reserve E, x, y, X for set;
reserve A, B, C, D for Subset of E^omega;
reserve a, a1, a2, b, c, c1, c2, d, ab, bc for Element of E^omega;
reserve e for Element of E;
reserve i, j, k, l, n, n1, n2, m for Nat;

theorem Th73:
  a in Lex(E) |^ len a
proof
  defpred P[Nat] means for a holds len a = $1 implies a in Lex(E) |^ $1;
A1: now
    let n;
    assume
A2: P[n];
    now
      let b;
      assume len b = n + 1;
      then consider c, e such that
A3:   len c = n and
A4:   b = c ^ <%e%> by Th7;
      <%e%> is Element of E^omega by A4,Th5;
      then e in E by Th6;
      then <%e%> in Lex(E) by Def4;
      then
A5:   <%e%> in Lex(E) |^ 1 by Th25;
      c in Lex(E) |^ n by A2,A3;
      hence b in Lex(E) |^ (n + 1) by A4,A5,Th40;
    end;
    hence P[n + 1];
  end;
A6: P[0]
  proof
    let a;
    assume len a = 0;
    then a = <%>E;
    then a in {<%>E} by TARSKI:def 1;
    hence thesis by Th24;
  end;
  for n holds P[n] from NAT_1:sch 2(A6, A1);
  hence thesis;
end;
