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

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

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

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

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

hence f . x is integer by A1, Def11; :: thesis: verum

end;

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

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

end;

end;

end;

assume for x being set holds f . x is integer ; :: thesis: f is integer-valued
then for x being set st x in dom f holds
f . x is integer ;
hence f is integer-valued by Def11; :: thesis: verum