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
  f is s.n.c. implies f|i is s.n.c.
proof
  assume
A1: f is s.n.c.;
  set f1 = f|i;
    let n,m be Nat;
    assume m>n+1;
    then LSeg(f,n) misses LSeg(f,m) by A1;
    then
A2: LSeg(f,n) /\ LSeg(f,m) = {};
    now
A3:     m <= m+1 by NAT_1:11;
A4:     n <= n+1 by NAT_1:11;
        now
          per cases;
          suppose
A5:         n in dom f1;
            now
              per cases;
              suppose
                n+1 in dom f1;
                then
A6:             LSeg(f,n)=LSeg(f1,n) by A5,TOPREAL3:17;
                now
                  per cases;
                  suppose
A7:                 m in dom f1;
                    now
                      per cases;
                      suppose
                        m+1 in dom f1;
                        hence LSeg(f1,n) /\ LSeg(f1,m) = {} by A2,A6,A7,
TOPREAL3:17;
                      end;
                      suppose
                        not m+1 in dom f1;
                        then not(1 <= m+1 & m+1<= len f1) by FINSEQ_3:25;
                        then LSeg(f1,m)={} by NAT_1:11,TOPREAL1:def 3;
                        hence LSeg(f1,n) /\ LSeg(f1,m) = {};
                      end;
                    end;
                    hence thesis;
                  end;
                  suppose
                    not m in dom f1;
                    then not(1 <= m & m <= len f1) by FINSEQ_3:25;
                    then not(1 <= m & m+1 <= len f1) by A3,XXREAL_0:2;
                    then LSeg(f1,m)={} by TOPREAL1:def 3;
                    hence thesis;
                  end;
                end;
                hence thesis;
              end;
              suppose
                not n+1 in dom f1;
                then not(1 <= n+1 & n+1<= len f1) by FINSEQ_3:25;
                then LSeg(f1,n)={} by NAT_1:11,TOPREAL1:def 3;
                hence thesis;
              end;
            end;
            hence thesis;
          end;
          suppose
            not n in dom f1;
            then not(1 <= n & n <= len f1) by FINSEQ_3:25;
            then not(1 <= n & n+1 <= len f1) by A4,XXREAL_0:2;
            then LSeg(f1,n)={} by TOPREAL1:def 3;
            hence thesis;
          end;
        end;
        hence thesis;
    end;
    hence thesis;
end;
