
theorem
  for S, T, T1, T2, Y being non empty TopSpace, f being Function of [:T1
  ,Y:],S, g being Function of [:T2,Y:],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 (
[#][:T1,Y:]) /\ ([#][:T2,Y:]) holds f.p = g.p) ex h being Function of [:T,Y:],S
  st h = f+*g & h is continuous
proof
  let S, T, T1, T2, Y be non empty TopSpace, f be Function of [:T1,Y:],S, g be
  Function of [:T2,Y:],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 ([#][:T1,Y:]) /\ ([#][:T2,Y:]) holds f.p = g .p;
A9: dom f = the carrier of [:T1,Y:] by FUNCT_2:def 1;
A10: Y is SubSpace of Y by TSEP_1:2;
  then
A11: [:T2,Y:] is SubSpace of [:T,Y:] by A2,BORSUK_3:21;
  set h = f+*g;
A12: the carrier of [:T2,Y:] = [:the carrier of T2,the carrier of Y:] by
BORSUK_1:def 2;
A13: dom h = dom f \/ dom g by FUNCT_4:def 1;
A14: rng h c= rng f \/ rng g by FUNCT_4:17;
A15: dom g = the carrier of [:T2,Y:] by FUNCT_2:def 1;
A16: the carrier of [:T1,Y:] = [:the carrier of T1,the carrier of Y:] by
BORSUK_1:def 2;
  then
A17: dom h = [:the carrier of T,the carrier of Y:] by A5,A12,A9,A15,A13,
ZFMISC_1:97;
A18: the carrier of [:T,Y:] = [:the carrier of T,the carrier of Y:] by
BORSUK_1:def 2;
  then reconsider h as Function of [:T,Y:],S by A17,A14,FUNCT_2:2,XBOOLE_1:1;
  take h;
  thus h = f+*g;
A19: [:T1,Y:] is SubSpace of [:T,Y:] by A1,A10,BORSUK_3:21;
  for P being Subset of S st P is closed holds h"P is closed
  proof
    reconsider M = [:F1,[#]Y:] as Subset of [:T,Y:];
    let P be Subset of S;
A20: now
      let x be object;
      thus x in h"P /\ [#][:T2,Y:] implies x in g"P
      proof
        assume
A21:    x in h"P /\ [#][:T2,Y:];
        then x in h"P by XBOOLE_0:def 4;
        then
A22:    h.x in P by FUNCT_1:def 7;
        g.x = h.x by A15,A21,FUNCT_4:13;
        hence thesis by A15,A21,A22,FUNCT_1:def 7;
      end;
      assume
A23:  x in g"P;
      then
A24:  x in dom g by FUNCT_1:def 7;
      g.x in P by A23,FUNCT_1:def 7;
      then
A25:  h.x in P by A24,FUNCT_4:13;
      x in dom h by A13,A24,XBOOLE_0:def 3;
      then x in h"P by A25,FUNCT_1:def 7;
      hence x in h"P /\ [#][:T2,Y:] by A23,XBOOLE_0:def 4;
    end;
A26: for x being set st x in [#][:T1,Y:] holds h.x = f.x
    proof
      let x be set such that
A27:  x in [#][:T1,Y:];
      now
        per cases;
        suppose
A28:      x in [#][:T2,Y:];
          then x in [#][:T1,Y:] /\ [#][:T2,Y:] by A27,XBOOLE_0:def 4;
          then f.x = g.x by A8;
          hence thesis by A15,A28,FUNCT_4:13;
        end;
        suppose
          not x in [#][:T2,Y:];
          hence thesis by A15,FUNCT_4:11;
        end;
      end;
      hence thesis;
    end;
    now
      let x be object;
      thus x in h"P /\ [#][:T1,Y:] implies x in f"P
      proof
        assume
A29:    x in h"P /\ [#][:T1,Y:];
        then x in h"P by XBOOLE_0:def 4;
        then
A30:    h.x in P by FUNCT_1:def 7;
        x in [#][:T1,Y:] by A29;
        then
A31:    x in dom f by FUNCT_2:def 1;
        f.x = h.x by A26,A29;
        hence thesis by A30,A31,FUNCT_1:def 7;
      end;
      assume
A32:  x in f"P;
      then x in dom f by FUNCT_1:def 7;
      then
A33:  x in dom h by A13,XBOOLE_0:def 3;
      f.x in P by A32,FUNCT_1:def 7;
      then h.x in P by A26,A32;
      then x in h"P by A33,FUNCT_1:def 7;
      hence x in h"P /\ [#][:T1,Y:] by A32,XBOOLE_0:def 4;
    end;
    then
A34: h"P /\ [#][:T1,Y:] = f"P by TARSKI:2;
    the carrier of T2 is Subset of T by A2,TSEP_1:1;
    then [#][:T2,Y:] c= [#][:T,Y:] by A18,A12,ZFMISC_1:95;
    then reconsider P2 = g"P as Subset of [:T,Y:] by XBOOLE_1:1;
    the carrier of T1 is Subset of T by A1,TSEP_1:1;
    then [#][:T1,Y:] c= [#][:T,Y:] by A18,A16,ZFMISC_1:95;
    then reconsider P1 = f"P as Subset of [:T,Y:] by XBOOLE_1:1;
    assume
A35: P is closed;
    then f"P is closed by A6,PRE_TOPC:def 6;
    then
A36: ex F01 being Subset of [:T,Y:] st F01 is closed & f"P = F01 /\ [#][:T1
    ,Y:] by A19,PRE_TOPC:13;
    h"P = h"P /\ ([#][:T1,Y:] \/ [#][:T2,Y:]) by A18,A9,A15,A13,A17,XBOOLE_1:28
      .= (h"P /\ [#][:T1,Y:]) \/ (h"P /\ [#][:T2,Y:]) by XBOOLE_1:23;
    then
A37: h"P = f"P \/ g"P by A34,A20,TARSKI:2;
    M is closed & [#][:T1,Y:] = [:F1,[#]Y:] by A3,Th15,BORSUK_3:1;
    then
A38: P1 is closed by A36;
    g"P is closed by A7,A35,PRE_TOPC:def 6;
    then
A39: ex F02 being Subset of [:T,Y:] st F02 is closed & g"P = F02 /\ [#][:T2
    ,Y:] by A11,PRE_TOPC:13;
    reconsider M = [:F2,[#]Y:] as Subset of [:T,Y:];
    M is closed & [#][:T2,Y:] = [:F2,[#]Y:] by A4,Th15,BORSUK_3:1;
    then P2 is closed by A39;
    hence thesis by A37,A38;
  end;
  hence thesis by PRE_TOPC:def 6;
end;
