
theorem
  for f be FinSequence of TOP-REAL 2 for i be Nat st
  f is unfolded s.n.c. one-to-one & len f >= 2 & f/.1 in LSeg (f,i) holds
  i = 1
proof
  let f be FinSequence of TOP-REAL 2, i be Nat;
  assume that
A1: f is unfolded s.n.c. one-to-one and
A2: 2 <= len f;
  1 <= len f by A2,XXREAL_0:2;
  then
A3: 1 in dom f by FINSEQ_3:25;
A4: 2 in dom f by A2,FINSEQ_3:25;
  assume
A5: f/.1 in LSeg (f,i);
  assume
A6: i <> 1;
  per cases by A6,XXREAL_0:1;
  suppose
A7: i > 1;
    1+1 <= len f by A2;
    then f/.1 in LSeg (f,1) by TOPREAL1:21;
    then
A8: f/.1 in LSeg (f,1) /\ LSeg (f,i) by A5,XBOOLE_0:def 4;
    then
A9: LSeg (f,1) meets LSeg (f,i);
    now per cases by XXREAL_0:1;
      suppose
A10:    i = 2;
        then
A11:    1 + 2 <= len f by A5,TOPREAL1:def 3;
        1 + 1 = 2;
        then f/.1 in { f/.2 } by A1,A8,A10,A11,TOPREAL1:def 6;
        then f/.1 = f/.2 by TARSKI:def 1;
        hence contradiction by A1,A3,A4,PARTFUN2:10;
      end;
      suppose i > 2;
        then 1 + 1 < i;
        hence contradiction by A1,A9,TOPREAL1:def 7;
      end;
      suppose i < 2;
        then i < 1+1;
        hence contradiction by A7,NAT_1:13;
      end;
    end;
    hence thesis;
  end;
  suppose i < 1;
    hence thesis by A5,TOPREAL1:def 3;
  end;
end;
