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

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

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

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

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

hence f . x is rational by A1, Def10; :: thesis: verum

end;

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

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

end;

end;

end;

assume for x being set holds f . x is rational ; :: thesis: f is rational-valued
then for x being set st x in dom f holds
f . x is rational ;
hence f is rational-valued by Def10; :: thesis: verum