defpred S_{1}[ object ] means $1 is Probability of S;

consider X being set such that

A1: for x being object holds

( x in X iff ( x in Funcs (S,REAL) & S_{1}[x] ) )
from XBOOLE_0:sch 1();

take X ; :: thesis: for x being object holds

( x in X iff x is Probability of S )

let x be object ; :: thesis: ( x in X iff x is Probability of S )

( x is Probability of S implies x in Funcs (S,REAL) ) by FUNCT_2:8;

hence ( x in X iff x is Probability of S ) by A1; :: thesis: verum

consider X being set such that

A1: for x being object holds

( x in X iff ( x in Funcs (S,REAL) & S

take X ; :: thesis: for x being object holds

( x in X iff x is Probability of S )

let x be object ; :: thesis: ( x in X iff x is Probability of S )

( x is Probability of S implies x in Funcs (S,REAL) ) by FUNCT_2:8;

hence ( x in X iff x is Probability of S ) by A1; :: thesis: verum