:: Fano-Desargues Parallelity Spaces
:: by Eugeniusz Kusak and Wojciech Leo\'nczuk
::
:: Received March 23, 1990
:: Copyright (c) 1990-2011 Association of Mizar Users

begin

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) - (f `1)) * ((g `2) - (h `2))) - (((g `1) - (h `1)) * ((e `2) - (f `2))) = 0. F & (((e `1) - (f `1)) * ((g `3) - (h `3))) - (((g `1) - (h `1)) * ((e `3) - (f `3))) = 0. F & (((e `2) - (f `2)) * ((g `3) - (h `3))) - (((g `2) - (h `2)) * ((e `3) - (f `3))) = 0. F ) iff ( ex K being Element of F st
( K * ((e `1) - (f `1)) = (g `1) - (h `1) & K * ((e `2) - (f `2)) = (g `2) - (h `2) & K * ((e `3) - (f `3)) = (g `3) - (h `3) ) or ( (e `1) - (f `1) = 0. F & (e `2) - (f `2) = 0. F & (e `3) - (f `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) - (f `1)) = (g `1) - (h `1) & K * ((e `2) - (f `2)) = (g `2) - (h `2) & K * ((e `3) - (f `3)) = (g `3) - (h `3) ) or ( (e `1) - (f `1) = 0. F & (e `2) - (f `2) = 0. F & (e `3) - (f `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) - (f `1)) = L * ((e `1) - (g `1)) & K * ((e `2) - (f `2)) = L * ((e `2) - (g `2)) & K * ((e `3) - (f `3)) = L * ((e `3) - (g `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, N, M, L, 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, N, M 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 = ((f `1) + (g `1)) - (e `1) & h `2 = ((f `2) + (g `2)) - (e `2) & h `3 = ((f `3) + (g `3)) - (e `3) )

proof end;

Lm8: for F being Field
for L, K, 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 L, K, 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 b, c, a, 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, p, b, c, 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;

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 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;

theorem :: PARSP_2:9

canceled;

theorem :: PARSP_2:10

canceled;

theorem :: PARSP_2:11

canceled;

theorem :: PARSP_2:12

canceled;

theorem Th13: :: PARSP_2:13

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 is_collinear means :Def2: :: PARSP_2:def 2
a,b '||' a,c;

end;

:: deftheorem Def2 defines is_collinear PARSP_2:def 2 :
for FdSp being FanodesSp
for a, b, c being Element of FdSp holds
( a,b,c is_collinear iff a,b '||' a,c );

theorem :: PARSP_2:14

canceled;

theorem Th15: :: PARSP_2:15

for FdSp being FanodesSp
for a, b, c being Element of FdSp st a,b,c is_collinear holds
( a,c,b is_collinear & c,b,a is_collinear & b,a,c is_collinear & b,c,a is_collinear & c,a,b is_collinear )

proof end;

theorem :: PARSP_2:16

canceled;

theorem Th17: :: PARSP_2:17

for FdSp being FanodesSp
for a, b, c, p, q, r being Element of FdSp st not a,b,c is_collinear & a,b '||' p,q & a,c '||' p,r & p <> q & p <> r holds
not p,q,r is_collinear

proof end;

theorem Th18: :: PARSP_2:18

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 is_collinear

proof end;

theorem Th19: :: PARSP_2:19

for FdSp being FanodesSp
for a, b, p, q, r being Element of FdSp st a <> b & a,b,p is_collinear & a,b,q is_collinear & a,b,r is_collinear holds
p,q,r is_collinear

proof end;

theorem Th20: :: PARSP_2:20

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 is_collinear

proof end;

theorem Th21: :: PARSP_2:21

for FdSp being FanodesSp
for a, b, c, d being Element of FdSp st a,b,c is_collinear & a,b,d is_collinear holds
a,b '||' c,d

proof end;

theorem Th22: :: PARSP_2:22

for FdSp being FanodesSp
for a, b, c, d being Element of FdSp st not a,b,c is_collinear & a,b '||' c,d holds
not a,b,d is_collinear

proof end;

theorem Th23: :: PARSP_2:23

for FdSp being FanodesSp
for a, b, c, d, x being Element of FdSp st not a,b,c is_collinear & a,b '||' c,d & c <> d & a,b,x is_collinear holds
not c,d,x is_collinear

proof end;

theorem :: PARSP_2:24

for FdSp being FanodesSp
for o, a, b, x being Element of FdSp holds
( o,a,b is_collinear or not o,a,x is_collinear or not o,b,x is_collinear or o = x )

proof end;

theorem :: PARSP_2:25

for FdSp being FanodesSp
for o, a, b, p, q being Element of FdSp st o <> a & o <> b & o,a,b is_collinear & o,a,p is_collinear & o,b,q is_collinear holds
a,b '||' p,q

proof end;

theorem :: PARSP_2:26

for FdSp being FanodesSp
for a, b, c, d, p, q being Element of FdSp st not a,b '||' c,d & a,b,p is_collinear & a,b,q is_collinear & c,d,p is_collinear & c,d,q is_collinear holds
p = q

proof end;

theorem :: PARSP_2:27

for FdSp being FanodesSp
for a, b, c, d being Element of FdSp st a <> b & a,b,c is_collinear & a,b '||' c,d holds
a,c '||' b,d

proof end;

theorem :: PARSP_2:28

for FdSp being FanodesSp
for a, b, c, d being Element of FdSp st a <> b & a,b,c is_collinear & a,b '||' c,d holds
c,b '||' c,d

proof end;

theorem :: PARSP_2:29

for FdSp being FanodesSp
for o, a, c, b, p, q being Element of FdSp st not o,a,c is_collinear & o,a,b is_collinear & o,c,p is_collinear & o,c,q is_collinear & a,c '||' b,p & a,c '||' b,q holds
p = q

proof end;

theorem :: PARSP_2:30

for FdSp being FanodesSp
for a, b, c, d being Element of FdSp st a <> b & a,b,c is_collinear & a,b,d is_collinear holds
a,c,d is_collinear

proof end;

theorem :: PARSP_2:31

for FdSp being FanodesSp
for a, b, c, d being Element of FdSp st a,b,c is_collinear & a,c,d is_collinear & a <> c holds
b,c,d is_collinear

proof end;

definition

let FdSp be FanodesSp;
let a, b, c, d be Element of FdSp;
pred parallelogram a,b,c,d means :Def3: :: PARSP_2:def 3
( not a,b,c is_collinear & a,b '||' c,d & a,c '||' b,d );

end;

:: deftheorem Def3 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 is_collinear & a,b '||' c,d & a,c '||' b,d ) );

theorem :: PARSP_2:32

canceled;

theorem :: PARSP_2:33

canceled;

theorem Th34: :: PARSP_2:34

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 Th35: :: PARSP_2:35

for FdSp being FanodesSp
for a, b, c, d being Element of FdSp st parallelogram a,b,c,d holds
( not a,b,c is_collinear & not b,a,d is_collinear & not c,d,a is_collinear & not d,c,b is_collinear )

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,b,c is_collinear & not b,a,d is_collinear & not c,d,a is_collinear & not d,c,b is_collinear & not a,c,b is_collinear & not b,a,c is_collinear & not b,c,a is_collinear & not c,a,b is_collinear & not c,b,a is_collinear & not b,d,a is_collinear & not a,b,d is_collinear & not a,d,b is_collinear & not d,a,b is_collinear & not d,b,a is_collinear & not c,a,d is_collinear & not a,c,d is_collinear & not a,d,c is_collinear & not d,a,c is_collinear & not d,c,a is_collinear & not d,b,c is_collinear & not b,c,d is_collinear & not b,d,c is_collinear & not c,b,d is_collinear & not c,d,b is_collinear )

proof end;

theorem Th37: :: PARSP_2:37

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 is_collinear or not c,d,x is_collinear )

proof end;

theorem Th38: :: PARSP_2:38

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

proof end;

theorem Th39: :: PARSP_2:39

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

proof end;

theorem Th40: :: PARSP_2:40

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

proof end;

theorem Th41: :: PARSP_2:41

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 Th42: :: PARSP_2:42

for FdSp being FanodesSp
for a, b, c being Element of FdSp st not a,b,c is_collinear holds
ex d being Element of FdSp st parallelogram a,b,c,d

proof end;

theorem Th43: :: PARSP_2:43

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 Th44: :: PARSP_2:44

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 Th45: :: PARSP_2:45

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

proof end;

theorem Th46: :: PARSP_2:46

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 is_collinear & c <> a & c <> b )

proof end;

theorem Th47: :: PARSP_2:47

for FdSp being FanodesSp
for a, p, b, q, c, r being Element of FdSp st parallelogram a,p,b,q & parallelogram a,p,c,r holds
b,c '||' q,r

proof end;

theorem Th48: :: PARSP_2:48

for FdSp being FanodesSp
for b, q, c, a, p, r being Element of FdSp st not b,q,c is_collinear & parallelogram a,p,b,q & parallelogram a,p,c,r holds
parallelogram b,q,c,r

proof end;

theorem Th49: :: PARSP_2:49

for FdSp being FanodesSp
for a, b, c, p, q, r being Element of FdSp st a,b,c is_collinear & b <> c & parallelogram a,p,b,q & parallelogram a,p,c,r holds
parallelogram b,q,c,r

proof end;

theorem Th50: :: PARSP_2:50

for FdSp being FanodesSp
for a, p, b, q, c, r, d, 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 Th51: :: PARSP_2:51

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_2:52

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 :Def4: :: 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 Def4 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 :: PARSP_2:53

canceled;

theorem :: PARSP_2:54

canceled;

theorem Th55: :: PARSP_2:55

for FdSp being FanodesSp
for a, b, c being Element of FdSp st a,a congr b,c holds
b = c

proof end;

theorem :: PARSP_2:56

for FdSp being FanodesSp
for a, b, c being Element of FdSp st a,b congr c,c holds
a = b

proof end;

theorem :: PARSP_2:57

for FdSp being FanodesSp
for a, b being Element of FdSp st a,b congr b,a holds
a = b

proof end;

theorem Th58: :: PARSP_2:58

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 Th59: :: PARSP_2:59

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 Th60: :: PARSP_2:60

for FdSp being FanodesSp
for a, b, c, d being Element of FdSp st a,b congr c,d & not a,b,c is_collinear holds
parallelogram a,b,c,d

proof end;

theorem Th61: :: PARSP_2:61

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 Th62: :: PARSP_2:62

for FdSp being FanodesSp
for a, b, c, d, r, s being Element of FdSp st a,b congr c,d & a,b,c is_collinear & parallelogram r,s,a,b holds
parallelogram r,s,c,d

proof end;

theorem :: PARSP_2:63

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_2:64

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_2:65

canceled;

theorem Th66: :: PARSP_2:66

for FdSp being FanodesSp
for a, b being Element of FdSp holds a,b congr a,b

proof end;

theorem Th67: :: PARSP_2:67

for FdSp being FanodesSp
for r, s, a, b, c, d 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_2:68

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

proof end;

theorem :: PARSP_2:69

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;