
theorem Th18:
  for q1,q,n1 being Element of NAT st
  q1 divides q|^n1 & q is prime & q1 is prime holds q = q1
proof
  let q1,q,n1 be Element of NAT;
  assume that
A1: q1 divides q|^n1 and
A2: q is prime;
  assume
A3: q1 is prime;
  then q1>1 by INT_2:def 4;
  then consider k be Element of NAT such that
A4: q1=q*k by A1,A2,PEPIN:32;
A5: k<>q1
  proof
    assume k=q1;
    then q=1 by A3,A4,XCMPLX_1:7;
    hence contradiction by A2,INT_2:def 4;
  end;
  k divides q1 by A4,NAT_D:def 3;
  then k=1 or k=q1 by A3,INT_2:def 4;
  hence thesis by A4,A5;
end;
