reserve i, j, k, l, m, n for Nat,
  a, b, c, t, u for object,
  X, Y, Z for set,
  D, D1, D2, Fml for non empty set;
reserve p, q, r, s for FinSequence;
 reserve R, R1, R2 for Rule;
 reserve A, A1, A2 for non empty set;
 reserve B, B1, B2 for set;
 reserve P, P1, P2 for Formula-sequence;
 reserve S, S1, S2 for Formula-finset;
 reserve C for Extension of B;
 reserve E for Extension of R;
 reserve P for non empty ProofSystem;
 reserve B, B1, B2 for Subset of P;
 reserve F for finite Subset of P;

theorem Th60:
  for X being non empty finite-character set
    ex Y being Element of X st for Z being Element of X holds not Y c< Z
proof
  let X be non empty finite-character set;
  for C being set st C c= X & C is c=-linear
      ex Y st Y in X & for Z st Z in C holds Z c= Y
  proof
    let C be set;
    assume A2: C c= X & C is c=-linear;
    take Y = union C;
    thus thesis by A2, Lm59, ZFMISC_1:74;
  end;
  then consider Y such that
      A5: Y in X and
      A6: for Z st Z in X & Z <> Y holds not Y c= Z
    by ORDERS_1:65;
  reconsider Y as Element of X by A5;
  take Y;
  let Z be Element of X;
  assume Y c< Z;
  hence thesis by A6;
end;
