let X be set ; :: thesis: N_Funcs X is Subset of (N_PFuncs X)
N_Funcs X c= N_PFuncs X

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in N_Funcs X or x in N_PFuncs X )
assume x in N_Funcs X ; :: thesis: x in N_PFuncs X
then x is Function of X,NAT by Def19;
hence x in N_PFuncs X by Def18; :: thesis: verum

end;

hence N_Funcs X is Subset of (N_PFuncs X) ; :: thesis: verum