
theorem Th11:
  for T being non empty TopSpace, T0 being T_0-TopSpace, f being
  continuous Function of T,T0 holds for p,q being Point of T holds [p,q] in
  Indiscernibility(T) implies f.p = f.q
proof
  let T be non empty TopSpace;
  let T0 be T_0-TopSpace;
  let f be continuous Function of T,T0;
  let p,q be Point of T;
  set p9 = f.p, q9 = f.q;
  assume that
A1: [p,q] in Indiscernibility(T) and
A2: not f.p = f.q;
  consider V being Subset of T0 such that
A3: V is open and
A4: p9 in V & not q9 in V or q9 in V & not p9 in V by A2,Def7;
  set A = f"V;
  [#]T0 <> {};
  then
A5: A is open by A3,TOPS_2:43;
  reconsider f as Function of the carrier of T, the carrier of T0;
  q in the carrier of T;
  then
A6: q in dom f by FUNCT_2:def 1;
  p in the carrier of T;
  then p in dom f by FUNCT_2:def 1;
  then not (p in A iff q in A) by A4,A6,FUNCT_1:def 7;
  hence contradiction by A1,A5,Def3;
end;
