 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 Th40:
  for N being unital non empty SubStr of <REAL,+> holds
  the_unity_wrt the multF of N = 0
proof
  let N be unital non empty SubStr of <REAL,+>;
  consider a be Element of N such that
A1: for b being Element of N holds a*b = b & b*a = b by Th6;
  carr(N) c= REAL by Th23;
  then reconsider x = a as Real;
  now
    let b be Element of N;
    a*b = op(N).(a,b) & b*a = op(N).(b,a);
    hence op(N).(a,b) = b & op(N).(b,a) = b by A1;
  end;
  then
A2: a is_a_unity_wrt op(N) by BINOP_1:3;
  x+0 = a*a by A1
    .= x+x by Th39;
  hence thesis by A2,BINOP_1:def 8;
end;
