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 such that

A3: b is UpperBound of X by A1;

A4: for x being Real st x in X holds

x <= b by A3, XXREAL_2:def 1;

r * b is LowerBound of r ** X

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 such that

A3: b is UpperBound of X by A1;

A4: for x being Real st x in X holds

x <= b by A3, XXREAL_2:def 1;

r * b is LowerBound of r ** X

proof

hence
r ** X is bounded_below
; :: thesis: verum
let y be ExtReal; :: according to XXREAL_2:def 2 :: thesis: ( not y in r ** X or 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, A4, XREAL_1:65; :: thesis: verum

end;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, A4, XREAL_1:65; :: thesis: verum