let i, j be Nat; for G being Matrix of (TOP-REAL 2) st G is Y_equal-in-column & 1 <= j & j <= width G & 1 <= i & i <= len G holds
(G * (i,j)) `2 = (G * (1,j)) `2
let G be Matrix of (TOP-REAL 2); ( G is Y_equal-in-column & 1 <= j & j <= width G & 1 <= i & i <= len G implies (G * (i,j)) `2 = (G * (1,j)) `2 )
assume that
A1:
G is Y_equal-in-column
and
A2:
1 <= j
and
A3:
j <= width G
and
A4:
1 <= i
and
A5:
i <= len G
; (G * (i,j)) `2 = (G * (1,j)) `2
j in Seg (width G)
by A2, A3, FINSEQ_1:1;
then A6:
Y_axis (Col (G,j)) is V8()
by A1;
reconsider c = Col (G,j) as FinSequence of (TOP-REAL 2) ;
A7:
i in dom G
by A4, A5, FINSEQ_3:25;
A8:
1 <= len G
by A4, A5, XXREAL_0:2;
then A9:
1 in dom G
by FINSEQ_3:25;
A10:
len c = len G
by MATRIX_0:def 8;
then
1 in dom c
by A8, FINSEQ_3:25;
then A11:
c /. 1 = c . 1
by PARTFUN1:def 6;
i in dom c
by A4, A5, A10, FINSEQ_3:25;
then A12:
c /. i = c . i
by PARTFUN1:def 6;
A13:
len (Y_axis (Col (G,j))) = len c
by GOBOARD1:def 2;
then A14:
1 in dom (Y_axis (Col (G,j)))
by A8, A10, FINSEQ_3:25;
A15:
i in dom (Y_axis (Col (G,j)))
by A4, A5, A10, A13, FINSEQ_3:25;
thus (G * (i,j)) `2 =
(c /. i) `2
by A7, A12, MATRIX_0:def 8
.=
(Y_axis (Col (G,j))) . i
by A15, GOBOARD1:def 2
.=
(Y_axis (Col (G,j))) . 1
by A6, A14, A15
.=
(c /. 1) `2
by A14, GOBOARD1:def 2
.=
(G * (1,j)) `2
by A9, A11, MATRIX_0:def 8
; verum