reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r for Real;
reserve p,p1,p2 for Prime;

theorem Th64:
  not p*p is_a_product_of_two_different_primes
  proof
    given p1,q1 being Prime such that
A1: p1 <> q1 and
A2: p*p = p1*q1;
A3: p > 1 by INT_2:def 4;
A4: p1*p1/p1 = p1 & p1*q1/p1 = q1 by XCMPLX_1:89;
A5: q1*q1/q1 = q1 & p1*q1/q1 = p1 by XCMPLX_1:89;
    p divides p*p;
    then p divides p1 or p divides q1 by A2,INT_5:7;
    then per cases by A3,INT_2:def 4;
    suppose p = p1;
      hence thesis by A1,A2,A4;
    end;
    suppose p = q1;
      hence thesis by A1,A2,A5;
    end;
  end;
