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
  st S1, S2 are_equivalent & S2, S3 are_equivalent holds S1, S3 are_equivalent
proof
  let S1, S2, S3 be non empty ManySortedSign;
  given f1,g1 be one-to-one Function such that
A1: S1, S2 are_equivalent_wrt f1, g1;
  given f2,g2 be one-to-one Function such that
A2: S2, S3 are_equivalent_wrt f2, g2;
  take f2*f1, g2*g1;
  thus thesis by A1,A2,Th27;
end;
