .=(a*d)/(b*d) + (c*b)/(b*d) by A1,Lm27 .=(a*d + c*b)/(b*d) by Th63; end; theorem Th118: :: REAL_2'96 a <> 0 implies a + b = a * (1 + b / a) proof assume A1:a<>0; hence a+b=a*(a/a+b/a) by Th115 .=a*(1+b/a) by A1,Lm17; end; :: 2 theorem :: REAL_2'91_1 a / (2 * b) + a / (2 * b) = a / b proof thus a/(2*b)+a/(2*b)=(a+a)/(2*b) by Th63 .=2*a/(2*b) by Th11 .=a/b by Lm27; end; :: 3 theorem :: REAL_2'91_2 a / (3 * b) + a / (3 * b) + a / (3 * b) = a / b proof thus a/(3*b)+a/(3*b)+a/(3*b)=(a+a)/(3*b)+a/(3*b) by Th63 .=(a+a+a)/(3*b) by Th63 .=3*a/(3*b) by Th12 .=a/b by Lm27; end; :: using operations '-' and '/' Lm35: -a/b=(-a)/b proof thus -a/b=-(a*b") by XCMPLX_0:def 9 .=(-a)*b" by Lm20 .=(-a)/b by XCMPLX_0:def 9; end; theorem Th121: :: REAL_1'40_2 a / c - b / c = (a - b) / c proof thus a/c - b/c = a/c+ -b/c by XCMPLX_0:def 8 .=a/c+(-b)/c by Lm35 .=(a+ -b)/c by Th63 .=(a-b)/c by XCMPLX_0:def 8; end; theorem :: TOPREAL6'4 a - a / 2 = a / 2 proof thus a - a/2 = a/2 + a/2 - a/2 by Lm30 .= a/2 + (a/2 - a/2) by Lm2 .= a/2 + 0 by Lm1 .= a/2; end; theorem :: REAL_2'100_4 (a - b - c) / d = a / d - b / d - c / d proof thus (a-b-c)/d=(a-b)/d-c/d by Th121 .=a/d-b/d-c/d by Th121; end; theorem :: REAL_2'82 b <> 0 & d <> 0 & b <> d & a / b = e / d implies a / b = (a - e) / (b - d) proof assume A1:b<>0 & d<>0 & b<>d & a/b=e/d; then A2:b-d<>0 by Th15; a*d=e*b by A1,Th96; then a*(b-d)=a*b-e*b by Th40; then a*(b-d)=(a-e)*b by Th40; hence thesis by A1,A2,Th95; end; :: using operations '-', '/' and '+' theorem :: REAL_2'100_2 (a + b - e) / d = a / d + b / d - e / d proof thus (a+b-e)/d=(a+b)/d-e/d by Th121 .=a/d+b/d-e/d by Th63; end; theorem :: REAL_2'100_3 (a - b + e) / d = a / d - b / d + e / d proof thus (a-b+e)/d=(a-b)/d+e/d by Th63 .=a/d-b/d+e/d by Th121; end; :: using operations '-', '/' and '*' theorem Th127: :: REAL_2'66_1 b <> 0 implies a / b - e = (a - e * b) / b proof assume A1:b<>0; thus a/b-e=a/b+-e by XCMPLX_0:def 8 .=(a+(-e)*b)/b by A1,Th114 .=(a+-e*b)/b by Lm20 .=(a-e*b)/b by XCMPLX_0:def 8; end; theorem :: REAL_2'66_2 b <> 0 implies c - a / b = (c * b - a) / b proof assume A1:b<>0; thus c-a/b = -(a/b-c) by Lm18 .=-(a-c*b)/b by A1,Th127 .=(-(a-c*b))/b by Lm35 .=(c*b-a)/b by Lm18; end; theorem :: REAL_2'93 c <> 0 implies a - b = c * (a / c - b / c) proof assume A1:c<>0; hence a-b=c*(a/c)-b by Lm15 .=c*(a/c)-c*(b/c) by A1,Lm15 .=c*(a/c-b/c) by Th40; end; theorem :: REAL_2'95 c <> 0 implies a - b = (a * c - b * c) / c proof assume A1:c<>0; thus a-b=a+-b by XCMPLX_0:def 8 .=(a*c+(-b)*c)/c by A1,Th116 .=(a*c+-b*c)/c by Lm20 .=(a*c-b*c)/c by XCMPLX_0:def 8; end; theorem :: REAL_1'41_2 b <> 0 & d <> 0 implies a / b - c / d = (a * d - c * b) / (b * d) proof assume A1: b<>0; assume A2: d<>0; thus a/b - c/d =a/b + -c/d by XCMPLX_0:def 8 .=a/b + (-c)/d by Lm35 .=(a*d + (-c)*b)/(b*d) by A1,A2,Th117 .=(a*d + -(c*b))/(b*d) by Lm20 .=(a*d - c*b)/(b*d) by XCMPLX_0:def 8; end; theorem :: REAL_2'97 a <> 0 implies a - b = a * (1 - b / a) proof assume A1:a<>0; thus a-b=a+-b by XCMPLX_0:def 8 .=a*(1+(-b)/a) by A1,Th118 .=a*(1+-b/a) by Lm35 .=a*(1-b/a) by XCMPLX_0:def 8; end; :: using operation '-', '/', '*' and '+' theorem :: POLYEQ_1'24 a <> 0 implies c = (a * c + b - b) / a proof assume A1:a <> 0; a*c = a*c+b-b by Lm8; hence thesis by A1,Lm26; end; :: using unary operation '-' theorem :: REAL_2'2_2 -a = -b implies a = b proof assume -a=-b; then a=--b;hence a=b; end; theorem Th135: :: REAL_1'22: :: right to left - requirements REAL -a = 0 implies a = 0 proof assume -a=0; then 0=a +0 by XCMPLX_0:def 6; hence thesis; end; theorem :: REAL_2'2_1 a + -b = 0 implies a = b proof assume a+-b=0; then a+(-b+b)=0+b by Th1; then a+0=0+b by XCMPLX_0:def 6; hence a=b; end; theorem :: REAL_2'11 a = a + b + -b by Lm9; theorem Th138: :: REAL_2'17_1 a = a + (b + -b) proof a=a+0; then a=a+(b-b) by Lm1; hence thesis by XCMPLX_0:def 8; end; theorem Th139: :: INT_1'3 a = (- b + a) + b proof a = 0 + a .= b + (- b) + a by XCMPLX_0:def 6; hence thesis by Th1; end; theorem Th140: :: REAL_2'6_1 - (a + b) = -a + -b proof -(a+b)+(a+b)=0 by XCMPLX_0:def 6; then -(a+b)+a+b=0 by Th1; then -(a+b)+a+0=0+-b by XCMPLX_0:def 6; then -(a+b)+(a+-a)=-b+-a by Th1; then -(a+b)+0=-b+-a by XCMPLX_0:def 6; hence thesis; end; theorem :: REAL_2'9_2 - (-a + b) = a + -b proof -(-a+b)=--a+-b by Th140 .=a-b by XCMPLX_0:def 8; hence thesis by XCMPLX_0:def 8; end; Lm36: -(a-b)=-a+b proof thus -(a-b)=-(a+-b) by XCMPLX_0:def 8 .=-a+--b by Th140 .=-a+b; end; theorem :: REAL_2'10_2 a+b=-(-a+-b) proof a+b=--a+b .=-(-a-b) by Lm36; hence thesis by XCMPLX_0:def 8; end; :: using unary and binary operation '-' theorem :: REAL_1'83 -(a - b) = b - a by Lm18; theorem :: REAL_2'5 - a - b = - b - a proof -a-b=-b+-a by XCMPLX_0:def 8; hence thesis by XCMPLX_0:def 8; end; Lm37: -(a+b)=-b-a proof -(a+b)+(a+b)=0 by XCMPLX_0:def 6; then -(a+b)+a+b=0 by Th1; then -(a+b)+a+0=0+-b by XCMPLX_0:def 6; then -(a+b)+(a+-a)=-b+-a by Th1; then -(a+b)+0=-b+-a by XCMPLX_0:def 6; hence thesis by XCMPLX_0:def 8; end; theorem :: REAL_2'17_4 a = - b - (- a - b) proof a=a+0; then a=a+(b-b) by Lm1; then a=a+(b+-b) by XCMPLX_0:def 8; then a=-b+(a+b) by Th1; then a=-b--(a+b) by Lm11; hence thesis by Lm37; end; :: binary '-' 4 times theorem :: REAL_2'26_1 - a - b - c = - a - c - b proof -a-b-c =-a+-b-c by XCMPLX_0:def 8 .=-a+-b+-c by XCMPLX_0:def 8; hence -a-b-c =-a+-c+-b by Th1 .=-a-c+-b by XCMPLX_0:def 8 .=-a-c-b by XCMPLX_0:def 8; end; theorem :: REAL_2'26_2 - a - b - c = - b - c - a proof -a-b-c =-a+-b-c by XCMPLX_0:def 8 .=-a+-b+-c by XCMPLX_0:def 8; hence -a-b-c =-b+-c+-a by Th1 .=-b-c+-a by XCMPLX_0:def 8 .=-b-c-a by XCMPLX_0:def 8; end; theorem :: REAL_2'26_4 - a - b - c = - c - b - a proof -a-b-c = -a+-b-c by XCMPLX_0:def 8 .=-a+-b+-c by XCMPLX_0:def 8; hence -a-b-c = -c+-b+-a by Th1 .=-c-b+-a by XCMPLX_0:def 8 .=-c-b-a by XCMPLX_0:def 8; end; theorem :: JGRAPH_6'1_2 (c - a) - (c - b) = - (a - b) proof thus (c-a)-(c-b)=b-a by Th23 .=b+-a by XCMPLX_0:def 8 .=-(a-b) by Lm36; end; :: 0 theorem :: REAL_1'19 0 - a = - a proof thus 0-a=0+ -a by XCMPLX_0:def 8 .=-a; end; :: using unary and binary operations '-' and '+' theorem :: REAL_2'10_3 a + b = a - - b by Lm11; theorem :: REAL_2'17_3 a = a - (b + -b) proof a=a-0; then a=a-(b-b) by Lm1; hence thesis by XCMPLX_0:def 8; end; theorem :: REAL_2'2_4 a - c = b + - c implies a = b proof assume a-c =b+-c; then a+-c =b+-c by XCMPLX_0:def 8; hence a=b by Th2; end; theorem :: REAL_2'2_6 c - a = c + - b implies a = b proof assume c-a=c+-b; then c+-a=c+-b by XCMPLX_0:def 8; then -a=-b by Th2; then --a=b; hence a=b; end; :: '+' 3 times theorem Th155: :: REAL_2'22_2 a + b - c = - c + a + b proof a+b-c =a+b+-c by XCMPLX_0:def 8; hence thesis by Th1; end; theorem :: REAL_2'23_2 a - b + c = - b + c + a proof a-b+c =a+-b+c by XCMPLX_0:def 8; hence thesis by Th1; end; :: binary '-' 2 times Lm38: a+b=-(-a-b) proof thus a+b=--a+b .=-(-a-b) by Lm36; end; theorem :: REAL_2'20_2 a - (- b - c) = a + b + c proof thus a-(-b-c)=a+-(-b-c) by XCMPLX_0:def 8 .=a+(b+c) by Lm38 .=a+b+c by Th1; end; :: binary '-' 3 times theorem :: REAL_2'20_1 a - b - c = - b - c + a proof thus -b-c+a=a+-b-c by Lm2 .=a-b-c by XCMPLX_0:def 8; end; theorem :: REAL_2'24_3 a - b - c = - c + a - b proof a-b-c =a+-b-c by XCMPLX_0:def 8; hence a-b-c =-c+a+-b by Th155 .=-c+a-b by XCMPLX_0:def 8; end; theorem :: REAL_2'24_4 a - b - c = - c - b + a proof a-b-c = a+-b-c by XCMPLX_0:def 8; hence a-b-c =-c+-b+a by Th155 .=-c-b+a by XCMPLX_0:def 8; end; :: using unary and binary operations '-' and '+' theorem :: REAL_2'6_2 - (a + b) = - b - a by Lm37; theorem :: REAL_2'8 - (a - b) = - a + b by Lm36; theorem Th163: :: REAL_2'9_1 -(-a+b)=a-b proof thus -(-a+b)=--a+-b by Th140 .=a-b by XCMPLX_0:def 8; end; theorem :: REAL_2'10_1 a + b = -(- a - b) by Lm38; theorem :: REAL_2'25_1 - a + b - c = - c + b - a proof thus -a+b-c = -c+b+-a by Th155 .= -c+b-a by XCMPLX_0:def 8; end; :: using unary and binary operations '-' and '+' (both '-' 2 times) theorem :: REAL_2'25_2 - a + b - c = - c - a + b proof thus -a+b-c = -c+-a+b by Th155 .= -c-a+b by XCMPLX_0:def 8; end; theorem :: REAL_2'27_1 - (a + b + c) = - a - b - c proof thus -(a+b+c)=-(a+b)-c by Lm37 .=-a-b-c by Lm37; end; theorem :: REAL_2'27_2 - (a + b - c) = - a - b + c proof thus -(a+b-c)=-(a+b)+c by Lm36 .=-a-b+c by Lm37; end; theorem :: REAL_2'27_3 - (a - b + c) = - a + b - c proof thus -(a-b+c)=-(a-b)-c by Lm37 .=-a+b-c by Lm36; end; theorem :: REAL_2'27_5 - (a - b - c) = - a + b + c proof thus -(a-b-c)=-(a-b)+c by Lm36 .=-a+b+c by Lm36; end; theorem :: REAL_2'27_4 - (- a + b + c) = a - b - c proof thus -(-a+b+c)=-(-a+b)-c by Lm37 .=a-b-c by Th163; end; theorem :: REAL_2'27_6 - (- a + b - c) = a - b + c proof thus -(-a+b-c)=-(-a+b)+c by Lm36 .=a-b+c by Th163; end; theorem :: REAL_2'27_7 - (- a - b + c) = a + b - c proof thus -(-a-b+c)=-(-a-b)-c by Lm37 .=a+b-c by Lm38; end; theorem :: REAL_2'27_8 - (- a - b - c) = a + b + c proof thus -(-a-b-c)=-(-a-b)+c by Lm36 .=a+b+c by Lm38; end; :: using unary operations '-' and '*' theorem :: REAL_1'21_1 (- a) * b = -(a * b) by Lm20; theorem :: REAL_1'21_2 (- a) * b = a * (- b) by Lm21; theorem :: REAL_2'49_1 (- a) * (- b) = a * b proof thus (-a)*(-b)=-a*(-b) by Lm20 .=--a*b by Lm20 .=a*b; end; theorem :: REAL_2'49_2 - a * (- b) = a * b proof thus -a*(-b)=--a*b by Lm20 .=a*b; end; theorem :: REAL_2'49_3 -(-a) * b = a * b proof thus -(-a)*b=--a*b by Lm20 .=a*b; end; theorem Th180: :: REAL_2'71_1 (-1) * a = -a proof thus (-1)*a=-1*a by Lm20 .=-a; end; theorem :: REAL_2'71_2 (- a) * (- 1) = a proof (-1)*a=-1*a by Lm20 .=-a; hence (-a)*(-1)=--a by Lm20 .=a; end; theorem Th182: :: REAL_2'38 b<>0 & a*b=-b implies a=-1 proof assume A1:b<>0 & a*b=-b; then a*(b*b")=(-b)*b" by Th4; then a*1=(-b)*b" by A1,XCMPLX_0:def 7; hence a=-b*b" by Lm20 .=-1 by A1,XCMPLX_0:def 7; end; theorem Th183: :: Thx a * a = 1 implies a = 1 or a = -1 proof assume A1: a*a=1; (a-1)*(a+1) = a*(a+1) - 1*(a+1) by Th40 .= a*a+a*1 - 1*(a+1) by Th8 .= a*a+a*1 - 1*a-1*1 by Lm4 .= a*a+(a*1 - 1*a)-1*1 by Lm2 .= a*a+0-1*1 by Lm1 .=0 by A1; then a-1=0 or a+1=0 by Th6; then a-1+1=1 or a+1-1=0-1; hence thesis by Lm7,Lm8; end; Lm39: a - 2*a = -a proof thus a - 2*a = 1 * a - 2*a .= (1-2) * a by Th40 .= -a by Th180; end; theorem :: TOPREAL6'3 -a + 2 * a = a proof thus -a + 2*a = -(a - 2*a) by Lm36 .= --a by Lm39 .= a; end; theorem :: REAL_2'85_1 (a - b) * c = (b - a) * (- c) proof thus (a-b)*c =(-(b-a))*c by Lm18 .=(b-a)*(-c) by Lm21; end; theorem :: REAL_2'85_2 (a - b) * c = - (b - a) * c proof (a-b)*c =(-(b-a))*c by Lm18 .=(b-a)*(-c) by Lm21; hence thesis by Lm20; end; theorem :: TOPREAL6'2 a - 2 * a = -a by Lm39; :: using unary operations '-' and '/' theorem :: REAL_1'39_1 -a / b = (-a) / b by Lm35; theorem Th189: :: REAL_1'39_2 a / (- b) = -a / b proof a/(-b)=(a*(-1))/((-b)*(-1)) by Lm27; then a/(-b)=(-a*1)/((-b)*(-1)) by Lm20 .= (-a)/((-(-b))*1) by Lm21 .= -a/b by Lm35; hence thesis; end; theorem :: REAL_2'58_1 - a / (- b) = a / b proof thus -a/(-b)=--a/b by Th189 .=a/b; end; theorem Th191: :: REAL_2'58_2 -(- a) / b = a / b proof thus -(-a)/b=--a/b by Lm35 .=a/b; end; theorem :: REAL_2'58_3 (- a) / (- b) = a / b proof -(-a)/b=a/b by Th191; hence thesis by Th189; end; theorem :: REAL_2'58 (-a) / b = a / (-b) proof thus (-a)/b=-a/b by Lm35 .=a/(-b) by Th189; end; theorem :: REAL_2'71_3 -a = a / (-1) proof thus a / (-1) = -a/1 by Th189 .=-a; end; theorem :: REAL_2'71 a = (- a) / (-1) proof thus (-a)/(-1)=-(-a)/1 by Th189 .=a; end; theorem :: REAL_2'34 a / b = - 1 implies a = - b & b = - a proof assume A1:a/b=-1; then b <> 0 by Th49; then a=(-1)*b by A1,Lm15; then a=-1*b by Lm20; hence thesis; end; theorem :: REAL_2'40 b <> 0 & b / a = - b implies a = -1 proof assume A1: b<>0 & b/a=-b; then -b <> 0 by Th135; then a <> 0 by A1,Th49; then b=(-b)*a by A1,Lm15; then b=-b*a by Lm20; hence thesis by A1,Th182; end; theorem :: REAL_2'45_2 a <> 0 implies (-a) / a = -1 proof assume A1:a<>0; thus (-a)/a=-a/a by Lm35 .=-1 by A1,Lm17; end; theorem :: REAL_2'45_3 a <> 0 implies a / (- a) = -1 proof assume A1:a<>0; thus a/(-a)=-a/a by Th189 .=-1 by A1,Lm17; end; Lm40: a<>0 & a=a" implies a=1 or a=-1 proof assume A1:a<>0; assume a=a"; then a*a=1 by A1,XCMPLX_0:def 7; hence thesis by Th183; end; theorem :: REAL_2'46_2 a <> 0 & a = 1 / a implies a = 1 or a = -1 proof assume a<>0; then a=a" implies a=1 or a=-1 by Lm40; hence thesis by Lm16; end; theorem :: REAL_2'83: b <> 0 & d <> 0 & b <> -d & a / b = e / d implies a / b = (a + e) / (b + d) proof assume A1:b<>0 & d<>0 & b<>-d & a/b=e/d; then A2:b+d<>0 by XCMPLX_0:def 6; a*d=e*b by A1,Th96; then a*(b+d)=a*b+e*b by Th8; then a*(b+d)=(a+e)*b by Th8; hence thesis by A1,A2,Th95; end; :: using operation '"' theorem :: REAL_2'33_1 a" = b" implies a = b by Lm29; theorem :: REAL_1'31 a" = 0 implies a = 0 by Lm22; :: using '"' and '*' theorem b <> 0 implies a = a*b*b" proof assume A1: b <> 0; a*(b*b") = a*b*b" by Th4; then a*1 = a*b*b" by A1,XCMPLX_0:def 7; hence a = a*b*b"; end; theorem :: REAL_1'24 a" * b" = (a * b)" by Lm13; theorem :: REAL_2'47_1 (a * b")" = a" * b by Lm28; theorem :: REAL_2'47_2 (a" * b")" = a * b proof thus (a"*b")"=a""*b"" by Lm13 .=a*b; end; theorem :: REAL_2'42_1: a <> 0 & b <> 0 implies a * b" <> 0 proof assume A1:a<>0 & b<>0; then a"<>0 & b"<>0 by Lm22; hence thesis by A1,Th6; end; theorem :: REAL_2'42_3 a <> 0 & b <> 0 implies a" * b" <> 0 proof assume a<>0 & b<>0; then a"<>0 & b"<>0 by Lm22; hence thesis by Th6; end; theorem :: REAL_2'30_2 a * b" = 1 implies a = b proof assume a*b"=1; then a/b = 1 by XCMPLX_0:def 9; hence thesis by Th58; end; theorem :: REAL_2'35_2 a * b = 1 implies a = b" proof assume A1: a*b=1; then b<>0; hence a=b" by A1,XCMPLX_0:def 7; end; :: using '"', '*', and '+' canceled; theorem Th213: a <> 0 & b <> 0 implies a" + b" = (a + b)*(a*b)" proof assume A1: a <> 0 & b <> 0; a" = a"*1 & b" = b"*1; then a" = a"*(b"*b) & b" = b"*(a"*a) by A1,XCMPLX_0:def 7; then a" = (a"*b")*b & b" = (a"*b")*a by Th4; then a" = (a*b)"*b & b" = (a*b)"*a by Lm13; hence a" + b" = (a + b)*(a*b)" by Th8; end; Lm41: (- a)" = -a" proof thus (-a)"=1/(-a) by Lm16 .=-1/a by Th189 .=-a" by Lm16; end; :: using '"', '*', and '-' theorem a <> 0 & b <> 0 implies a" - b" = (b - a)*(a*b)" proof assume A1: a <> 0 & b <> 0; then A2: -b <> 0 by Th135; thus a" - b" = a" + -(b") by XCMPLX_0:def 8 .= a" + (-b)" by Lm41 .= (a + -b)*(a*-b)" by A1,A2,Th213 .= (a + -b)*(-a*b)" by Lm20 .= (a + -b)*-((a*b)") by Lm41 .= -((a + -b)*(a*b)") by Lm20 .= (-(a + -b))*(a*b)" by Lm20 .= (-(a - b))*(a*b)" by XCMPLX_0:def 8 .= (b - a)*(a*b)" by Lm18; end; :: using '"' and '/' theorem :: REAL_1'81 (a / b)" = b / a by Lm24; theorem (a"/b") = b/a proof thus (a"/b") = a"*b"" by XCMPLX_0:def 9 .= b/a by XCMPLX_0:def 9; end; theorem :: REAL_1'33_1 1 / a = a" by Lm16; theorem :: REAL_1'33_2 1 / a" = a by Lm34; theorem :: REAL_2'36_21 (1 / a)" = a proof 1/a=a" implies (1/a)"=a; hence thesis by Lm16; end; theorem :: REAL_2'33_3 1 / a = b" implies a = b proof 1/a=1/b implies a=b by Th59; hence thesis by Lm16; end; :: using '"', '*', and '/' theorem a/b" = a*b proof thus a/b" = a*b"" by XCMPLX_0:def 9 .= a*b; end; theorem a"*(c/b) = c/(a*b) proof thus a"*(c/b) = a"*(c*b") by XCMPLX_0:def 9 .= c*(a"*b") by Th4 .= c*(a*b)" by Lm13 .= c/(a*b) by XCMPLX_0:def 9; end; theorem a"/b = (a*b)" proof thus a"/b = a"*b" by XCMPLX_0:def 9 .= (a*b)" by Lm13; end; :: both unary operations theorem :: REAL_2'45_1 (- a)" = -a" by Lm41; theorem :: REAL_2'46_1 a <> 0 & a = a" implies a = 1 or a = -1 by Lm40; begin :: additional :: from JORDAN4 theorem a+b+c-b=a+c proof a+b+c-b=a+b-b+c by Lm2; hence a+b+c-b=a+c by Lm8; end; theorem a-b+c+b=a+c proof a-b+c+b=a+c-b+b by Lm2; hence a-b+c+b=a+c by Lm7; end; theorem a+b-c-b=a-c proof thus a+b-c-b=b+(a-c)-b by Lm2 .=a-c by Lm8; end; theorem a-b-c+b=a-c proof thus a-b-c+b=a-b-(c-b) by Lm5 .=a-c by Th22; end; theorem a-a-b=-b proof thus a-a-b=a-a+-b by XCMPLX_0:def 8 .=-b by Th25; end; theorem -a+a-b=-b proof thus -a+a-b=-a+a+-b by XCMPLX_0:def 8 .=-b by Th138; end; theorem a-b-a=-b proof thus a-b-a=a+-b-a by XCMPLX_0:def 8 .=-b by Lm8; end; theorem -a-b+a=-b proof thus -a-b+a=-a+-b+a by XCMPLX_0:def 8 .=-b by Th139; end;

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