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 Th132; :: thesis: verum