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;

theorem
  B = B (O) B1 implies X (O) B c= X (O) B1
proof
  assume
A1: B = B (O) B1;
  let x be object;
  assume x in X (O) B;
  then consider x1,b1 being Point of T such that
A2: x=x1+b1 and
A3: x1 in X (-) B and
A4: b1 in B;
  consider x2 being Point of T such that
A5: x1=x2 & B+x2 c= X by A3;
  consider x3,b2 being Point of T such that
A6: b1=x3+b2 and
A7: x3 in B (-) B1 and
A8: b2 in B1 by A1,A4;
  consider x4 being Point of T such that
A9: x3=x4 and
A10: B1+x4 c= B by A7;
  B1+x4+x2 c= B+x2 by A10,Th3;
  then B1+x3+x1 c= X by A5,A9;
  then B1+(x3+x1) c= X by Th16;
  then x1+x3 in X (-) B1;
  then x1+x3+b2 in (X (-) B1) (+) B1 by A8;
  hence thesis by A2,A6,RLVECT_1:def 3;
end;
