reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve ff for Cardinal-Function;
reserve F,G for Cardinal-Function;

theorem Th45:
  X is c=-linear implies
  ex Y st Y c= X & union Y = union X & for Z st Z c= Y & Z <> {}
  ex Z1 st Z1 in Z & for Z2 st Z2 in Z holds Z1 c= Z2
proof
  assume
A1: X is c=-linear;
  consider R such that
A2: R well_orders X by WELLORD2:17;
A3: R|_2 X is well-ordering by A2,WELLORD2:16;
A4: field(R|_2 X) = X by A2,WELLORD2:16;
  R|_2 X, RelIncl order_type_of R|_2 X are_isomorphic by A3,WELLORD2:def 2;
  then RelIncl order_type_of R|_2 X, R|_2 X are_isomorphic by WELLORD1:40;
  then consider f such that
A5: f is_isomorphism_of RelIncl order_type_of R|_2 X, R|_2 X;
  field RelIncl order_type_of R|_2 X = order_type_of R|_2 X by WELLORD2:def 1;
  then
A6: dom f = order_type_of R|_2 X by A5;
A7: rng f = X by A4,A5;
  defpred P[object] means for A,B st B in A & $1 = A holds f.B c= f.A;
  consider Y such that
A8: for x being set holds x in Y iff x in dom f & P[x] from XFAMILY:sch 1;
  take Z = f.:Y;
  thus Z c= X by A7,RELAT_1:111;
  thus union Z c= union X by A7,RELAT_1:111,ZFMISC_1:77;
  thus union X c= union Z
  proof
    let x be object;
    assume x in union X;
    then consider Z1 such that
A9: x in Z1 and
A10: Z1 in X by TARSKI:def 4;
    consider y being object such that
A11: y in dom f and
A12: Z1 = f.y by A7,A10,FUNCT_1:def 3;
    reconsider y as Ordinal by A6,A11;
    defpred P[Ordinal] means $1 c= y & x in f.$1;
A13: ex A st P[A] by A9,A12;
    consider A such that
A14: P[A] & for B st P[B] holds A c= B from ORDINAL1:sch 1(A13);
A15: A in dom f by A6,A11,A14,ORDINAL1:12;
    now
      let B,C;
      assume that
A16:  C in B and
A17:  A = B;
A18:  C in dom f by A6,A11,A14,A16,A17,ORDINAL1:10;
A19:  not C c= y or not x in f.C by A14,A16,A17,ORDINAL1:5;
A20:  f.A in X by A7,A15,FUNCT_1:def 3;
      f.C in X by A7,A18,FUNCT_1:def 3;
      then f.C,f.A are_c=-comparable by A1,A20;
      then f.C c= f.A or f.A c= f.C;
      hence f.C c= f.B by A14,A16,A17,A19,ORDINAL1:def 2;
    end;
    then A in Y by A8,A15;
    then f.A in Z by A15,FUNCT_1:def 6;
    hence thesis by A14,TARSKI:def 4;
  end;
  let V be set;
  assume that
A21: V c= Z and
A22: V <> {};
  set x = the Element of V;
  x in Z by A21,A22;
  then consider y being object such that
A23: y in dom f and
A24: y in Y and
A25: x = f.y by FUNCT_1:def 6;
  reconsider y as Ordinal by A6,A23;
  defpred P[Ordinal] means $1 in Y & f.$1 in V;
  y = y;
  then
A26: ex A st P[A] by A22,A24,A25;
  consider A such that
A27: P[A] & for B st P[B] holds A c= B from ORDINAL1:sch 1(A26);
  take Z1 = f.A;
  thus Z1 in V by A27;
  let Z2;
  assume
A28: Z2 in V;
  then consider y being object such that
A29: y in dom f and
A30: y in Y and
A31: Z2 = f.y by A21,FUNCT_1:def 6;
  reconsider y as Ordinal by A6,A29;
  A c< y iff A c= y & A <> y;
  then A = y or A in y by A27,A28,A30,A31,ORDINAL1:11;
  hence thesis by A8,A30,A31;
end;
