
theorem Th38:
  for a, b be Element of Gauss_INT_Ring, aa, bb be G_INTEG st a = aa & b = bb
  holds aa is_associated_to bb implies a is_associated_to b
  proof
    let a, b be Element of Gauss_INT_Ring, aa, bb be G_INTEG such that
    A1: a = aa & b = bb;
    assume aa is_associated_to bb;
    then A2: aa Divides bb & bb Divides aa;
    then consider cca be G_INTEG such that
    A3: bb = aa * cca;
    consider ccb be G_INTEG such that
    A4: aa = bb * ccb by A2;
    reconsider ca = cca as Element of Gauss_INT_Ring by Th3;
    reconsider cb = ccb as Element of Gauss_INT_Ring by Th3;
    b = a*ca by A1,A3,Th6;
    then A5: a divides b by GCD_1:def 1;
    a = b*cb by A1,A4,Th6;
    then b divides a by GCD_1:def 1;
    hence thesis by A5,GCD_1:def 3;
  end;
