let r be Real; :: thesis: for X being non empty Subset of REAL st X is bounded_below & r <= 0 holds
r ** X is bounded_above
let X be non empty Subset of REAL; :: thesis: ( X is bounded_below & r <= 0 implies r ** X is bounded_above )
assume that
A1: X is bounded_below and
A2: r <= 0 ; :: thesis: r ** X is bounded_above
consider b being Real such that
A3: b is LowerBound of X by A1;
A4: for x being Real st x in X holds
b <= x by A3, XXREAL_2:def 2;
r * b is UpperBound of r ** X

proof

let y be ExtReal; :: according to XXREAL_2:def 1 :: thesis: ( not y in r ** X or y <= r * b )
assume y in r ** X ; :: thesis: y <= r * b
then y in { (r * x) where x is Real : x in X } by Th8;
then consider x being Real such that
A5: y = r * x and
A6: x in X ;
b <= x by A4, A6;
hence y <= r * b by A2, A5, XREAL_1:65; :: thesis: verum

end;

hence r ** X is bounded_above ; :: thesis: verum