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)*;

theorem Th6:
  for w1,w2 being Element of (the carrier of S)* holds w1 <= w2 &
  w2 <= w1 implies w1 = w2
proof
  let w1,w2 be Element of (the carrier of S)*;
  assume that
A1: w1 <= w2 and
A2: w2 <= w1;
  len w1 = len w2 by A1;
  then
A3: dom w1 = dom w2 by FINSEQ_3:29;
  for i being object st i in dom w1 holds w1.i = w2.i
  proof
    let i be object such that
A4: i in dom w1;
    reconsider s3 = w1.i, s4 = w2.i as Element of S by A3,A4,PARTFUN1:4;
    s3 <= s4 & s4 <= s3 by A1,A2,A3,A4;
    hence thesis by ORDERS_2:2;
  end;
  hence thesis by A3,FUNCT_1:2;
end;
