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

theorem Th21:
  for f being s.c.c. FinSequence of TOP-REAL 2, n st 1 <= n holds
  f/^n is s.n.c.
proof
  let f be s.c.c. FinSequence of TOP-REAL 2, n such that
A1: 1 <= n;
  let i,j be Nat such that
A2: i+1 < j;
  per cases;
  suppose
    i < 1;
    then LSeg(f/^n,i) = {} by TOPREAL1:def 3;
    then LSeg(f/^n,i) /\ LSeg(f/^n,j) = {};
    hence thesis by XBOOLE_0:def 7;
  end;
  suppose
    n > len f;
    then f/^n = <*>the carrier of TOP-REAL2 by RFINSEQ:def 1;
    then not(1 <= i & i+1 <= len(f/^n));
    then LSeg(f/^n,i) = {} by TOPREAL1:def 3;
    then LSeg(f/^n,i) /\ LSeg(f/^n,j) = {};
    hence thesis by XBOOLE_0:def 7;
  end;
  suppose
    len(f/^n) <= j;
    then len(f/^n) < j+1 by NAT_1:13;
    then LSeg(f/^n,j) = {} by TOPREAL1:def 3;
    then LSeg(f/^n,i) /\ LSeg(f/^n,j) = {};
    hence thesis by XBOOLE_0:def 7;
  end;
  suppose that
A3: n <= len f and
A4: 1 <= i and
A5: j < len(f/^n);
A6: j < len f - n by A3,A5,RFINSEQ:def 1;
    then
A7: j+1 <= len f - n by INT_1:7;
    1+1 <= i+1 by A4,XREAL_1:6;
    then 1+1 <= j by A2,XXREAL_0:2;
    then 1 < j by NAT_1:13;
    then
A8: LSeg(f,j+n) = LSeg(f/^n,j) by A7,SPPOL_2:5;
    i+1+n < j+n by A2,XREAL_1:6;
    then
A9: i+n+1 < j+n;
    j <= j+1 by NAT_1:11;
    then i+1 <= j+1 by A2,XXREAL_0:2;
    then
A10: i+1 <= len f - n by A7,XXREAL_0:2;
    1+1 <= i+n by A1,A4,XREAL_1:7;
    then
A11: 1 < i+n by NAT_1:13;
    j+n < len f by A6,XREAL_1:20;
    then LSeg(f,i+n) misses LSeg(f,j+n) by A11,A9,GOBOARD5:def 4;
    hence thesis by A4,A8,A10,SPPOL_2:5;
  end;
end;
