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;

theorem Th20:
  w1 <= w2 implies (the Sorts of A)#.w1 c= (the Sorts of A)#.w2
proof
  set iw1 = (the Sorts of A) * w1, iw2 = (the Sorts of A) * w2;
  assume
A1: w1 <= w2;
  then
A2: len w1 = len w2;
  let x be object;
  assume x in (the Sorts of A)#.w1;
  then x in product(iw1) by FINSEQ_2:def 5;
  then consider g being Function such that
A3: x = g and
A4: dom g = dom iw1 and
A5: for y being object st y in dom iw1 holds g.y in iw1.y by CARD_3:def 5;
A6: dom iw1 = dom w1 by Lm1
    .= dom w2 by A2,FINSEQ_3:29
    .= dom iw2 by Lm1;
  for y being object st y in dom iw2 holds g.y in iw2.y
  proof
    let y be object such that
A7: y in dom iw2;
A8: y in dom w1 by A6,A7,Lm1;
    then
A9: iw1.y = (the Sorts of A).(w1.y) by FUNCT_1:13;
A10: y in dom w2 by A7,Lm1;
    then
A11: iw2.y = (the Sorts of A).(w2.y) by FUNCT_1:13;
    reconsider s1 = w1.y, s2 = w2.y as SortSymbol of S by A8,A10,PARTFUN1:4;
    s1 <= s2 by A1,A8;
    then
A12: (the Sorts of A).(w1.y) c= (the Sorts of A).(w2.y) by Def17;
    g.y in iw1.y by A5,A6,A7;
    hence thesis by A9,A11,A12;
  end;
  then g in product(iw2) by A4,A6,CARD_3:def 5;
  hence thesis by A3,FINSEQ_2:def 5;
end;
