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 Th52:
  a is_a_unity_wrt o implies {a} is_a_unity_wrt o.:^2 & o.:^2 is
  having_a_unity & the_unity_wrt o.:^2 = {a}
proof
  assume
A1: a is_a_unity_wrt o;
  now
    let x be Subset of D;
    thus (o.:^2).({a},x) = o.:[:{a},x:] by Th44
      .= D /\ x by A1,Th51
      .= x by XBOOLE_1:28;
    thus (o.:^2).(x,{a}) = o.:[:x,{a}:] by Th44
      .= D /\ x by A1,Th51
      .= x by XBOOLE_1:28;
  end;
  hence
A2: {a} is_a_unity_wrt o.:^2 by BINOP_1:3;
  hence ex A being Subset of D st A is_a_unity_wrt o.:^2;
  thus thesis by A2,BINOP_1:def 8;
end;
