let f be non empty FinSequence of (TOP-REAL 2); :: thesis: for j being Element of NAT st 1 <= j & j <= width (GoB f) holds
ex k, i being Element of NAT st
( k in dom f & [i,j] in Indices (GoB f) & f /. k = (GoB f) * i,j )
let j be Element of NAT ; :: thesis: ( 1 <= j & j <= width (GoB f) implies ex k, i being Element of NAT st
( k in dom f & [i,j] in Indices (GoB f) & f /. k = (GoB f) * i,j ) )
assume that
A1: 1 <= j and
A2: j <= width (GoB f) ; :: thesis: ex k, i being Element of NAT st
( k in dom f & [i,j] in Indices (GoB f) & f /. k = (GoB f) * i,j )
A3: j in Seg (width (GoB f)) by A1, A2, FINSEQ_1:3;
A4: GoB f = GoB (Incr (X_axis f)),(Incr (Y_axis f)) by GOBOARD2:def 3;
then len (Incr (Y_axis f)) = width (GoB f) by GOBOARD2:def 1;
then j in dom (Incr (Y_axis f)) by A1, A2, FINSEQ_3:27;
then (Incr (Y_axis f)) . j in rng (Incr (Y_axis f)) by FUNCT_1:def 5;
then (Incr (Y_axis f)) . j in rng (Y_axis f) by GOBOARD2:def 2;
then consider k being Nat such that
A5: k in dom (Y_axis f) and
A6: (Y_axis f) . k = (Incr (Y_axis f)) . j by FINSEQ_2:11;
A7: len (X_axis f) = len f by GOBOARD1:def 3
.= len (Y_axis f) by GOBOARD1:def 4 ;
then k in dom (X_axis f) by A5, FINSEQ_3:31;
then (X_axis f) . k in rng (X_axis f) by FUNCT_1:def 5;
then (X_axis f) . k in rng (Incr (X_axis f)) by GOBOARD2:def 2;
then consider i being Nat such that
A8: i in dom (Incr (X_axis f)) and
A9: (X_axis f) . k = (Incr (X_axis f)) . i by FINSEQ_2:11;
reconsider k = k, i = i as Element of NAT by ORDINAL1:def 13;
k in dom (X_axis f) by A5, A7, FINSEQ_3:31;
then A10: (X_axis f) . k = (f /. k) `1 by GOBOARD1:def 3;
take k ; :: thesis: ex i being Element of NAT st
( k in dom f & [i,j] in Indices (GoB f) & f /. k = (GoB f) * i,j )
take i ; :: thesis: ( k in dom f & [i,j] in Indices (GoB f) & f /. k = (GoB f) * i,j )
len (Y_axis f) = len f by GOBOARD1:def 4;
hence k in dom f by A5, FINSEQ_3:31; :: thesis: ( [i,j] in Indices (GoB f) & f /. k = (GoB f) * i,j )
len (GoB (Incr (X_axis f)),(Incr (Y_axis f))) = len (Incr (X_axis f)) by GOBOARD2:def 1;
then i in dom (GoB (Incr (X_axis f)),(Incr (Y_axis f))) by A8, FINSEQ_3:31;
then [i,j] in [:(dom (GoB f)),(Seg (width (GoB f))):] by A4, A3, ZFMISC_1:106;
hence A11: [i,j] in Indices (GoB f) by MATRIX_1:def 5; :: thesis: f /. k = (GoB f) * i,j
(Y_axis f) . k = (f /. k) `2 by A5, GOBOARD1:def 4;
hence f /. k = |[((Incr (X_axis f)) . i),((Incr (Y_axis f)) . j)]| by A6, A9, A10, EUCLID:57
.= (GoB f) * i,j by A4, A11, GOBOARD2:def 1 ;
:: thesis: verum