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

theorem
  X is c=-linear implies X\/{union X\/x} is c=-linear
 proof
  assume that
   A1: X is c=-linear;
  set U=union X;
  set Ux=U\/x;
  A2: U c=Ux by XBOOLE_1:7;
  let x1,x2 be set such that
   A3: x1 in X\/{Ux} & x2 in X\/{Ux};
  per cases by A3,XBOOLE_0:def 3;
  suppose x1 in X & x2 in X;
   hence thesis by A1;
  end;
  suppose A4: x1 in {Ux} & x2 in {Ux};
   then x1=Ux by TARSKI:def 1;
   hence thesis by A4,TARSKI:def 1;
  end;
  suppose x1 in X & x2 in {Ux};
   then x1 c=U & x2=Ux by TARSKI:def 1,ZFMISC_1:74;
   then x1 c=x2 by A2;
   hence thesis;
  end;
  suppose x2 in X & x1 in {Ux};
   then x2 c=U & x1=Ux by TARSKI:def 1,ZFMISC_1:74;
   then x2 c=x1 by A2;
   hence thesis;
  end;
 end;
