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
  for S1, S2, S3 being non empty ManySortedSign
  for f1, g1, f2, g2 being Function for C1 being non-empty MSAlgebra over S1
  for C2 being non-empty MSAlgebra over S2
  for C3 being non-empty MSAlgebra over S3
  st C1, C2 are_similar_wrt f1, g1 & C2, C3 are_similar_wrt f2, g2
  holds C1, C3 are_similar_wrt f2*f1, g2*g1
proof
  let S1, S2, S3 be non empty ManySortedSign;
  let f1,g1, f2,g2 be Function;
  let C1 be non-empty MSAlgebra over S1;
  let C2 be non-empty MSAlgebra over S2;
  let C3 be non-empty MSAlgebra over S3;
  assume that
A1: f1, g1 form_embedding_of C1, C2 and
A2: f1", g1" form_embedding_of C2, C1 and
A3: f2, g2 form_embedding_of C2, C3 and
A4: f2", g2" form_embedding_of C3, C2;
  thus f2*f1, g2*g1 form_embedding_of C1, C3 by A1,A3,Th35;
A5: f1 is one-to-one by A1;
A6: g1 is one-to-one by A1;
A7: f2 is one-to-one by A3;
A8: g2 is one-to-one by A3;
A9: (f2*f1)" = f1"*f2" by A5,A7,FUNCT_1:44;
  (g2*g1)" = g1"*g2" by A6,A8,FUNCT_1:44;
  hence thesis by A2,A4,A9,Th35;
end;
