
theorem
  for FT1,FT2 being non empty RelStr, h being Function of FT1, FT2, n
  being Nat,x being Element of FT1,y being Element of FT2 st h is
being_homeomorphism & y=h.x holds for v being Element of FT2 holds h".v in U_FT
  (x,n) iff v in U_FT(y,n)
proof
  let FT1,FT2 be non empty RelStr, h be Function of FT1, FT2, n be Nat,
   x be Element of FT1,y be Element of FT2;
  assume that
A1: h is being_homeomorphism and
A2: y=h.x;
  x in the carrier of FT1;
  then
A3: x in dom h by FUNCT_2:def 1;
  consider g being Function of FT2, FT1 such that
A4: g=h" and
A5: g is being_homeomorphism by A1,Th3;
  h is one-to-one onto by A1;
  then x=g.y by A2,A4,A3,FUNCT_1:34;
  hence thesis by A4,A5,Th4;
end;
