reserve X for non empty set;
reserve e,e1,e2,e19,e29 for Equivalence_Relation of X,
  x,y,x9,y9 for set;
reserve A for non empty set,
  L for lower-bounded LATTICE;
reserve T,L1 for Sequence,
  O,O1,O2,O3,C for Ordinal;

theorem Th39:
  for d be distance_function of A,L for S being ExtensionSeq of A,
  d for k,l being Nat st k <= l holds (S.k)`2 c= (S.l)`2
proof
  let d be distance_function of A,L;
  let S be ExtensionSeq of A,d;
  let k be Nat;
  defpred X[Nat] means k <= $1 implies (S.k)`2 c= (S.$1)`2;
A1: for i being Nat st X[i] holds X[i+1]
  proof
    let i be Nat;
    assume that
A2: k <= i implies (S.k)`2 c= (S.i)`2 and
A3: k <= i+1;
    per cases by A3,NAT_1:8;
    suppose
      k = i+1;
      hence thesis;
    end;
    suppose
A4:   k <= i;
      consider A9 being non empty set, d9 being distance_function of A9,L, Aq
      being non empty set, dq being distance_function of Aq,L such that
A5:   Aq, dq is_extension_of A9,d9 and
A6:   S.i = [A9,d9] and
A7:   S.(i+1) = [Aq,dq] by Def20;
      consider q being QuadrSeq of d9 such that
      Aq = NextSet(d9) and
A8:   dq = NextDelta(q) by A5;
A9:   (S.i)`2 c= ConsecutiveDelta(q,DistEsti(d9)) by Th31,A6;
      (S.(i+1))`2 = ConsecutiveDelta(q,DistEsti(d9)) by A7,A8;
      hence thesis by A2,A4,A9;
    end;
  end;
A10: X[ 0 ] by NAT_1:3;
  thus for l being Nat holds X[l] from NAT_1:sch 2(A10, A1);
end;
