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 Th38:
  for f being Function of X,Y
  holds canon_image(Y,1)*f is Function of X, free_magma Y
proof
  let f be Function of X,Y;
  A1: dom f = X by FUNCT_2:def 1;
  dom canon_image(Y,1) = Y by Lm3; then
  rng f c= dom canon_image(Y,1); then
  A2: dom(canon_image(Y,1)*f) = X by A1,RELAT_1:27;
  rng(canon_image(Y,1)*f) c= rng canon_image(Y,1) by RELAT_1:26;
  hence thesis by A2,FUNCT_2:2;
end;
