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

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

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

end;

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

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

end;

end;

end;

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