reserve x,y,y1,y2 for object;

theorem
  for GF be add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr, V be Abelian
  add-associative right_zeroed right_complementable vector-distributive
  scalar-distributive scalar-associative scalar-unital non empty
  ModuleStr over GF, V1 be Subset of V st V1 is linearly-closed for v,u being
  Element of V st v in V1 & u in V1 holds v - u in V1
proof
  let GF be add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr, V be Abelian
  add-associative right_zeroed right_complementable vector-distributive
  scalar-distributive scalar-associative scalar-unital non empty
  ModuleStr over GF, V1 be Subset of V;
  assume
A1: V1 is linearly-closed;
  let v,u be Element of V;
  assume that
A2: v in V1 and
A3: u in V1;
  - u in V1 by A1,A3,Th2;
  hence thesis by A1,A2;
end;
