reserve T1,T2,T3 for TopSpace,
  A1 for Subset of T1, A2 for Subset of T2, A3 for Subset of T3;
reserve n,k for Nat;
reserve M,N for non empty TopSpace;

theorem Th10:
  M is Hausdorff & M,N are_homeomorphic implies N is Hausdorff
proof
  assume
A1: M is Hausdorff;
  assume M,N are_homeomorphic;
  then consider f be Function of N,M such that
A2: f is being_homeomorphism by T_0TOPSP:def 1;
A3: dom f = [#]N & rng f = [#]M & f is one-to-one & f is continuous &
  f" is continuous by A2,TOPS_2:def 5;
  for p, q being Point of N st p <> q holds
  ex N1, N2 being Subset of N st
  N1 is open & N2 is open & p in N1 & q in N2 & N1 misses N2
  proof
    let p, q be Point of N;
    assume p <> q;
    then consider M1, M2 be Subset of M such that
A4: M1 is open & M2 is open & f.p in M1 & f.q in M2 & M1 misses M2
    by A1,A3,PRE_TOPC:def 10;
    reconsider N1 = f"M1 as Subset of N;
    reconsider N2 = f"M2 as Subset of N;
    take N1, N2;
    thus N1 is open & N2 is open by A4,A3,TOPS_2:43;
    thus p in N1 & q in N2 by A4,FUNCT_2:38;
    thus N1 misses N2 by A4,FUNCT_1:71;
  end;
  hence N is Hausdorff by PRE_TOPC:def 10;
end;
