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
  C c= B implies X (O) B c= (X (-) C) (+) B
proof
  assume
A1: C c= B;
  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 and
A6: B+x2 c= X by A3;
  C+x2 c= B+x2 by A1,Th3;
  then C+x2 c= X by A6;
  then x1 in X (-) C by A5;
  hence thesis by A2,A4;
end;
