
theorem Th10:
  for S1,S2 being TopStruct st the TopStruct of S1 = the TopStruct
  of S2 for T1,T2 being non empty TopRelStr st the TopRelStr of T1 = the
  TopRelStr of T2 holds ContMaps(S1,T1) = ContMaps(S2,T2)
proof
  let S1,S2 be TopStruct;
  assume
A1: the TopStruct of S1 = the TopStruct of S2;
  let T1,T2 be non empty TopRelStr;
  assume
A2: the TopRelStr of T1 = the TopRelStr of T2;
  then the RelStr of T1 = the RelStr of T2;
  then T1 |^ the carrier of S1 = T2 |^ the carrier of S2 by A1,WAYBEL27:15;
  then reconsider
  C2 = ContMaps(S2,T2) as full SubRelStr of T1 |^ the carrier of S1
  by WAYBEL24:def 3;
  reconsider C1 = ContMaps(S1,T1) as full SubRelStr of T1 |^ the carrier of S1
  by WAYBEL24:def 3;
  the carrier of ContMaps(S1,T1) = the carrier of ContMaps(S2,T2)
  proof
    thus the carrier of ContMaps(S1,T1) c= the carrier of ContMaps(S2,T2)
    proof
      let x be object;
      assume x in the carrier of ContMaps(S1,T1);
      then consider f being Function of S1,T1 such that
A3:   x=f and
A4:   f is continuous by WAYBEL24:def 3;
      reconsider f2=f as Function of S2,T2 by A1,A2;
      f2 is continuous
      proof
        let P2 be Subset of T2;
        reconsider P1=P2 as Subset of T1 by A2;
        assume P2 is closed;
        then [#]T2 \ P2 is open;
        then [#]T2 \ P2 in the topology of T2;
        then ([#]T1) \ P1 is open by A2;
        then P1 is closed;
        then f" P1 is closed by A4;
        then [#]S1 \ (f" P1) is open;
        then [#]S1 \ (f2" P2) in the topology of S2 by A1;
        then [#]S2 \ (f2" P2) is open by A1;
        hence thesis;
      end;
      hence thesis by A3,WAYBEL24:def 3;
    end;
    let x be object;
    assume x in the carrier of ContMaps(S2,T2);
    then consider f being Function of S2,T2 such that
A5: x=f and
A6: f is continuous by WAYBEL24:def 3;
    reconsider f1=f as Function of S1,T1 by A1,A2;
    f1 is continuous
    proof
      let P1 be Subset of T1;
      reconsider P2=P1 as Subset of T2 by A2;
      assume P1 is closed;
      then [#]T1 \ P1 is open;
      then ([#]T1) \ P2 in the topology of T2 by A2;
      then ([#]T2) \ P2 is open by A2;
      then P2 is closed;
      then f" P2 is closed by A6;
      then [#]S2 \ (f" P2) is open;
      then [#]S2 \ (f1" P1) in the topology of S1 by A1;
      then [#]S1 \ (f1" P1) is open by A1;
      hence thesis;
    end;
    hence thesis by A5,WAYBEL24:def 3;
  end;
  then the RelStr of C1 = the RelStr of C2 by YELLOW_0:57;
  hence thesis;
end;
