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 divides j iff |.i.| divides j
  proof
    thus i divides j implies |.i.| divides j
    proof
      assume
A1:   i divides j;
      then consider a being Integer such that
A2:   j = i * a;
      per cases;
      suppose i >= 0;
        hence thesis by A1,ABSVALUE:def 1;
      end;
      suppose i < 0;
        then
A3:     -i = |.i.| by ABSVALUE:def 1;
        take -a;
        thus thesis by A2,A3;
      end;
    end;
    assume
A4: |.i.| divides j;
    then consider a being Integer such that
A5: j = |.i.| * a;
    per cases;
    suppose i >= 0;
      hence thesis by A4,ABSVALUE:def 1;
    end;
    suppose i < 0;
      then
A6:   -i = |.i.| by ABSVALUE:def 1;
      take -a;
      thus thesis by A5,A6;
    end;
  end;
