reserve x,y, X,Y,Z for set,
        D for non empty set,
        n,k for Nat,
        i,i1,i2 for Integer;

theorem Th9:
  X is non empty finite c=-linear implies union X in X
 proof
  assume X is non empty finite c=-linear;
  then consider U be set such that
   A1: U in X and
   A2: for x st x in X holds x c=U by FINSET_1:12;
  A3: union X c=U
  proof
   let x be object;
   assume x in union X;
   then consider y such that
    A4: x in y and
    A5: y in X by TARSKI:def 4;
   y c=U by A2,A5;
   hence thesis by A4;
  end;
  U c=union X by A1,ZFMISC_1:74;
  hence thesis by A1,A3,XBOOLE_0:def 10;
 end;
