let T, S be non empty TopSpace; :: thesis: for X being set
for T1, T2 being SubSpace of T
for f being Function of T1,S
for g being Function of T2,S st ([#] T1) \/ ([#] T2) = [#] T & ([#] T1) /\ ([#] T2) = X & T1 is compact & T2 is compact & T is T_2 & f is continuous & g is continuous & f | X tolerates g | X holds
f +* g is continuous Function of T,S

let X be set ; :: thesis: for T1, T2 being SubSpace of T
for f being Function of T1,S
for g being Function of T2,S st ([#] T1) \/ ([#] T2) = [#] T & ([#] T1) /\ ([#] T2) = X & T1 is compact & T2 is compact & T is T_2 & f is continuous & g is continuous & f | X tolerates g | X holds
f +* g is continuous Function of T,S

let T1, T2 be SubSpace of T; :: thesis: for f being Function of T1,S
for g being Function of T2,S st ([#] T1) \/ ([#] T2) = [#] T & ([#] T1) /\ ([#] T2) = X & T1 is compact & T2 is compact & T is T_2 & f is continuous & g is continuous & f | X tolerates g | X holds
f +* g is continuous Function of T,S

let f be Function of T1,S; :: thesis: for g being Function of T2,S st ([#] T1) \/ ([#] T2) = [#] T & ([#] T1) /\ ([#] T2) = X & T1 is compact & T2 is compact & T is T_2 & f is continuous & g is continuous & f | X tolerates g | X holds
f +* g is continuous Function of T,S

let g be Function of T2,S; :: thesis: ( ([#] T1) \/ ([#] T2) = [#] T & ([#] T1) /\ ([#] T2) = X & T1 is compact & T2 is compact & T is T_2 & f is continuous & g is continuous & f | X tolerates g | X implies f +* g is continuous Function of T,S )
assume that
A1: ([#] T1) \/ ([#] T2) = [#] T and
A2: ([#] T1) /\ ([#] T2) = X and
A3: T1 is compact and
A4: T2 is compact and
A5: T is T_2 and
A6: f is continuous and
A7: g is continuous and
A8: f | X tolerates g | X ; :: thesis: f +* g is continuous Function of T,S
set h = f +* g;
A9: dom g = [#] T2 by FUNCT_2:def 1;
rng (f +* g) c= (rng f) \/ (rng g) by FUNCT_4:17;
then A10: rng (f +* g) c= the carrier of S by XBOOLE_1:1;
A11: dom f = [#] T1 by FUNCT_2:def 1;
then dom (f +* g) = the carrier of T by ;
then reconsider h = f +* g as Function of T,S by ;
for P being Subset of S st P is closed holds
h " P is closed
proof
let P be Subset of S; :: thesis: ( P is closed implies h " P is closed )
[#] T1 c= [#] T by ;
then reconsider P1 = f " P as Subset of T by XBOOLE_1:1;
[#] T2 c= [#] T by ;
then reconsider P2 = g " P as Subset of T by XBOOLE_1:1;
A12: dom h = (dom f) \/ (dom g) by FUNCT_4:def 1;
A13: now :: thesis: for x being object holds
( ( x in (h " P) /\ ([#] T2) implies x in g " P ) & ( x in g " P implies x in (h " P) /\ ([#] T2) ) )
let x be object ; :: thesis: ( ( x in (h " P) /\ ([#] T2) implies x in g " P ) & ( x in g " P implies x in (h " P) /\ ([#] T2) ) )
thus ( x in (h " P) /\ ([#] T2) implies x in g " P ) :: thesis: ( x in g " P implies x in (h " P) /\ ([#] T2) )
proof
assume A14: x in (h " P) /\ ([#] T2) ; :: thesis: x in g " P
then x in h " P by XBOOLE_0:def 4;
then A15: h . x in P by FUNCT_1:def 7;
g . x = h . x by ;
hence x in g " P by ; :: thesis: verum
end;
assume A16: x in g " P ; :: thesis: x in (h " P) /\ ([#] T2)
then A17: x in dom g by FUNCT_1:def 7;
g . x in P by ;
then A18: h . x in P by ;
x in dom h by ;
then x in h " P by ;
hence x in (h " P) /\ ([#] T2) by ; :: thesis: verum
end;
A19: for x being set st x in [#] T1 holds
h . x = f . x
proof
let x be set ; :: thesis: ( x in [#] T1 implies h . x = f . x )
assume A20: x in [#] T1 ; :: thesis: h . x = f . x
A21: dom g = the carrier of T2 by FUNCT_2:def 1;
per cases ( x in [#] T2 or not x in [#] T2 ) ;
suppose A22: x in [#] T2 ; :: thesis: h . x = f . x
then x in ([#] T1) /\ ([#] T2) by ;
then A23: x in X by A2;
then A24: ( (f | X) . x = f . x & (g | X) . x = g . x ) by FUNCT_1:49;
( x in dom (f | X) & x in dom (g | X) ) by ;
then x in (dom (f | X)) /\ (dom (g | X)) by XBOOLE_0:def 4;
then (f | X) . x = (g | X) . x by ;
hence h . x = f . x by ; :: thesis: verum
end;
suppose not x in [#] T2 ; :: thesis: h . x = f . x
hence h . x = f . x by ; :: thesis: verum
end;
end;
end;
now :: thesis: for x being object holds
( ( x in (h " P) /\ ([#] T1) implies x in f " P ) & ( x in f " P implies x in (h " P) /\ ([#] T1) ) )
let x be object ; :: thesis: ( ( x in (h " P) /\ ([#] T1) implies x in f " P ) & ( x in f " P implies x in (h " P) /\ ([#] T1) ) )
thus ( x in (h " P) /\ ([#] T1) implies x in f " P ) :: thesis: ( x in f " P implies x in (h " P) /\ ([#] T1) )
proof
assume A25: x in (h " P) /\ ([#] T1) ; :: thesis: x in f " P
then x in h " P by XBOOLE_0:def 4;
then A26: h . x in P by FUNCT_1:def 7;
f . x = h . x by ;
hence x in f " P by ; :: thesis: verum
end;
assume A27: x in f " P ; :: thesis: x in (h " P) /\ ([#] T1)
then x in dom f by FUNCT_1:def 7;
then A28: x in dom h by ;
f . x in P by ;
then h . x in P by ;
then x in h " P by ;
hence x in (h " P) /\ ([#] T1) by ; :: thesis: verum
end;
then A29: (h " P) /\ ([#] T1) = f " P by TARSKI:2;
assume A30: P is closed ; :: thesis: h " P is closed
then f " P is closed by A6;
then f " P is compact by ;
then A31: P1 is compact by Th19;
g " P is closed by ;
then g " P is compact by ;
then A32: P2 is compact by Th19;
h " P = (h " P) /\ (([#] T1) \/ ([#] T2)) by
.= ((h " P) /\ ([#] T1)) \/ ((h " P) /\ ([#] T2)) by XBOOLE_1:23 ;
then h " P = (f " P) \/ (g " P) by ;
hence h " P is closed by A5, A31, A32; :: thesis: verum
end;
hence f +* g is continuous Function of T,S by PRE_TOPC:def 6; :: thesis: verum