
theorem Th3:
  for FT1,FT2 being non empty RelStr, h being Function of FT1, FT2
  st h is being_homeomorphism ex g being Function of FT2, FT1 st g=h" & g is
  being_homeomorphism
proof
  let FT1,FT2 be non empty RelStr, h be Function of FT1, FT2;
  assume
A1: h is being_homeomorphism;
  then
A2: h is one-to-one onto;
  then
A3: rng h = the carrier of FT2 by FUNCT_2:def 3;
  then reconsider g2=h" as Function of FT2, FT1 by A2,FUNCT_2:25;
A4: for y being Element of FT2 holds g2.:U_FT(y)=Im(the InternalRel of FT1,
  g2.y)
  proof
    let y be Element of FT2;
    reconsider x = g2.y as Element of FT1;
    y=h.x & h.:U_FT x= Im(the InternalRel of FT2,h.x) by A1,A3,
FUNCT_1:35;
    hence thesis by A2,Th1;
  end;
  rng g2=dom h by A2,FUNCT_1:33
    .=the carrier of FT1 by FUNCT_2:def 1;
  then
A5: g2 is onto by FUNCT_2:def 3;
  g2 is one-to-one by A2,FUNCT_1:40;
  then g2 is being_homeomorphism by A5,A4;
  hence thesis;
end;
