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

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

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

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

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

hence f . x is real by A1, Def9; :: thesis: verum

end;

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

hence f . x is real by FUNCT_1:def 4; :: thesis: verum

end;

end;

end;

assume for x being set holds f . x is real ; :: thesis: f is real-valued
then for x being set st x in dom f holds
f . x is real ;
hence f is real-valued by Def9; :: thesis: verum