
theorem
  for S, T, T1, T2, Y being non empty TopSpace, f being Function of [:Y,
  T1:],S, g being Function of [:Y,T2:],S, F1, F2 being closed Subset of T st T1
is SubSpace of T & T2 is SubSpace of T & F1 = [#]T1 & F2 = [#]T2 & ([#]T1) \/ (
[#]T2) = [#]T & f is continuous & g is continuous & (for p being set st p in (
[#][:Y,T1:]) /\ ([#][:Y,T2:]) holds f.p = g.p) ex h being Function of [:Y,T:],S
  st h = f+*g & h is continuous
proof
  let S, T, T1, T2, Y be non empty TopSpace, f be Function of [:Y,T1:],S, g be
  Function of [:Y,T2:],S, F1, F2 be closed Subset of T;
  assume that
A1: T1 is SubSpace of T and
A2: T2 is SubSpace of T and
A3: F1 = [#]T1 and
A4: F2 = [#]T2 and
A5: ([#]T1) \/ ([#]T2) = [#]T and
A6: f is continuous and
A7: g is continuous and
A8: for p being set st p in ([#][:Y,T1:]) /\ ([#][:Y,T2:]) holds f.p = g .p;
A9: dom f = the carrier of [:Y,T1:] by FUNCT_2:def 1;
  set h = f+*g;
A10: the carrier of [:Y,T2:] = [:the carrier of Y,the carrier of T2:] by
BORSUK_1:def 2;
A11: [:Y,T2:] is SubSpace of [:Y,T:] by A2,BORSUK_3:15;
A12: rng h c= rng f \/ rng g by FUNCT_4:17;
A13: dom g = the carrier of [:Y,T2:] by FUNCT_2:def 1;
A14: dom h = dom f \/ dom g by FUNCT_4:def 1;
A15: the carrier of [:Y,T1:] = [:the carrier of Y,the carrier of T1:] by
BORSUK_1:def 2;
  then
A16: dom h = [:the carrier of Y,the carrier of T:] by A5,A10,A9,A13,A14,
ZFMISC_1:97;
A17: the carrier of [:Y,T:] = [:the carrier of Y,the carrier of T:] by
BORSUK_1:def 2;
  then reconsider h as Function of [:Y,T:],S by A16,A12,FUNCT_2:2,XBOOLE_1:1;
  take h;
  thus h = f+*g;
A18: [:Y,T1:] is SubSpace of [:Y,T:] by A1,BORSUK_3:15;
  for P being Subset of S st P is closed holds h"P is closed
  proof
    reconsider M = [:[#]Y,F1:] as Subset of [:Y,T:];
    let P be Subset of S;
A19: now
      let x be object;
      thus x in h"P /\ [#][:Y,T2:] implies x in g"P
      proof
        assume
A20:    x in h"P /\ [#][:Y,T2:];
        then x in h"P by XBOOLE_0:def 4;
        then
A21:    h.x in P by FUNCT_1:def 7;
        g.x = h.x by A13,A20,FUNCT_4:13;
        hence thesis by A13,A20,A21,FUNCT_1:def 7;
      end;
      assume
A22:  x in g"P;
      then
A23:  x in dom g by FUNCT_1:def 7;
      g.x in P by A22,FUNCT_1:def 7;
      then
A24:  h.x in P by A23,FUNCT_4:13;
      x in dom h by A14,A23,XBOOLE_0:def 3;
      then x in h"P by A24,FUNCT_1:def 7;
      hence x in h"P /\ [#][:Y,T2:] by A22,XBOOLE_0:def 4;
    end;
A25: for x being set st x in [#][:Y,T1:] holds h.x = f.x
    proof
      let x be set such that
A26:  x in [#][:Y,T1:];
      now
        per cases;
        suppose
A27:      x in [#][:Y,T2:];
          then x in [#][:Y,T1:] /\ [#][:Y,T2:] by A26,XBOOLE_0:def 4;
          then f.x = g.x by A8;
          hence thesis by A13,A27,FUNCT_4:13;
        end;
        suppose
          not x in [#][:Y,T2:];
          hence thesis by A13,FUNCT_4:11;
        end;
      end;
      hence thesis;
    end;
    now
      let x be object;
      thus x in h"P /\ [#][:Y,T1:] implies x in f"P
      proof
        assume
A28:    x in h"P /\ [#][:Y,T1:];
        then x in h"P by XBOOLE_0:def 4;
        then
A29:    h.x in P by FUNCT_1:def 7;
        x in [#][:Y,T1:] by A28;
        then
A30:    x in dom f by FUNCT_2:def 1;
        f.x = h.x by A25,A28;
        hence thesis by A29,A30,FUNCT_1:def 7;
      end;
      assume
A31:  x in f"P;
      then x in dom f by FUNCT_1:def 7;
      then
A32:  x in dom h by A14,XBOOLE_0:def 3;
      f.x in P by A31,FUNCT_1:def 7;
      then h.x in P by A25,A31;
      then x in h"P by A32,FUNCT_1:def 7;
      hence x in h"P /\ [#][:Y,T1:] by A31,XBOOLE_0:def 4;
    end;
    then
A33: h"P /\ [#][:Y,T1:] = f"P by TARSKI:2;
    the carrier of T2 is Subset of T by A2,TSEP_1:1;
    then [#][:Y,T2:] c= [#][:Y,T:] by A17,A10,ZFMISC_1:95;
    then reconsider P2 = g"P as Subset of [:Y,T:] by XBOOLE_1:1;
    the carrier of T1 is Subset of T by A1,TSEP_1:1;
    then [#][:Y,T1:] c= [#][:Y,T:] by A17,A15,ZFMISC_1:95;
    then reconsider P1 = f"P as Subset of [:Y,T:] by XBOOLE_1:1;
    assume
A34: P is closed;
    then f"P is closed by A6,PRE_TOPC:def 6;
    then
A35: ex F01 being Subset of [:Y,T:] st F01 is closed & f"P = F01 /\ [#][:Y,
    T1:] by A18,PRE_TOPC:13;
    h"P = h"P /\ ([#][:Y,T1:] \/ [#][:Y,T2:]) by A17,A9,A13,A14,A16,XBOOLE_1:28
      .= (h"P /\ [#][:Y,T1:]) \/ (h"P /\ [#][:Y,T2:]) by XBOOLE_1:23;
    then
A36: h"P = f"P \/ g"P by A33,A19,TARSKI:2;
    M is closed & [#][:Y,T1:] = [:[#]Y,F1:] by A3,Th15,BORSUK_3:1;
    then
A37: P1 is closed by A35;
    g"P is closed by A7,A34,PRE_TOPC:def 6;
    then
A38: ex F02 being Subset of [:Y,T:] st F02 is closed & g"P = F02 /\ [#][:Y,
    T2:] by A11,PRE_TOPC:13;
    reconsider M = [:[#]Y,F2:] as Subset of [:Y,T:];
    M is closed & [#][:Y,T2:] = [:[#]Y,F2:] by A4,Th15,BORSUK_3:1;
    then P2 is closed by A38;
    hence thesis by A36,A37;
  end;
  hence thesis by PRE_TOPC:def 6;
end;
