
theorem Th37:
  for a, b be Element of Gauss_INT_Ring, aa, bb be G_INTEG st a = aa & b = bb
  holds a is_associated_to b implies aa is_associated_to bb
  proof
    let a, b be Element of Gauss_INT_Ring, aa, bb be G_INTEG such that
    A1: a = aa & b = bb;
    assume a is_associated_to b;
    then A2: a divides b & b divides a by GCD_1:def 3;
    then consider ca be Element of Gauss_INT_Ring such that
    A3: b = a * ca by GCD_1:def 1;
    consider cb be Element of Gauss_INT_Ring such that
    A4: a = b * cb by A2,GCD_1:def 1;
    reconsider cca = ca as G_INTEG by Th2;
    reconsider ccb = cb as G_INTEG by Th2;
    bb = aa*cca by A1,A3,Th6;
    then A5: aa Divides bb;
    aa = bb*ccb by A1,A4,Th6;
    then bb Divides aa;
    hence thesis by A5;
  end;
