reserve p,p1,p2,q for Point of TOP-REAL 2,
  f,f1,f2,g,g1,g2 for FinSequence of TOP-REAL 2,
  r,s for Real,

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

theorem
  (for n,m st m>n+1 & n in dom f & n+1 in dom f & m in dom f & m+1 in
  dom f holds LSeg(f,n) misses LSeg(f,m)) implies f is s.n.c.
proof
  assume
A1: for n,m st m>n+1 & n in dom f & n+1 in dom f & m in dom f & m+1 in
  dom f holds LSeg(f,n) misses LSeg(f,m);
  let n,m be Nat such that
A2: m>n+1;
A3: n <= n+1 & m <= m+1 by NAT_1:11;
  per cases;
  suppose
    n in dom f & n+1 in dom f & m in dom f & m+1 in dom f;
    hence thesis by A1,A2;
  end;
  suppose
    not(n in dom f & n+1 in dom f & m in dom f & m+1 in dom f);
    then
    not(1 <= n & n <= len f & 1 <= n+1 & n+1<= len f & 1 <= m & m <= len f
    & 1 <= m+1 & m+1<= len f) by FINSEQ_3:25;
    then not(1 <= n & n+1 <= len f & 1 <= m & m+1 <= len f) by A3,XXREAL_0:2;
    then LSeg(f,m)={} or LSeg(f,n)={} by TOPREAL1:def 3;
    hence thesis;
  end;
end;
