:: Fano-Desargues Parallelity Spaces
:: by Eugeniusz Kusak and Wojciech Leo\'nczuk
::
:: Received March 23, 1990
:: Copyright (c) 1990-2025 Association of Mizar Users


Lm1: for F being non empty right_complementable add-associative right_zeroed addLoopStr
for a, b being Element of F st a - b = 0. F holds
a = b
proof end;

Lm2: for F being non empty right_complementable add-associative right_zeroed addLoopStr
for x, y being Element of F holds
( ( x + (- y) = 0. F implies x = y ) & ( x = y implies x + (- y) = 0. F ) & ( x - y = 0. F implies x = y ) & ( x = y implies x - y = 0. F ) )
proof end;

theorem Th1: :: PARSP_2:1
for F being Field holds MPS F is ParSp
proof end;

Lm3: for F being Field
for e, f, g, h being Element of [: the carrier of F, the carrier of F, the carrier of F:] holds
( ( (((e `1_3) - (f `1_3)) * ((g `2_3) - (h `2_3))) - (((g `1_3) - (h `1_3)) * ((e `2_3) - (f `2_3))) = 0. F & (((e `1_3) - (f `1_3)) * ((g `3_3) - (h `3_3))) - (((g `1_3) - (h `1_3)) * ((e `3_3) - (f `3_3))) = 0. F & (((e `2_3) - (f `2_3)) * ((g `3_3) - (h `3_3))) - (((g `2_3) - (h `2_3)) * ((e `3_3) - (f `3_3))) = 0. F ) iff ( ex K being Element of F st
( K * ((e `1_3) - (f `1_3)) = (g `1_3) - (h `1_3) & K * ((e `2_3) - (f `2_3)) = (g `2_3) - (h `2_3) & K * ((e `3_3) - (f `3_3)) = (g `3_3) - (h `3_3) ) or ( (e `1_3) - (f `1_3) = 0. F & (e `2_3) - (f `2_3) = 0. F & (e `3_3) - (f `3_3) = 0. F ) ) )
proof end;

theorem Th2: :: PARSP_2:2
for F being Field
for a, b, c, d being Element of (MPS F) holds
( a,b '||' c,d iff ex e, f, g, h being Element of [: the carrier of F, the carrier of F, the carrier of F:] st
( [[a,b],[c,d]] = [[e,f],[g,h]] & ( ex K being Element of F st
( K * ((e `1_3) - (f `1_3)) = (g `1_3) - (h `1_3) & K * ((e `2_3) - (f `2_3)) = (g `2_3) - (h `2_3) & K * ((e `3_3) - (f `3_3)) = (g `3_3) - (h `3_3) ) or ( (e `1_3) - (f `1_3) = 0. F & (e `2_3) - (f `2_3) = 0. F & (e `3_3) - (f `3_3) = 0. F ) ) ) )
proof end;

theorem Th3: :: PARSP_2:3
for F being Field
for a, b, c being Element of (MPS F)
for e, f, g being Element of [: the carrier of F, the carrier of F, the carrier of F:] st not a,b '||' a,c & [[a,b],[a,c]] = [[e,f],[e,g]] holds
( e <> f & e <> g & f <> g )
proof end;

theorem Th4: :: PARSP_2:4
for F being Field
for a, b, c being Element of (MPS F)
for e, f, g being Element of [: the carrier of F, the carrier of F, the carrier of F:]
for K, L being Element of F st not a,b '||' a,c & [[a,b],[a,c]] = [[e,f],[e,g]] & K * ((e `1_3) - (f `1_3)) = L * ((e `1_3) - (g `1_3)) & K * ((e `2_3) - (f `2_3)) = L * ((e `2_3) - (g `2_3)) & K * ((e `3_3) - (f `3_3)) = L * ((e `3_3) - (g `3_3)) holds
( K = 0. F & L = 0. F )
proof end;

Lm4: for F being non empty right_complementable add-associative right_zeroed addLoopStr
for a, b, c being Element of F holds (b + a) - (c + a) = b - c
proof end;

Lm5: for F being Field
for K, L, R being Element of F holds (K - L) - (R - L) = K - R
proof end;

Lm6: for F being Field
for K, L, M, N, S being Element of F st (K * (N - M)) - (L * (N - S)) = S - M holds
(K + (- (1_ F))) * (N - M) = (L + (- (1_ F))) * (N - S)
proof end;

Lm7: for F being Field
for K, L, M, N being Element of F st K - L = N - M holds
M = (L + N) - K
proof end;

theorem Th5: :: PARSP_2:5
for F being Field
for a, b, c, d being Element of (MPS F)
for e, f, g, h being Element of [: the carrier of F, the carrier of F, the carrier of F:] st not a,b '||' a,c & a,b '||' c,d & a,c '||' b,d & [[a,b],[c,d]] = [[e,f],[g,h]] holds
( h `1_3 = ((f `1_3) + (g `1_3)) - (e `1_3) & h `2_3 = ((f `2_3) + (g `2_3)) - (e `2_3) & h `3_3 = ((f `3_3) + (g `3_3)) - (e `3_3) )
proof end;

Lm8: for F being Field
for K, L, R being Element of F st (L * K) - (L * R) = R + K holds
(L - (1_ F)) * K = (L + (1_ F)) * R
proof end;

Lm9: for F being Field
for K, L, N, R, S being Element of F st L * (K - N) = R - S & S = (K + N) - R holds
(L - (1_ F)) * (R - N) = (L + (1_ F)) * (R - K)
proof end;

Lm10: for F being Field
for K, L, M, N, R, S being Element of F st K = (L + M) - N & R = (L + S) - N holds
(1_ F) * (M - S) = K - R
proof end;

theorem Th6: :: PARSP_2:6
for F being Field holds
not for a, b, c being Element of (MPS F) holds a,b '||' a,c
proof end;

theorem Th7: :: PARSP_2:7
for F being Field
for a, b, c, d being Element of (MPS F) st (1_ F) + (1_ F) <> 0. F & b,c '||' a,d & a,b '||' c,d & a,c '||' b,d holds
a,b '||' a,c
proof end;

theorem Th8: :: PARSP_2:8
for F being Field
for a, b, c, p, q, r being Element of (MPS F) st not a,p '||' a,b & not a,p '||' a,c & a,p '||' b,q & a,p '||' c,r & a,b '||' p,q & a,c '||' p,r holds
b,c '||' q,r
proof end;

:: 2. DEFINITION OF A FANO-DESARGUES SPACE
definition
let IT be ParSp;
attr IT is FanodesSp-like means :Def1: :: PARSP_2:def 1
( not for a, b, c being Element of IT holds a,b '||' a,c & ( for a, b, c, d being Element of IT st b,c '||' a,d & a,b '||' c,d & a,c '||' b,d holds
a,b '||' a,c ) & ( for a, b, c, p, q, r being Element of IT st not a,p '||' a,b & not a,p '||' a,c & a,p '||' b,q & a,p '||' c,r & a,b '||' p,q & a,c '||' p,r holds
b,c '||' q,r ) );
end;

:: deftheorem Def1 defines FanodesSp-like PARSP_2:def 1 :
for IT being ParSp holds
( IT is FanodesSp-like iff ( not for a, b, c being Element of IT holds a,b '||' a,c & ( for a, b, c, d being Element of IT st b,c '||' a,d & a,b '||' c,d & a,c '||' b,d holds
a,b '||' a,c ) & ( for a, b, c, p, q, r being Element of IT st not a,p '||' a,b & not a,p '||' a,c & a,p '||' b,q & a,p '||' c,r & a,b '||' p,q & a,c '||' p,r holds
b,c '||' q,r ) ) );

registration
cluster non empty strict ParSp-like FanodesSp-like for ParStr ;
existence
ex b1 being ParSp st
( b1 is strict & b1 is FanodesSp-like )
proof end;
end;

definition
mode FanodesSp is FanodesSp-like ParSp;
end;

:: 3. AXIOMS OF A FANO-DESARGUES PARALLELITY SPACE
theorem Th9: :: PARSP_2:9
for FdSp being FanodesSp
for p, q being Element of FdSp st p <> q holds
ex r being Element of FdSp st not p,q '||' p,r
proof end;

definition
let FdSp be FanodesSp;
let a, b, c be Element of FdSp;
pred a,b,c are_collinear means :: PARSP_2:def 2
a,b '||' a,c;
end;

:: deftheorem defines are_collinear PARSP_2:def 2 :
for FdSp being FanodesSp
for a, b, c being Element of FdSp holds
( a,b,c are_collinear iff a,b '||' a,c );

theorem Th10: :: PARSP_2:10
for FdSp being FanodesSp
for a, b, c being Element of FdSp st a,b,c are_collinear holds
( a,c,b are_collinear & c,b,a are_collinear & b,a,c are_collinear & b,c,a are_collinear & c,a,b are_collinear ) by PARSP_1:24;

theorem Th11: :: PARSP_2:11
for FdSp being FanodesSp
for a, b, c, p, q, r being Element of FdSp st not a,b,c are_collinear & a,b '||' p,q & a,c '||' p,r & p <> q & p <> r holds
not p,q,r are_collinear by PARSP_1:30;

theorem Th12: :: PARSP_2:12
for FdSp being FanodesSp
for a, b, c being Element of FdSp st ( a = b or b = c or c = a ) holds
a,b,c are_collinear by PARSP_1:25;

theorem Th13: :: PARSP_2:13
for FdSp being FanodesSp
for a, b, p, q, r being Element of FdSp st a <> b & a,b,p are_collinear & a,b,q are_collinear & a,b,r are_collinear holds
p,q,r are_collinear
proof end;

theorem Th14: :: PARSP_2:14
for FdSp being FanodesSp
for p, q being Element of FdSp st p <> q holds
ex r being Element of FdSp st not p,q,r are_collinear
proof end;

theorem Th15: :: PARSP_2:15
for FdSp being FanodesSp
for a, b, c, d being Element of FdSp st a,b,c are_collinear & a,b,d are_collinear holds
a,b '||' c,d by PARSP_1:35;

theorem Th16: :: PARSP_2:16
for FdSp being FanodesSp
for a, b, c, d being Element of FdSp st not a,b,c are_collinear & a,b '||' c,d holds
not a,b,d are_collinear
proof end;

theorem Th17: :: PARSP_2:17
for FdSp being FanodesSp
for a, b, c, d, x being Element of FdSp st not a,b,c are_collinear & a,b '||' c,d & c <> d & a,b,x are_collinear holds
not c,d,x are_collinear by Th16, PARSP_1:26;

theorem :: PARSP_2:18
for FdSp being FanodesSp
for a, b, o, x being Element of FdSp holds
( o,a,b are_collinear or not o,a,x are_collinear or not o,b,x are_collinear or o = x )
proof end;

theorem :: PARSP_2:19
for FdSp being FanodesSp
for a, b, p, q, o being Element of FdSp st o <> a & o <> b & o,a,b are_collinear & o,a,p are_collinear & o,b,q are_collinear holds
a,b '||' p,q
proof end;

theorem :: PARSP_2:20
for FdSp being FanodesSp
for a, b, c, d, p, q being Element of FdSp st not a,b '||' c,d & a,b,p are_collinear & a,b,q are_collinear & c,d,p are_collinear & c,d,q are_collinear holds
p = q
proof end;

theorem :: PARSP_2:21
for FdSp being FanodesSp
for a, b, c, d being Element of FdSp st a <> b & a,b,c are_collinear & a,b '||' c,d holds
a,c '||' b,d
proof end;

theorem :: PARSP_2:22
for FdSp being FanodesSp
for a, b, c, d being Element of FdSp st a <> b & a,b,c are_collinear & a,b '||' c,d holds
c,b '||' c,d
proof end;

theorem :: PARSP_2:23
for FdSp being FanodesSp
for a, b, c, p, q, o being Element of FdSp st not o,a,c are_collinear & o,a,b are_collinear & o,c,p are_collinear & o,c,q are_collinear & a,c '||' b,p & a,c '||' b,q holds
p = q by PARSP_1:34;

theorem :: PARSP_2:24
for FdSp being FanodesSp
for a, b, c, d being Element of FdSp st a <> b & a,b,c are_collinear & a,b,d are_collinear holds
a,c,d are_collinear by PARSP_1:def 12;

theorem :: PARSP_2:25
for FdSp being FanodesSp
for a, b, c, d being Element of FdSp st a,b,c are_collinear & a,c,d are_collinear & a <> c holds
b,c,d are_collinear
proof end;

definition
let FdSp be FanodesSp;
let a, b, c, d be Element of FdSp;
pred parallelogram a,b,c,d means :: PARSP_2:def 3
( not a,b,c are_collinear & a,b '||' c,d & a,c '||' b,d );
end;

:: deftheorem defines parallelogram PARSP_2:def 3 :
for FdSp being FanodesSp
for a, b, c, d being Element of FdSp holds
( parallelogram a,b,c,d iff ( not a,b,c are_collinear & a,b '||' c,d & a,c '||' b,d ) );

theorem Th26: :: PARSP_2:26
for FdSp being FanodesSp
for a, b, c, d being Element of FdSp st parallelogram a,b,c,d holds
( a <> b & b <> c & c <> a & a <> d & b <> d & c <> d )
proof end;

theorem Th27: :: PARSP_2:27
for FdSp being FanodesSp
for a, b, c, d being Element of FdSp st parallelogram a,b,c,d holds
( not a,b,c are_collinear & not b,a,d are_collinear & not c,d,a are_collinear & not d,c,b are_collinear )
proof end;

theorem Th28: :: PARSP_2:28
for FdSp being FanodesSp
for a, b, c, d being Element of FdSp st parallelogram a,b,c,d holds
( not a,b,c are_collinear & not b,a,d are_collinear & not c,d,a are_collinear & not d,c,b are_collinear & not a,c,b are_collinear & not b,a,c are_collinear & not b,c,a are_collinear & not c,a,b are_collinear & not c,b,a are_collinear & not b,d,a are_collinear & not a,b,d are_collinear & not a,d,b are_collinear & not d,a,b are_collinear & not d,b,a are_collinear & not c,a,d are_collinear & not a,c,d are_collinear & not a,d,c are_collinear & not d,a,c are_collinear & not d,c,a are_collinear & not d,b,c are_collinear & not b,c,d are_collinear & not b,d,c are_collinear & not c,b,d are_collinear & not c,d,b are_collinear )
proof end;

theorem Th29: :: PARSP_2:29
for FdSp being FanodesSp
for a, b, c, d, x being Element of FdSp holds
( not parallelogram a,b,c,d or not a,b,x are_collinear or not c,d,x are_collinear )
proof end;

theorem Th30: :: PARSP_2:30
for FdSp being FanodesSp
for a, b, c, d being Element of FdSp st parallelogram a,b,c,d holds
parallelogram a,c,b,d by Th28;

theorem Th31: :: PARSP_2:31
for FdSp being FanodesSp
for a, b, c, d being Element of FdSp st parallelogram a,b,c,d holds
parallelogram c,d,a,b by PARSP_1:23, Th28;

theorem Th32: :: PARSP_2:32
for FdSp being FanodesSp
for a, b, c, d being Element of FdSp st parallelogram a,b,c,d holds
parallelogram b,a,d,c by PARSP_1:23, Th28;

theorem Th33: :: PARSP_2:33
for FdSp being FanodesSp
for a, b, c, d being Element of FdSp st parallelogram a,b,c,d holds
( parallelogram a,c,b,d & parallelogram c,d,a,b & parallelogram b,a,d,c & parallelogram c,a,d,b & parallelogram d,b,c,a & parallelogram b,d,a,c & parallelogram d,c,b,a )
proof end;

theorem Th34: :: PARSP_2:34
for FdSp being FanodesSp
for a, b, c being Element of FdSp st not a,b,c are_collinear holds
ex d being Element of FdSp st parallelogram a,b,c,d
proof end;

theorem Th35: :: PARSP_2:35
for FdSp being FanodesSp
for a, b, c, p, q being Element of FdSp st parallelogram a,b,c,p & parallelogram a,b,c,q holds
p = q
proof end;

theorem Th36: :: PARSP_2:36
for FdSp being FanodesSp
for a, b, c, d being Element of FdSp st parallelogram a,b,c,d holds
not a,d '||' b,c
proof end;

theorem Th37: :: PARSP_2:37
for FdSp being FanodesSp
for a, b, c, d being Element of FdSp st parallelogram a,b,c,d holds
not parallelogram a,b,d,c by Th36;

theorem Th38: :: PARSP_2:38
for FdSp being FanodesSp
for a, b being Element of FdSp st a <> b holds
ex c being Element of FdSp st
( a,b,c are_collinear & c <> a & c <> b )
proof end;

theorem Th39: :: PARSP_2:39
for FdSp being FanodesSp
for a, b, c, p, q, r being Element of FdSp st parallelogram a,p,b,q & parallelogram a,p,c,r holds
b,c '||' q,r
proof end;

theorem Th40: :: PARSP_2:40
for FdSp being FanodesSp
for a, b, c, p, q, r being Element of FdSp st not b,q,c are_collinear & parallelogram a,p,b,q & parallelogram a,p,c,r holds
parallelogram b,q,c,r
proof end;

theorem Th41: :: PARSP_2:41
for FdSp being FanodesSp
for a, b, c, p, q, r being Element of FdSp st a,b,c are_collinear & b <> c & parallelogram a,p,b,q & parallelogram a,p,c,r holds
parallelogram b,q,c,r
proof end;

theorem Th42: :: PARSP_2:42
for FdSp being FanodesSp
for a, b, c, d, p, q, r, s being Element of FdSp st parallelogram a,p,b,q & parallelogram a,p,c,r & parallelogram b,q,d,s holds
c,d '||' r,s
proof end;

theorem Th43: :: PARSP_2:43
for FdSp being FanodesSp
for a, b being Element of FdSp st a <> b holds
ex c, d being Element of FdSp st parallelogram a,b,c,d
proof end;

theorem :: PARSP_2:44
for FdSp being FanodesSp
for a, d being Element of FdSp st a <> d holds
ex b, c being Element of FdSp st parallelogram a,b,c,d
proof end;

definition
let FdSp be FanodesSp;
let a, b, r, s be Element of FdSp;
pred a,b congr r,s means :: PARSP_2:def 4
( ( a = b & r = s ) or ex p, q being Element of FdSp st
( parallelogram p,q,a,b & parallelogram p,q,r,s ) );
end;

:: deftheorem defines congr PARSP_2:def 4 :
for FdSp being FanodesSp
for a, b, r, s being Element of FdSp holds
( a,b congr r,s iff ( ( a = b & r = s ) or ex p, q being Element of FdSp st
( parallelogram p,q,a,b & parallelogram p,q,r,s ) ) );

theorem Th45: :: PARSP_2:45
for FdSp being FanodesSp
for a, b, c being Element of FdSp st a,a congr b,c holds
b = c by Th26;

theorem :: PARSP_2:46
for FdSp being FanodesSp
for a, b, c being Element of FdSp st a,b congr c,c holds
a = b by Th26;

theorem :: PARSP_2:47
for FdSp being FanodesSp
for a, b being Element of FdSp st a,b congr b,a holds
a = b by Th37;

theorem Th48: :: PARSP_2:48
for FdSp being FanodesSp
for a, b, c, d being Element of FdSp st a,b congr c,d holds
a,b '||' c,d
proof end;

theorem Th49: :: PARSP_2:49
for FdSp being FanodesSp
for a, b, c, d being Element of FdSp st a,b congr c,d holds
a,c '||' b,d
proof end;

theorem Th50: :: PARSP_2:50
for FdSp being FanodesSp
for a, b, c, d being Element of FdSp st a,b congr c,d & not a,b,c are_collinear holds
parallelogram a,b,c,d by Th48, Th49;

theorem Th51: :: PARSP_2:51
for FdSp being FanodesSp
for a, b, c, d being Element of FdSp st parallelogram a,b,c,d holds
a,b congr c,d
proof end;

theorem Th52: :: PARSP_2:52
for FdSp being FanodesSp
for a, b, c, d, r, s being Element of FdSp st a,b congr c,d & a,b,c are_collinear & parallelogram r,s,a,b holds
parallelogram r,s,c,d
proof end;

theorem :: PARSP_2:53
for FdSp being FanodesSp
for a, b, c, x, y being Element of FdSp st a,b congr c,x & a,b congr c,y holds
x = y
proof end;

theorem :: PARSP_2:54
for FdSp being FanodesSp
for a, b, c being Element of FdSp ex d being Element of FdSp st a,b congr c,d
proof end;

theorem :: PARSP_2:55
for FdSp being FanodesSp
for a, b being Element of FdSp holds a,b congr a,b
proof end;

theorem :: PARSP_2:56
for FdSp being FanodesSp
for a, b, c, d, r, s being Element of FdSp st r,s congr a,b & r,s congr c,d holds
a,b congr c,d
proof end;

theorem :: PARSP_2:57
for FdSp being FanodesSp
for a, b, c, d being Element of FdSp st a,b congr c,d holds
c,d congr a,b ;

theorem :: PARSP_2:58
for FdSp being FanodesSp
for a, b, c, d being Element of FdSp st a,b congr c,d holds
b,a congr d,c
proof end;