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 Th8:
  1 <= j & j <= width GoB f implies ex k,i st k in dom f & [i,j]
  in Indices GoB f & f/.k = (GoB f)*(i,j)
proof
  assume that
A1: 1 <= j and
A2: j <= width GoB f;
A3: j in Seg width GoB f by A1,A2,FINSEQ_1:1;
A4: 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 A1,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
A5: k in dom Y_axis f and
A6: (Y_axis f).k = (Incr Y_axis f).j by FINSEQ_2:10;
A7: len X_axis f = len f by GOBOARD1:def 1
    .= len Y_axis f by GOBOARD1:def 2;
  then k in dom X_axis f by A5,FINSEQ_3:29;
  then (X_axis f).k in rng X_axis f by FUNCT_1:def 3;
  then (X_axis f).k in rng Incr X_axis f by SEQ_4:def 21;
  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_2:10;
  reconsider k,i as Nat;
  k in dom X_axis f by A5,A7,FINSEQ_3:29;
  then
A10: (X_axis f).k = (f/.k)`1 by GOBOARD1:def 1;
  take k,i;
  len Y_axis f = len f by GOBOARD1:def 2;
  hence k in dom f by A5,FINSEQ_3:29;
  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_3:29;
  then [i,j] in [:dom GoB f, Seg width GoB f:] by A4,A3,ZFMISC_1:87;
  hence
A11: [i,j] in Indices GoB f by MATRIX_0:def 4;
  (Y_axis f).k = (f/.k)`2 by A5,GOBOARD1:def 2;
  hence f/.k = |[Incr(X_axis f).i,Incr(Y_axis f).j]| by A6,A9,A10,EUCLID:53
    .= (GoB f)*(i,j) by A4,A11,GOBOARD2:def 1;
end;
