
theorem
  for T1,S1,T2,S2 being non empty TopSpace for R1 being Refinement of T1
  ,S1, R2 being Refinement of T2,S2 for f being Function of T1,T2, g being
  Function of S1,S2 for h being Function of R1,R2 st h = f & h = g holds f is
  continuous & g is continuous implies h is continuous
proof
  let T1,S1,T2,S2 be non empty TopSpace;
  let R1 be Refinement of T1,S1, R2 be Refinement of T2,S2;
  let f be Function of T1,T2, g be Function of S1,S2;
  let h be Function of R1,R2 such that
A1: h = f and
A2: h = g;
A3: [#]S2 <> {};
  reconsider K = (the topology of T2) \/ the topology of S2 as prebasis of R2
  by YELLOW_9:def 6;
A4: [#]T2 <> {};
  assume
A5: f is continuous;
  assume
A6: g is continuous;
  now
    let A be Subset of R2;
    assume
A7: A in K;
    per cases by A7,XBOOLE_0:def 3;
    suppose
A8:   A in the topology of T2;
      then reconsider A1 = A as Subset of T2;
      A1 is open by A8;
      then f"A1 is open by A4,A5,TOPS_2:43;
      hence h"A is open by A1,Th20;
    end;
    suppose
A9:   A in the topology of S2;
      then reconsider A1 = A as Subset of S2;
      A1 is open by A9;
      then
A10:  g"A1 is open by A3,A6,TOPS_2:43;
      R1 is Refinement of S1,T1 by YELLOW_9:55;
      hence h"A is open by A10,A2,Th20;
    end;
  end;
  hence thesis by YELLOW_9:36;
end;
