let F be Field; :: thesis: 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 ) )
let a, b, c, d be Element of (MPS F); :: thesis: 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 ) )
let e, f, g, h be Element of [:the carrier of F,the carrier of F,the carrier of F:]; :: thesis: ( not a,b '||' a,c & a,b '||' c,d & a,c '||' b,d & [a,b,c,d] = [e,f,g,h] implies ( h `1 = ((f `1 ) + (g `1 )) - (e `1 ) & h `2 = ((f `2 ) + (g `2 )) - (e `2 ) & h `3 = ((f `3 ) + (g `3 )) - (e `3 ) ) )
assume that
A1: not a,b '||' a,c and
A2: a,b '||' c,d and
A3: a,c '||' b,d and
A4: [a,b,c,d] = [e,f,g,h] ; :: thesis: ( h `1 = ((f `1 ) + (g `1 )) - (e `1 ) & h `2 = ((f `2 ) + (g `2 )) - (e `2 ) & h `3 = ((f `3 ) + (g `3 )) - (e `3 ) )
A5: e = [(e `1 ),(e `2 ),(e `3 )] by MCART_1:48;
consider m, n, o, w being Element of [:the carrier of F,the carrier of F,the carrier of F:] such that
A6: [a,c,b,d] = [m,n,o,w] and
A7: ( ex L being Element of F st
( L * ((m `1 ) - (n `1 )) = (o `1 ) - (w `1 ) & L * ((m `2 ) - (n `2 )) = (o `2 ) - (w `2 ) & L * ((m `3 ) - (n `3 )) = (o `3 ) - (w `3 ) ) or ( (m `1 ) - (n `1 ) = 0. F & (m `2 ) - (n `2 ) = 0. F & (m `3 ) - (n `3 ) = 0. F ) ) by A3, Th2;
A8: b = f by A4, MCART_1:33;
then A9: o = f by A6, MCART_1:33;
d = h by A4, MCART_1:33;
then A10: w = h by A6, MCART_1:33;
c = g by A4, MCART_1:33;
then A11: n = g by A6, MCART_1:33;
A12: a = e by A4, MCART_1:33;
then A13: [a,b,a,c] = [e,f,e,g] by A4, A8, MCART_1:33;
consider i, j, k, l being Element of [:the carrier of F,the carrier of F,the carrier of F:] such that
A14: [a,b,c,d] = [i,j,k,l] and
A15: ( ex K being Element of F st
( K * ((i `1 ) - (j `1 )) = (k `1 ) - (l `1 ) & K * ((i `2 ) - (j `2 )) = (k `2 ) - (l `2 ) & K * ((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;
A16: ( e = i & f = j ) by A4, A14, MCART_1:33;
A17: ( g = k & h = l ) by A4, A14, MCART_1:33;
A18: e = m by A12, A6, MCART_1:33;
f = [(f `1 ),(f `2 ),(f `3 )] by MCART_1:48;
then ( e `1 <> f `1 or e `2 <> f `2 or e `3 <> f `3 ) by A1, A13, A5, Th3;
then consider K being Element of F such that
A19: K * ((e `1 ) - (f `1 )) = (g `1 ) - (h `1 ) and
A20: K * ((e `2 ) - (f `2 )) = (g `2 ) - (h `2 ) and
A21: K * ((e `3 ) - (f `3 )) = (g `3 ) - (h `3 ) by A15, A16, A17, Lm2;
g = [(g `1 ),(g `2 ),(g `3 )] by MCART_1:48;
then ( e `1 <> g `1 or e `2 <> g `2 or e `3 <> g `3 ) by A1, A13, A5, Th3;
then consider L being Element of F such that
A22: L * ((e `1 ) - (g `1 )) = (f `1 ) - (h `1 ) and
A23: L * ((e `2 ) - (g `2 )) = (f `2 ) - (h `2 ) and
A24: L * ((e `3 ) - (g `3 )) = (f `3 ) - (h `3 ) by A7, A18, A11, A9, A10, Lm2;
(K * ((e `2 ) - (f `2 ))) - (L * ((e `2 ) - (g `2 ))) = (g `2 ) - (f `2 ) by A20, A23, Lm5;
then A25: (K + (- (1_ F))) * ((e `2 ) - (f `2 )) = (L + (- (1_ F))) * ((e `2 ) - (g `2 )) by Lm6;
(K * ((e `3 ) - (f `3 ))) - (L * ((e `3 ) - (g `3 ))) = (g `3 ) - (f `3 ) by A21, A24, Lm5;
then A26: (K + (- (1_ F))) * ((e `3 ) - (f `3 )) = (L + (- (1_ F))) * ((e `3 ) - (g `3 )) by Lm6;
(K * ((e `1 ) - (f `1 ))) - (L * ((e `1 ) - (g `1 ))) = (g `1 ) - (f `1 ) by A19, A22, Lm5;
then (K + (- (1_ F))) * ((e `1 ) - (f `1 )) = (L + (- (1_ F))) * ((e `1 ) - (g `1 )) by Lm6;
then A27: K + (- (1_ F)) = 0. F by A1, A13, A25, A26, Th4;
then ((e `2 ) - (f `2 )) * (1_ F) = (g `2 ) - (h `2 ) by A20, Lm2;
then A28: (e `2 ) - (f `2 ) = (g `2 ) - (h `2 ) by VECTSP_1:def 19;
((e `3 ) - (f `3 )) * (1_ F) = (g `3 ) - (h `3 ) by A21, A27, Lm2;
then A29: (e `3 ) - (f `3 ) = (g `3 ) - (h `3 ) by VECTSP_1:def 19;
((e `1 ) - (f `1 )) * (1_ F) = (g `1 ) - (h `1 ) by A19, A27, Lm2;
then (e `1 ) - (f `1 ) = (g `1 ) - (h `1 ) by VECTSP_1:def 19;
hence ( h `1 = ((f `1 ) + (g `1 )) - (e `1 ) & h `2 = ((f `2 ) + (g `2 )) - (e `2 ) & h `3 = ((f `3 ) + (g `3 )) - (e `3 ) ) by A28, A29, Lm7; :: thesis: verum