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 A1, A9, FUNCT_4:def 1;

then reconsider h = f +* g as Function of T,S by A10, FUNCT_2:def 1, RELSET_1:4;

for P being Subset of S st P is closed holds

h " P is closed

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 A1, A9, FUNCT_4:def 1;

then reconsider h = f +* g as Function of T,S by A10, FUNCT_2:def 1, RELSET_1:4;

for P being Subset of S st P is closed holds

h " P is closed

proof

hence
f +* g is continuous Function of T,S
by PRE_TOPC:def 6; :: thesis: verum
let P be Subset of S; :: thesis: ( P is closed implies h " P is closed )

[#] T1 c= [#] T by A1, XBOOLE_1:7;

then reconsider P1 = f " P as Subset of T by XBOOLE_1:1;

[#] T2 c= [#] T by A1, XBOOLE_1:7;

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;

h . x = f . x

assume A30: P is closed ; :: thesis: h " P is closed

then f " P is closed by A6;

then f " P is compact by A3, Th8;

then A31: P1 is compact by Th19;

g " P is closed by A7, A30;

then g " P is compact by A4, Th8;

then A32: P2 is compact by Th19;

h " P = (h " P) /\ (([#] T1) \/ ([#] T2)) by A11, A9, A12, RELAT_1:132, XBOOLE_1:28

.= ((h " P) /\ ([#] T1)) \/ ((h " P) /\ ([#] T2)) by XBOOLE_1:23 ;

then h " P = (f " P) \/ (g " P) by A29, A13, TARSKI:2;

hence h " P is closed by A5, A31, A32; :: thesis: verum

end;[#] T1 c= [#] T by A1, XBOOLE_1:7;

then reconsider P1 = f " P as Subset of T by XBOOLE_1:1;

[#] T2 c= [#] T by A1, XBOOLE_1:7;

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) ) )

A19:
for x being set st x in [#] T1 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) )

then A17: x in dom g by FUNCT_1:def 7;

g . x in P by A16, FUNCT_1:def 7;

then A18: h . x in P by A17, FUNCT_4:13;

x in dom h by A12, A17, XBOOLE_0:def 3;

then x in h " P by A18, FUNCT_1:def 7;

hence x in (h " P) /\ ([#] T2) by A16, XBOOLE_0:def 4; :: thesis: verum

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

proof

assume A16:
x in g " P
; :: thesis: x in (h " P) /\ ([#] T2)
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 A9, A14, FUNCT_4:13;

hence x in g " P by A9, A14, A15, FUNCT_1:def 7; :: thesis: verum

end;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 A9, A14, FUNCT_4:13;

hence x in g " P by A9, A14, A15, FUNCT_1:def 7; :: thesis: verum

then A17: x in dom g by FUNCT_1:def 7;

g . x in P by A16, FUNCT_1:def 7;

then A18: h . x in P by A17, FUNCT_4:13;

x in dom h by A12, A17, XBOOLE_0:def 3;

then x in h " P by A18, FUNCT_1:def 7;

hence x in (h " P) /\ ([#] T2) by A16, XBOOLE_0:def 4; :: thesis: verum

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;

end;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 )
;

end;

suppose A22:
x in [#] T2
; :: thesis: h . x = f . x

then
x in ([#] T1) /\ ([#] T2)
by A20, XBOOLE_0:def 4;

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 A11, A20, A21, A22, A23, RELAT_1:57;

then x in (dom (f | X)) /\ (dom (g | X)) by XBOOLE_0:def 4;

then (f | X) . x = (g | X) . x by A8, PARTFUN1:def 4;

hence h . x = f . x by A9, A22, A24, FUNCT_4:13; :: thesis: verum

end;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 A11, A20, A21, A22, A23, RELAT_1:57;

then x in (dom (f | X)) /\ (dom (g | X)) by XBOOLE_0:def 4;

then (f | X) . x = (g | X) . x by A8, PARTFUN1:def 4;

hence h . x = f . x by A9, A22, A24, FUNCT_4:13; :: thesis: verum

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) ) )

then A29:
(h " P) /\ ([#] T1) = f " P
by TARSKI:2;( ( 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) )

then x in dom f by FUNCT_1:def 7;

then A28: x in dom h by A12, XBOOLE_0:def 3;

f . x in P by A27, FUNCT_1:def 7;

then h . x in P by A19, A27;

then x in h " P by A28, FUNCT_1:def 7;

hence x in (h " P) /\ ([#] T1) by A27, XBOOLE_0:def 4; :: thesis: verum

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

proof

assume A27:
x in f " P
; :: thesis: x in (h " P) /\ ([#] T1)
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 A19, A25;

hence x in f " P by A11, A25, A26, FUNCT_1:def 7; :: thesis: verum

end;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 A19, A25;

hence x in f " P by A11, A25, A26, FUNCT_1:def 7; :: thesis: verum

then x in dom f by FUNCT_1:def 7;

then A28: x in dom h by A12, XBOOLE_0:def 3;

f . x in P by A27, FUNCT_1:def 7;

then h . x in P by A19, A27;

then x in h " P by A28, FUNCT_1:def 7;

hence x in (h " P) /\ ([#] T1) by A27, XBOOLE_0:def 4; :: thesis: verum

assume A30: P is closed ; :: thesis: h " P is closed

then f " P is closed by A6;

then f " P is compact by A3, Th8;

then A31: P1 is compact by Th19;

g " P is closed by A7, A30;

then g " P is compact by A4, Th8;

then A32: P2 is compact by Th19;

h " P = (h " P) /\ (([#] T1) \/ ([#] T2)) by A11, A9, A12, RELAT_1:132, XBOOLE_1:28

.= ((h " P) /\ ([#] T1)) \/ ((h " P) /\ ([#] T2)) by XBOOLE_1:23 ;

then h " P = (f " P) \/ (g " P) by A29, A13, TARSKI:2;

hence h " P is closed by A5, A31, A32; :: thesis: verum