let r be Real; :: thesis: for X being non empty Subset of REAL st X is bounded_below & r <= 0 holds
upper_bound (r ** X) = r * (lower_bound X)
let X be non empty Subset of REAL ; :: thesis: ( X is bounded_below & r <= 0 implies upper_bound (r ** X) = r * (lower_bound X) )
assume A1: ( X is bounded_below & r <= 0 ) ; :: thesis: upper_bound (r ** X) = r * (lower_bound X)
A2: for a being real number st a in r ** X holds
r * (lower_bound X) >= a

proof

let a be real number ; :: thesis: ( a in r ** X implies r * (lower_bound X) >= a )
assume a in r ** X ; :: thesis: r * (lower_bound X) >= a
then a in { (r * x) where x is Real : x in X } by Th8;
then consider x being Real such that
A3: ( a = r * x & x in X ) ;
lower_bound X <= x by A1, A3, SEQ_4:def 5;
hence r * (lower_bound X) >= a by A1, A3, XREAL_1:67; :: thesis: verum

end;

A4: for b being real number st ( for a being real number st a in r ** X holds
a <= b ) holds
r * (lower_bound X) <= b

proof

let b be real number ; :: thesis: ( ( for a being real number st a in r ** X holds
a <= b ) implies r * (lower_bound X) <= b )
assume A5: for a being real number st a in r ** X holds
a <= b ; :: thesis: r * (lower_bound X) <= b
consider x being Real such that
A6: x in X by SUBSET_1:10;
r * x in { (r * y) where y is Real : y in X } by A6;
then A7: r * x in r ** X by Th8;

now

per cases ( r = 0 or r < 0 ) by A1;

suppose r = 0 ; :: thesis: r * (lower_bound X) <= b

hence r * (lower_bound X) <= b by A5, A7; :: thesis: verum

end;

suppose A8: r < 0 ; :: thesis: r * (lower_bound X) <= b

for z being real number st z in X holds
z >= b / r

proof

let z be real number ; :: thesis: ( z in X implies z >= b / r )
assume z in X ; :: thesis: z >= b / r
then r * z in { (r * y) where y is Real : y in X } ;
then r * z in r ** X by Th8;
then r * z <= b by A5;
hence z >= b / r by A8, XREAL_1:80; :: thesis: verum

end;

then lower_bound X >= b / r by SEQ_4:60;
then r * (lower_bound X) <= (b / r) * r by A8, XREAL_1:67;
hence r * (lower_bound X) <= b by A8, XCMPLX_1:88; :: thesis: verum

end;

end;

end;

hence r * (lower_bound X) <= b ; :: thesis: verum

end;

r ** X is non empty Subset of REAL by Th7;
hence upper_bound (r ** X) = r * (lower_bound X) by A2, A4, SEQ_4:63; :: thesis: verum