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;
reserve X for set;
reserve x,y,Y for set;
reserve n,m,p for Nat;

theorem Th25:
  for X being non empty set, w being Element of free_magma_carrier X
  holds w in [:free_magma(X,w`2),{w`2}:]
proof
  let X be non empty set;
  let w be Element of free_magma_carrier X;
  w in free_magma_carrier X; then
  w in union rng disjoin((free_magma_seq X)|NATPLUS) by CARD_3:def 4; then
  consider Y be set such that
  A1: w in Y & Y in rng disjoin((free_magma_seq X)|NATPLUS) by TARSKI:def 4;
  consider n be object such that
  A2: n in dom disjoin((free_magma_seq X)|NATPLUS) &
  Y = disjoin((free_magma_seq X)|NATPLUS).n by A1,FUNCT_1:def 3;
  A3: n in dom((free_magma_seq X)|NATPLUS) by A2,CARD_3:def 3; then
  A4: ((free_magma_seq X)|NATPLUS).n = (free_magma_seq X).n by FUNCT_1:47;
  reconsider n as Nat by A3;
  w in [:((free_magma_seq X)|NATPLUS).n,{n}:] by A2,A1,A3,CARD_3:def 3; then
  w`2 in {n} by MCART_1:10; then
  w`2 = n by TARSKI:def 1;
  hence w in [:free_magma(X,w`2),{w`2}:] by A4,A2,A1,A3,CARD_3:def 3;
end;
