let X, X1 be set ; :: thesis: for f1, f2 being PartFunc of REAL,REAL st X c= dom f1 & X1 c= dom f2 & f1 | X is continuous & f2 | X1 is continuous holds

( (f1 + f2) | (X /\ X1) is continuous & (f1 - f2) | (X /\ X1) is continuous & (f1 (#) f2) | (X /\ X1) is continuous )

let f1, f2 be PartFunc of REAL,REAL; :: thesis: ( X c= dom f1 & X1 c= dom f2 & f1 | X is continuous & f2 | X1 is continuous implies ( (f1 + f2) | (X /\ X1) is continuous & (f1 - f2) | (X /\ X1) is continuous & (f1 (#) f2) | (X /\ X1) is continuous ) )

assume ( X c= dom f1 & X1 c= dom f2 ) ; :: thesis: ( not f1 | X is continuous or not f2 | X1 is continuous or ( (f1 + f2) | (X /\ X1) is continuous & (f1 - f2) | (X /\ X1) is continuous & (f1 (#) f2) | (X /\ X1) is continuous ) )

then A1: X /\ X1 c= (dom f1) /\ (dom f2) by XBOOLE_1:27;

assume ( f1 | X is continuous & f2 | X1 is continuous ) ; :: thesis: ( (f1 + f2) | (X /\ X1) is continuous & (f1 - f2) | (X /\ X1) is continuous & (f1 (#) f2) | (X /\ X1) is continuous )

then ( f1 | (X /\ X1) is continuous & f2 | (X /\ X1) is continuous ) by Th16, XBOOLE_1:17;

hence ( (f1 + f2) | (X /\ X1) is continuous & (f1 - f2) | (X /\ X1) is continuous & (f1 (#) f2) | (X /\ X1) is continuous ) by A1, Th18; :: thesis: verum

( (f1 + f2) | (X /\ X1) is continuous & (f1 - f2) | (X /\ X1) is continuous & (f1 (#) f2) | (X /\ X1) is continuous )

let f1, f2 be PartFunc of REAL,REAL; :: thesis: ( X c= dom f1 & X1 c= dom f2 & f1 | X is continuous & f2 | X1 is continuous implies ( (f1 + f2) | (X /\ X1) is continuous & (f1 - f2) | (X /\ X1) is continuous & (f1 (#) f2) | (X /\ X1) is continuous ) )

assume ( X c= dom f1 & X1 c= dom f2 ) ; :: thesis: ( not f1 | X is continuous or not f2 | X1 is continuous or ( (f1 + f2) | (X /\ X1) is continuous & (f1 - f2) | (X /\ X1) is continuous & (f1 (#) f2) | (X /\ X1) is continuous ) )

then A1: X /\ X1 c= (dom f1) /\ (dom f2) by XBOOLE_1:27;

assume ( f1 | X is continuous & f2 | X1 is continuous ) ; :: thesis: ( (f1 + f2) | (X /\ X1) is continuous & (f1 - f2) | (X /\ X1) is continuous & (f1 (#) f2) | (X /\ X1) is continuous )

then ( f1 | (X /\ X1) is continuous & f2 | (X /\ X1) is continuous ) by Th16, XBOOLE_1:17;

hence ( (f1 + f2) | (X /\ X1) is continuous & (f1 - f2) | (X /\ X1) is continuous & (f1 (#) f2) | (X /\ X1) is continuous ) by A1, Th18; :: thesis: verum