
theorem
  for L being 1-sorted, R being (Relation of the carrier of L), C being
  strict_chain of R holds Strict_Chains (R,C) is_inductive_wrt RelIncl
Strict_Chains (R,C) & ex D being set st D is_maximal_in RelIncl Strict_Chains (
  R,C) & C c= D
proof
  let L be 1-sorted, R be (Relation of the carrier of L), C be strict_chain of
  R;
  set X = Strict_Chains (R,C);
A1: field RelIncl X = X by WELLORD2:def 1;
  thus
A2: X is_inductive_wrt RelIncl X
  proof
    let Y be set such that
A3: Y c= X and
A4: RelIncl X |_2 Y is being_linear-order;
    per cases;
    suppose
A5:   Y is empty;
      take C;
      thus thesis by A5,Def5;
    end;
    suppose
A6:   Y is non empty;
      take Z = union Y;
      Z c= the carrier of L
      proof
        let z be object;
        assume z in Z;
        then consider A being set such that
A7:     z in A and
A8:     A in Y by TARSKI:def 4;
        A is strict_chain of R by A3,A8,Def5;
        hence thesis by A7;
      end;
      then reconsider S = Z as Subset of L;
A9:   S is strict_chain of R
      proof
        RelIncl X |_2 Y is connected by A4;
        then
A10:    RelIncl X |_2 Y is_connected_in field (RelIncl X |_2 Y) by
RELAT_2:def 14;
A11:    (RelIncl X |_2 Y) = RelIncl Y by A3,WELLORD2:7;
        let x, y be set;
A12:    field RelIncl Y = Y by WELLORD2:def 1;
        assume x in S;
        then consider A being set such that
A13:    x in A and
A14:    A in Y by TARSKI:def 4;
A15:    A is strict_chain of R by A3,A14,Def5;
        assume y in S;
        then consider B being set such that
A16:    y in B and
A17:    B in Y by TARSKI:def 4;
A18:    B is strict_chain of R by A3,A17,Def5;
        per cases;
        suppose
          A <> B;
          then [A,B] in RelIncl Y or [B,A] in RelIncl Y by A14,A17,A10,A11,A12,
RELAT_2:def 6;
          then A c= B or B c= A by A14,A17,WELLORD2:def 1;
          hence thesis by A13,A16,A15,A18,Def3;
        end;
        suppose
          A = B;
          hence thesis by A13,A16,A15,Def3;
        end;
      end;
      C c= Z
      proof
        let c be object;
        assume
A19:    c in C;
        consider y being object such that
A20:    y in Y by A6;
        reconsider y as set by TARSKI:1;
        C c= y by A3,A20,Def5;
        hence thesis by A19,A20,TARSKI:def 4;
      end;
      hence
A21:  Z in X by A9,Def5;
      let y be set;
      assume
A22:  y in Y;
      then y c= Z by ZFMISC_1:74;
      hence thesis by A3,A21,A22,WELLORD2:def 1;
    end;
  end;
A23: RelIncl X is_transitive_in X by WELLORD2:20;
A24: RelIncl X is_antisymmetric_in X by WELLORD2:21;
  RelIncl X is_reflexive_in X by WELLORD2:19;
  then RelIncl X partially_orders X by A23,A24;
  then consider D being set such that
A25: D is_maximal_in RelIncl X by A1,A2,ORDERS_1:63;
  take D;
  thus D is_maximal_in RelIncl X by A25;
  D in field RelIncl X by A25;
  hence thesis by A1,Def5;
end;
