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 B1 implies X (&) (B1,B2) = X
proof
  assume
A1: 0.TOP-REAL n in B1;
  thus X (&) (B1,B2) c= X
  proof
    let x be object;
    assume
A2: x in X (&) (B1,B2);
    per cases by A2,XBOOLE_0:def 3;
    suppose
      x in X;
      hence thesis;
    end;
    suppose
      x in X (*) (B1,B2);
      then x in X (-) B1 by XBOOLE_0:def 4;
      then consider y being Point of TOP-REAL n such that
A3:   x=y and
A4:   B1+y c= X;
      0.TOP-REAL n + y in {z+y where z is Point of TOP-REAL n:z in B1} by A1;
      then x in B1+y by A3;
      hence thesis by A4;
    end;
  end;
  thus thesis by XBOOLE_1:7;
end;
