let x be object ; :: according to VALUED_2:def 3 :: thesis: ( x in PFuncs (X,Y) implies x is ext-real-valued Function )
assume x in PFuncs (X,Y) ; :: thesis: x is ext-real-valued Function
then consider f being Function such that
A1: x = f and
A2: ( dom f c= X & rng f c= Y ) by PARTFUN1:def 3;
reconsider f = f as PartFunc of X,Y by A2, RELSET_1:4;
f is set ;
hence x is ext-real-valued Function by A1; :: thesis: verum