let T, T1, T2, S be non empty TopSpace; :: thesis:
let f be Function of T1,S; :: thesis:
let g be Function of T2,S; :: thesis:
assume that
A1: T1 is SubSpace of T and
A2: T2 is SubSpace of T and
A3: ([#] T1) \/ ([#] T2) = [#] T and
A4: T1 is compact and
A5: T2 is compact and
A6: T is T_2 and
A7: f is continuous and
A8: g is continuous and
A9: for p being set st p in ([#] T1) /\ ([#] T2) holds
f . p = g . p ; :: thesis:
set h = f +* g;
A10: dom g = [#] T2 by FUNCT_2:def 1;
A11: dom f = [#] T1 by FUNCT_2:def 1;
then A12: dom (f +* g) = the carrier of T by A3, A10, FUNCT_4:def 1;
rng (f +* g) c= (rng f) \/ (rng g) by FUNCT_4:17;
then reconsider h = f +* g as Function of T,S by A12, FUNCT_2:2, XBOOLE_1:1;
take h ; :: thesis:
thus h = f +* g ; :: thesis:
for P being Subset of S st P is closed holds
h " P is closed

let P be Subset of S; :: thesis:
reconsider P3 = f " P as Subset of T1 ;
reconsider P4 = g " P as Subset of T2 ;
[#] T1 c= [#] T by A3, XBOOLE_1:7;
then reconsider P1 = f " P as Subset of T by XBOOLE_1:1;
[#] T2 c= [#] T by A3, XBOOLE_1:7;
then reconsider P2 = g " P as Subset of T by XBOOLE_1:1;
A13: dom h = (dom f) \/ (dom g) by FUNCT_4:def 1;

A14: now :: thesis:

let x be object ; :: thesis:
thus ( x in (h " P) /\ ([#] T2) implies x in g " P ) :: thesis:

assume A15: x in (h " P) /\ ([#] T2) ; :: thesis:
then x in h " P by XBOOLE_0:def 4;
then A16: h . x in P by FUNCT_1:def 7;
g . x = h . x by A10, A15, FUNCT_4:13;
hence x in g " P by A10, A15, A16, FUNCT_1:def 7; :: thesis:

end;

assume A17: x in g " P ; :: thesis:
then A18: x in dom g by FUNCT_1:def 7;
g . x in P by A17, FUNCT_1:def 7;
then A19: h . x in P by A18, FUNCT_4:13;
x in dom h by A13, A18, XBOOLE_0:def 3;
then x in h " P by A19, FUNCT_1:def 7;
hence x in (h " P) /\ ([#] T2) by A17, XBOOLE_0:def 4; :: thesis:

end;

A20: for x being set st x in [#] T1 holds
h . x = f . x

let x be set ; :: thesis:
assume A21: x in [#] T1 ; :: thesis:

suppose A22: x in [#] T2 ; :: thesis:

then x in ([#] T1) /\ ([#] T2) by A21, XBOOLE_0:def 4;
then f . x = g . x by A9;
hence h . x = f . x by A10, A22, FUNCT_4:13; :: thesis:

end;

suppose not x in [#] T2 ; :: thesis:

hence h . x = f . x by A10, FUNCT_4:11; :: thesis:

end;

end;

end;

hence h . x = f . x ; :: thesis:

end;

let x be object ; :: thesis:
thus ( x in (h " P) /\ ([#] T1) implies x in f " P ) :: thesis:

assume A23: x in (h " P) /\ ([#] T1) ; :: thesis:
then x in h " P by XBOOLE_0:def 4;
then A24: h . x in P by FUNCT_1:def 7;
f . x = h . x by A20, A23;
hence x in f " P by A11, A23, A24, FUNCT_1:def 7; :: thesis:

end;

assume A25: x in f " P ; :: thesis:
then x in dom f by FUNCT_1:def 7;
then A26: x in dom h by A13, XBOOLE_0:def 3;
f . x in P by A25, FUNCT_1:def 7;
then h . x in P by A20, A25;
then x in h " P by A26, FUNCT_1:def 7;
hence x in (h " P) /\ ([#] T1) by A25, XBOOLE_0:def 4; :: thesis:

end;

then A27: (h " P) /\ ([#] T1) = f " P by TARSKI:2;
assume A28: P is closed ; :: thesis:
then P3 is closed by A7;
then P3 is compact by A4, COMPTS_1:8;
then A29: P1 is compact by A1, COMPTS_1:19;
P4 is closed by A8, A28;
then P4 is compact by A5, COMPTS_1:8;
then A30: P2 is compact by A2, COMPTS_1:19;
h " P = (h " P) /\ (([#] T1) \/ ([#] T2)) by A11, A10, A13, RELAT_1:132, XBOOLE_1:28
.= ((h " P) /\ ([#] T1)) \/ ((h " P) /\ ([#] T2)) by XBOOLE_1:23 ;
then h " P = (f " P) \/ (g " P) by A27, A14, TARSKI:2;
hence h " P is closed by A6, A29, A30; :: thesis:

end;

hence h is continuous ; :: thesis: