let f be Function; :: thesis: ( f is INT -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 INT -valued )

assume A1: f is INT -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

then f . x in rng f by FUNCT_1:def 3;
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;

assume A2: for x being object holds f . x is integer ; :: thesis: f is INT -valued
let y be object ; :: according to TARSKI:def 3,RELAT_1:def 19 :: thesis: ( not y in rng f or y in INT )
assume y in rng f ; :: thesis: y in INT
then consider x being object such that
x in dom f and
A3: f . x = y by FUNCT_1:def 3;
f . x is integer by A2;
hence y in INT by A3, INT_1:def 2; :: thesis: verum