 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 Th36:
  the_unity_wrt the multF of G in the carrier of H &
  G is uniquely-decomposable implies H is uniquely-decomposable
proof
  assume
A1: the_unity_wrt op(G) in carr(H);
  assume that
A2: op(G) is having_a_unity and
A3: for a,b being Element of G st op(G).(a,b) = the_unity_wrt op(G)
  holds a = b & b = the_unity_wrt op(G);
A4: G is unital by A2;
  then H is unital by A1,Th30;
  hence op(H) is having_a_unity;
  let a,b be Element of H;
  carr(H) c= carr(G) by Th23;
  then reconsider a9 = a, b9 = b as Element of G;
A5: op(H).(a,b) = a*b .= a9*b9 by Th25
    .= op(G).(a9,b9);
  the_unity_wrt op(G) = the_unity_wrt op(H) by A1,A4,Th30;
  hence thesis by A3,A5;
end;
