let i, j be Nat; for G being Matrix of (TOP-REAL 2) st G is X_equal-in-line & 1 <= j & j <= width G & 1 <= i & i <= len G holds
(G * (i,j)) `1 = (G * (i,1)) `1
let G be Matrix of (TOP-REAL 2); ( G is X_equal-in-line & 1 <= j & j <= width G & 1 <= i & i <= len G implies (G * (i,j)) `1 = (G * (i,1)) `1 )
assume that
A1:
G is X_equal-in-line
and
A2:
1 <= j
and
A3:
j <= width G
and
A4:
1 <= i
and
A5:
i <= len G
; (G * (i,j)) `1 = (G * (i,1)) `1
i in dom G
by A4, A5, FINSEQ_3:25;
then A6:
X_axis (Line (G,i)) is constant
by A1;
reconsider c = Line (G,i) as FinSequence of (TOP-REAL 2) ;
A7:
j in Seg (width G)
by A2, A3, FINSEQ_1:1;
A8:
1 <= width G
by A2, A3, XXREAL_0:2;
then A9:
1 in Seg (width G)
by FINSEQ_1:1;
A10:
len c = width G
by MATRIX_0:def 7;
then
1 in dom c
by A8, FINSEQ_3:25;
then A11:
c /. 1 = c . 1
by PARTFUN1:def 6;
j in dom c
by A2, A3, A10, FINSEQ_3:25;
then A12:
c /. j = c . j
by PARTFUN1:def 6;
A13:
len (X_axis (Line (G,i))) = len c
by GOBOARD1:def 1;
then A14:
1 in dom (X_axis (Line (G,i)))
by A8, A10, FINSEQ_3:25;
A15:
j in dom (X_axis (Line (G,i)))
by A2, A3, A10, A13, FINSEQ_3:25;
thus (G * (i,j)) `1 =
(c /. j) `1
by A7, A12, MATRIX_0:def 7
.=
(X_axis (Line (G,i))) . j
by A15, GOBOARD1:def 1
.=
(X_axis (Line (G,i))) . 1
by A6, A14, A15
.=
(c /. 1) `1
by A14, GOBOARD1:def 1
.=
(G * (i,1)) `1
by A9, A11, MATRIX_0:def 7
; verum