  reserve n,m,i for Nat,
          p,q for Point of TOP-REAL n,
          r,s for Real,
          R for real-valued FinSequence;
reserve T1,T2,S1,S2 for non empty TopSpace,
        t1 for Point of T1, t2 for Point of T2,
        pn,qn for Point of TOP-REAL n,
        pm,qm for Point of TOP-REAL m;

theorem Th14:
  for f be Function of T1,T2 for g be Function of S1,S2 st
    f is being_homeomorphism & g is being_homeomorphism holds
  [:f,g:] is being_homeomorphism
proof
  let f be Function of T1,T2;
  let g be Function of S1,S2 such that
A1: f is being_homeomorphism
  and
A2: g is being_homeomorphism;
A3: rng g=[#]S2 by A2,TOPS_2:def 5;
A4: g" is continuous by A2,TOPS_2:def 5;
A5: f" is continuous by A1,TOPS_2:def 5;
A6:the carrier of [:T2,S2:] = [:the carrier of T2,the carrier of S2:]
    by BORSUK_1:def 2;
  set fg=[:f,g:];
A7: rng f = [#]T2 by A1,TOPS_2:def 5;
  then
A8:rng fg = [#][:T2,S2:] by A3, FUNCT_3:67,A6;
  reconsider F=f,G=g,FG=fg as Function;
A9:FG" = [:F",G":] by A1, A2,FUNCTOR0:6;
A10:F"=f" by A1,TOPS_2:def 4;
A11:rng fg = [:rng f, rng g:] by FUNCT_3:67;
  then fg is onto by A6,A7,A3,FUNCT_2:def 3;
  then
A12:FG" = fg" by A1, A2,TOPS_2:def 4;
A13: now
    let P be Subset of [:T2,S2:];
    thus P is open implies fg"P is open by BORSUK_2:10, A1, A2;
    thus fg"P is open implies P is open
    proof
      assume
A14:    fg"P is open;
      [:f",g":]"(fg"P) = (fg")"(fg"P) by A10, A2,TOPS_2:def 4,A9,A12
                      .= fg.:(fg"P) by TOPS_2:54,A1, A2,A8
                      .=P by FUNCT_1:77,A7,A3,A11,A6;
      hence thesis by BORSUK_2:10,A5,A4,A14;
    end;
  end;
A15: dom g=[#]S1 by A2,TOPS_2:def 5;
A16:the carrier of [:T1,S1:] = [:the carrier of T1,the carrier of S1:]
    by BORSUK_1:def 2;
  dom f = [#]T1 by A1,TOPS_2:def 5;
  then dom fg = [#][:T1,S1:] by A15, FUNCT_3:def 8,A16;
  hence thesis by A13,TOPGRP_1:26,A1, A2,A8;
end;
