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;
