let X, Y be set ; :: thesis: for f being real-valued Function st f | X is bounded_below & f | Y is bounded_below holds
f | (X \/ Y) is bounded_below
let f be real-valued Function; :: thesis: ( f | X is bounded_below & f | Y is bounded_below implies f | (X \/ Y) is bounded_below )
assume that
A1: f | X is bounded_below and
A2: f | Y is bounded_below ; :: thesis: f | (X \/ Y) is bounded_below
consider r1 being real number such that
A3: for c being set st c in X /\ (dom f) holds
r1 <= f . c by A1, Th88;
consider r2 being real number such that
A4: for c being set st c in Y /\ (dom f) holds
r2 <= f . c by A2, Th88;

now

take r = (- (abs r1)) - (abs r2); :: thesis: for c being set st c in (X \/ Y) /\ (dom f) holds
r <= f . c
let c be set ; :: thesis: ( c in (X \/ Y) /\ (dom f) implies r <= f . c )
assume A5: c in (X \/ Y) /\ (dom f) ; :: thesis: r <= f . c
then A6: c in dom f by XBOOLE_0:def 4;
A7: c in X \/ Y by A5, XBOOLE_0:def 4;

now

per cases ( c in X or c in Y ) by A7, XBOOLE_0:def 3;

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

then c in X /\ (dom f) by A6, XBOOLE_0:def 4;
then A8: r1 <= f . c by A3;
A9: 0 <= abs r2 by COMPLEX1:132;
- (abs r1) <= r1 by ABSVALUE:11;
then - (abs r1) <= f . c by A8, XXREAL_0:2;
then r <= (f . c) - 0 by A9, XREAL_1:15;
hence r <= f . c ; :: thesis: verum

end;

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

then c in Y /\ (dom f) by A6, XBOOLE_0:def 4;
then A10: r2 <= f . c by A4;
A11: 0 <= abs r1 by COMPLEX1:132;
- (abs r2) <= r2 by ABSVALUE:11;
then - (abs r2) <= f . c by A10, XXREAL_0:2;
then (- (abs r2)) - (abs r1) <= (f . c) - 0 by A11, XREAL_1:15;
hence r <= f . c ; :: thesis: verum

end;

end;

end;

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

end;

hence f | (X \/ Y) is bounded_below by Th88; :: thesis: verum