let T1, S1, T2, S2 be non empty TopSpace; for R1 being Refinement of T1,S1
for R2 being Refinement of T2,S2
for f being Function of T1,T2
for g being Function of S1,S2
for h being Function of R1,R2 st h = f & h = g & f is continuous & g is continuous holds
h is continuous
let R1 be Refinement of T1,S1; for R2 being Refinement of T2,S2
for f being Function of T1,T2
for g being Function of S1,S2
for h being Function of R1,R2 st h = f & h = g & f is continuous & g is continuous holds
h is continuous
let R2 be Refinement of T2,S2; for f being Function of T1,T2
for g being Function of S1,S2
for h being Function of R1,R2 st h = f & h = g & f is continuous & g is continuous holds
h is continuous
let f be Function of T1,T2; for g being Function of S1,S2
for h being Function of R1,R2 st h = f & h = g & f is continuous & g is continuous holds
h is continuous
let g be Function of S1,S2; for h being Function of R1,R2 st h = f & h = g & f is continuous & g is continuous holds
h is continuous
let h be Function of R1,R2; ( h = f & h = g & f is continuous & g is continuous implies h is continuous )
assume that
A1:
h = f
and
A2:
h = g
; ( not f is continuous or not g is continuous or h is continuous )
A3:
[#] S2 <> {}
;
reconsider K = the topology of T2 \/ the topology of S2 as prebasis of R2 by YELLOW_9:def 6;
A4:
[#] T2 <> {}
;
assume A5:
f is continuous
; ( not g is continuous or h is continuous )
assume A6:
g is continuous
; h is continuous
hence
h is continuous
by YELLOW_9:36; verum