reserve A,O for non empty set,
  R for Order of A,
  Ol for Equivalence_Relation of O,
  f for Function of O,A*,
  g for Function of O,A;
reserve S for OverloadedRSSign;
reserve S0 for non empty non void ManySortedSign;
reserve S for non empty Poset;
reserve s1,s2 for Element of S;
reserve w1,w2 for Element of (the carrier of S)*;
reserve S for OrderSortedSign;
reserve o,o1,o2 for OperSymbol of S;
reserve w1 for Element of (the carrier of S)*;
reserve SM for monotone OrderSortedSign,
  o,o1,o2 for OperSymbol of SM,
  w1 for Element of (the carrier of SM)*;
reserve SR for regular monotone OrderSortedSign,
  o,o1,o3,o4 for OperSymbol of SR,
  w1 for Element of (the carrier of SR)*;
reserve R for non empty Poset;
reserve z for non empty set;
reserve s1,s2 for SortSymbol of S,
  o,o1,o2,o3 for OperSymbol of S,
  w1,w2 for Element of (the carrier of S)*;
reserve CH for ManySortedFunction of ConstOSSet(S,z)# * the Arity of S,
  ConstOSSet(S,z) * the ResultSort of S;
reserve A for OSAlgebra of S;
reserve M for MSAlgebra over S0;
reserve A for OSAlgebra of S;

theorem Th21:
  for w1,w2,w3 being Element of (the carrier of S)* holds w1 <= w2
  & w2 <= w3 implies w1 <= w3
proof
  let w1,w2,w3 be Element of (the carrier of S)*;
  assume that
A1: w1 <= w2 and
A2: w2 <= w3;
A3: len w1 = len w2 by A1;
  then
A4: dom w1 = dom w2 by FINSEQ_3:29;
A5: len w2 = len w3 by A2;
  then
A6: dom w2 = dom w3 by FINSEQ_3:29;
  for i being set st i in dom w1 for s1,s2 st s1 = w1.i & s2 = w3.i holds
  s1 <= s2
  proof
    let i be set such that
A7: i in dom w1;
    reconsider s3 = w1.i, s4 = w2.i, s5 = w3.i as SortSymbol of S by A4,A6,A7,
PARTFUN1:4;
A8: s3 <= s4 & s4 <= s5 by A1,A2,A4,A7;
    let s1,s2;
    assume s1 = w1.i & s2 = w3.i;
    hence thesis by A8,ORDERS_2:3;
  end;
  hence thesis by A3,A5;
end;
