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 <= width G implies G*(i,j)`1 <= G* (
  len G,k)`1
proof
  assume that
A1: [i,j] in Indices G and
A2: 1 <= k & k <= width G;
A3: 1 <= i by A1,MATRIX_0:32;
A4: i <= len G by A1,MATRIX_0:32;
  then
A5: i < len G or i = len G by XXREAL_0:1;
  1 <= j & j <= width G by A1,MATRIX_0:32;
  then G*(i,j)`1 = G*(i,1)`1 by A3,A4,GOBOARD5:2
    .= G*(i,k)`1 by A2,A3,A4,GOBOARD5:2;
  hence thesis by A2,A3,A5,GOBOARD5:3;
end;
