reserve S for non empty non void ManySortedSign;
reserve X for non-empty ManySortedSet of S;
reserve x,y,z for set, i,j for Nat;
reserve
  A0 for (X,S)-terms non-empty MSAlgebra over S,
  A1 for all_vars_including (X,S)-terms MSAlgebra over S,
  A2 for all_vars_including inheriting_operations (X,S)-terms MSAlgebra over S,
  A for all_vars_including inheriting_operations free_in_itself
  (X,S)-terms MSAlgebra over S;

theorem
  for A being non-empty MSAlgebra over S
  for s being SortSymbol of S
  for t1,t2,t3 being Element of TermAlg S,s
  holds A |= t1 '=' t2 & A |= t2 '=' t3 implies A |= t1 '=' t3
  proof
    let A be non-empty MSAlgebra over S;
    let s be SortSymbol of S;
    let t1,t2,t3 be Element of TermAlg S,s;
    assume
A1: A |= t1 '=' t2 & A |= t2 '=' t3;
    let h be ManySortedFunction of TermAlg S, A such that
A2: h is_homomorphism TermAlg S, A;
    h.s.t1 = h.s.t2 & h.s.t2 = h.s.t3 by A1,A2;
    hence thesis;
  end;
