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_1:44;
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_1:29;
then A9: o = f by A6, MCART_1:29;
d = h by A4, MCART_1:29;
then A10: w = h by A6, MCART_1:29;
c = g by A4, MCART_1:29;
then A11: n = g by A6, MCART_1:29;
A12: a = e by A4, MCART_1:29;
then A13: [a,b,a,c] = [e,f,e,g] by A4, A8, MCART_1:29;
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_1:29;
A17: ( g = k & h = l ) by A4, A14, MCART_1:29;
A18: e = m by A12, A6, MCART_1:29;
f = [(f `1),(f `2),(f `3)] by MCART_1:44;
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_1:44;
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 8;
((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 8;
((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 8;
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