let F be ext-real-membered set ; :: thesis: for f being ExtReal holds f ** F = { (f * w) where w is Element of ExtREAL : w in F }
let f be ExtReal; :: thesis: f ** F = { (f * w) where w is Element of ExtREAL : w in F }
thus f ** F c= { (f * w) where w is Element of ExtREAL : w in F } :: according to XBOOLE_0:def 10 :: thesis: { (f * w) where w is Element of ExtREAL : w in F } c= f ** F

proof

let e be object ; :: according to TARSKI:def 3 :: thesis: ( not e in f ** F or e in { (f * w) where w is Element of ExtREAL : w in F } )
assume e in f ** F ; :: thesis: e in { (f * w) where w is Element of ExtREAL : w in F }
then consider w1, w2 being Element of ExtREAL such that
A1: e = w1 * w2 and
A2: w1 in {f} and
A3: w2 in F ;
w1 = f by A2, TARSKI:def 1;
hence e in { (f * w) where w is Element of ExtREAL : w in F } by A1, A3; :: thesis: verum

end;

let e be object ; :: according to TARSKI:def 3 :: thesis: ( not e in { (f * w) where w is Element of ExtREAL : w in F } or e in f ** F )
assume e in { (f * w) where w is Element of ExtREAL : w in F } ; :: thesis: e in f ** F
then ex w being Element of ExtREAL st
( e = f * w & w in F ) ;
hence e in f ** F by Th186; :: thesis: verum