reserve G for Go-board,
  i,j,k,m,n for Nat;

theorem Th28:
  for f,g being FinSequence of TOP-REAL 2 st j < len f holds LSeg(
  f^'g,j) = LSeg(f,j)
proof
  let f,g be FinSequence of TOP-REAL 2 such that
A1: j < len f;
  per cases;
  suppose
A2: j < 1;
    hence LSeg(f^'g,j) = {} by TOPREAL1:def 3
      .= LSeg(f,j) by A2,TOPREAL1:def 3;
  end;
  suppose
A3: 1 <= j;
    then
A4: (f^'g)/.j = f/.j by A1,FINSEQ_6:159;
A5: j+1 <= len f by A1,NAT_1:13;
    then
A6: (f^'g)/.(j+1) = f/.(j+1) by FINSEQ_6:159,NAT_1:11;
    len f <= len(f^'g) by Th7;
    then j+1 <= len (f^'g) by A5,XXREAL_0:2;
    hence LSeg(f^'g,j) = LSeg((f^'g)/.j,(f^'g)/.(j+1)) by A3,TOPREAL1:def 3
      .= LSeg(f,j) by A3,A4,A5,A6,TOPREAL1:def 3;
  end;
end;
