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;

theorem
  for G being unital non empty multMagma, H being non empty SubStr of
  G st the_unity_wrt the multF of G in the carrier of H holds .:(H,X) is
  MonoidalSubStr of .:(G,X)
proof
  let G be unital non empty multMagma, H be non empty SubStr of G;
  assume
A1: the_unity_wrt the multF of G in carr(H);
  then reconsider G9 = G, H9 = H as unital non empty multMagma by MONOID_0:30;
A2: the_unity_wrt op(H9) = the_unity_wrt op(G9) by A1,MONOID_0:30;
A3: op(.:(H,X)) = (op(H), carr(H)).:X by Th17;
  op(H) c= op(G) & op(.:(G,X)) = (op(G), carr(G)).:X by Th17,MONOID_0:def 23;
  then
A4: op(.:(H,X)) c= op(.:(G,X)) by A3,Th16;
  1..:(G9,X) = X --> the_unity_wrt op(G) & 1..:(H9,X) = X -->
  the_unity_wrt op (H) by Th22;
  hence thesis by A2,A4,MONOID_0:def 25;
end;
