
theorem
  for S1,S2,T1,T2 being non empty TopSpace st the TopStruct of S1 = the
TopStruct of S2 & the TopStruct of T1 = the TopStruct of T2 holds oContMaps(S1,
  T1) = oContMaps(S2, T2)
proof
  let S1,S2,T1,T2 be non empty TopSpace;
  assume that
A1: the TopStruct of S1 = the TopStruct of S2 and
A2: the TopStruct of T1 = the TopStruct of T2;
A3: oContMaps(S2, T2) = ContMaps(S2, Omega T2) by WAYBEL26:def 1;
  Omega T1 = Omega T2 by A2,WAYBEL25:13;
  then reconsider
  oCM2 = oContMaps(S2, T2) as full SubRelStr of (Omega T1) |^ the
  carrier of S1 by A3,A1,WAYBEL24:def 3;
  oContMaps(S1, T1) = ContMaps(S1, Omega T1) by WAYBEL26:def 1;
  then reconsider
  oCM1 = oContMaps(S1, T1) as full SubRelStr of (Omega T1) |^ the
  carrier of S1 by WAYBEL24:def 3;
  the carrier of oCM1 = the carrier of oCM2
  proof
    thus the carrier of oCM1 c= the carrier of oCM2
    proof
      let x be object;
A4:   the TopStruct of Omega T2 = the TopStruct of T2 by WAYBEL25:def 2;
      assume x in the carrier of oCM1;
      then x in the carrier of ContMaps(S1, Omega T1) by WAYBEL26:def 1;
      then consider f being Function of S1, Omega T1 such that
A5:   x = f and
A6:   f is continuous by WAYBEL24:def 3;
A7:   the TopStruct of Omega T1 = the TopStruct of T1 by WAYBEL25:def 2;
      then reconsider f1=f as Function of S2, Omega T2 by A4,A1,A2;
      for P1 being Subset of Omega T2 st P1 is closed holds f1" P1 is closed
      proof
        let P1 be Subset of Omega T2;
        reconsider P = P1 as Subset of (Omega T1) by A2,A7,A4;
        assume P1 is closed;
        then P is closed by A2,A7,A4,TOPS_3:79;
        then f"P is closed by A6,PRE_TOPC:def 6;
        hence thesis by A1,TOPS_3:79;
      end;
      then f1 is continuous by PRE_TOPC:def 6;
      then x in the carrier of ContMaps(S2, Omega T2) by A5,WAYBEL24:def 3;
      hence thesis by WAYBEL26:def 1;
    end;
    let x be object;
A8: the TopStruct of Omega T1 = the TopStruct of T1 by WAYBEL25:def 2;
    assume x in the carrier of oCM2;
    then x in the carrier of ContMaps(S2, Omega T2) by WAYBEL26:def 1;
    then consider f being Function of S2, Omega T2 such that
A9: x = f and
A10: f is continuous by WAYBEL24:def 3;
A11: the TopStruct of Omega T2 = the TopStruct of T2 by WAYBEL25:def 2;
    then reconsider f1=f as Function of S1, Omega T1 by A8,A1,A2;
    for P1 being Subset of Omega T1 st P1 is closed holds f1" P1 is closed
    proof
      let P1 be Subset of Omega T1;
      reconsider P = P1 as Subset of (Omega T2) by A2,A11,A8;
      assume P1 is closed;
      then P is closed by A2,A11,A8,TOPS_3:79;
      then f"P is closed by A10,PRE_TOPC:def 6;
      hence thesis by A1,TOPS_3:79;
    end;
    then f1 is continuous by PRE_TOPC:def 6;
    then x in the carrier of ContMaps(S1, Omega T1) by A9,WAYBEL24:def 3;
    hence thesis by WAYBEL26:def 1;
  end;
  hence thesis by YELLOW_0:57;
end;
