let X be set ; :: thesis: C_Funcs X is Subset of (C_PFuncs X)
C_Funcs X c= C_PFuncs X

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in C_Funcs X or x in C_PFuncs X )
assume x in C_Funcs X ; :: thesis: x in C_PFuncs X
then x is Function of X,COMPLEX by Def9;
hence x in C_PFuncs X by Def8; :: thesis: verum

end;

hence C_Funcs X is Subset of (C_PFuncs X) ; :: thesis: verum