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 Th27:
  for S1, S2, S3 being non empty ManySortedSign, f1,g1, f2,g2 being Function
  st S1, S2 are_equivalent_wrt f1, g1 & S2, S3 are_equivalent_wrt f2, g2
  holds S1, S3 are_equivalent_wrt f2*f1, g2*g1
proof
  let S1, S2, S3 be non empty ManySortedSign;
  let f1,g1, f2,g2 be Function such that
A1: f1 is one-to-one and
A2: g1 is one-to-one and
A3: f1, g1 form_morphism_between S1, S2 and
A4: f1", g1" form_morphism_between S2, S1 and
A5: f2 is one-to-one and
A6: g2 is one-to-one and
A7: f2, g2 form_morphism_between S2, S3 and
A8: f2", g2" form_morphism_between S3, S2;
  thus f2*f1 is one-to-one & g2*g1 is one-to-one by A1,A2,A5,A6;
A9: (f2*f1)" = f1"*f2" by A1,A5,FUNCT_1:44;
  (g2*g1)" = g1"*g2" by A2,A6,FUNCT_1:44;
  hence thesis by A3,A4,A7,A8,A9,PUA2MSS1:29;
end;
