 reserve x,y,X,Y for set;
reserve G for non empty multMagma,
  D for set,
  a,b,c,r,l for Element of G;
reserve M for non empty multLoopStr;
reserve H for non empty SubStr of G,
  N for non empty MonoidalSubStr of G;

theorem
  for N being unital non empty SubStr of <REAL,*> holds
    the_unity_wrt the multF of N = 0 or the_unity_wrt the multF of N = 1
proof
  let N be unital non empty SubStr of <REAL,*>;
  set e = the_unity_wrt op(N);
  carr(N) c= carr(<REAL,*>) by Th23;
  then reconsider x = e as Real;
  e is_a_unity_wrt op(N) by Th4;
  then e = e*e by BINOP_1:3
    .= x*x by Th50;
  then x*1 = x*x;
  hence thesis by XCMPLX_1:5;
end;
