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

theorem Th23:
  for f being circular s.c.c. FinSequence of TOP-REAL 2 st len f >
4 for i,j being Nat st 1 < i & i < j & j <= len f holds f/.i <> f/.j
proof
  let f be circular s.c.c. FinSequence of TOP-REAL 2 such that
A1: len f > 4;
  let i,j be Nat such that
A2: 1 < i and
A3: i < j and
A4: j <= len f;
  per cases by A4,XXREAL_0:1;
  suppose
    j < len f;
    hence thesis by A1,A2,A3,GOBOARD7:35;
  end;
  suppose
    j = len f;
    then
A5: f/.j = f/.1 by FINSEQ_6:def 1;
    i < len f by A3,A4,XXREAL_0:2;
    hence thesis by A1,A2,A5,GOBOARD7:35;
  end;
end;
