let X, Y be non empty TopSpace; for X1, X2 being non empty open SubSpace of X st X1 meets X2 holds
for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st f1 | (X1 meet X2) = f2 | (X1 meet X2) holds
f1 union f2 is continuous Function of (X1 union X2),Y
let X1, X2 be non empty open SubSpace of X; ( X1 meets X2 implies for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st f1 | (X1 meet X2) = f2 | (X1 meet X2) holds
f1 union f2 is continuous Function of (X1 union X2),Y )
assume A1:
X1 meets X2
; for f1 being continuous Function of X1,Y
for f2 being continuous Function of X2,Y st f1 | (X1 meet X2) = f2 | (X1 meet X2) holds
f1 union f2 is continuous Function of (X1 union X2),Y
let f1 be continuous Function of X1,Y; for f2 being continuous Function of X2,Y st f1 | (X1 meet X2) = f2 | (X1 meet X2) holds
f1 union f2 is continuous Function of (X1 union X2),Y
let f2 be continuous Function of X2,Y; ( f1 | (X1 meet X2) = f2 | (X1 meet X2) implies f1 union f2 is continuous Function of (X1 union X2),Y )
assume
f1 | (X1 meet X2) = f2 | (X1 meet X2)
; f1 union f2 is continuous Function of (X1 union X2),Y
then
( (f1 union f2) | X1 = f1 & (f1 union f2) | X2 = f2 )
by A1, Th128;
hence
f1 union f2 is continuous Function of (X1 union X2),Y
by Th116; verum