let X, Y be set ; :: thesis: for f being real-valued Function st f | X is bounded_above & f | Y is bounded_above holds
f | (X \/ Y) is bounded_above
let f be real-valued Function; :: thesis: ( f | X is bounded_above & f | Y is bounded_above implies f | (X \/ Y) is bounded_above )
assume A1: ( f | X is bounded_above & f | Y is bounded_above ) ; :: thesis: f | (X \/ Y) is bounded_above
then consider r1 being real number such that
A2: for c being set st c in X /\ (dom f) holds
f . c <= r1 by Def9;
consider r2 being real number such that
A3: for c being set st c in Y /\ (dom f) holds
f . c <= r2 by A1, Def9;

now

take r = (abs r1) + (abs r2); :: thesis: for c being set st c in (X \/ Y) /\ (dom f) holds
f . c <= r
let c be set ; :: thesis: ( c in (X \/ Y) /\ (dom f) implies f . c <= r )
assume c in (X \/ Y) /\ (dom f) ; :: thesis: f . c <= r
then A4: ( c in X \/ Y & c in dom f ) by XBOOLE_0:def 4;

now

per cases ( c in X or c in Y ) by A4, XBOOLE_0:def 3;

suppose c in X ; :: thesis: f . c <= r

then c in X /\ (dom f) by A4, XBOOLE_0:def 4;
then A5: f . c <= r1 by A2;
r1 <= abs r1 by ABSVALUE:11;
then A6: f . c <= abs r1 by A5, XXREAL_0:2;
0 <= abs r2 by COMPLEX1:132;
then (f . c) + 0 <= r by A6, XREAL_1:9;
hence f . c <= r ; :: thesis: verum

end;

suppose c in Y ; :: thesis: f . c <= r

then c in Y /\ (dom f) by A4, XBOOLE_0:def 4;
then A7: f . c <= r2 by A3;
r2 <= abs r2 by ABSVALUE:11;
then A8: f . c <= abs r2 by A7, XXREAL_0:2;
0 <= abs r1 by COMPLEX1:132;
then 0 + (f . c) <= r by A8, XREAL_1:9;
hence f . c <= r ; :: thesis: verum

end;

end;

end;

hence f . c <= r ; :: thesis: verum

end;

hence f | (X \/ Y) is bounded_above by Def9; :: thesis: verum