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;
reserve M for non empty multMagma;
reserve M,N for non empty multMagma,
      f for Function of M, N,
      H for non empty multSubmagma of N,
      R for compatible Equivalence_Relation of M;
reserve f for Function of X,Y;
reserve g for Function of Y,Z;

theorem
  f is one-to-one implies free_magmaF f is one-to-one
proof
  assume A1: f is one-to-one; then
  A2: f"*f = id dom f by FUNCT_1:39;
  set Y9 = rng f;
  dom f = X by FUNCT_2:def 1; then
  reconsider f1=f as Function of X, Y9 by FUNCT_2:1;
  reconsider f2=f1" as Function of Y9, X by A1,FUNCT_2:25;
  f2*f1 = id X by A2,FUNCT_2:def 1; then
  (free_magmaF f2)*(free_magmaF f1) = free_magmaF(id X) by Th44; then
  (free_magmaF f2)*(free_magmaF f) = free_magmaF(id X) by Th45; then
  (free_magmaF f2)*(free_magmaF f) = id dom(free_magmaF f) by Th46;
  hence free_magmaF f is one-to-one by FUNCT_1:31;
end;
