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;
  thus X (@) (B1,B2) c= X
  by XBOOLE_0:def 5;
  let x be object;
  assume
A2: x in X;
  not x in (X (*) (B1,B2))
  proof
    assume x in (X (*) (B1,B2));
    then x in X` (-) B2 by XBOOLE_0:def 4;
    then consider y being Point of TOP-REAL n such that
A3: x=y and
A4: 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 A3;
    hence contradiction by A2,A4,XBOOLE_0:def 5;
  end;
  hence thesis by A2,XBOOLE_0:def 5;
end;
