reserve x,y,z, X,Y,Z for set,
  n for Element of NAT;
reserve A for set,
  D for non empty set,
  a,b,c,l,r for Element of D,
  o,o9 for BinOp of D,
  f,g,h for Function of A,D;
reserve G for non empty multMagma;
reserve A for non empty set,
  a for Element of A,
  p for FinSequence of A,
  m1,m2 for Multiset of A;
reserve p,q for FinSequence of A;
reserve fm for Element of finite-MultiSet_over A;
reserve a,b,c for Element of D;

theorem Th53:
  o is having_a_unity implies o.:^2 is having_a_unity & {
the_unity_wrt o} is_a_unity_wrt o.:^2 & the_unity_wrt o.:^2 = {the_unity_wrt o}
proof
  given a such that
A1: a is_a_unity_wrt o;
  a = the_unity_wrt o by A1,BINOP_1:def 8;
  hence thesis by A1,Th52;
end;
