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

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

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

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

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

hence f . x is ext-natural by A1; :: thesis: verum

end;

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

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

end;

end;

end;

thus ( ( for x being object holds f . x is ext-natural ) implies f is ext-natural-valued ) ; :: thesis: verum