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

theorem Th8:
  for m,n being Integer st
  m divides n*p holds m divides n or ex z be Integer st m = z*p & z divides n
  proof
    let m,n be Integer;
    assume m divides n*p;
    then |.m.| divides |.n*p.| by INT_2:16;
    then |.m.| divides |.n.|*|.p.| by COMPLEX1:65;
    then per cases by Th7;
    suppose |.m.| divides |.n.|;
      hence thesis by INT_2:16;
    end;
    suppose ex z being Nat st |.m.| = z*|.p.| & z divides |.n.|;
      then consider z being Nat such that
A1:   |.m.| = z*|.p.| and
A2:   z divides |.n.|;
      now
        per cases;
        case
A3:       m >= 0;
          take z;
          thus m = z*p by A1,A3,ABSVALUE:def 1;
          thus z divides n by A2,Th4;
        end;
        case m < 0;
          then
A4:       |.m.| = -m by ABSVALUE:def 1;
          take Z = -z;
          thus m = -|.m.| by A4
          .= Z*p by A1;
          Z divides |.n.| by A2,INT_2:10;
          hence Z divides n by Th4;
        end;
      end;
      hence thesis;
    end;
  end;
