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_3:31;
then A4: Indices (GoB v1,v2) = [:(dom v1),(Seg (len v2)):] by A1, MATRIX_1:def 5;
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

let n be Element of NAT ; :: according to GOBOARD1:def 8 :: 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_1:def 8;
A8: len (Y_axis l) = len l by GOBOARD1:def 4;
then A9: dom (Y_axis l) = dom l by FINSEQ_3:31;

now

let i, j be Element of 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_1:106;
then (GoB v1,v2) * n,j = |[r,s2]| by Def1;
then A13: ((GoB v1,v2) * n,j) `2 = s2 by EUCLID:56;
l /. j = l . j by A9, A11, PARTFUN1:def 8;
then l /. j = (GoB v1,v2) * n,j by A8, A7, A11, MATRIX_1:def 8;
then A14: (Y_axis l) . j = s2 by A11, A13, GOBOARD1:def 4;
[n,i] in Indices (GoB v1,v2) by A1, A3, A4, A6, A8, A7, A10, ZFMISC_1:106;
then (GoB v1,v2) * n,i = |[r,s1]| by Def1;
then A15: ((GoB v1,v2) * n,i) `2 = s1 by EUCLID:56;
l /. i = l . i by A9, A10, PARTFUN1:def 8;
then l /. i = (GoB v1,v2) * n,i by A8, A7, A10, MATRIX_1:def 8;
then (Y_axis l) . i = s1 by A10, A15, GOBOARD1:def 4;
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;

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

end;

A16: dom v1 = Seg (len v1) by FINSEQ_1:def 3;
thus GoB v1,v2 is X_increasing-in-column :: thesis: verum

proof

let n be Element of NAT ; :: according to GOBOARD1:def 9 :: 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 3;
then A19: dom (X_axis c) = dom c by FINSEQ_3:31;
A20: len c = len (GoB v1,v2) by MATRIX_1:def 9;
then A21: dom c = dom (GoB v1,v2) by FINSEQ_3:31;
A22: dom (X_axis c) = Seg (len (X_axis c)) by FINSEQ_1:def 3;

now

let i, j be Element of 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_1:106;
then (GoB v1,v2) * j,n = |[s2,r]| by Def1;
then A26: ((GoB v1,v2) * j,n) `1 = s2 by EUCLID:56;
c /. j = c . j by A19, A24, PARTFUN1:def 8;
then c /. j = (GoB v1,v2) * j,n by A19, A21, A24, MATRIX_1:def 9;
then A27: (X_axis c) . j = s2 by A24, A26, GOBOARD1:def 3;
[i,n] in Indices (GoB v1,v2) by A1, A3, A4, A17, A19, A21, A23, ZFMISC_1:106;
then (GoB v1,v2) * i,n = |[s1,r]| by Def1;
then A28: ((GoB v1,v2) * i,n) `1 = s1 by EUCLID:56;
c /. i = c . i by A19, A23, PARTFUN1:def 8;
then c /. i = (GoB v1,v2) * i,n by A19, A21, A23, MATRIX_1:def 9;
then (X_axis c) . i = s1 by A23, A28, GOBOARD1:def 3;
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;

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

end;