reserve r,lambda for Real,
  i,j,n for Nat;
reserve p,p1,p2,q1,q2 for Point of TOP-REAL 2,
  P, P1 for Subset of TOP-REAL 2;
reserve T for TopSpace;

theorem Th22:
  for f being FinSequence of TOP-REAL n holds len f = 0 or len f =
  1 iff L~f = {}
proof
  let f be FinSequence of TOP-REAL n;
  thus (len f = 0 or len f = 1) implies L~f = {}
  proof
    set L = { LSeg(f,i) where i is Nat : 1 <= i & i+1 <= len f };
    set x = the Element of L;
    assume
A1: len f = 0 or len f = 1;
    now
      per cases by A1;
      suppose
A2:     len f = 0;
        now
          assume L <> {};
          then x in L;
          then
          ex i being Nat st x = LSeg(f,i) & 1 <= i & i+1 <= len f;
          hence contradiction by A2;
        end;
        hence thesis by ZFMISC_1:2;
      end;
      suppose
A3:     len f = 0+1;
        now
          assume L <> {};
          then x in L;
          then ex i being Nat st x = LSeg(f,i) & 1 <= i & i+1 <=
          len f;
          hence contradiction by A3,XREAL_1:6;
        end;
        hence thesis by ZFMISC_1:2;
      end;
    end;
    hence thesis;
  end;
  set L = { LSeg(f,i) where i is Nat : 1 <= i & i+1 <= len f };
  assume
A4: L~f = {};
  assume that
A5: len f <> 0 and
A6: len f <> 1;
  now
    assume len f <= 1;
    then len f < 0+1 by A6,XXREAL_0:1;
    hence contradiction by A5,NAT_1:13;
  end;
  then
A7: len f >= 1+1 by NAT_1:13;
  then LSeg(f,1) in L;
  then LSeg(f,1) = {} by A4,XBOOLE_1:3,ZFMISC_1:74;
  hence contradiction by A7,Th21;
end;
