reserve a,b,c,h for Integer;
reserve k,m,n for Nat;
reserve i,j,z for Integer;
reserve p for Prime;

theorem
  i <> 1 & i <> -1 & not i is prime & not -i is prime implies
  multiples(i) misses SetPrimes
  proof
    assume that
A1: i <> 1 and
A2: i <> -1 and
A3: not i is prime and
A4: not -i is prime;
    assume multiples(i) meets SetPrimes;
    then consider x being object such that
A5: x in multiples(i) and
A6: x in SetPrimes by XBOOLE_0:3;
    reconsider m = x as Multiple of i by A5,Th61;
A7: i divides m by Def15;
A8: m is prime by A6,NEWTON:def 6;
    per cases;
    suppose i >= 0;
      then i in NAT by INT_1:3;
      hence contradiction by A1,A3,A8,Def15;
    end;
    suppose i < 0;
      then -i in NAT by INT_1:3;
      then -i = 1 or -i = x by A7,A8,INT_2:10;
      hence contradiction by A2,A4,A6,NEWTON:def 6;
    end;
  end;
