reserve M,N for non empty multMagma,
  f for Function of M, N;
reserve M for multMagma;
reserve N,K for multSubmagma of M;
reserve M,N for non empty multMagma,
  A for Subset of M,
  f,g for Function of M,N,
  X for stable Subset of M,
  Y for stable Subset of N;

theorem Th14:
  for M being multMagma, A being Subset of M
  holds A is empty iff the_submagma_generated_by A is empty
proof
  let M be multMagma;
  let A be Subset of M;
  hereby
    assume A1: A is empty;
    then
for v,w being Element of M st v in A & w in A holds v*w in A; then
    reconsider A9=A as stable Subset of M by Def10;
    reconsider N=multMagma(# A9, the_mult_induced_by A9 #)
    as strict multSubmagma of M by Def9;
    the_submagma_generated_by A is multSubmagma of N by Def12; then
    the carrier of the_submagma_generated_by A c= the carrier of N by Def9;
    hence the_submagma_generated_by A is empty by A1;
  end;
  assume the_submagma_generated_by A is empty; then
  the carrier of the_submagma_generated_by A = {}; then
  A c= {} by Def12;
  hence A is empty;
end;
