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, MEASURE6:77;
A4: -- X is bounded_below by A1, MEASURE6:77;
then lower_bound ((-- X) ++ (-- Y)) = (lower_bound (-- X)) + (lower_bound (-- Y)) by A3, SEQ_4:142;
then A5: lower_bound ((-- X) ++ (-- Y)) = (- (upper_bound (-- (-- X)))) + (lower_bound (-- Y)) by A4, MEASURE6:79
.= (- (upper_bound X)) + (- (upper_bound (-- (-- Y)))) by A3, MEASURE6:79
.= - ((upper_bound X) + (upper_bound Y)) ;
A6: (-- X) ++ (-- Y) = -- (X ++ Y) by Th52;
then a: -- (X ++ Y) is bounded_below by A4, A3, SEQ_4:141;
X ++ Y c= REAL by MEMBERED:3;
then reconsider XY = X ++ Y as Subset of REAL ;
-- XY is bounded_below by a;
then - (upper_bound (-- (-- XY))) = - ((upper_bound X) + (upper_bound Y)) by A6, A5, MEASURE6:79;
then upper_bound (-- (-- XY)) = (upper_bound X) + (upper_bound Y) ;
then upper_bound XY = (upper_bound X) + (upper_bound Y) ;
hence upper_bound (X ++ Y) = (upper_bound X) + (upper_bound Y) ; :: thesis: verum