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;
reserve v,v1,v2,w,w1,w2 for Element of free_magma X;
reserve X,Y,Z for non empty set;

theorem
  for R being compatible Equivalence_Relation of free_magma(X) holds
  (free_magma X)./.R =
  the_submagma_generated_by (nat_hom R).: (canon_image(X,1) .: X)
proof
  let R be compatible Equivalence_Relation of free_magma(X);
  set A = canon_image(X,1) .: X;
  reconsider A as Subset of free_magma X;
  A1: the carrier of the_submagma_generated_by A
  = the carrier of free_magma X by Th36;
  the carrier of (free_magma X)./.R = rng nat_hom R by FUNCT_2:def 3; then
  the carrier of (free_magma X)./.R = (nat_hom R) .: dom(nat_hom R)
  by RELAT_1:113; then
  the carrier of (free_magma X)./.R =
  (nat_hom R).: the carrier of the_submagma_generated_by A
  by A1,FUNCT_2:def 1; then
  the carrier of (free_magma X)./.R =
  the carrier of the_submagma_generated_by (nat_hom R).: A by Th15;
  hence thesis by Th7;
end;
