### The Mizar article:

### Axioms of Incidency

**by****Wojciech A. Trybulec**

- Received April 14, 1989
Copyright (c) 1989 Association of Mizar Users

- MML identifier: INCSP_1
- [ MML identifier index ]

environ vocabulary RELAT_1, BOOLE, INCSP_1; notation TARSKI, XBOOLE_0, ENUMSET1, ZFMISC_1, SUBSET_1, DOMAIN_1, RELSET_1; constructors DOMAIN_1, RELSET_1, MEMBERED, XBOOLE_0; clusters SUBSET_1, MEMBERED, ZFMISC_1, XBOOLE_0; requirements SUBSET, BOOLE; definitions TARSKI; theorems ENUMSET1, TARSKI, ZFMISC_1, RELSET_1, XBOOLE_0, XBOOLE_1; begin definition struct IncProjStr (# Points, Lines -> non empty set, Inc -> Relation of the Points, the Lines #); end; definition struct (IncProjStr) IncStruct (# Points, Lines, Planes -> non empty set, Inc -> Relation of the Points,the Lines, Inc2 -> Relation of the Points,the Planes, Inc3 -> Relation of the Lines,the Planes #); end; definition let S be IncProjStr; mode POINT of S is Element of the Points of S; mode LINE of S is Element of the Lines of S; end; definition let S be IncStruct; mode PLANE of S is Element of the Planes of S; end; reserve S for IncStruct; reserve A,B,C,D for POINT of S; reserve L for LINE of S; reserve P for PLANE of S; reserve F,G for Subset of the Points of S; :: Definitions of predicates: on, is_collinear, is_coplanar definition let S be IncProjStr; let A be POINT of S, L be LINE of S; pred A on L means :Def1: [A,L] in the Inc of S; end; definition let S; let A be POINT of S, P be PLANE of S; pred A on P means :Def2: [A,P] in the Inc2 of S; end; definition let S; let L be LINE of S, P be PLANE of S; pred L on P means :Def3: [L,P] in the Inc3 of S; end; definition let S be IncProjStr; let F be Subset of the Points of S, L be LINE of S; pred F on L means :Def4: for A being POINT of S st A in F holds A on L; end; definition let S; let F be Subset of the Points of S, P be PLANE of S; pred F on P means :Def5: for A st A in F holds A on P; end; definition let S be IncProjStr; let F be Subset of the Points of S; attr F is linear means :Def6: ex L being LINE of S st F on L; synonym F is_collinear; end; definition let S be IncStruct; let F be Subset of the Points of S; attr F is planar means :Def7: ex P be PLANE of S st F on P; synonym F is_coplanar; end; :: Definitional theorems of predicates: on, is_collinear, is_coplanar canceled 10; theorem Th11: {A,B} on L iff A on L & B on L proof thus {A,B} on L implies A on L & B on L proof assume A1: {A,B} on L; A in {A,B} & B in {A,B} by TARSKI:def 2; hence thesis by A1,Def4; end; assume A2: A on L & B on L; let C be POINT of S; assume C in {A,B}; hence thesis by A2,TARSKI:def 2; end; theorem Th12: {A,B,C} on L iff A on L & B on L & C on L proof thus {A,B,C} on L implies A on L & B on L & C on L proof assume A1: {A,B,C} on L; A in {A,B,C} & B in {A,B,C} & C in {A,B,C} by ENUMSET1:14; hence thesis by A1,Def4; end; assume A2: A on L & B on L & C on L; let D be POINT of S; assume D in {A,B,C}; hence thesis by A2,ENUMSET1:def 1; end; theorem Th13: {A,B} on P iff A on P & B on P proof thus {A,B} on P implies A on P & B on P proof assume A1: {A,B} on P; A in {A,B} & B in {A,B} by TARSKI:def 2; hence thesis by A1,Def5; end; assume A2: A on P & B on P; let C be POINT of S; assume C in {A,B}; hence thesis by A2,TARSKI:def 2; end; theorem Th14: {A,B,C} on P iff A on P & B on P & C on P proof thus {A,B,C} on P implies A on P & B on P & C on P proof assume A1: {A,B,C} on P; A in {A,B,C} & B in {A,B,C} & C in {A,B,C} by ENUMSET1:14; hence thesis by A1,Def5; end; assume A2: A on P & B on P & C on P; let D be POINT of S; assume D in {A,B,C}; hence thesis by A2,ENUMSET1:def 1; end; theorem Th15: {A,B,C,D} on P iff A on P & B on P & C on P & D on P proof thus {A,B,C,D} on P implies A on P & B on P & C on P & D on P proof assume A1: {A,B,C,D} on P; A in {A,B,C,D} & B in {A,B,C,D} & C in {A,B,C,D} & D in {A,B,C,D} by ENUMSET1:19; hence thesis by A1,Def5; end; assume A2: A on P & B on P & C on P & D on P; let E be POINT of S; assume E in {A,B,C,D}; hence thesis by A2,ENUMSET1:18; end; theorem Th16: G c= F & F on L implies G on L proof assume that A1: G c= F and A2: F on L; let A be POINT of S; assume A in G; hence thesis by A1,A2,Def4; end; theorem Th17: G c= F & F on P implies G on P proof assume that A1: G c= F and A2: F on P; let A be POINT of S; assume A in G; hence thesis by A1,A2,Def5; end; theorem Th18: F on L & A on L iff F \/ {A} on L proof thus F on L & A on L implies F \/ {A} on L proof assume A1: F on L & A on L; let C be POINT of S; assume C in F \/ {A}; then C in F or C in {A} by XBOOLE_0:def 2; hence C on L by A1,Def4,TARSKI:def 1; end; assume A2: F \/ {A} on L; F c= F \/ {A} by XBOOLE_1:7; hence F on L by A2,Th16; {A} c= F \/ {A} by XBOOLE_1:7; then {A,A} c= F \/ {A} by ENUMSET1:69; then {A,A} on L by A2,Th16; hence A on L by Th11; end; theorem Th19: F on P & A on P iff F \/ {A} on P proof thus F on P & A on P implies F \/ {A} on P proof assume A1: F on P & A on P; let C be POINT of S; assume C in F \/ {A}; then C in F or C in {A} by XBOOLE_0:def 2; hence C on P by A1,Def5,TARSKI:def 1; end; assume A2: F \/ {A} on P; F c= F \/ {A} by XBOOLE_1:7; hence F on P by A2,Th17; {A} c= F \/ {A} by XBOOLE_1:7; then {A,A} c= F \/ {A} by ENUMSET1:69; then {A,A} on P by A2,Th17; hence A on P by Th13; end; theorem Th20: F \/ G on L iff F on L & G on L proof thus F \/ G on L implies F on L & G on L proof assume A1: F \/ G on L; F c= F \/ G & G c= F \/ G by XBOOLE_1:7; hence thesis by A1,Th16; end; assume A2: F on L & G on L; let C be POINT of S; assume C in F \/ G; then C in F or C in G by XBOOLE_0:def 2; hence thesis by A2,Def4; end; theorem Th21: F \/ G on P iff F on P & G on P proof thus F \/ G on P implies F on P & G on P proof assume A1: F \/ G on P; F c= F \/ G & G c= F \/ G by XBOOLE_1:7; hence thesis by A1,Th17; end; assume A2: F on P & G on P; let C be POINT of S; assume C in F \/ G; then C in F or C in G by XBOOLE_0:def 2; hence thesis by A2,Def5; end; theorem G c= F & F is_collinear implies G is_collinear proof assume that A1: G c= F and A2: F is_collinear; consider L such that A3: F on L by A2,Def6; take L; let A be POINT of S; assume A in G; hence thesis by A1,A3,Def4; end; theorem G c= F & F is_coplanar implies G is_coplanar proof assume that A1: G c= F and A2: F is_coplanar; consider P such that A3: F on P by A2,Def7; take P; let A be POINT of S; assume A in G; hence thesis by A1,A3,Def5; end; reconsider Po = {0,1,2,3} as non empty set by ENUMSET1:19; reserve a,b,c for Element of Po; reconsider _Zero = 0, One = 1, Two = 2, Three = 3 as Element of Po by ENUMSET1:19; {_Zero,One} in {{a,b}: a <> b }; then reconsider Li = {{a,b}: a <> b } as non empty set; {_Zero,One,Two} in {{a,b,c}: a <> b & a <> c & b <> c }; then reconsider Pl = {{a,b,c}: a <> b & a <> c & b <> c } as non empty set; reserve k,l for Element of Li; reserve p,q for Element of Pl; set i1 = {[a,l]: a in l}; i1 c= [:Po,Li:] proof let x be set; assume x in i1; then ex a,l st x =[a,l] & a in l; hence x in [:Po,Li:]; end; then reconsider i1 as Relation of Po,Li by RELSET_1:def 1; set i2 = {[a,p]: a in p}; i2 c= [:Po,Pl:] proof let x be set; assume x in i2; then ex a,p st x =[a,p] & a in p; hence x in [:Po,Pl:]; end; then reconsider i2 as Relation of Po,Pl by RELSET_1:def 1; set i3 = {[l,p]: l c= p}; i3 c= [:Li,Pl:] proof let x be set; assume x in i3; then ex l,p st x =[l,p] & l c= p; hence x in [:Li,Pl:]; end; then reconsider i3 as Relation of Li,Pl by RELSET_1:def 1; :: Introduction of mode IncSpace Lm1: now set S = IncStruct (# Po,Li,Pl,i1,i2,i3 #); A1: [a,l] in i1 implies a in l proof assume [a,l] in i1; then consider b,k such that A2: [a,l] = [b,k] and A3: b in k; a = b & l = k by A2,ZFMISC_1:33; hence thesis by A3; end; A4: [a,p] in i2 implies a in p proof assume [a,p] in i2; then consider b,q such that A5: [a,p] = [b,q] and A6: b in q; a = b & p = q by A5,ZFMISC_1:33; hence thesis by A6; end; A7: [l,p] in i3 implies l c= p proof assume [l,p] in i3; then consider k,q such that A8: [l,p] = [k,q] and A9: k c= q; l = k & p = q by A8,ZFMISC_1:33; hence thesis by A9; end; thus for L being LINE of S ex A,B being POINT of S st A <> B & {A,B} on L proof let L be LINE of S; reconsider l = L as Element of Li; l in Li; then consider a,b such that A10: l = {a,b} and A11: a <> b; reconsider A = a, B = b as POINT of S; take A,B; thus A <> B by A11; a in l & b in l by A10,TARSKI:def 2; then [a,l] in i1 & [b,l] in i1; then A on L & B on L by Def1; hence {A,B} on L by Th11; end; thus A12: for A,B being POINT of S ex L being LINE of S st {A,B} on L proof let A,B be POINT of S; reconsider a = A ,b = B as Element of Po; A13: now assume a <> b; then {a,b} in Li; then consider l such that A14: l = {a,b}; reconsider L = l as LINE of S; a in l & b in l by A14,TARSKI:def 2; then [a,l] in i1 & [b,l] in i1; then A on L & B on L by Def1; then {A,B} on L by Th11; hence thesis; end; now assume A15: a = b; for a ex c st a <> c proof let a; A16: now assume a = 0; then a <> One; hence thesis; end; now assume a = 1 or a = 2 or a = 3; then a <> _Zero; hence thesis; end; hence thesis by A16,ENUMSET1:18; end; then consider c being Element of Po such that A17: a <> c; {a,c} in Li by A17; then consider l such that A18: l = {a,c}; reconsider L = l as LINE of S; a in l by A18,TARSKI:def 2; then [a,l] in i1; then A on L & B on L by A15,Def1; then {A,B} on L by Th11; hence thesis; end; hence thesis by A13; end; thus for A,B being (POINT of S), K,L being LINE of S st A <> B & {A,B} on K & {A,B} on L holds K = L proof let A,B be (POINT of S), K,L be LINE of S; assume that A19: A <> B and A20: {A,B} on K and A21: {A,B} on L; reconsider k = K,l = L as Element of Li; A22: A on K & B on K by A20,Th11; A23: A on L & B on L by A21,Th11; reconsider a = A, b = B as Element of Po; [a,k] in i1 & [b,k] in i1 by A22,Def1; then A24: a in k & b in k by A1; [a,l] in i1 & [b,l] in i1 by A23,Def1; then A25: a in l & b in l by A1; k in Li; then consider x1,x2 being Element of Po such that A26: k = {x1,x2} and x1 <> x2; A27: (a = x1 or a = x2) & (b = x1 or b = x2) by A24,A26,TARSKI:def 2; l in Li; then consider x3,x4 being Element of Po such that A28: l = {x3,x4} and x3 <> x4; (a = x3 or a = x4) & (b = x3 or b = x4) by A25,A28,TARSKI:def 2; hence thesis by A19,A26,A27,A28; end; thus for P being PLANE of S ex A being POINT of S st A on P proof let P be PLANE of S; reconsider p = P as Element of Pl; p in Pl; then consider a,b,c such that A29: p = {a,b,c} and a <> b & a <> c & b <> c; reconsider A = a as POINT of S; take A; a in p by A29,ENUMSET1:def 1; then [a,p] in i2; hence A on P by Def2; end; thus for A,B,C being POINT of S ex P being PLANE of S st {A,B,C} on P proof let A,B,C be POINT of S; reconsider a = A, b = B, c = C as Element of Po; A30: now assume a <> b & a <> c & b <> c; then {a,b,c} in Pl; then consider p such that A31: p = {a,b,c}; reconsider P = p as PLANE of S; a in p & b in p & c in p by A31,ENUMSET1:def 1; then [a,p] in i2 & [b,p] in i2 & [c,p] in i2; then A on P & B on P & C on P by Def2; then {A,B,C} on P by Th14; hence thesis; end; A32: now assume A33: a = b & a = c & b = c; for a ex b,c st a <> b & a <> c & b <> c proof let a; A34: now assume a = 0 or a = 1; then a <> Two & a <> Three & Two <> Three; hence thesis; end; now assume a = 2 or a = 3; then a <> _Zero & a <> One & _Zero <> One; hence thesis; end; hence thesis by A34,ENUMSET1:18; end; then consider x,y being Element of Po such that A35: a <> x & a <> y & x <> y; {a,x,y} in Pl by A35; then consider p such that A36: p = {a,x,y}; reconsider P = p as PLANE of S; a in p by A36,ENUMSET1:def 1; then [a,p] in i2; then A on P by Def2; then {A,B,C} on P by A33,Th14; hence thesis; end; now assume A37: (a = b & a <> c) or (a = c & a <> b) or (b = c & a <> b ) ; then consider x,y being Element of Po such that A38: (x = a or x = b or x = c) & (y = a or y = b or y = c) and A39: x <> y; for a,b ex c st a <> c & b <> c proof let a,b; A40: now assume a = 0 & (b = 0 or b = 1 or b = 2); then a <> Three & b <> Three; hence thesis; end; A41: now assume a = 0 & b = 3; then a <> One & b <> One; hence thesis; end; A42: now assume (a = 1 or a = 2 or a = 3) & (b = 1 or b = 2 or b = 3); then a <> _Zero & b <> _Zero; hence thesis; end; A43: now assume (a = 1 or a = 2) & b = 0; then a <> Three & b <> Three; hence thesis; end; now assume a=3 & b = 0; then a <> Two & b <> Two; hence thesis; end; hence thesis by A40,A41,A42,A43,ENUMSET1:18; end; then consider z being Element of Po such that A44: x <> z & y <> z; {x,y,z} in Pl by A39,A44; then consider p such that A45: p = {x,y,z}; reconsider P = p as PLANE of S; a in p & b in p & c in p by A37,A38,A39,A45,ENUMSET1:def 1; then [a,p] in i2 & [b,p] in i2 & [c,p] in i2; then A on P & B on P & C on P by Def2; then {A,B,C} on P by Th14; hence thesis; end; hence thesis by A30,A32; end; thus for A,B,C being (POINT of S), P,Q being PLANE of S st not {A,B,C} is_collinear & {A,B,C} on P & {A,B,C} on Q holds P = Q proof let A,B,C be (POINT of S), P,Q be PLANE of S; assume that A46: not {A,B,C} is_collinear and A47: {A,B,C} on P and A48: {A,B,C} on Q; reconsider p = P,q = Q as Element of Pl; A49: now assume A50: A = B or A = C or B = C; consider K being LINE of S such that A51: {A,B} on K by A12; consider L being LINE of S such that A52: {A,C} on L by A12; A53: A on K & B on K & A on L & C on L by A51,A52,Th11; not {A,B,C} on K & not {A,B,C} on L by A46,Def6; hence contradiction by A50,A53,Th12; end; A54: A on P & B on P & C on P by A47,Th14; reconsider a = A, b = B, c = C as Element of Po; [a,p] in i2 & [b,p] in i2 & [c,p] in i2 by A54,Def2; then A55: a in p & b in p & c in p by A4; p in Pl; then consider x1,x2,x3 being Element of Po such that A56: p = {x1,x2,x3} and x1 <> x2 & x1 <> x3 & x2 <> x3; A57: (a = x1 or a = x2 or a = x3) & (b = x1 or b = x2 or b = x3) & (c = x1 or c = x2 or c = x3) by A55,A56,ENUMSET1:def 1; A on Q & B on Q & C on Q by A48,Th14; then [a,q] in i2 & [b,q] in i2 & [c,q] in i2 by Def2; then A58: a in q & b in q & c in q by A4; q in Pl; then consider x1,x2,x3 being Element of Po such that A59: q = {x1,x2,x3} and x1 <> x2 & x1 <> x3 & x2 <> x3; (a = x1 or a = x2 or a = x3) & (b = x1 or b = x2 or b = x3) & (c = x1 or c = x2 or c = x3) by A58,A59,ENUMSET1:def 1; hence thesis by A49,A56,A57,A59,ENUMSET1:98,99,100,102; end; thus for L being (LINE of S), P being PLANE of S st ex A,B being POINT of S st A <> B & {A,B} on L & {A,B} on P holds L on P proof let L be (LINE of S), P be PLANE of S; given A,B being POINT of S such that A60: A <> B and A61: {A,B} on L and A62: {A,B} on P; reconsider l = L as Element of Li; reconsider p = P as Element of Pl; reconsider a = A, b = B as Element of Po; A on L & B on L by A61,Th11; then [a,l] in i1 & [b,l] in i1 by Def1; then A63: a in l & b in l by A1; A on P & B on P by A62,Th13; then [a,p] in i2 & [b,p] in i2 by Def2; then A64: a in p & b in p by A4; now let x be set; assume A65: x in l; l in Li; then consider x1,x2 being Element of Po such that A66: l = {x1,x2} and x1 <> x2; (a = x1 or a = x2) & (b = x1 or b = x2) by A63,A66,TARSKI:def 2; hence x in p by A60,A64,A65,A66,TARSKI:def 2; end; then l c= p by TARSKI:def 3; then [l,p] in i3; hence L on P by Def3; end; thus for A being (POINT of S), P,Q being PLANE of S st A on P & A on Q ex B being POINT of S st A <> B & B on P & B on Q proof let A be (POINT of S), P,Q be PLANE of S; assume that A67: A on P and A68: A on Q; reconsider a = A as Element of Po; reconsider p = P,q = Q as Element of Pl; p in Pl; then consider x1,x2,x3 being Element of Po such that A69: p = {x1,x2,x3} and A70: x1 <> x2 & x1 <> x3 & x2 <> x3; q in Pl; then consider y1,y2,y3 being Element of Po such that A71: q = {y1,y2,y3} and A72: y1 <> y2 & y1 <> y3 & y2 <> y3; A73: x1 in p & x2 in p & x3 in p by A69,ENUMSET1:def 1; A74: y1 in q & y2 in q & y3 in q by A71,ENUMSET1:def 1; [a,p] in i2 by A67,Def2; then a in p by A4; then a = x1 or a = x2 or a = x3 by A69,ENUMSET1:def 1; then consider z1,z2 being Element of Po such that A75: z1 in p & z2 in p and A76: z1 <> a & z2 <> a and A77: z1 <> z2 by A70, A73; [a,q] in i2 by A68,Def2; then a in q by A4; then a = y1 or a = y2 or a = y3 by A71,ENUMSET1:def 1; then consider z3,z4 being Element of Po such that A78: z3 in q & z4 in q and A79: z3 <> a & z4 <> a and A80: z3 <> z4 by A72, A74; now assume A81: z1 <> z3 & z1 <> z4 & z2 <> z3 & z2 <> z4; per cases by ENUMSET1:18; suppose A82: a = 0; then A83: (z1 = 1 or z1 = 2 or z1 = 3) & (z2 = 1 or z2 = 2 or z2 = 3) by A76,ENUMSET1:18; (z3 = 1 or z3 = 2 or z3 = 3) & (z4 = 1 or z4 = 2 or z4 = 3) by A79,A82,ENUMSET1:18; hence contradiction by A77,A80,A81,A83; suppose A84: a = 1; then A85: (z1 = 0 or z1 = 2 or z1 = 3) & (z2 = 0 or z2 = 2 or z2 = 3) by A76,ENUMSET1:18; (z3 = 0 or z3 = 2 or z3 = 3) & (z4 = 0 or z4 = 2 or z4 = 3) by A79,A84,ENUMSET1:18; hence contradiction by A77,A80,A81,A85; suppose A86: a = 2; then A87: (z1 = 0 or z1 = 1 or z1 = 3) & (z2 = 0 or z2 = 1 or z2 = 3) by A76,ENUMSET1:18; (z3 = 0 or z3 = 1 or z3 = 3) & (z4 = 0 or z4 = 1 or z4 = 3) by A79,A86,ENUMSET1:18; hence contradiction by A77,A80,A81,A87; suppose A88: a = 3; then A89: (z1 = 0 or z1 = 1 or z1 = 2) & (z2 = 0 or z2 = 1 or z2 = 2) by A76,ENUMSET1:18; (z3 = 0 or z3 = 1 or z3 = 2) & (z4 = 0 or z4 = 1 or z4 = 2) by A79,A88,ENUMSET1:18; hence contradiction by A77,A80,A81,A89; end; then consider z being Element of Po such that A90: z in p & z in q and A91: a <> z by A75,A76,A78; reconsider B = z as POINT of S; take B; thus A <> B by A91; [z,p] in i2 & [z,q] in i2 by A90; hence B on P & B on Q by Def2; end; thus ex A,B,C,D being POINT of S st not {A,B,C,D} is_coplanar proof reconsider Three = 3 as Element of Po by ENUMSET1:19; reconsider A = _Zero, B = One, C = Two, D = Three as POINT of S; take A,B,C,D; assume {A,B,C,D} is_coplanar; then consider P being PLANE of S such that A92: {A,B,C,D} on P by Def7; reconsider p = P as Element of Pl; p in Pl; then consider a,b,c such that A93: p = {a,b,c} and a <> b & a <> c & b <> c; A on P & B on P & C on P & D on P by A92,Th15; then [_Zero,p] in i2 & [One,p] in i2 & [Two,p] in i2 & [Three,p] in i2 by Def2; then _Zero in p & One in p & Two in p & Three in p by A4; then (_Zero = a or _Zero = b or _Zero = c) & (One = a or One = b or One = c) & (Two = a or Two = b or Two = c) & (Three = a or Three = b or Three = c) by A93,ENUMSET1:def 1; hence contradiction; end; thus for A being (POINT of S), L being (LINE of S), P being PLANE of S st A on L & L on P holds A on P proof let A be (POINT of S), L be (LINE of S), P be PLANE of S; reconsider a = A as Element of Po; reconsider l = L as Element of Li; reconsider p = P as Element of Pl; assume A on L & L on P; then [a,l] in i1 & [l,p] in i3 by Def1,Def3; then a in l & l c= p by A1,A7; then [a,p] in i2; hence A on P by Def2; end; end; definition let IT be IncStruct; attr IT is IncSpace-like means :Def8: (for L being LINE of IT ex A,B being POINT of IT st A <> B & {A,B} on L) & (for A,B being POINT of IT ex L being LINE of IT st {A,B} on L) & (for A,B being (POINT of IT), K,L being LINE of IT st A <> B & {A,B} on K & {A,B} on L holds K = L) & (for P being PLANE of IT ex A being POINT of IT st A on P) & (for A,B,C being POINT of IT ex P being PLANE of IT st {A,B,C} on P) & (for A,B,C being (POINT of IT), P,Q being PLANE of IT st not {A,B,C} is_collinear & {A,B,C} on P & {A,B,C} on Q holds P = Q) & (for L being (LINE of IT), P being PLANE of IT st ex A,B being POINT of IT st A <> B & {A,B} on L & {A,B} on P holds L on P) & (for A being (POINT of IT), P,Q being PLANE of IT st A on P & A on Q ex B being POINT of IT st A <> B & B on P & B on Q) & (ex A,B,C,D being POINT of IT st not {A,B,C,D} is_coplanar) & (for A being (POINT of IT), L being (LINE of IT), P being PLANE of IT st A on L & L on P holds A on P); end; definition cluster strict IncSpace-like IncStruct; existence proof IncStruct (# Po,Li,Pl,i1,i2,i3 #) is IncSpace-like by Def8,Lm1; hence thesis; end; end; definition mode IncSpace is IncSpace-like IncStruct; end; reserve S for IncSpace; reserve A,B,C,D,E for POINT of S; reserve K,L,L1,L2 for LINE of S; reserve P,P1,P2,Q for PLANE of S; reserve F for Subset of the Points of S; :: Axioms of Incidency canceled 11; theorem Th35: F on L & L on P implies F on P proof assume that A1: F on L and A2: L on P; let A be POINT of S; assume A in F; then A on L by A1,Def4; hence thesis by A2,Def8; end; :: Collinearity of points & coplanariy of points & lines theorem Th36: {A,A,B} is_collinear proof consider K such that A1: {A,B} on K by Def8; take K; thus thesis by A1,ENUMSET1:70; end; theorem Th37: {A,A,B,C} is_coplanar proof consider P such that A1: {A,B,C} on P by Def8; take P; thus thesis by A1,ENUMSET1:71; end; theorem Th38: {A,B,C} is_collinear implies {A,B,C,D} is_coplanar proof given L such that A1: {A,B,C} on L; assume A2: not {A,B,C,D} is_coplanar; consider P such that A3: {A,B,D} on P by Def8; {A,B} \/ {C} on L & {A,B} \/ {D} on P by A1,A3,ENUMSET1:43; then A <> B & {A,B} on L & {A,B} on P by A2,Th18,Th21,Th37; then L on P by Def8; then {A,B,C} on P by A1,Th35; then A on P & B on P & C on P & D on P by A3,Th14; then {A,B,C,D} on P by Th15; hence contradiction by A2,Def7; end; theorem Th39: A <> B & {A,B} on L & not C on L implies not {A,B,C} is_collinear proof assume that A1: A <> B & {A,B} on L and A2: not C on L; given K such that A3: {A,B,C} on K; {A,B} \/ {C} on K by A3,ENUMSET1:43; then {A,B} on K by Th20; then L = K by A1,Def8; hence contradiction by A2,A3,Th12; end; theorem Th40: not {A,B,C} is_collinear & {A,B,C} on P & not D on P implies not {A,B,C,D} is_coplanar proof assume that A1: not {A,B,C} is_collinear & {A,B,C} on P and A2: not D on P; given Q such that A3: {A,B,C,D} on Q; {A,B,C} \/ {D} on Q by A3,ENUMSET1:46; then {A,B,C} on Q by Th19; then P = Q by A1,Def8; hence contradiction by A2,A3,Th15; end; theorem not(ex P st K on P & L on P) implies K <> L proof assume that A1: not(ex P st K on P & L on P) and A2: K = L; consider A,B such that A3: A <> B and A4: {A,B} on K by Def8; consider C,D such that A5: C <> D and A6: {C,D} on L by Def8; A on K & B on K & C on K by A2,A4,A6,Th11; then {A,B,C} on K by Th12; then {A,B,C} is_collinear by Def6; then {A,B,C,D} is_coplanar by Th38; then consider Q such that A7: {A,B,C,D} on Q by Def7; A on Q & B on Q & C on Q & D on Q by A7,Th15; then {A,B} on Q & {C,D} on Q by Th13; then K on Q & L on Q by A3,A4,A5,A6,Def8; hence contradiction by A1; end; Lm2: ex B st A <> B & B on L proof consider B,C such that A1: B <> C & {B,C} on L by Def8; (A <> C or A <> B) & B on L & C on L by A1,Th11; hence thesis; end; theorem not(ex P st L on P & L1 on P & L2 on P) & (ex A st A on L & A on L1 & A on L2) implies L <> L1 proof assume A1: not(ex P st L on P & L1 on P & L2 on P); given A such that A2: A on L and A3: A on L1 and A4: A on L2; assume A5: L = L1; consider B such that A6: A <> B and A7: B on L by Lm2; consider C such that A8: A <> C and A9: C on L1 by Lm2; consider D such that A10: A <> D and A11: D on L2 by Lm2; {A,B,C} on L by A3,A5,A7,A9,Th12; then {A,B,C} is_collinear by Def6; then {A,B,C,D} is_coplanar by Th38; then consider Q such that A12: {A,B,C,D} on Q by Def7; {A,B} \/ {C,D} on Q by A12,ENUMSET1:45; then {A,B} on Q & {A,B} on L by A2,A7,Th11,Th21; then A13: L on Q by A6,Def8; {A,C,B} on L1 by A3,A5,A7,A9,Th12; then {A,C} \/ {B} on L1 & A on Q & C on Q by A12,Th15,ENUMSET1:43; then {A,C} on L1 & {A,C} on Q by Th13,Th20; then A14: L1 on Q by A8,Def8; A on Q & D on Q by A12,Th15; then {A,D} on L2 & {A,D} on Q by A4,A11,Th11,Th13; then L2 on Q by A10,Def8; hence contradiction by A1,A13,A14; end; theorem L1 on P & L2 on P & not L on P & L1 <> L2 implies not(ex Q st L on Q & L1 on Q & L2 on Q) proof assume that A1: L1 on P and A2: L2 on P and A3: not L on P and A4: L1 <> L2; given Q such that A5: L on Q & L1 on Q & L2 on Q; consider A,B such that A6: A <> B and A7: {A,B} on L1 by Def8; consider C,C1 being POINT of S such that A8: C <> C1 and A9: {C,C1} on L2 by Def8; consider D,E such that A10: D <> E and A11: {D,E} on L by Def8; {C,C1} on Q & {D,E} on Q by A5,A9,A11,Th35; then C on Q & C1 on Q & D on Q & E on Q by Th13; then {A,B} on Q & {C,D} on Q & {C,E} on Q & {C1,D} on Q & {C1,E} on Q by A5,A7,Th13,Th35; then {A,B} \/ {C,D} on Q & {A,B} \/ {C,E} on Q & {A,B} \/ {C1,D} on Q & {A,B} \/ {C1,E} on Q by Th21; then {A,B,C,D} on Q & {A,B,C,E} on Q & {A,B,C1,D} on Q & {A,B,C1,E} on Q by ENUMSET1:45; then A12: {A,B,C,D} is_coplanar & {A,B,C,E} is_coplanar & {A,B,C1,D} is_coplanar & {A,B,C1,E} is_coplanar by Def7; now assume C on L1 & C1 on L1; then {C,C1} on L1 by Th11; hence contradiction by A4,A8,A9,Def8; end; then A13: not {A,B,C} is_collinear or not {A,B,C1} is_collinear by A6,A7, Th39; {C,C1} on P by A2,A9,Th35; then {A,B} on P & C on P & C1 on P by A1,A7,Th13,Th35; then {A,B} \/ {C} on P & {A,B} \/ {C1} on P by Th19; then A14: {A,B,C} on P & {A,B,C1} on P by ENUMSET1:43; not {D,E} on P by A3,A10,A11,Def8; then not D on P or not E on P by Th13; hence contradiction by A12,A13,A14,Th40; end; :: Lines & planes theorem Th44: ex P st A on P & L on P proof consider B,C such that A1: B <> C and A2: {B,C} on L by Def8; consider P such that A3: {B,C,A} on P by Def8; take P; thus A on P by A3,Th14; {B,C} \/ {A} on P by A3,ENUMSET1:43; then {B,C} on P by Th19; hence L on P by A1,A2,Def8; end; theorem Th45: (ex A st A on K & A on L) implies (ex P st K on P & L on P) proof given A such that A1: A on K & A on L; consider B such that A2: A <> B and A3: B on K by Lm2; consider C such that A4: A <> C and A5: C on L by Lm2; consider P such that A6: {A,B,C} on P by Def8; take P; A on P & B on P & C on P by A6,Th14; then {A,B} on K & {A,C} on L & {A,B} on P & {A,C} on P by A1,A3,A5,Th11,Th13; hence thesis by A2,A4,Def8; end; theorem A <> B implies ex L st for K holds {A,B} on K iff K = L proof assume A1: A <> B; consider L such that A2: {A,B} on L by Def8; take L; thus {A,B} on K iff L = K by A1,A2,Def8; end; theorem not {A,B,C} is_collinear implies ex P st for Q holds {A,B,C} on Q iff P = Q proof assume A1: not {A,B,C} is_collinear; consider P such that A2: {A,B,C} on P by Def8; take P; thus {A,B,C} on Q iff P = Q by A1,A2,Def8; end; theorem Th48: not A on L implies ex P st for Q holds A on Q & L on Q iff P = Q proof assume A1: not A on L; consider B,C such that A2: B <> C and A3: {B,C} on L by Def8; consider P such that A4: {B,C,A} on P by Def8; take P; let Q; thus A on Q & L on Q implies P = Q proof assume that A5: A on Q and A6: L on Q; {B,C} on Q by A3,A6,Th35; then B on Q & C on Q by Th13; then A7: {B,C,A} on Q by A5,Th14; not {B,C,A} is_collinear by A1,A2,A3,Th39; hence P = Q by A4,A7,Def8; end; {B,C} \/ {A} on P by A4,ENUMSET1:43; then A on P & {B,C} on P by Th19; hence P = Q implies A on Q & L on Q by A2,A3,Def8; end; theorem Th49: K <>L & (ex A st A on K & A on L) implies ex P st for Q holds K on Q & L on Q iff P = Q proof assume that A1: K <> L and A2: ex A st A on K & A on L; consider A such that A3: A on K and A4: A on L by A2; consider B such that A5: A <> B and A6: B on K by Lm2; consider C such that A7: A <> C and A8: C on L by Lm2; consider P such that A9: {A,B,C} on P by Def8; take P; let Q; thus K on Q & L on Q implies P = Q proof assume A10: K on Q & L on Q; {A,C} on L by A4,A8,Th11; then not {A,C} on K by A1,A7,Def8; then not C on K & {A,B} on K by A3,A6,Th11; then A11: not {A,B,C} is_collinear by A5,Th39; A on Q & B on Q & C on Q by A3,A6,A8,A10,Def8; then {A,B,C} on Q by Th14; hence P = Q by A9,A11,Def8; end; A on P & B on P & C on P by A9,Th14; then {A,B} on K & {A,C} on L & {A,B} on P & {A,C} on P by A3,A4,A6,A8,Th11,Th13; hence P = Q implies K on Q & L on Q by A5,A7,Def8; end; :: Definitions of functions: Line, Plane definition let S; let A,B; assume A1: A <> B; func Line(A,B) -> LINE of S means :Def9: {A,B} on it; correctness by A1,Def8; end; definition let S; let A,B,C; assume A1: not {A,B,C} is_collinear; func Plane(A,B,C) -> PLANE of S means :Def10: {A,B,C} on it; correctness by A1,Def8; end; definition let S; let A,L; assume A1: not A on L; func Plane(A,L) -> PLANE of S means :Def11: A on it & L on it; existence by Th44; uniqueness proof let P,Q; assume A2: A on P & L on P & A on Q & L on Q; consider P1 such that A3: for P2 holds A on P2 & L on P2 iff P1 = P2 by A1,Th48; P1 = P & P1 = Q by A2,A3; hence thesis; end; end; definition let S; let K,L; assume that A1: K <> L and A2: (ex A st A on K & A on L); func Plane(K,L) -> PLANE of S means :Def12: K on it & L on it; existence by A2,Th45; uniqueness proof let P,Q; assume A3: K on P & L on P & K on Q & L on Q; consider P1 such that A4: for P2 holds K on P2 & L on P2 iff P1 = P2 by A1,A2,Th49 ; P = P1 & Q = P1 by A3,A4; hence thesis; end; end; :: Definitional theorems of functions: Line, Plane canceled 7; theorem A <> B implies Line(A,B) = Line(B,A) proof assume A1: A <> B; then {A,B} on Line(A,B) & {A,B} on Line(B,A) by Def9; hence thesis by A1,Def8; end; theorem Th58: not {A,B,C} is_collinear implies Plane(A,B,C) = Plane(A,C,B) proof assume A1: not {A,B,C} is_collinear; then not {A,C,B} is_collinear by ENUMSET1:98; then {A,C,B} on Plane(A,C,B) by Def10; then {A,B,C} on Plane(A,B,C) & {A,B,C} on Plane(A,C,B) by A1,Def10,ENUMSET1:98; hence thesis by A1,Def8; end; theorem Th59: not {A,B,C} is_collinear implies Plane(A,B,C) = Plane(B,A,C) proof assume A1: not {A,B,C} is_collinear; then not {B,A,C} is_collinear by ENUMSET1:99; then {B,A,C} on Plane(B,A,C) by Def10; then {A,B,C} on Plane(A,B,C) & {A,B,C} on Plane(B,A,C) by A1,Def10,ENUMSET1:99; hence thesis by A1,Def8; end; theorem not {A,B,C} is_collinear implies Plane(A,B,C) = Plane(B,C,A) proof assume A1: not {A,B,C} is_collinear; then A2: not {B,C,A} is_collinear by ENUMSET1:100; thus Plane(A,B,C) = Plane(B,A,C) by A1,Th59 .= Plane(B,C,A) by A2,Th58; end; theorem Th61: not {A,B,C} is_collinear implies Plane(A,B,C) = Plane(C,A,B) proof assume A1: not {A,B,C} is_collinear; then A2: not {C,A,B} is_collinear by ENUMSET1:100; thus Plane(A,B,C) = Plane(A,C,B) by A1,Th58 .= Plane(C,A,B) by A2,Th59; end; theorem not {A,B,C} is_collinear implies Plane(A,B,C) = Plane(C,B,A) proof assume A1: not {A,B,C} is_collinear; then A2: not {C,B,A} is_collinear by ENUMSET1:102; thus Plane(A,B,C) = Plane(C,A,B) by A1,Th61 .= Plane(C,B,A) by A2,Th58; end; canceled; theorem K <> L & (ex A st A on K & A on L) implies Plane(K,L) = Plane(L,K) proof assume A1: K <> L; given A such that A2: A on K & A on L; consider B such that A3: A <> B and A4: B on K by Lm2; consider C such that A5: A <> C and A6: C on L by Lm2; set P1 = Plane(K,L); set P2 = Plane(L,K); K on P1 & L on P1 & K on P2 & L on P2 by A1,A2,Def12; then A on P1 & B on P1 & C on P1 & A on P2 & B on P2 & C on P2 by A2,A4,A6,Def8; then A7: {A,B,C} on P1 & {A,B,C} on P2 by Th14; now assume {A,B,C} is_collinear; then consider L1 such that A8: {A,B,C} on L1 by Def6; A on L1 & B on L1 & C on L1 by A8,Th12; then {A,B} on L1 & {A,B} on K & {A,C} on L1 & {A,C} on L by A2,A4,A6,Th11; then K = L1 & L = L1 by A3,A5,Def8; hence contradiction by A1; end; hence thesis by A7,Def8; end; theorem Th65: A <> B & C on Line(A,B) implies {A,B,C} is_collinear proof assume A <> B; then {A,B} on Line(A,B) by Def9; then A1: A on Line(A,B) & B on Line(A,B) by Th11; assume C on Line(A,B); then {A,B,C} on Line(A,B) by A1,Th12; hence thesis by Def6; end; theorem A <> B & A <> C & {A,B,C} is_collinear implies Line(A,B) = Line(A,C) proof assume A1: A <> B; then A2: {A,B} on Line(A,B) by Def9; assume A3: A <> C; assume {A,B,C} is_collinear; then A on Line(A,B) & C on Line(A,B) by A1,A2,Th11,Th39; then {A,C} on Line(A,B) by Th11; hence thesis by A3,Def9; end; theorem not {A,B,C} is_collinear implies Plane(A,B,C) = Plane(C,Line(A,B)) proof assume A1: not {A,B,C} is_collinear; then A <> B by Th36; then A2: {A,B} on Line(A,B) by Def9; then A on Line(A,B) & B on Line(A,B) by Th11; then C on Line(A,B) implies {A,B,C} on Line(A,B) by Th12; then C on Plane(C,Line(A,B)) & Line(A,B) on Plane(C,Line(A,B)) by A1,Def6, Def11; then C on Plane(C,Line(A,B)) & {A,B} on Plane(C,Line(A,B)) by A2,Th35; then {A,B} \/ {C} on Plane(C,Line(A,B)) by Th19; then {A,B,C} on Plane(C,Line(A,B)) by ENUMSET1:43; hence thesis by A1,Def10; end; theorem not {A,B,C} is_collinear & D on Plane(A,B,C) implies {A,B,C,D} is_coplanar proof assume not {A,B,C} is_collinear & D on Plane(A,B,C); then {A,B,C} on Plane(A,B,C) & D on Plane(A,B,C) by Def10; then {A,B,C} \/ {D} on Plane(A,B,C) by Th19; then {A,B,C,D} on Plane(A,B,C) by ENUMSET1:46; hence thesis by Def7; end; theorem not C on L & {A,B} on L & A <> B implies Plane(C,L) = Plane(A,B,C) proof assume that A1: not C on L and A2: {A,B} on L and A3: A <> B; set P1 = Plane(C,L); C on P1 & L on P1 by A1,Def11; then C on P1 & {A,B} on P1 by A2,Th35; then {A,B} \/ {C} on P1 by Th19; then A4: {A,B,C} on P1 by ENUMSET1:43; not {A,B,C} is_collinear by A1,A2,A3,Th39; hence thesis by A4,Def10; end; theorem not {A,B,C} is_collinear implies Plane(A,B,C) = Plane(Line(A,B),Line(A,C)) proof set P2 = Plane(Line(A,B),Line(A,C)); set L1 = Line(A,B); set L2 = Line(A,C); assume A1: not {A,B,C} is_collinear; then not {A,C,B} is_collinear by ENUMSET1:98; then A2: A <> B & A <> C by A1,Th36; then A3: {A,B} on L1 & {A,C} on L2 by Def9; then A4: A on L1 & A on L2 by Th11; {A,C} on L2 by A2,Def9; then C on L2 by Th11; then L1 <> L2 by A1,A2,Th65; then L1 on P2 & L2 on P2 by A4,Def12; then {A,B} on P2 & {A,C} on P2 by A3,Th35; then {A,B} on P2 & C on P2 by Th13; then {A,B} \/ {C} on P2 by Th19; then {A,B,C} on P2 by ENUMSET1:43; hence thesis by A1,Def10; end; Lm3: ex A,B,C,D st A on P & not {A,B,C,D} is_coplanar proof consider A such that A1: A on P by Def8; consider A1,B1,C1,D1 being POINT of S such that A2: not {A1,B1,C1,D1} is_coplanar by Def8; now assume A3: not A1 on P; A4: now assume A5: A on Line(A1,B1); set Q = Plane(A1,B1,C1); A6: A1 <> B1 by A2,Th37; then A7: {A1,B1} on Line(A1,B1) by Def9; then A1 on Line(A1,B1) by Th11; then A8: {A,A1} on Line(A1,B1) by A5,Th11; A9: not {A1,B1,C1} is_collinear by A2,Th38; then A10: {A1,B1,C1} on Q by Def10; then D1 on Q implies {A1,B1,C1} \/ {D1} on Q by Th19; then A11: D1 on Q implies {A1,B1,C1,D1} on Q by ENUMSET1:46; {A1,B1} on Line (A1,B1) by A6,Def9; then C1 on Line(A1,B1) implies {A1,B1} \/ {C1} on Line(A1,B1) by Th18; then C1 on Line(A1,B1) implies {A1,B1,C1} on Line(A1,B1) by ENUMSET1:43; then A12: not {A,A1,C1} is_collinear by A1,A3,A8,A9,Def6,Th39; {A1,B1} \/ {C1} on Q by A10,ENUMSET1:43; then {A1,B1} on Q by Th21; then Line(A1,B1) on Q & {A1,B1,C1} on Q by A6,A7,A9,Def8,Def10; then A on Q & A1 on Q & C1 on Q by A5,Def8,Th14; then {A,A1,C1} on Plane(A1,B1,C1) by Th14; then not {A,A1,C1,D1} is_coplanar by A2,A11,A12,Def7,Th40; hence thesis by A1; end; now assume A13: not A on Line(A1,B1); set Q = Plane(A1,B1,A); A14: A1 <> B1 by A2,Th37; then {A1,B1} on Line(A1,B1) by Def9; then A15: not {A1,B1,A} is_collinear by A13,A14,Th39; then A16: {A1,B1,A} on Q by Def10; then {A1,B1} \/ {A} on Q by ENUMSET1:43; then {A1,B1} on Q by Th19; then {C1,D1} on Q implies {A1,B1} \/ {C1,D1} on Q by Th21; then {C1,D1} on Q implies {A1,B1,C1,D1} on Q by ENUMSET1:45; then not C1 on Q or not D1 on Q by A2,Def7,Th13; then not {A1,B1,A,C1} is_coplanar or not {A1,B1,A,D1} is_coplanar by A15,A16,Th40; then not {A,A1,B1,C1} is_coplanar or not {A,A1,B1,D1} is_coplanar by ENUMSET1:110; hence thesis by A1; end; hence thesis by A4; end; hence thesis by A2; end; :: The fourth axiom of incidency theorem Th71: ex A,B,C st {A,B,C} on P & not {A,B,C} is_collinear proof consider A1,B1,C1,D1 being POINT of S such that A1: A1 on P and A2: not {A1,B1,C1,D1} is_coplanar by Lm3; not {A1,B1,C1,D1} on P by A2,Def7; then not {B1,C1,D1,A1} on P by ENUMSET1:111; then not {B1,C1,D1} \/ {A1} on P by ENUMSET1:46; then not {B1,C1,D1} on P by A1,Th19; then not B1 on P or not C1 on P or not D1 on P by Th14; then consider X being POINT of S such that A3: X = B1 or X = C1 or X = D1 and A4: not X on P; not {B1,C1,A1,D1} is_coplanar & not {B1,D1,A1,C1} is_coplanar & not {C1,D1,A1,B1} is_coplanar by A2,ENUMSET1:110,112,118; then B1 <> C1 & B1 <> D1 & C1 <> D1 by Th37; then consider Y,Z being POINT of S such that A5: (Y = B1 or Y = C1 or Y = D1) & (Z = B1 or Z = C1 or Z = D1) & Y <> X & Z <> X & Y <> Z by A3; A6: now assume {A1,X,Y,Z} is_coplanar; then {A1,D1,B1,C1} is_coplanar or {A1,D1,C1,B1} is_coplanar by A2,A3,A5, ENUMSET1:104; hence contradiction by A2,ENUMSET1:105,107; end; set P1 = Plane(X,Y,A1), P2 = Plane(X,Z,A1); not {A1,X,Y} is_collinear by A6,Th38; then not {X,Y,A1} is_collinear by ENUMSET1:100; then A7: {X,Y,A1} on P1 by Def10; then A8: A1 on P1 by Th14; then consider B such that A9: A1 <> B and A10: B on P1 and A11: B on P by A1,Def8; not {X,Z,A1,Y} is_coplanar by A6,ENUMSET1:112; then not {X,Z,A1} is_collinear by Th38; then A12: {X,Z,A1} on P2 by Def10; then A13: A1 on P2 by Th14; then consider C such that A14: A1 <> C and A15: C on P and A16: C on P2 by A1,Def8 ; take A1,B,C; thus {A1,B,C} on P by A1,A11,A15,Th14; given K such that A17: {A1,B,C} on K; A18: B on K by A17,Th12; consider E such that A19: B <> E and A20: E on K by Lm2; {A1,C,B} on K by A17,ENUMSET1:98; then {A1,B} \/ {C} on K & {A1,C} \/ {B} on K by A17,ENUMSET1:43; then A21: {A1,B} on K & {A1,C} on K & {A1,B} on P1 & {A1,C} on P2 by A8,A10,A13,A16,Th13,Th20; then K on P1 & K on P2 by A9,A14,Def8; then A22: B on P2 & E on P1 & E on P2 by A18,A20,Def8; A23: now assume {X,B,E} is_collinear; then consider L such that A24: {X,B,E} on L by Def6; A25: {E,B,X} on L by A24,ENUMSET1:102; {A1,B} on P by A1,A11,Th13; then K on P by A9,A21,Def8; then E on P & {E,B} \/ {X} on L by A20,A25,Def8,ENUMSET1:43; then {E,B} on P & {E,B} on L by A11,Th13,Th18; then L on P & X on L by A19,A24,Def8,Th12; hence contradiction by A4,Def8; end; X on P1 & X on P2 by A7,A12,Th14; then {X,B,E} on P1 & {X,B,E} on P2 by A10,A22,Th14; then P1 = P2 by A23,Def8; then Z on P1 by A12,Th14; then {X,Y,A1} \/ {Z} on P1 by A7,Th19; then {X,Y,A1,Z} on P1 by ENUMSET1:46; then {X,Y,A1,Z} is_coplanar by Def7; hence contradiction by A6,ENUMSET1:110; end; :: Fundamental existence theorems theorem ex A,B,C,D st A on P & not {A,B,C,D} is_coplanar by Lm3; theorem ex B st A <> B & B on L by Lm2; theorem Th74: A <> B implies ex C st C on P & not {A,B,C} is_collinear proof assume A1: A <> B; consider L such that A2: {A,B} on L by Def8; consider C,D,E such that A3: {C,D,E} on P and A4: not {C,D,E} is_collinear by Th71; not {C,D,E} on L by A4,Def6; then not C on L or not D on L or not E on L by Th12; then (not {A,B,C} is_collinear or not {A,B,D} is_collinear or not {A,B,E} is_collinear) & C on P & D on P & E on P by A1,A2,A3,Th14,Th39; hence thesis; end; theorem Th75: not {A,B,C} is_collinear implies ex D st not {A,B,C,D} is_coplanar proof assume A1: not {A,B,C} is_collinear; consider P such that A2: {A,B,C} on P by Def8; consider A1,B1,C1,D1 being POINT of S such that A3: not {A1,B1,C1,D1} is_coplanar by Def8; not {A1,B1,C1,D1} on P by A3,Def7; then not A1 on P or not B1 on P or not C1 on P or not D1 on P by Th15; then not {A,B,C,A1} is_coplanar or not {A,B,C,B1} is_coplanar or not {A,B,C,C1} is_coplanar or not {A,B,C,D1} is_coplanar by A1,A2,Th40; hence thesis; end; theorem Th76: ex B,C st {B,C} on P & not {A,B,C} is_collinear proof A1: now assume A on P; then consider B such that A2: A <> B and A3: B on P & B on P by Def8; consider C such that A4: C on P and A5: not {A,B,C} is_collinear by A2,Th74 ; {B,C} on P by A3,A4,Th13; hence thesis by A5; end; now assume A6: not A on P; consider B,C,D such that A7: {B,C,D} on P and A8: not {B,C,D} is_collinear by Th71; take B, C; {B,C} \/ {D} on P by A7,ENUMSET1:43; hence A9: {B,C} on P by Th19; assume {A,B,C} is_collinear; then consider K such that A10: {A,B,C} on K by Def6; {B,C,A} on K by A10,ENUMSET1:100; then A11: {B,C} \/ {A} on K by ENUMSET1:43; then B <> C & {B,C} on K by A8,Th20,Th36; then K on P & A on K by A9,A11,Def8,Th18; hence contradiction by A6,Def8; end; hence thesis by A1; end; theorem Th77: A <> B implies (ex C,D st not {A,B,C,D} is_coplanar) proof assume A1: A <> B; consider P; consider C such that C on P and A2: not {A,B,C} is_collinear by A1,Th74; ex D st not {A,B,C,D} is_coplanar by A2,Th75; hence thesis; end; theorem ex B,C,D st not {A,B,C,D} is_coplanar proof consider L; consider B such that A1: A <> B and B on L by Lm2; ex C,D st not {A,B,C,D} is_coplanar by A1,Th77; hence thesis; end; theorem ex L st not A on L & L on P proof consider B,C such that A1: {B,C} on P and A2: not {A,B,C} is_collinear by Th76; consider L such that A3: {B,C} on L by Def8; take L; A on L implies {B,C} \/ {A} on L by A3,Th18; then A on L implies {B,C,A} on L by ENUMSET1:43; then A on L implies {A,B,C} on L by ENUMSET1:100; hence not A on L by A2,Def6; not {B,C,A} is_collinear by A2,ENUMSET1:100; then B <> C by Th36; hence L on P by A1,A3,Def8; end; theorem Th80: A on P implies (ex L,L1,L2 st L1 <> L2 & L1 on P & L2 on P & not L on P & A on L & A on L1 & A on L2) proof assume A1: A on P; consider B,C such that A2: {B,C} on P and A3: not {A,B,C} is_collinear by Th76; consider D such that A4: not {A,B,C,D} is_coplanar by A3,Th75; take L3 = Line(A,D),L1 = Line(A,B),L2 = Line(A,C); A5: not {A,C,B} is_collinear by A3,ENUMSET1:98; then A6: A <> C by Th36; then A7: {A,C} on L2 by Def9; then B on L2 implies {A,C} \/ {B} on L2 by Th18; then A8: B on L2 implies {A,C,B} on L2 by ENUMSET1:43; A9: A <> B by A3,Th36; then A10: {A,B} on L1 by Def9; hence L1 <> L2 by A5,A8,Def6,Th11; A11: {A} \/ {B,C} on P by A1,A2,Th19; then A12: {A,B,C} on P by ENUMSET1:42; then {A,B} \/ {C} on P by ENUMSET1:43; then {A,B} on P by Th21; hence L1 on P by A9,A10,Def8; {A,C,B} on P by A11,ENUMSET1:42; then {A,C} \/ {B} on P by ENUMSET1:43; then {A,C} on P by Th19; hence L2 on P by A6,A7,Def8; not {A,D,B,C} is_coplanar by A4,ENUMSET1:105; then A <> D by Th37; then A13: {A,D} on L3 by Def9; then L3 on P implies {A,D} on P by Th35; then L3 on P implies D on P by Th13; then L3 on P implies {A,B,C} \/ {D} on P by A12,Th19; then L3 on P implies {A,B,C,D} on P by ENUMSET1:46; hence not L3 on P by A4,Def7; thus thesis by A7,A10,A13,Th11; end; theorem ex L,L1,L2 st A on L & A on L1 & A on L2 & not(ex P st L on P & L1 on P & L2 on P) proof consider P such that A1: {A,A,A} on P by Def8; A on P by A1,Th14; then consider L,L1,L2 such that A2: L1 <> L2 and A3: L1 on P & L2 on P & not L on P and A4: A on L & A on L1 & A on L2 by Th80; take L,L1,L2; thus A on L & A on L1 & A on L2 by A4; given Q such that A5: L on Q & L1 on Q & L2 on Q; consider B such that A6: A <> B and A7: B on L1 by Lm2; consider C such that A8: A <> C and A9: C on L2 by Lm2; A10: {A,B} on L1 & {A,C} on L2 by A4,A7,A9,Th11; then {A,B} on P & {A,B} on Q & C on P & C on Q by A3,A5,A9,Def8,Th35; then {A,B} \/ {C} on P & {A,B} \/ {C} on Q by Th19; then A11: {A,B,C} on P & {A,B,C} on Q by ENUMSET1:43; now given K such that A12: {A,B,C} on K; {A,C,B} on K by A12,ENUMSET1:98; then {A,C} \/ {B} on K & {A,B} \/ {C} on K by A12,ENUMSET1:43; then {A,B} on K & {A,C} on K by Th18; then K = L1 & K = L2 by A6,A8,A10,Def8; hence contradiction by A2; end; then not {A,B,C} is_collinear by Def6; hence contradiction by A3,A5,A11,Def8; end; theorem ex P st A on P & not L on P proof consider B such that A1: A <> B and A2: B on L by Lm2; consider C,D such that A3: not {A,B,C,D} is_coplanar by A1,Th77; take P = Plane(A,C,D); A4: not {A,C,D,B} is_coplanar by A3,ENUMSET1:105; then not {A,C,D} is_collinear by Th38; then A5: {A,C,D} on P by Def10; hence A on P by Th14; B on P implies {A,C,D} \/ {B} on P by A5,Th19; then B on P implies {A,C,D,B} on P by ENUMSET1:46; hence not L on P by A2,A4,Def7,Def8; end; theorem ex A st A on P & not A on L proof consider A,B such that A1: A <> B and A2: {A,B} on L by Def8; consider C such that A3: C on P and A4: not {A,B,C} is_collinear by A1,Th74 ; take C; thus C on P by A3; C on L implies {A,B} \/ {C} on L by A2,Th18; then C on L implies {A,B,C} on L by ENUMSET1:43; hence thesis by A4,Def6; end; theorem ex K st not(ex P st L on P & K on P) proof consider A,B such that A1: A <> B and A2: {A,B} on L by Def8; consider C,D such that A3: not {A,B,C,D} is_coplanar by A1,Th77; take K = Line(C,D); given P such that A4: L on P & K on P; not {C,D,A,B} is_coplanar by A3,ENUMSET1:118; then C <> D by Th37; then {C,D} on K by Def9; then {A,B} on P & {C,D} on P by A2,A4,Th35; then {A,B} \/ {C,D} on P by Th21; then {A,B,C,D} on P by ENUMSET1:45; hence thesis by A3,Def7; end; theorem ex P,Q st P <> Q & L on P & L on Q proof consider A,B such that A1: A <> B and A2: {A,B} on L by Def8; consider C,D such that A3: not {A,B,C,D} is_coplanar by A1,Th77; take P = Plane (A,B,C), Q = Plane(A,B,D); not {A,B,D,C} is_coplanar by A3,ENUMSET1:103; then not {A,B,C} is_collinear & not {A,B,D} is_collinear by A3,Th38; then A4: {A,B,C} on P & {A,B,D} on Q by Def10; then {A,B,C} on P & D on Q by Th14; then P = Q implies {A,B,C} \/ {D} on P by Th19; then P = Q implies {A,B,C,D} on P by ENUMSET1:46; hence P <> Q by A3,Def7; {A,B} \/ {C} on P & {A,B} \/ {D} on Q by A4,ENUMSET1:43; then {A,B} on P & {A,B} on Q by Th21; hence thesis by A1,A2,Def8; end; :: Intersection of lines and planes canceled; theorem not L on P & {A,B} on L & {A,B} on P implies A = B by Def8; theorem P <> Q implies not(ex A st A on P & A on Q) or (ex L st for B holds B on P & B on Q iff B on L) proof assume A1: P <> Q; given A such that A2: A on P & A on Q; consider C such that A3: A <> C & C on P & C on Q by A2,Def8; take L = Line(A,C); {A,C} on L & {A,C} on P & {A,C} on Q by A2,A3,Def9,Th13; then A4:L on P & L on Q by A3,Def8; let B; thus B on P & B on Q implies B on L proof assume that A5: B on P & B on Q and A6: not B on L; consider P1 such that A7: for P2 holds B on P2 & L on P2 iff P1 = P2 by A6,Th48; P = P1 & Q = P1 by A4,A5,A7; hence contradiction by A1; end; thus B on L implies B on P & B on Q by A4,Def8; end;

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