let j1, j2, i be Element of NAT ; for G being Matrix of (TOP-REAL 2) st G is Y_increasing-in-line & 1 <= j1 & j1 < j2 & j2 <= width G & 1 <= i & i <= len G holds
(G * i,j1) `2 < (G * i,j2) `2
let G be Matrix of (TOP-REAL 2); ( G is Y_increasing-in-line & 1 <= j1 & j1 < j2 & j2 <= width G & 1 <= i & i <= len G implies (G * i,j1) `2 < (G * i,j2) `2 )
assume that
A1:
G is Y_increasing-in-line
and
A2:
1 <= j1
and
A3:
j1 < j2
and
A4:
j2 <= width G
and
A5:
1 <= i
and
A6:
i <= len G
; (G * i,j1) `2 < (G * i,j2) `2
i in dom G
by A5, A6, FINSEQ_3:27;
then A7:
Y_axis (Line G,i) is increasing
by A1, GOBOARD1:def 8;
reconsider c = Line G,i as FinSequence of (TOP-REAL 2) ;
A8:
j1 <= width G
by A3, A4, XXREAL_0:2;
then A9:
j1 in Seg (width G)
by A2, FINSEQ_1:3;
A10:
1 <= j2
by A2, A3, XXREAL_0:2;
then A11:
j2 in Seg (width G)
by A4, FINSEQ_1:3;
A12:
len c = width G
by MATRIX_1:def 8;
then
j1 in dom c
by A2, A8, FINSEQ_3:27;
then A13:
c /. j1 = c . j1
by PARTFUN1:def 8;
j2 in dom c
by A4, A10, A12, FINSEQ_3:27;
then A14:
c /. j2 = c . j2
by PARTFUN1:def 8;
A15:
len (Y_axis (Line G,i)) = len c
by GOBOARD1:def 4;
then A16:
j1 in dom (Y_axis (Line G,i))
by A2, A8, A12, FINSEQ_3:27;
A17: (G * i,j1) `2 =
(c /. j1) `2
by A9, A13, MATRIX_1:def 8
.=
(Y_axis (Line G,i)) . j1
by A16, GOBOARD1:def 4
;
A18:
j2 in dom (Y_axis (Line G,i))
by A4, A10, A12, A15, FINSEQ_3:27;
then (Y_axis (Line G,i)) . j2 =
(c /. j2) `2
by GOBOARD1:def 4
.=
(G * i,j2) `2
by A11, A14, MATRIX_1:def 8
;
hence
(G * i,j1) `2 < (G * i,j2) `2
by A3, A7, A16, A17, A18, SEQM_3:def 1; verum