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

theorem Th24:
  for f being circular s.c.c. FinSequence of TOP-REAL 2, n st 1 <=
  n & 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: 1 <= n 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
    let c1,c2 being Element of NAT such that
A3: c1 in dom(f/^n) and
A4: c2 in dom(f/^n) and
A5: (f/^n)/.c1 = (f/^n)/.c2;
A6: (f/^n)/.c1 = f/.(c1+n) by A3,FINSEQ_5:27;
A7: n <= len f by A3,RELAT_1:38,RFINSEQ:def 1;
    then len(f/^n) = len f - n by RFINSEQ:def 1;
    then
A8: len(f/^n) + n = len f;
    len(f/^n) = len f - n by A7,RFINSEQ:def 1;
    then
A9: len(f/^n) + n = len f;
    c1 <= len(f/^n) by A3,FINSEQ_3:25;
    then
A10: c1+n <= len f by A9,XREAL_1:6;
    0+1 <= c1 by A3,FINSEQ_3:25;
    then
A11: 1+0 < c1+n by A1,XREAL_1:8;
A12: (f/^n)/.c2 = f/.(c2+n) by A4,FINSEQ_5:27;
    c2 <= len(f/^n) by A4,FINSEQ_3:25;
    then
A13: c2+n <= len f by A8,XREAL_1:6;
    0+1 <= c2 by A4,FINSEQ_3:25;
    then
A14: 1+0 < c2+n by A1,XREAL_1:8;
    assume
A15: c1 <> c2;
    per cases by A15,XXREAL_0:1;
    suppose
      c1 < c2;
      then c1+n < c2+n by XREAL_1:6;
      hence thesis by A2,A5,A11,A13,A6,A12,Th23;
    end;
    suppose
      c2 < c1;
      then c2+n < c1+n by XREAL_1:6;
      hence thesis by A2,A5,A14,A10,A6,A12,Th23;
    end;
  end;
  hence thesis by PARTFUN2:9;
end;
