let i, j be Nat; for f being non empty FinSequence of (TOP-REAL 2) st 1 <= i & i <= len (GoB f) & 1 <= j & j <= width (GoB f) holds
ex k being Nat st
( k in dom f & (f /. k) `2 = ((GoB f) * (i,j)) `2 )
let f be non empty FinSequence of (TOP-REAL 2); ( 1 <= i & i <= len (GoB f) & 1 <= j & j <= width (GoB f) implies ex k being Nat st
( k in dom f & (f /. k) `2 = ((GoB f) * (i,j)) `2 ) )
assume that
A1:
( 1 <= i & i <= len (GoB f) )
and
A2:
( 1 <= j & j <= width (GoB f) )
; ex k being Nat st
( k in dom f & (f /. k) `2 = ((GoB f) * (i,j)) `2 )
A3:
GoB f = GoB ((Incr (X_axis f)),(Incr (Y_axis f)))
by GOBOARD2:def 2;
then
len (Incr (Y_axis f)) = width (GoB f)
by GOBOARD2:def 1;
then
j in dom (Incr (Y_axis f))
by A2, FINSEQ_3:25;
then
(Incr (Y_axis f)) . j in rng (Incr (Y_axis f))
by FUNCT_1:def 3;
then
(Incr (Y_axis f)) . j in rng (Y_axis f)
by SEQ_4:def 21;
then consider k being Nat such that
A4:
k in dom (Y_axis f)
and
A5:
(Y_axis f) . k = (Incr (Y_axis f)) . j
by FINSEQ_2:10;
[i,j] in Indices (GoB f)
by A1, A2, MATRIX_0:30;
then A6:
(GoB f) * (i,j) = |[((Incr (X_axis f)) . i),((Incr (Y_axis f)) . j)]|
by A3, GOBOARD2:def 1;
reconsider k = k as Nat ;
take
k
; ( k in dom f & (f /. k) `2 = ((GoB f) * (i,j)) `2 )
len (Y_axis f) = len f
by GOBOARD1:def 2;
hence
k in dom f
by A4, FINSEQ_3:29; (f /. k) `2 = ((GoB f) * (i,j)) `2
thus (f /. k) `2 =
(Incr (Y_axis f)) . j
by A4, A5, GOBOARD1:def 2
.=
((GoB f) * (i,j)) `2
by A6, EUCLID:52
; verum