reserve P for Subset of TOP-REAL 2,
  f,f1,f2,g for FinSequence of TOP-REAL 2,
  p,p1,p2,q,q1,q2 for Point of TOP-REAL 2,
  r1,r2,r19,r29 for Real,
  i,j,k,n for Nat;

theorem Th35:
  f is s.n.c. implies Rev f is s.n.c.
proof
  assume
A1: f is s.n.c.;
  let i,j be Nat such that
A2: i+1 < j;
  per cases;
  suppose
A3: i = 0 or j+1 > len(Rev f);
    now
      per cases by A3;
      case
        i = 0;
        hence LSeg(Rev f,i) = {} by TOPREAL1:def 3;
      end;
      case
        j+1 > len(Rev f);
        hence LSeg(Rev f,j) = {} by TOPREAL1:def 3;
      end;
    end;
    then LSeg(Rev f,i) /\ LSeg(Rev f,j) = {};
    hence thesis;
  end;
  suppose that
    i <> 0 and
A4: j+1 <= len(Rev f);
A5: j <= j+1 by NAT_1:11;
    i <= i+1 by NAT_1:11;
    then
A6: i < j by A2,XXREAL_0:2;
A7: len Rev f = len f by FINSEQ_5:def 3;
    then reconsider j9 = len f - j as Element of NAT by A4,A5,INT_1:5
,XXREAL_0:2;
    j <= len f by A4,A7,A5,XXREAL_0:2;
    then reconsider i9 = len f - i as Element of NAT by A6,INT_1:5,XXREAL_0:2;
A8: j9+j = len f;
    len f - (i+1) > j9 by A2,XREAL_1:10;
    then i9 - 1 + 1 > j9+1 by XREAL_1:6;
    then
A9: LSeg(f,i9) misses LSeg(f,j9) by A1;
    i9+i = len f;
    hence LSeg(Rev f,i) /\ LSeg(Rev f,j) = LSeg(f,i9) /\ LSeg(Rev f,j) by Th2
      .= {} by A9,A8,Th2;
  end;
end;
