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

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

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

end;

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

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

end;

end;

end;

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