
theorem Th12:
  for T being non empty TopSpace, T0 being T_0-TopSpace, f being
  continuous Function of T,T0 holds for p being Point of T holds f.:Class(
  Indiscernibility(T),p) = {f.p}
proof
  let T be non empty TopSpace;
  let T0 be T_0-TopSpace;
  let f be continuous Function of T,T0;
  let p be Point of T;
  set R = Indiscernibility(T);
  for y being object holds y in f.:Class(R,p) iff y in {f.p}
  proof
    let y be object;
    hereby
      assume y in f.:Class(R,p);
      then consider x being object such that
A1:   x in the carrier of T and
A2:   x in Class(R,p) and
A3:   y = f.x by FUNCT_2:64;
      [x,p] in R by A2,EQREL_1:19;
      then f.x = f.p by A1,Th11;
      hence y in {f.p} by A3,TARSKI:def 1;
    end;
    assume y in {f.p};
    then
A4: y = f.p by TARSKI:def 1;
    p in Class(R,p) by EQREL_1:20;
    hence thesis by A4,FUNCT_2:35;
  end;
  hence thesis by TARSKI:2;
end;
