reserve p, q for Point of TOP-REAL 2,
  r for Real,
  h for non constant standard special_circular_sequence,
  g for FinSequence of TOP-REAL 2,
  f for non empty FinSequence of TOP-REAL 2,
  I, i1, i, j, k for Nat;

theorem Th9:
  1 <= i & i <= len GoB f & 1 <= j & j <= width GoB f implies ex k
  st k in dom f & [i,j] in Indices GoB f & (f/.k)`1 = (GoB f)*(i,j)`1
proof
  assume that
A1: 1 <= i and
A2: i <= len GoB f and
A3: 1 <= j and
A4: j <= width GoB f;
A5: 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 A3,A4,FINSEQ_3:25;
  then j in Seg len Incr Y_axis f by FINSEQ_1:def 3;
  then
A6: j in Seg width GoB(Incr X_axis f,Incr Y_axis f) by GOBOARD2:def 1;
  len Incr X_axis f = len GoB f by A5,GOBOARD2:def 1;
  then i in dom Incr X_axis f by A1,A2,FINSEQ_3:25;
  then (Incr X_axis f).i in rng Incr X_axis f by FUNCT_1:def 3;
  then (Incr X_axis f).i in rng X_axis f by SEQ_4:def 21;
  then consider k being Nat such that
A7: k in dom X_axis f and
A8: (X_axis f).k = (Incr X_axis f).i by FINSEQ_2:10;
  reconsider k as Nat;
  take k;
  len X_axis f = len f by GOBOARD1:def 1;
  hence k in dom f by A7,FINSEQ_3:29;
  i in dom GoB f by A1,A2,FINSEQ_3:25;
  then [i,j] in [:dom GoB f, Seg width GoB f:] by A5,A6,ZFMISC_1:87;
  hence [i,j] in Indices GoB f by MATRIX_0:def 4;
  then
A9: (GoB f)*(i,j) = |[Incr(X_axis f).i,Incr(Y_axis f).j]| by A5,GOBOARD2:def 1;
  thus (f/.k)`1 = Incr(X_axis f).i by A7,A8,GOBOARD1:def 1
    .= (GoB f)*(i,j)`1 by A9,EUCLID:52;
end;
