reserve n for Element of NAT,
  V for Subset of TOP-REAL n,
  s,s1,s2,t,t1,t2 for Point of TOP-REAL n,
  C for Simple_closed_curve,
  P for Subset of TOP-REAL 2,
  a,p ,p1,p2,q,q1,q2 for Point of TOP-REAL 2;

theorem
  p1 <> p2 & p1 in C & p2 in C implies (p1,p2, C-separate q1,q2 implies
  q1,q2, C-separate p1,p2)
proof
  assume that
A1: p1 <> p2 and
A2: p1 in C and
A3: p2 in C and
A4: p1,p2, C-separate q1,q2;
  per cases;
  suppose
    q1 = p1;
    hence thesis by Th27;
  end;
  suppose
    q2 = p2;
    hence thesis by Th28;
  end;
  suppose
    q1 = p2;
    hence thesis by Th30;
  end;
  suppose
    p1 = q2;
    hence thesis by Th29;
  end;
  suppose that
A5: q1 <> p1 & q2 <> p2 & q1 <> p2 & q2 <> p1;
    let A be Subset of TOP-REAL 2;
    assume A is_an_arc_of q1,q2 & A c= C;
    then consider B being non empty Subset of TOP-REAL 2 such that
A6: B is_an_arc_of q1,q2 and
A7: A \/ B = C and
    A /\ B = {q1,q2} by Th15;
    assume
A8: A misses {p1,p2};
    then not p2 in A by ZFMISC_1:49;
    then
A9: p2 in B by A3,A7,XBOOLE_0:def 3;
    not p1 in A by A8,ZFMISC_1:49;
    then p1 in B by A2,A7,XBOOLE_0:def 3;
    then consider P being non empty Subset of TOP-REAL 2 such that
A10: P is_an_arc_of p1,p2 and
A11: P c= B and
A12: P misses {q1,q2} by A1,A5,A6,A9,JORDAN16:23;
    B c= C by A7,XBOOLE_1:7;
    then P c= C by A11;
    hence thesis by A4,A10,A12;
  end;
end;
