let j, i be Element of NAT ; :: thesis: 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); :: thesis: ( 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 & j <= width G ) and
A3: ( 1 <= i & i <= len G ) ; :: thesis: (G * i,j) `1 = (G * i,1) `1
i in dom G by A3, FINSEQ_3:27;
then A4: X_axis (Line G,i) is constant by A1, GOBOARD1:def 6;
reconsider c = Line G,i as FinSequence of (TOP-REAL 2) ;
A5: j in Seg (width G) by A2, FINSEQ_1:3;
A6: 1 <= width G by A2, XXREAL_0:2;
then A7: 1 in Seg (width G) by FINSEQ_1:3;
A8: len c = width G by MATRIX_1:def 8;
then 1 in dom c by A6, FINSEQ_3:27;
then A9: c /. 1 = c . 1 by PARTFUN1:def 8;
j in dom c by A2, A8, FINSEQ_3:27;
then A10: c /. j = c . j by PARTFUN1:def 8;
A11: len (X_axis (Line G,i)) = len c by GOBOARD1:def 3;
then A12: 1 in dom (X_axis (Line G,i)) by A6, A8, FINSEQ_3:27;
A13: j in dom (X_axis (Line G,i)) by A2, A8, A11, FINSEQ_3:27;
thus (G * i,j) `1 = (c /. j) `1 by A5, A10, MATRIX_1:def 8
.= (X_axis (Line G,i)) . j by A13, GOBOARD1:def 3
.= (X_axis (Line G,i)) . 1 by A4, A12, A13, GOBOARD1:def 2
.= (c /. 1) `1 by A12, GOBOARD1:def 3
.= (G * i,1) `1 by A7, A9, MATRIX_1:def 8 ; :: thesis: verum