let s be Real; :: thesis: [.s,+infty.[ = { g where g is Real : s <= g }
thus [.s,+infty.[ c= { g where g is Real : s <= g } :: according to XBOOLE_0:def 10 :: thesis: { g where g is Real : s <= g } c= [.s,+infty.[

proof

let x be Real; :: according to MEMBERED:def 9 :: thesis: ( not x in [.s,+infty.[ or x in { g where g is Real : s <= g } )
assume x in [.s,+infty.[ ; :: thesis: x in { g where g is Real : s <= g }
then A1: s <= x by Th3;
thus x in { g where g is Real : s <= g } by A1; :: thesis: verum

end;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { g where g is Real : s <= g } or x in [.s,+infty.[ )
assume x in { g where g is Real : s <= g } ; :: thesis: x in [.s,+infty.[
then consider g being Real such that
A2: x = g and
A3: s <= g ;
g in REAL by XREAL_0:def 1;
then g < +infty by XXREAL_0:9;
hence x in [.s,+infty.[ by A2, A3, Th3; :: thesis: verum