reserve X,Y,Z,X1,X2,X3,X4,X5,X6 for set, x,y for object;
reserve a,b,c for object, X,Y,Z,x,y,z for set;
reserve A,B,C,D for Ordinal;
reserve F,G for Function;
reserve L,L1 for Sequence;

theorem
  (for a st a in X holds a is Sequence) & X is c=-linear implies
  union X is Sequence
proof
  assume that
A1: for a st a in X holds a is Sequence and
A2: X is c=-linear;
  union X is Relation-like Function-like
  proof
    thus for a being object st a in union X
     ex b,c being object st [b,c] = a
    proof let a be object;
      assume a in union X;
      then consider x such that
A3:   a in x and
A4:   x in X by TARSKI:def 4;
      reconsider x as Sequence by A1,A4;
      for a being object st a in x
       ex b,c being object st [b,c] = a by RELAT_1:def 1;
      hence thesis by A3;
    end;
    let a,b,c be object;
    assume that
A5: [a,b] in union X and
A6: [a,c] in union X;
    consider y such that
A7: [a,c] in y and
A8: y in X by A6,TARSKI:def 4;
    reconsider y as Sequence by A1,A8;
    consider x such that
A9: [a,b] in x and
A10: x in X by A5,TARSKI:def 4;
    reconsider x as Sequence by A1,A10;
    x,y are_c=-comparable by A2,A10,A8;
    then x c= y or y c= x;
    hence thesis by A9,A7,FUNCT_1:def 1;
  end;
  then reconsider F = union X as Function;
  defpred P[object,object] means $1 in X & for L st L = $1 holds dom L = $2;
A11: for a,b,c being object st P[a,b] & P[a,c] holds b = c
  proof
    let a,b,c be object;
    assume that
A12: a in X and
A13: for L st L = a holds dom L = b and
    a in X and
A14: for L st L = a holds dom L = c;
    reconsider a as Sequence by A1,A12;
    dom a = b by A13;
    hence thesis by A14;
  end;
  consider G such that
A15: for a,b being object holds [a,b] in G iff a in X & P[a,b]
    from FUNCT_1:sch 1(A11);
A16: for a,b being object holds [a,b] in G implies b is Ordinal
  proof let a,b be object;
    assume
A17: [a,b] in G;
    then a in X by A15;
    then reconsider a as Sequence by A1;
    dom a = b by A15,A17;
    hence thesis;
  end;
  a in rng G implies a is Ordinal
  proof
    assume a in rng G;
    then consider b being object such that
A18: b in dom G & a = G.b by FUNCT_1:def 3;
    [b,a] in G by A18,FUNCT_1:1;
    hence thesis by A16;
  end;
  then consider A such that
A19: rng G c= A by Th20;
  defpred P[Ordinal] means for B st B in rng G holds B c= $1;
  for B st B in rng G holds B c= A by A19,Def2;
  then
A20: ex A st P[A];
  consider A such that
A21: P[A] & for C st P[C] holds A c= C from OrdinalMin(A20);
  dom F = A
  proof
    thus for a be object holds  a in dom F implies a in A
    proof let a be object;
      assume a in dom F;
      then consider b being object such that
A22:  [a,b] in F by XTUPLE_0:def 12;
      consider x such that
A23:  [a,b] in x and
A24:  x in X by A22,TARSKI:def 4;
      reconsider x as Sequence by A1,A24;
      for L st L = x holds dom L = dom x;
      then [x,dom x] in G by A15,A24;
      then x in dom G & dom x = G.x by FUNCT_1:1;
      then dom x in rng G by FUNCT_1:def 3;
      then
A25:  dom x c= A by A21;
      a in dom x by A23,XTUPLE_0:def 12;
      hence thesis by A25;
    end;
    let a be object;
    assume
A26: a in A;
    then reconsider a9 = a as Ordinal by Th9;
    now
      assume
A27:  for L st L in X holds not a9 in dom L;
      now
        let B;
        assume B in rng G;
        then consider c being object such that
A28:    c in dom G & B = G.c by FUNCT_1:def 3;
A29:    [c,B] in G by A28,FUNCT_1:1;
        then c in X by A15;
        then reconsider c as Sequence by A1;
        c in X & dom c = B by A15,A29;
        hence B c= a9 by A27,Th12;
      end;
      then
A30:  A c= a9 by A21;
      a9 c= A by A26,Def2;
      then A: A = a by A30,XBOOLE_0:def 10;
      reconsider aa = a as set by TARSKI:1;
      not aa in aa;
      hence contradiction by A,A26;
    end;
    then consider L such that
A31: L in X & a in dom L;
    L c= F & ex b being object st [a,b] in L
      by A31,XTUPLE_0:def 12,ZFMISC_1:74;
    hence thesis by XTUPLE_0:def 12;
  end;
  hence thesis by Def7;
end;
