let X, Y be non empty TopSpace; for X1, X2 being non empty open SubSpace of X
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st X = X1 union X2 & (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 holds
f1 union f2 is continuous Function of X,Y
let X1, X2 be non empty open SubSpace of X; for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st X = X1 union X2 & (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 holds
f1 union f2 is continuous Function of X,Y
let f1 be continuous Function of X1,Y; for f2 being continuous Function of X2,Y st X = X1 union X2 & (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 holds
f1 union f2 is continuous Function of X,Y
let f2 be continuous Function of X2,Y; ( X = X1 union X2 & (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 implies f1 union f2 is continuous Function of X,Y )
assume that
A1:
X = X1 union X2
and
A2:
( (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 )
; f1 union f2 is continuous Function of X,Y
reconsider g = f1 union f2 as Function of X,Y by A1;
( g | X1 = f1 & g | X2 = f2 )
by A1, A2, Def5;
hence
f1 union f2 is continuous Function of X,Y
by A1, Th122; verum