let X, Y be non empty Subset of REAL ; :: thesis: ( X is bounded_above & Y is bounded_above implies upper_bound (X + Y) = (upper_bound X) + (upper_bound Y) )
assume that
A1: X is bounded_above and
A2: Y is bounded_above ; :: thesis: upper_bound (X + Y) = (upper_bound X) + (upper_bound Y)
A3: - Y is bounded_below by A2, PSCOMP_1:15;
A4: - X is bounded_below by A1, PSCOMP_1:15;
then lower_bound ((- X) + (- Y)) = (lower_bound (- X)) + (lower_bound (- Y)) by A3, COMPLSP1:96;
then A5: lower_bound ((- X) + (- Y)) = (- (upper_bound (- (- X)))) + (lower_bound (- Y)) by A4, PSCOMP_1:17
.= (- (upper_bound X)) + (- (upper_bound (- (- Y)))) by A3, PSCOMP_1:17
.= - ((upper_bound X) + (upper_bound Y)) ;
A6: (- X) + (- Y) = - (X + Y) by Th52;
then A7: not - (X + Y) is empty by COMPLSP1:94;
- (X + Y) is bounded_below by A6, A4, A3, COMPLSP1:95;
then - (upper_bound (- (- (X + Y)))) = - ((upper_bound X) + (upper_bound Y)) by A6, A7, A5, PSCOMP_1:17;
hence upper_bound (X + Y) = (upper_bound X) + (upper_bound Y) ; :: thesis: verum