reserve S for non empty non void ManySortedSign,
  V for non-empty ManySortedSet of the carrier of S,
  A for non-empty MSAlgebra over S,
  X for non empty Subset of S-Terms V,
  t for Element of X;
reserve S for non empty non void ManySortedSign,
  A for non-empty finite-yielding MSAlgebra over S,
  V for Variables of A,
  X for SetWithCompoundTerm of S,V;

theorem Th39:
  for S1, S2 being non empty ManySortedSign for f, g being Function
  for C1 being non-empty MSAlgebra over S1
  for C2 being non-empty MSAlgebra over S2 st C1, C2 are_similar_wrt f, g
  holds C2, C1 are_similar_wrt f", g"
proof
  let S1, S2 be non empty ManySortedSign;
  let f,g be Function;
  let C1 be non-empty MSAlgebra over S1;
  let C2 be non-empty MSAlgebra over S2;
  assume that
A1: f, g form_embedding_of C1, C2 and
A2: f", g" form_embedding_of C2, C1;
A3: f is one-to-one by A1;
  f" " = f by A3,FUNCT_1:43;
  hence f", g" form_embedding_of C2, C1 &
  f" ", g" " form_embedding_of C1, C2 by A1,A2,FUNCT_1:43;
end;
