let X be set ; :: thesis: ( X is real-membered implies X c= REAL )
assume A1: X is real-membered ; :: thesis: X c= REAL
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in X or x in REAL )
assume x in X ; :: thesis: x in REAL
then x is real by A1;
hence x in REAL ; :: thesis: verum