thus ( f is natural-functions-valued implies for x being object st x in dom f holds
f . x is natural-valued Function ) :: thesis: ( ( for x being object st x in dom f holds
f . x is natural-valued Function ) implies f is natural-functions-valued )

proof

assume A11: rng f is natural-functions-membered ; :: according to VALUED_2:def 25 :: thesis: for x being object st x in dom f holds
f . x is natural-valued Function
let x be object ; :: thesis: ( x in dom f implies f . x is natural-valued Function )
assume x in dom f ; :: thesis: f . x is natural-valued Function
then f . x in rng f by FUNCT_1:def 3;
hence f . x is natural-valued Function by A11; :: thesis: verum

end;

assume A12: for x being object st x in dom f holds
f . x is natural-valued Function ; :: thesis: f is natural-functions-valued
let y be object ; :: according to VALUED_2:def 7,VALUED_2:def 25 :: thesis: ( y in rng f implies y is natural-valued Function )
assume y in rng f ; :: thesis: y is natural-valued Function
then ex x being object st
( x in dom f & f . x = y ) by FUNCT_1:def 3;
hence y is natural-valued Function by A12; :: thesis: verum