let F be Field; :: thesis: 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
let b, c, a, d be Element of (MPS F); :: thesis: ( (1_ F) + (1_ F) <> 0. F & b,c '||' a,d & a,b '||' c,d & a,c '||' b,d implies a,b '||' a,c )
assume that
A1: (1_ F) + (1_ F) <> 0. F and
A2: b,c '||' a,d and
A3: a,b '||' c,d and
A4: a,c '||' b,d ; :: thesis: a,b '||' a,c
assume A5: not a,b '||' a,c ; :: thesis: contradiction
consider i, j, k, l being Element of [:the carrier of F,the carrier of F,the carrier of F:] such that
A6: [b,c,a,d] = [i,j,k,l] and
A7: ( ex L being Element of F st
( L * ((i `1 ) - (j `1 )) = (k `1 ) - (l `1 ) & L * ((i `2 ) - (j `2 )) = (k `2 ) - (l `2 ) & L * ((i `3 ) - (j `3 )) = (k `3 ) - (l `3 ) ) or ( (i `1 ) - (j `1 ) = 0. F & (i `2 ) - (j `2 ) = 0. F & (i `3 ) - (j `3 ) = 0. F ) ) by A2, Th2;
A8: ( b = i & c = j ) by A6, MCART_1:33;
A9: ( a = k & d = l ) by A6, MCART_1:33;
consider e, f, g, h being Element of [:the carrier of F,the carrier of F,the carrier of F:] such that
A10: [a,b,c,d] = [e,f,g,h] and
( 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 ) ) by A3, Th2;
A11: b = f by A10, MCART_1:33;
A12: d = h by A10, MCART_1:33;
A13: c = g by A10, MCART_1:33;
A14: a = e by A10, MCART_1:33;
then A15: [a,b,a,c] = [e,f,e,g] by A10, A11, MCART_1:33;
( f = [(f `1 ),(f `2 ),(f `3 )] & g = [(g `1 ),(g `2 ),(g `3 )] ) by MCART_1:48;
then ( i `1 <> j `1 or i `2 <> j `2 or i `3 <> j `3 ) by A5, A11, A13, A15, A8, Th3;
then consider L being Element of F such that
A16: L * ((f `1 ) - (g `1 )) = (e `1 ) - (h `1 ) and
A17: L * ((f `2 ) - (g `2 )) = (e `2 ) - (h `2 ) and
A18: L * ((f `3 ) - (g `3 )) = (e `3 ) - (h `3 ) by A14, A11, A13, A12, A7, A8, A9, Lm2;
h `2 = ((f `2 ) + (g `2 )) - (e `2 ) by A3, A4, A5, A10, Th5;
then A19: (L - (1_ F)) * ((e `2 ) - (g `2 )) = (L + (1_ F)) * ((e `2 ) - (f `2 )) by A17, Lm9;
h `3 = ((f `3 ) + (g `3 )) - (e `3 ) by A3, A4, A5, A10, Th5;
then A20: (L - (1_ F)) * ((e `3 ) - (g `3 )) = (L + (1_ F)) * ((e `3 ) - (f `3 )) by A18, Lm9;
h `1 = ((f `1 ) + (g `1 )) - (e `1 ) by A3, A4, A5, A10, Th5;
then (L - (1_ F)) * ((e `1 ) - (g `1 )) = (L + (1_ F)) * ((e `1 ) - (f `1 )) by A16, Lm9;
then ( L + (1_ F) = 0. F & L - (1_ F) = 0. F ) by A5, A15, A19, A20, Th4;
then (L + (1_ F)) - (L - (1_ F)) = (0. F) + (- (0. F)) by RLVECT_1:def 14;
then (L + (1_ F)) - (L - (1_ F)) = 0. F by RLVECT_1:16;
then (L + (1_ F)) + (- (L - (1_ F))) = 0. F by RLVECT_1:def 14;
then (L + (1_ F)) + ((1_ F) + (- L)) = 0. F by RLVECT_1:47;
then ((L + (1_ F)) + (1_ F)) + (- L) = 0. F by RLVECT_1:def 6;
then (((1_ F) + (1_ F)) + L) + (- L) = 0. F by RLVECT_1:def 6;
then ((1_ F) + (1_ F)) + (L + (- L)) = 0. F by RLVECT_1:def 6;
then ((1_ F) + (1_ F)) + (0. F) = 0. F by RLVECT_1:16;
hence contradiction by A1, RLVECT_1:10; :: thesis: verum