let X, Y be non empty TopSpace; :: thesis: for X1, X2 being non empty closed 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 closed SubSpace of X; :: thesis: 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; :: thesis: 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; :: thesis: ( 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 ) ; :: thesis: 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, Th121; :: thesis: verum

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 closed SubSpace of X; :: thesis: 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; :: thesis: 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; :: thesis: ( 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 ) ; :: thesis: 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, Th121; :: thesis: verum