
theorem Th40:
  for x, y be G_INTEG holds
  x is_associated_to y iff ex c be G_INTEG st c is g_int_unit & x = c*y
  proof
    let x, y be G_INTEG;
    hereby assume A1:x is_associated_to y;
      reconsider x1 = x, y1 = y as Element of Gauss_INT_Ring by Th3;
      consider c1 be Element of Gauss_INT_Ring such that
      A2: c1 is unital & y1 * c1 = x1 by A1,Th38,GCD_1:18;
      reconsider c = c1 as G_INTEG by Th2;
      A3: c is g_int_unit by A2,Th39;
      c1 * y1 = c * y by Th6;
      hence ex c be G_INTEG st c is g_int_unit & x = c*y by A2,A3;
    end;
    given c0 be G_INTEG such that
    A4: c0 is g_int_unit & x = c0*y;
    reconsider xx = x as Element of Gauss_INT_Ring by Th3;
    reconsider yy = y as Element of Gauss_INT_Ring by Th3;
    reconsider c = c0 as Element of Gauss_INT_Ring by Th3;
    A5: c is unital by A4,Th39;
    c * yy = c0 * y by Th6;
    hence x is_associated_to y by A4,A5,Th37,GCD_1:18;
  end;
