### The Mizar article:

### From Double Loops to Fields

**by****Wojciech Skaba, and****Michal Muzalewski**

- Received September 27, 1990
Copyright (c) 1990 Association of Mizar Users

- MML identifier: ALGSTR_2
- [ MML identifier index ]

environ vocabulary VECTSP_1, FUNCT_1, RLVECT_1, ALGSTR_1, VECTSP_2, ARYTM_1, LATTICES, BINOP_1, ALGSTR_2; notation REAL_1, BINOP_1, STRUCT_0, RLVECT_1, VECTSP_1, VECTSP_2, ALGSTR_1; constructors BINOP_1, ALGSTR_1, VECTSP_2, REAL_1, MEMBERED, XBOOLE_0; clusters VECTSP_1, ALGSTR_1, VECTSP_2, MEMBERED, ZFMISC_1, XBOOLE_0; theorems VECTSP_1, ALGSTR_1, RLVECT_1, VECTSP_2; begin :: DOUBLE LOOPS reserve L for non empty doubleLoopStr; Lm1: for a,b being Element of F_Real for x,y being Real holds a = x & b = y implies a * b = x * y proof let a,b be Element of F_Real; let x,y be Real; assume a=x & b=y; hence a * b = multreal.(x,y) by VECTSP_1:def 10,def 15 .= x * y by VECTSP_1:def 14; end; Lm2: 0 = 0.F_Real by RLVECT_1:def 2,VECTSP_1:def 15; Lm3: for a,b being Element of F_Real st a<>0.F_Real ex x being Element of F_Real st a*x=b proof let a,b be Element of F_Real such that A1: a<>0.F_Real; reconsider p=a, q=b as Real by VECTSP_1:def 15; consider r be Real such that A2: p*r = q by A1,Lm2,ALGSTR_1:32; reconsider x=r as Element of F_Real by VECTSP_1:def 15; a*x = b by A2,Lm1; hence thesis; end; Lm4: for a,b being Element of F_Real st a<>0.F_Real ex x being Element of F_Real st x*a=b proof let a,b be Element of F_Real; assume a<>0.F_Real; then consider x being Element of F_Real such that A1: a*x=b by Lm3; thus thesis by A1; end; Lm5: for a,x,y being Element of F_Real st a<>0.F_Real holds a*x=a*y implies x=y by VECTSP_1:33; Lm6: for a,x,y being Element of F_Real st a<>0.F_Real holds x*a=y*a implies x=y by VECTSP_1:33; Lm7: for a being Element of F_Real holds a*0.F_Real = 0.F_Real by VECTSP_1:44; Lm8: for a being Element of F_Real holds 0.F_Real*a = 0.F_Real by VECTSP_1:44; :: Below is the basic definition of the mode of DOUBLE LOOP. :: The F_Real example in accordance with the many theorems proved above :: is used to prove the existence. definition cluster F_Real -> multLoop_0-like; coherence by Lm3,Lm4,Lm5,Lm6,Lm7,Lm8,ALGSTR_1:34; end; :: In the following part of this article the negation and minus functions :: are defined. This is the only definition of both functions in this article :: while some of their features are independently proved :: for various structures. definition let L be add-left-cancelable add-right-invertible (non empty LoopStr); let a be Element of L; canceled 6; func -a -> Element of L means :Def7: a+it = 0.L; existence by ALGSTR_1:def 9; uniqueness by ALGSTR_1:def 6; end; definition let L be add-left-cancelable add-right-invertible (non empty LoopStr); let a,b be Element of L; func a-b -> Element of L equals :Def8: a+ -b; correctness; end; definition cluster strict Abelian add-associative commutative associative distributive non degenerated left_zeroed right_zeroed Loop-like well-unital multLoop_0-like (non empty doubleLoopStr); existence proof take F_Real; thus thesis; end; end; definition mode doubleLoop is left_zeroed right_zeroed Loop-like well-unital multLoop_0-like (non empty doubleLoopStr); end; definition mode leftQuasi-Field is Abelian add-associative right-distributive non degenerated doubleLoop; end; reserve a,b,c,x,y,z for Element of L; :: The following theorem shows that the basic set of axioms of the :: left quasi-field may be replaced with the following one, :: by just removing a few and adding some other axioms. canceled 11; theorem L is leftQuasi-Field iff (for a holds a + 0.L = a) & (for a ex x st a+x = 0.L) & (for a,b,c holds (a+b)+c = a+(b+c)) & (for a,b holds a+b = b+a) & 0.L <> 1_ L & (for a holds a * 1_ L = a) & (for a holds 1_ L * a = a) & (for a,b st a<>0.L ex x st a*x=b) & (for a,b st a<>0.L ex x st x*a=b) & (for a,x,y st a<>0.L holds a*x=a*y implies x=y) & (for a,x,y st a<>0.L holds x*a=y*a implies x=y) & (for a holds a*0.L = 0.L) & (for a holds 0.L*a = 0.L) & (for a,b,c holds a*(b+c) = a*b + a*c) proof thus L is leftQuasi-Field implies (for a holds a + 0.L = a) & (for a ex x st a+x = 0.L) & (for a,b,c holds (a+b)+c = a+(b+c)) & (for a,b holds a+b = b+a) & 0.L <> 1_ L & (for a holds a * 1_ L = a) & (for a holds 1_ L * a = a) & (for a,b st a<>0.L ex x st a*x=b) & (for a,b st a<>0.L ex x st x*a=b) & (for a,x,y st a<>0.L holds a*x=a*y implies x=y) & (for a,x,y st a<>0.L holds x*a=y*a implies x=y) & (for a holds a*0.L = 0.L) & (for a holds 0.L*a = 0.L) & (for a,b,c holds a*(b+c) = a*b + a*c) by ALGSTR_1:7,34,RLVECT_1:def 5,def 6,def 7,VECTSP_1:def 11,def 21,VECTSP_2: def 2; assume A1: (for a holds a + 0.L = a) & (for a ex x st a+x = 0.L) & (for a,b,c holds (a+b)+c = a+(b+c)) & (for a,b holds a+b = b+a) & 0.L <> 1_ L & (for a holds a * 1_ L = a) & (for a holds 1_ L * a = a) & (for a,b st a<>0.L ex x st a*x=b) & (for a,b st a<>0.L ex x st x*a=b) & (for a,x,y st a<>0.L holds a*x=a*y implies x=y) & (for a,x,y st a<>0.L holds x*a=y*a implies x=y) & (for a holds a*0.L = 0.L) & (for a holds 0.L*a = 0.L) & (for a,b,c holds a*(b+c) = a*b + a*c); now thus A2:for a holds 0.L + a = a proof let a; thus 0.L + a = a + 0.L by A1 .= a by A1; end; thus A3: for a,b ex x st a+x=b proof let a,b; consider y such that A4: a+y = 0.L by A1; take x = y+b; thus a+x = 0.L + b by A1,A4 .= b by A2; end; thus for a,b ex x st x+a=b proof let a,b; consider x such that A5: a+x=b by A3; take x; thus thesis by A1,A5; end; thus A6: for a,x,y holds a+x=a+y implies x=y proof let a,x,y; consider z such that A7: z+a = 0.L by A1,ALGSTR_1:3; assume a+x = a+y; then (z+a)+x = z+(a+y) by A1 .= (z+a)+y by A1; hence x = 0.L + y by A2,A7 .= y by A2; end; thus for a,x,y holds x+a=y+a implies x=y proof let a,x,y; assume x+a = y+a; then a+x= y+a by A1 .= a+y by A1; hence thesis by A6; end; end; hence thesis by A1,ALGSTR_1:7,34,def 5,RLVECT_1:def 5,def 6,def 7, VECTSP_1:def 11,def 21,VECTSP_2:def 2; end; canceled; theorem Th14: for G being Abelian right-distributive doubleLoop, a,b being Element of G holds a*(-b) = -(a*b) proof let G be Abelian right-distributive doubleLoop, a,b be Element of G; a*b + a*(-b) = a*(b+ -b) by VECTSP_1:def 11 .= a*0.G by Def7 .= 0.G by ALGSTR_1:34; hence thesis by Def7; end; theorem Th15: for G being Abelian add-left-cancelable add-right-invertible (non empty LoopStr), a being Element of G holds -(-a) = a proof let G be Abelian add-left-cancelable add-right-invertible (non empty LoopStr), a be Element of G; -a+a = 0.G by Def7; hence thesis by Def7; end; theorem for G being Abelian right-distributive doubleLoop holds (-1_ G)*(-1_ G) = 1_ G proof let G be Abelian right-distributive doubleLoop; thus (-1_ G)*(-1_ G) = -((-1_ G)*1_ G) by Th14 .= -(-1_ G) by VECTSP_2:def 2 .= 1_ G by Th15; end; theorem for G being Abelian right-distributive doubleLoop, a,x,y being Element of G holds a*(x-y) = a*x - a*y proof let G be Abelian right-distributive doubleLoop, a,x,y be Element of G; thus a*(x-y) = a*(x+ -y) by Def8 .= a*x + a*(-y) by VECTSP_1:def 11 .= a*x + -(a*y) by Th14 .= a*x - a*y by Def8; end; :: RIGHT QUASI-FIELD :: The next contemplated algebraic structure is so called right quasi-field. :: This structure is defined as a DOUBLE LOOP augmented with three axioms. :: The reasoning is similar to that of left quasi-field. definition mode rightQuasi-Field is Abelian add-associative left-distributive non degenerated doubleLoop; end; canceled; theorem L is rightQuasi-Field iff (for a holds a + 0.L = a) & (for a ex x st a+x = 0.L) & (for a,b,c holds (a+b)+c = a+(b+c)) & (for a,b holds a+b = b+a) & 0.L <> 1_ L & (for a holds a * 1_ L = a) & (for a holds 1_ L * a = a) & (for a,b st a<>0.L ex x st a*x=b) & (for a,b st a<>0.L ex x st x*a=b) & (for a,x,y st a<>0.L holds a*x=a*y implies x=y) & (for a,x,y st a<>0.L holds x*a=y*a implies x=y) & (for a holds a*0.L = 0.L) & (for a holds 0.L*a = 0.L) & (for a,b,c holds (b+c)*a = b*a + c*a) proof thus L is rightQuasi-Field implies (for a holds a + 0.L = a) & (for a ex x st a+x = 0.L) & (for a,b,c holds (a+b)+c = a+(b+c)) & (for a,b holds a+b = b+a) & 0.L <> 1_ L & (for a holds a * 1_ L = a) & (for a holds 1_ L * a = a) & (for a,b st a<>0.L ex x st a*x=b) & (for a,b st a<>0.L ex x st x*a=b) & (for a,x,y st a<>0.L holds a*x=a*y implies x=y) & (for a,x,y st a<>0.L holds x*a=y*a implies x=y) & (for a holds a*0.L = 0.L) & (for a holds 0.L*a = 0.L) & (for a,b,c holds (b+c)*a = b*a + c*a) by ALGSTR_1:7,34,RLVECT_1:def 5,def 6,def 7,VECTSP_1:def 12,def 21,VECTSP_2:def 2; assume A1: (for a holds a + 0.L = a) & (for a ex x st a+x = 0.L) & (for a,b,c holds (a+b)+c = a+(b+c)) & (for a,b holds a+b = b+a) & 0.L <> 1_ L & (for a holds a * 1_ L = a) & (for a holds 1_ L * a = a) & (for a,b st a<>0.L ex x st a*x=b) & (for a,b st a<>0.L ex x st x*a=b) & (for a,x,y st a<>0.L holds a*x=a*y implies x=y) & (for a,x,y st a<>0.L holds x*a=y*a implies x=y) & (for a holds a*0.L = 0.L) & (for a holds 0.L*a = 0.L) & (for a,b,c holds (b+c)*a = b*a + c*a); now thus A2:for a holds 0.L + a = a proof let a; thus 0.L + a = a + 0.L by A1 .= a by A1; end; thus for a,b ex x st a+x=b proof let a,b; consider y such that A3: a+y = 0.L by A1; take x = y+b; thus a+x = 0.L + b by A1,A3 .= b by A2; end; thus for a,b ex x st x+a=b proof let a,b; consider y such that A4: y+a = 0.L by A1,ALGSTR_1:3; take x = b+y; thus x+a = b + 0.L by A1,A4 .= b by A1; end; thus for a,x,y holds a+x=a+y implies x=y proof let a,x,y; consider z such that A5: z+a = 0.L by A1,ALGSTR_1:3; assume a+x = a+y; then (z+a)+x = z+(a+y) by A1 .= (z+a)+y by A1; hence x = 0.L + y by A2,A5 .= y by A2; end; thus for a,x,y holds x+a=y+a implies x=y proof let a,x,y; consider z such that A6: a+z = 0.L by A1; assume x+a = y+a; then x+(a+z) = (y+a)+z by A1 .= y+(a+z) by A1; hence x = y + 0.L by A1,A6 .= y by A1; end; end; hence thesis by A1,ALGSTR_1:7,34,def 5,RLVECT_1:def 5,def 6,def 7, VECTSP_1:def 12,def 21,VECTSP_2:def 2; end; :: Below, the three features concerned with the - function, :: numbered 20..22 are proved. Where necessary, a few additional :: facts are included. They are independent of the similar proofs :: performed for the left quasi-field. reserve G for left-distributive doubleLoop, a,b,x,y for Element of G; canceled; theorem Th21: (-b)*a = -(b*a) proof b*a + (-b)*a = (b+(-b))*a by VECTSP_1:def 12 .= 0.G*a by Def7 .= 0.G by ALGSTR_1:34; hence thesis by Def7; end; canceled; theorem for G being Abelian left-distributive doubleLoop holds (-1_ G)*(-1_ G) = 1_ G proof let G be Abelian left-distributive doubleLoop; thus (-1_ G)*(-1_ G) = -(1_ G*(-1_ G)) by Th21 .= -(-1_ G) by VECTSP_2:def 2 .= 1_ G by Th15; end; theorem (x-y)*a = x*a - y*a proof thus (x-y)*a = (x+(-y))*a by Def8 .= x*a + (-y)*a by VECTSP_1:def 12 .= x*a + (-(y*a)) by Th21 .= x*a - y*a by Def8; end; :: DOUBLE SIDED QUASI-FIELD :: The next contemplated algebraic structure is so called double sided :: quasi-field. This structure is also defined as a DOUBLE LOOP augmented :: with four axioms, while its relevance to left/right quasi-field is :: independently contemplated. :: The reasoning is similar to that of left/right quasi-field. definition mode doublesidedQuasi-Field is Abelian add-associative distributive non degenerated doubleLoop; end; reserve a,b,c,x,y,z for Element of L; canceled; theorem L is doublesidedQuasi-Field iff (for a holds a + 0.L = a) & (for a ex x st a+x = 0.L) & (for a,b,c holds (a+b)+c = a+(b+c)) & (for a,b holds a+b = b+a) & 0.L <> 1_ L & (for a holds a * 1_ L = a) & (for a holds 1_ L * a = a) & (for a,b st a<>0.L ex x st a*x=b) & (for a,b st a<>0.L ex x st x*a=b) & (for a,x,y st a<>0.L holds a*x=a*y implies x=y) & (for a,x,y st a<>0.L holds x*a=y*a implies x=y) & (for a holds a*0.L = 0.L) & (for a holds 0.L*a = 0.L) & (for a,b,c holds a*(b+c) = a*b + a*c) & (for a,b,c holds (b+c)*a = b*a + c*a) proof thus L is doublesidedQuasi-Field implies (for a holds a + 0.L = a) & (for a ex x st a+x = 0.L) & (for a,b,c holds (a+b)+c = a+(b+c)) & (for a,b holds a+b = b+a) & 0.L <> 1_ L & (for a holds a * 1_ L = a) & (for a holds 1_ L * a = a) & (for a,b st a<>0.L ex x st a*x=b) & (for a,b st a<>0.L ex x st x*a=b) & (for a,x,y st a<>0.L holds a*x=a*y implies x=y) & (for a,x,y st a<>0.L holds x*a=y*a implies x=y) & (for a holds a*0.L = 0.L) & (for a holds 0.L*a = 0.L) & (for a,b,c holds a*(b+c) = a*b + a*c) & (for a,b,c holds (b+c)*a = b*a + c*a) by ALGSTR_1:7,34,RLVECT_1:def 5,def 6,def 7,VECTSP_1:def 18,def 21,VECTSP_2 :def 2; assume A1: (for a holds a + 0.L = a) & (for a ex x st a+x = 0.L) & (for a,b,c holds (a+b)+c = a+(b+c)) & (for a,b holds a+b = b+a) & 0.L <> 1_ L & (for a holds a * 1_ L = a) & (for a holds 1_ L * a = a) & (for a,b st a<>0.L ex x st a*x=b) & (for a,b st a<>0.L ex x st x*a=b) & (for a,x,y st a<>0.L holds a*x=a*y implies x=y) & (for a,x,y st a<>0.L holds x*a=y*a implies x=y) & (for a holds a*0.L = 0.L) & (for a holds 0.L*a = 0.L) & (for a,b,c holds a*(b+c) = a*b + a*c) & (for a,b,c holds (b+c)*a = b*a + c*a); now thus A2:for a holds 0.L + a = a proof let a; thus 0.L + a = a + 0.L by A1 .= a by A1; end; thus for a,b ex x st a+x=b proof let a,b; consider y such that A3: a+y = 0.L by A1; take x = y+b; thus a+x = 0.L + b by A1,A3 .= b by A2; end; thus for a,b ex x st x+a=b proof let a,b; consider y such that A4: y+a = 0.L by A1,ALGSTR_1:3; take x = b+y; thus x+a = b + 0.L by A1,A4 .= b by A1; end; thus for a,x,y holds a+x=a+y implies x=y proof let a,x,y; consider z such that A5: z+a = 0.L by A1,ALGSTR_1:3; assume a+x = a+y; then (z+a)+x = z+(a+y) by A1 .= (z+a)+y by A1; hence x = 0.L + y by A2,A5 .= y by A2; end; thus for a,x,y holds x+a=y+a implies x=y proof let a,x,y; consider z such that A6: a+z = 0.L by A1; assume x+a = y+a; then x+(a+z) = (y+a)+z by A1 .= y+(a+z) by A1; hence x = y + 0.L by A1,A6 .= y by A1; end; end; hence thesis by A1,ALGSTR_1:7,34,def 5,RLVECT_1:def 5,def 6,def 7, VECTSP_1:def 18,def 21,VECTSP_2:def 2; end; :: SKEW FIELD :: A Skew-Field is defined as a double sided quasi-field extended :: with the associativity of multiplication. definition mode _Skew-Field is associative doublesidedQuasi-Field; end; Lm9: 0.L <> 1_ L & (for a holds a * 1_ L = a) & (for a st a<>0.L ex x st a*x = 1_ L) & (for a,b,c holds (a*b)*c = a*(b*c)) & (for a holds a*0.L = 0.L) & (for a holds 0.L*a = 0.L) implies (a*b = 1_ L implies b*a = 1_ L) proof assume A1: 0.L <> 1_ L & (for a holds a * 1_ L = a) & (for a st a<>0.L ex x st a*x = 1_ L) & (for a,b,c holds (a*b)*c = a*(b*c)) & (for a holds a*0.L = 0.L) & (for a holds 0.L*a = 0.L); thus a*b = 1_ L implies b*a = 1_ L proof assume A2: a*b = 1_ L; then b<>0.L by A1; then consider x such that A3: b * x = 1_ L by A1; thus b*a = (b*a) * (b*x) by A1,A3 .= ((b*a) * b) * x by A1 .= (b * 1_ L) * x by A1,A2 .= 1_ L by A1,A3; end; end; Lm10: 0.L <> 1_ L & (for a holds a * 1_ L = a) & (for a st a<>0.L ex x st a*x = 1_ L) & (for a,b,c holds (a*b)*c = a*(b*c)) & (for a holds a*0.L = 0.L) & (for a holds 0.L*a = 0.L) implies 1_ L*a = a*1_ L proof assume A1: 0.L <> 1_ L & (for a holds a * 1_ L = a) & (for a st a<>0.L ex x st a*x = 1_ L) & (for a,b,c holds (a*b)*c = a*(b*c)) & (for a holds a*0.L = 0.L) & (for a holds 0.L*a = 0.L); A2: a<>0.L implies 1_ L*a = a*1_ L proof assume a<>0.L; then consider x such that A3: a * x = 1_ L by A1; thus 1_ L*a = a * (x*a) by A1,A3 .= a*1_ L by A1,A3,Lm9; end; a=0.L implies 1_ L*a = a*1_ L proof assume A4: a=0.L; hence 1_ L*a = 0.L by A1 .= a*1_ L by A1,A4; end; hence thesis by A2; end; Lm11: 0.L <> 1_ L & (for a holds a * 1_ L = a) & (for a st a<>0.L ex x st a*x = 1_ L) & (for a,b,c holds (a*b)*c = a*(b*c)) & (for a holds a*0.L = 0.L) & (for a holds 0.L*a = 0.L) implies for a st a<>0.L ex x st x*a = 1_ L proof assume A1: 0.L <> 1_ L & (for a holds a * 1_ L = a) & (for a st a<>0.L ex x st a*x = 1_ L) & (for a,b,c holds (a*b)*c = a*(b*c)) & (for a holds a*0.L = 0.L) & (for a holds 0.L*a = 0.L); let a; assume a<>0.L; then consider x such that A2: a * x = 1_ L by A1; x*a=1_ L by A1,A2,Lm9; hence thesis; end; :: The following theorem shows that the basic set of axioms of the :: skew field may be replaced with the following one, :: by just removing a few and adding some other axioms. :: A few theorems proved earlier are highly utilized. canceled 5; theorem Th32: L is _Skew-Field iff (for a holds a + 0.L = a) & (for a ex x st a+x = 0.L) & (for a,b,c holds (a+b)+c = a+(b+c)) & (for a,b holds a+b = b+a) & 0.L <> 1_ L & (for a holds a * 1_ L = a) & (for a st a<>0.L ex x st a*x = 1_ L) & (for a holds a*0.L = 0.L) & (for a holds 0.L*a = 0.L) & (for a,b,c holds (a*b)*c = a*(b*c)) & (for a,b,c holds a*(b+c) = a*b + a*c) & (for a,b,c holds (b+c)*a = b*a + c*a) proof thus L is _Skew-Field implies (for a holds a + 0.L = a) & (for a ex x st a+x = 0.L) & (for a,b,c holds (a+b)+c = a+(b+c)) & (for a,b holds a+b = b+a) & 0.L <> 1_ L & (for a holds a * 1_ L = a) & (for a st a<>0.L ex x st a*x = 1_ L) & (for a holds a*0.L = 0.L) & (for a holds 0.L*a = 0.L) & (for a,b,c holds (a*b)*c = a*(b*c)) & (for a,b,c holds a*(b+c) = a*b + a*c) & (for a,b,c holds (b+c)*a = b*a + c*a) by ALGSTR_1:7,34,RLVECT_1:def 5,def 6,def 7,VECTSP_1:def 16,def 18,def 21,VECTSP_2:def 2; assume A1: (for a holds a + 0.L = a) & (for a ex x st a+x = 0.L) & (for a,b,c holds (a+b)+c = a+(b+c)) & (for a,b holds a+b = b+a) & 0.L <> 1_ L & (for a holds a * 1_ L = a) & (for a st a<>0.L ex x st a*x = 1_ L) & (for a holds a*0.L = 0.L) & (for a holds 0.L*a = 0.L) & (for a,b,c holds (a*b)*c = a*(b*c)) & (for a,b,c holds a*(b+c) = a*b + a*c) & (for a,b,c holds (b+c)*a = b*a + c*a); now thus A2:for a holds 0.L + a = a proof let a; thus 0.L + a = a + 0.L by A1 .= a by A1; end; thus for a,b ex x st a+x=b proof let a,b; consider y such that A3: a+y = 0.L by A1; take x = y+b; thus a+x = 0.L + b by A1,A3 .= b by A2; end; thus for a,b ex x st x+a=b proof let a,b; consider y such that A4: y+a = 0.L by A1,ALGSTR_1:3; take x = b+y; thus x+a = b + 0.L by A1,A4 .= b by A1; end; thus for a,x,y holds a+x=a+y implies x=y proof let a,x,y; consider z such that A5: z+a = 0.L by A1,ALGSTR_1:3; assume a+x = a+y; then (z+a)+x = z+(a+y) by A1 .= (z+a)+y by A1; hence x = 0.L + y by A2,A5 .= y by A2; end; thus for a,x,y holds x+a=y+a implies x=y proof let a,x,y; consider z such that A6: a+z = 0.L by A1; assume x+a = y+a; then x+(a+z) = (y+a)+z by A1 .= y+(a+z) by A1; hence x = y + 0.L by A1,A6 .= y by A1; end; thus A7: for a holds 1_ L * a = a proof let a; thus 1_ L*a = a*1_ L by A1,Lm10 .= a by A1; end; thus for a,b st a<>0.L ex x st a*x=b proof let a,b; assume a<>0.L; then consider y such that A8: a*y = 1_ L by A1; take x = y*b; thus a*x = 1_ L * b by A1,A8 .= b by A7; end; thus for a,b st a<>0.L ex x st x*a=b proof let a,b; assume a<>0.L; then consider y such that A9: y*a = 1_ L by A1,Lm11; take x = b*y; thus x*a = b * 1_ L by A1,A9 .= b by A1; end; thus for a,x,y st a<>0.L holds a*x=a*y implies x=y proof let a,x,y; assume a<>0.L; then consider z such that A10: z*a = 1_ L by A1,Lm11; assume a*x = a*y; then (z*a)*x = z*(a*y) by A1 .= (z*a)*y by A1; hence x = 1_ L * y by A7,A10 .= y by A7; end; thus for a,x,y st a<>0.L holds x*a=y*a implies x=y proof let a,x,y; assume a<>0.L; then consider z such that A11: a*z = 1_ L by A1; assume x*a = y*a; then x*(a*z) = (y*a)*z by A1 .= y*(a*z) by A1; hence x = y * 1_ L by A1,A11 .= y by A1; end; end; hence thesis by A1,ALGSTR_1:7,34,def 5,RLVECT_1:def 5,def 6,def 7, VECTSP_1:def 16,def 18,def 21,VECTSP_2:def 2; end; :: FIELD :: A _Field is defined as a Skew-Field with the axiom of the commutativity :: of multiplication. definition mode _Field is commutative _Skew-Field; end; canceled; theorem L is _Field iff (for a holds a + 0.L = a) & (for a ex x st a+x = 0.L) & (for a,b,c holds (a+b)+c = a+(b+c)) & (for a,b holds a+b = b+a) & 0.L <> 1_ L & (for a holds a * 1_ L = a) & (for a st a<>0.L ex x st a*x = 1_ L) & (for a holds a*0.L = 0.L) & (for a,b,c holds (a*b)*c = a*(b*c)) & (for a,b,c holds a*(b+c) = a*b + a*c) & (for a,b holds a*b = b*a) proof thus L is _Field implies (for a holds a + 0.L = a) & (for a ex x st a+x = 0.L) & (for a,b,c holds (a+b)+c = a+(b+c)) & (for a,b holds a+b = b+a) & 0.L <> 1_ L & (for a holds a * 1_ L = a) & (for a st a<>0.L ex x st a*x = 1_ L) & (for a holds a*0.L = 0.L) & (for a,b,c holds (a*b)*c = a*(b*c)) & (for a,b,c holds a*(b+c) = a*b + a*c) & (for a,b holds a*b = b*a) by Th32,VECTSP_1:def 17; assume A1: (for a holds a + 0.L = a) & (for a ex x st a+x = 0.L) & (for a,b,c holds (a+b)+c = a+(b+c)) & (for a,b holds a+b = b+a) & 0.L <> 1_ L & (for a holds a * 1_ L = a) & (for a st a<>0.L ex x st a*x = 1_ L) & (for a holds a*0.L = 0.L) & (for a,b,c holds (a*b)*c = a*(b*c)) & (for a,b,c holds a*(b+c) = a*b + a*c) & (for a,b holds a*b = b*a); A2: for a holds 0.L*a = 0.L proof let a; thus 0.L*a = a*0.L by A1 .= 0.L by A1; end; for a,b,c holds (b+c)*a = b*a + c*a proof let a,b,c; thus (b+c)*a = a*(b+c) by A1 .= a*b + a*c by A1 .= b*a + a*c by A1 .= b*a + c*a by A1; end; hence thesis by A1,A2,Th32,VECTSP_1:def 17; end;

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