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) c= X
proof
  assume
A1: 0.TOP-REAL n in B1;
  let x be object;
  assume 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
A2: x=y and
A3: 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 A2;
  hence thesis by A3;
end;
