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

theorem Th22:
  for f being circular s.c.c. FinSequence of TOP-REAL 2, n st n <
  len f & len f > 4 holds f|n is one-to-one
proof
  let f be circular s.c.c. FinSequence of TOP-REAL 2,n such that
A1: n < len f and
A2: len f > 4;
  for c1,c2 being Element of NAT st c1 in dom(f|n) & c2 in dom(f|n) & (f|n
  )/.c1 = (f|n)/.c2 holds c1 = c2
  proof
A3: len(f|n) <= n by FINSEQ_5:17;
A4: len(f|n) <= n by FINSEQ_5:17;
    let c1,c2 being Element of NAT such that
A5: c1 in dom(f|n) and
A6: c2 in dom(f|n) and
A7: (f|n)/.c1 = (f|n)/.c2;
A8: 1 <= c1 by A5,FINSEQ_3:25;
    c1 <= len(f|n) by A5,FINSEQ_3:25;
    then c1 <= n by A3,XXREAL_0:2;
    then
A9: c1 < len f by A1,XXREAL_0:2;
A10: 1 <= c2 by A6,FINSEQ_3:25;
A11: (f|n)/.c1 = f/.c1 by A5,FINSEQ_4:70;
    c2 <= len(f|n) by A6,FINSEQ_3:25;
    then c2 <= n by A4,XXREAL_0:2;
    then
A12: c2 < len f by A1,XXREAL_0:2;
A13: (f|n)/.c2 = f/.c2 by A6,FINSEQ_4:70;
    assume
A14: c1 <> c2;
    per cases by A14,XXREAL_0:1;
    suppose
      c1 < c2;
      hence thesis by A2,A7,A8,A12,A11,A13,GOBOARD7:35;
    end;
    suppose
      c2 < c1;
      hence thesis by A2,A7,A10,A9,A11,A13,GOBOARD7:35;
    end;
  end;
  hence thesis by PARTFUN2:9;
end;
