reserve i,j,k,n for Nat,
  D for non empty set,
  f, g for FinSequence of D;
reserve G for Go-board,
  f, g for FinSequence of TOP-REAL 2,
  p for Point of TOP-REAL 2,
  r, s for Real,
  x for set;

theorem
  [i,j] in Indices G & 1 <= k & k <= len G implies G*(i,j)`2 <= G* (k,
  width G)`2
proof
  assume that
A1: [i,j] in Indices G and
A2: 1 <= k & k <= len G;
A3: 1 <= j by A1,MATRIX_0:32;
A4: j <= width G by A1,MATRIX_0:32;
  then
A5: j < width G or j = width G by XXREAL_0:1;
  1 <= i & i <= len G by A1,MATRIX_0:32;
  then G*(i,j)`2 = G*(1,j)`2 by A3,A4,GOBOARD5:1
    .= G*(k,j)`2 by A2,A3,A4,GOBOARD5:1;
  hence thesis by A2,A3,A5,GOBOARD5:4;
end;
