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 Th15:
  for M,N being non empty multMagma, f being Function of M,N,
      X being Subset of M st f is multiplicative holds
  f.:the carrier of the_submagma_generated_by X =
  the carrier of the_submagma_generated_by f.:X
proof
  let M,N be non empty multMagma;
  let f be Function of M,N;
  let X be Subset of M;
  assume A1: f is multiplicative;
  set X9 = the_submagma_generated_by X;
  set A = f.:the carrier of X9;
  the carrier of X9 is stable Subset of M by Th8; then
  reconsider A as stable Subset of N by A1,Th11;
  set Y9 = the_submagma_generated_by f.:X;
  set B = f"the carrier of Y9;
  the carrier of Y9 is stable Subset of N by Th8; then
  reconsider B as stable Subset of M by A1,Th12;
  A2: f.:X c= the carrier of Y9 & for N1 being strict multSubmagma of N
  st f.:X c= the carrier of N1
  holds Y9 is multSubmagma of N1 by Def12;
  reconsider N1 = multMagma(# A, the_mult_induced_by A #) as
  strict multSubmagma of N by Def9;
  X c= the carrier of X9 by Def12; then
  Y9 is multSubmagma of N1 by A2,RELAT_1:123; then
  A3: the carrier of Y9 c= A by Def9;
  A4: X c= the carrier of X9 & for M1 being strict multSubmagma of M
  st X c= the carrier of M1
  holds X9 is multSubmagma of M1 by Def12;
  reconsider M1 = multMagma(# B, the_mult_induced_by B #) as
  strict multSubmagma of M by Def9;
  A5: f.:(f"the carrier of Y9) c= the carrier of Y9 by FUNCT_1:75;
  f.:X c= the carrier of the_submagma_generated_by f.:X by Def12; then
  A6: f"(f.:X) c= f"the carrier of the_submagma_generated_by f.:X
  by RELAT_1:143;
  X c= the carrier of M; then
  X c= dom f by FUNCT_2:def 1; then
  X c= f"(f.:X) by FUNCT_1:76; then
  X9 is multSubmagma of M1 by A4,A6,XBOOLE_1:1; then
  the carrier of X9 c= B by Def9; then
  A c= f.:(f"the carrier of Y9) by RELAT_1:123; then
  A c= the carrier of Y9 by A5;
  hence thesis by A3,XBOOLE_0:def 10;
end;
