let a be Integer; :: thesis: for X being Subset of INT holds a ** X c= INT
let X be Subset of INT ; :: thesis: a ** X c= INT
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in a ** X or x in INT )
assume A1: x in a ** X ; :: thesis: x in INT
then reconsider x' = x as Real ;
consider y being Real such that
A2: ( y in X & x' = a * y ) by A1, INTEGRA2:def 2;
reconsider y' = y as Integer by A2;
x' = a * y' by A2;
hence x in INT by INT_1:def 2; :: thesis: verum