reserve T for non empty Abelian
  add-associative right_zeroed right_complementable RLSStruct,
  X,Y,Z,B,C,B1,B2 for Subset of T,
  x,y,p for Point of T;
reserve t,s,r1 for Real;
reserve n for Element of NAT;
reserve X,Y,B1,B2 for Subset of TOP-REAL n;
reserve x,y for Point of TOP-REAL n;

theorem
  0.TOP-REAL n in B2 implies (X (*) (B1,B2)) /\ X = {}
proof
  assume
A1: 0.TOP-REAL n in B2;
  now
    given x being object such that
A2: x in (X (*) (B1,B2)) /\ X;
A3: x in X by A2,XBOOLE_0:def 4;
    x in (X (*) (B1,B2)) by A2,XBOOLE_0:def 4;
    then x in X` (-) B2 by XBOOLE_0:def 4;
    then consider y being Point of TOP-REAL n such that
A4: x=y and
A5: B2+y c= X`;
    0.TOP-REAL n + y in {z+y where z is Point of TOP-REAL n :z in B2} by A1;
    then x in B2+y by A4;
    hence contradiction by A3,A5,XBOOLE_0:def 5;
  end;
  hence thesis by XBOOLE_0:def 1;
end;
