let r be Real; :: thesis: for X being non empty Subset of REAL st X is bounded_above & r <= 0 holds
r ** X is bounded_below
let X be non empty Subset of REAL ; :: thesis: ( X is bounded_above & r <= 0 implies r ** X is bounded_below )
assume that
A1: X is bounded_above and
A2: r <= 0 ; :: thesis: r ** X is bounded_below
consider b being real number such that
A3: for x being real number st x in X holds
x <= b by A1, SEQ_4:def 1;
for y being real number st y in r ** X holds
r * b <= y

proof

let y be real number ; :: thesis: ( y in r ** X implies r * b <= y )
assume y in r ** X ; :: thesis: r * b <= y
then y in { (r * x) where x is Real : x in X } by Th8;
then ex x being Real st
( y = r * x & x in X ) ;
hence r * b <= y by A2, A3, XREAL_1:67; :: thesis: verum

end;

hence r ** X is bounded_below by SEQ_4:def 2; :: thesis: verum