let x be object ; :: according to VALUED_2:def 4 :: thesis: ( x in Funcs (X,Y) implies x is real-valued Function )

assume x in Funcs (X,Y) ; :: thesis: x is real-valued Function

then consider f being Function such that

A1: x = f and

A2: ( dom f = X & rng f c= Y ) by FUNCT_2:def 2;

reconsider f = f as PartFunc of X,Y by A2, RELSET_1:4;

f is set ;

hence x is real-valued Function by A1; :: thesis: verum

assume x in Funcs (X,Y) ; :: thesis: x is real-valued Function

then consider f being Function such that

A1: x = f and

A2: ( dom f = X & rng f c= Y ) by FUNCT_2:def 2;

reconsider f = f as PartFunc of X,Y by A2, RELSET_1:4;

f is set ;

hence x is real-valued Function by A1; :: thesis: verum