 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 H1,H2 being non empty MonoidalSubStr of M st
  the carrier of H1 c= the carrier of H2 holds
    H1 is MonoidalSubStr of H2
proof
  let H1,H2 be non empty MonoidalSubStr of M such that
A1: carr(H1) c= carr(H2);
  H1 is SubStr of M & H2 is SubStr of M by Th21;
  then H1 is SubStr of H2 by A1,Th28;
  hence op(H1) c= op(H2) by Def23;
  un(H1) = un(M) by Def25;
  hence thesis by Def25;
end;
