set M = GoB (v1,v2);

A1: width (GoB (v1,v2)) = len v2 by Def1;

A2: len (GoB (v1,v2)) = len v1 by Def1;

then A3: dom (GoB (v1,v2)) = dom v1 by FINSEQ_3:29;

then A4: Indices (GoB (v1,v2)) = [:(dom v1),(Seg (len v2)):] by A1, MATRIX_0:def 4;

A5: dom v2 = Seg (len v2) by FINSEQ_1:def 3;

thus GoB (v1,v2) is Y_increasing-in-line :: thesis: GoB (v1,v2) is X_increasing-in-column

thus GoB (v1,v2) is X_increasing-in-column :: thesis: verum

A1: width (GoB (v1,v2)) = len v2 by Def1;

A2: len (GoB (v1,v2)) = len v1 by Def1;

then A3: dom (GoB (v1,v2)) = dom v1 by FINSEQ_3:29;

then A4: Indices (GoB (v1,v2)) = [:(dom v1),(Seg (len v2)):] by A1, MATRIX_0:def 4;

A5: dom v2 = Seg (len v2) by FINSEQ_1:def 3;

thus GoB (v1,v2) is Y_increasing-in-line :: thesis: GoB (v1,v2) is X_increasing-in-column

proof

A16:
dom v1 = Seg (len v1)
by FINSEQ_1:def 3;
let n be Nat; :: according to GOBOARD1:def 6 :: thesis: ( not n in dom (GoB (v1,v2)) or Y_axis (Line ((GoB (v1,v2)),n)) is increasing )

reconsider l = Line ((GoB (v1,v2)),n) as FinSequence of (TOP-REAL 2) ;

set y = Y_axis l;

assume A6: n in dom (GoB (v1,v2)) ; :: thesis: Y_axis (Line ((GoB (v1,v2)),n)) is increasing

A7: ( len l = width (GoB (v1,v2)) & dom (Y_axis l) = Seg (len (Y_axis l)) ) by FINSEQ_1:def 3, MATRIX_0:def 7;

A8: len (Y_axis l) = len l by GOBOARD1:def 2;

then A9: dom (Y_axis l) = dom l by FINSEQ_3:29;

end;reconsider l = Line ((GoB (v1,v2)),n) as FinSequence of (TOP-REAL 2) ;

set y = Y_axis l;

assume A6: n in dom (GoB (v1,v2)) ; :: thesis: Y_axis (Line ((GoB (v1,v2)),n)) is increasing

A7: ( len l = width (GoB (v1,v2)) & dom (Y_axis l) = Seg (len (Y_axis l)) ) by FINSEQ_1:def 3, MATRIX_0:def 7;

A8: len (Y_axis l) = len l by GOBOARD1:def 2;

then A9: dom (Y_axis l) = dom l by FINSEQ_3:29;

now :: thesis: for i, j being Nat st i in dom (Y_axis l) & j in dom (Y_axis l) & i < j holds

(Y_axis l) . i < (Y_axis l) . j

hence
Y_axis (Line ((GoB (v1,v2)),n)) is increasing
by SEQM_3:def 1; :: thesis: verum(Y_axis l) . i < (Y_axis l) . j

let i, j be Nat; :: thesis: ( i in dom (Y_axis l) & j in dom (Y_axis l) & i < j implies (Y_axis l) . i < (Y_axis l) . j )

assume that

A10: i in dom (Y_axis l) and

A11: j in dom (Y_axis l) and

A12: i < j ; :: thesis: (Y_axis l) . i < (Y_axis l) . j

reconsider r = v1 . n, s1 = v2 . i, s2 = v2 . j as Real ;

[n,j] in Indices (GoB (v1,v2)) by A1, A3, A4, A6, A8, A7, A11, ZFMISC_1:87;

then (GoB (v1,v2)) * (n,j) = |[r,s2]| by Def1;

then A13: ((GoB (v1,v2)) * (n,j)) `2 = s2 by EUCLID:52;

l /. j = l . j by A9, A11, PARTFUN1:def 6;

then l /. j = (GoB (v1,v2)) * (n,j) by A8, A7, A11, MATRIX_0:def 7;

then A14: (Y_axis l) . j = s2 by A11, A13, GOBOARD1:def 2;

[n,i] in Indices (GoB (v1,v2)) by A1, A3, A4, A6, A8, A7, A10, ZFMISC_1:87;

then (GoB (v1,v2)) * (n,i) = |[r,s1]| by Def1;

then A15: ((GoB (v1,v2)) * (n,i)) `2 = s1 by EUCLID:52;

l /. i = l . i by A9, A10, PARTFUN1:def 6;

then l /. i = (GoB (v1,v2)) * (n,i) by A8, A7, A10, MATRIX_0:def 7;

then (Y_axis l) . i = s1 by A10, A15, GOBOARD1:def 2;

hence (Y_axis l) . i < (Y_axis l) . j by A5, A1, A8, A7, A10, A11, A12, A14, SEQM_3:def 1; :: thesis: verum

end;assume that

A10: i in dom (Y_axis l) and

A11: j in dom (Y_axis l) and

A12: i < j ; :: thesis: (Y_axis l) . i < (Y_axis l) . j

reconsider r = v1 . n, s1 = v2 . i, s2 = v2 . j as Real ;

[n,j] in Indices (GoB (v1,v2)) by A1, A3, A4, A6, A8, A7, A11, ZFMISC_1:87;

then (GoB (v1,v2)) * (n,j) = |[r,s2]| by Def1;

then A13: ((GoB (v1,v2)) * (n,j)) `2 = s2 by EUCLID:52;

l /. j = l . j by A9, A11, PARTFUN1:def 6;

then l /. j = (GoB (v1,v2)) * (n,j) by A8, A7, A11, MATRIX_0:def 7;

then A14: (Y_axis l) . j = s2 by A11, A13, GOBOARD1:def 2;

[n,i] in Indices (GoB (v1,v2)) by A1, A3, A4, A6, A8, A7, A10, ZFMISC_1:87;

then (GoB (v1,v2)) * (n,i) = |[r,s1]| by Def1;

then A15: ((GoB (v1,v2)) * (n,i)) `2 = s1 by EUCLID:52;

l /. i = l . i by A9, A10, PARTFUN1:def 6;

then l /. i = (GoB (v1,v2)) * (n,i) by A8, A7, A10, MATRIX_0:def 7;

then (Y_axis l) . i = s1 by A10, A15, GOBOARD1:def 2;

hence (Y_axis l) . i < (Y_axis l) . j by A5, A1, A8, A7, A10, A11, A12, A14, SEQM_3:def 1; :: thesis: verum

thus GoB (v1,v2) is X_increasing-in-column :: thesis: verum

proof

let n be Nat; :: according to GOBOARD1:def 7 :: thesis: ( not n in Seg (width (GoB (v1,v2))) or X_axis (Col ((GoB (v1,v2)),n)) is increasing )

reconsider c = Col ((GoB (v1,v2)),n) as FinSequence of (TOP-REAL 2) ;

set x = X_axis c;

assume A17: n in Seg (width (GoB (v1,v2))) ; :: thesis: X_axis (Col ((GoB (v1,v2)),n)) is increasing

A18: len (X_axis c) = len c by GOBOARD1:def 1;

then A19: dom (X_axis c) = dom c by FINSEQ_3:29;

A20: len c = len (GoB (v1,v2)) by MATRIX_0:def 8;

then A21: dom c = dom (GoB (v1,v2)) by FINSEQ_3:29;

A22: dom (X_axis c) = Seg (len (X_axis c)) by FINSEQ_1:def 3;

end;reconsider c = Col ((GoB (v1,v2)),n) as FinSequence of (TOP-REAL 2) ;

set x = X_axis c;

assume A17: n in Seg (width (GoB (v1,v2))) ; :: thesis: X_axis (Col ((GoB (v1,v2)),n)) is increasing

A18: len (X_axis c) = len c by GOBOARD1:def 1;

then A19: dom (X_axis c) = dom c by FINSEQ_3:29;

A20: len c = len (GoB (v1,v2)) by MATRIX_0:def 8;

then A21: dom c = dom (GoB (v1,v2)) by FINSEQ_3:29;

A22: dom (X_axis c) = Seg (len (X_axis c)) by FINSEQ_1:def 3;

now :: thesis: for i, j being Nat st i in dom (X_axis c) & j in dom (X_axis c) & i < j holds

(X_axis c) . i < (X_axis c) . j

hence
X_axis (Col ((GoB (v1,v2)),n)) is increasing
by SEQM_3:def 1; :: thesis: verum(X_axis c) . i < (X_axis c) . j

let i, j be Nat; :: thesis: ( i in dom (X_axis c) & j in dom (X_axis c) & i < j implies (X_axis c) . i < (X_axis c) . j )

assume that

A23: i in dom (X_axis c) and

A24: j in dom (X_axis c) and

A25: i < j ; :: thesis: (X_axis c) . i < (X_axis c) . j

reconsider r = v2 . n, s1 = v1 . i, s2 = v1 . j as Real ;

[j,n] in Indices (GoB (v1,v2)) by A1, A3, A4, A17, A19, A21, A24, ZFMISC_1:87;

then (GoB (v1,v2)) * (j,n) = |[s2,r]| by Def1;

then A26: ((GoB (v1,v2)) * (j,n)) `1 = s2 by EUCLID:52;

c /. j = c . j by A19, A24, PARTFUN1:def 6;

then c /. j = (GoB (v1,v2)) * (j,n) by A19, A21, A24, MATRIX_0:def 8;

then A27: (X_axis c) . j = s2 by A24, A26, GOBOARD1:def 1;

[i,n] in Indices (GoB (v1,v2)) by A1, A3, A4, A17, A19, A21, A23, ZFMISC_1:87;

then (GoB (v1,v2)) * (i,n) = |[s1,r]| by Def1;

then A28: ((GoB (v1,v2)) * (i,n)) `1 = s1 by EUCLID:52;

c /. i = c . i by A19, A23, PARTFUN1:def 6;

then c /. i = (GoB (v1,v2)) * (i,n) by A19, A21, A23, MATRIX_0:def 8;

then (X_axis c) . i = s1 by A23, A28, GOBOARD1:def 1;

hence (X_axis c) . i < (X_axis c) . j by A16, A2, A18, A20, A22, A23, A24, A25, A27, SEQM_3:def 1; :: thesis: verum

end;assume that

A23: i in dom (X_axis c) and

A24: j in dom (X_axis c) and

A25: i < j ; :: thesis: (X_axis c) . i < (X_axis c) . j

reconsider r = v2 . n, s1 = v1 . i, s2 = v1 . j as Real ;

[j,n] in Indices (GoB (v1,v2)) by A1, A3, A4, A17, A19, A21, A24, ZFMISC_1:87;

then (GoB (v1,v2)) * (j,n) = |[s2,r]| by Def1;

then A26: ((GoB (v1,v2)) * (j,n)) `1 = s2 by EUCLID:52;

c /. j = c . j by A19, A24, PARTFUN1:def 6;

then c /. j = (GoB (v1,v2)) * (j,n) by A19, A21, A24, MATRIX_0:def 8;

then A27: (X_axis c) . j = s2 by A24, A26, GOBOARD1:def 1;

[i,n] in Indices (GoB (v1,v2)) by A1, A3, A4, A17, A19, A21, A23, ZFMISC_1:87;

then (GoB (v1,v2)) * (i,n) = |[s1,r]| by Def1;

then A28: ((GoB (v1,v2)) * (i,n)) `1 = s1 by EUCLID:52;

c /. i = c . i by A19, A23, PARTFUN1:def 6;

then c /. i = (GoB (v1,v2)) * (i,n) by A19, A21, A23, MATRIX_0:def 8;

then (X_axis c) . i = s1 by A23, A28, GOBOARD1:def 1;

hence (X_axis c) . i < (X_axis c) . j by A16, A2, A18, A20, A22, A23, A24, A25, A27, SEQM_3:def 1; :: thesis: verum