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

theorem Th20:
  for f being s.c.c. FinSequence of TOP-REAL 2, n st n < len f
  holds f|n is s.n.c.
proof
  let f be s.c.c. FinSequence of TOP-REAL 2, n such that
A1: n < len f;
  let i,j be Nat such that
A2: i+1 < j;
A3: len(f|n) <= n by FINSEQ_5:17;
  per cases;
  suppose
    n < j+1;
    then len(f|n) < j+1 by A3,XXREAL_0:2;
    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
    len(f|n) < j+1;
    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
A4: j+1 <= n and
A5: j+1 <= len(f|n);
    j+1 < len f by A1,A4,XXREAL_0:2;
    then
A6: LSeg(f,i) misses LSeg(f,j) by A2,GOBOARD5:def 4;
    j <= j+1 by NAT_1:11;
    then
A7: i+1 <= j+1 by A2,XXREAL_0:2;
    LSeg(f,j) = LSeg(f|n,j) by A5,SPPOL_2:3;
    hence thesis by A5,A6,A7,SPPOL_2:3,XXREAL_0:2;
  end;
end;
