let f be Function; :: thesis: ( f is natural-valued iff for x being set holds f . x is natural )

hereby :: thesis: ( ( for x being set holds f . x is natural ) implies f is natural-valued )

assume A1: f is natural-valued ; :: thesis: for x being set holds f . b₂ is natural
let x be set ; :: thesis: f . b₁ is natural

per cases ( x in dom f or not x in dom f ) ;

suppose x in dom f ; :: thesis: f . b₁ is natural

hence f . x is natural by A1, Def12; :: thesis: verum

end;

suppose not x in dom f ; :: thesis: f . b₁ is natural

hence f . x is natural by FUNCT_1:def 2; :: thesis: verum

end;

end;

end;

assume for x being set holds f . x is natural ; :: thesis: f is natural-valued
then for x being set st x in dom f holds
f . x is natural ;
hence f is natural-valued by Def12; :: thesis: verum