 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 Th33:
  G is associative implies H is associative
proof
  assume
A1: G is associative;
  now
    let a,b,c be Element of H;
    carr(H) c= carr(G) by Th23;
    then reconsider
    a9 = a, b9 = b, c9 = c, ab = a*b, bc = b*c as Element of G;
    thus a*b*c = ab*c9 by Th25
      .= a9*b9*c9 by Th25
      .= a9*(b9*c9) by A1
      .= a9*bc by Th25
      .= a*(b*c) by Th25;
  end;
  hence thesis;
end;
