reserve x,y,y1,y2 for object;

theorem Th1:
  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 <> {} & V1 is linearly-closed
  holds 0.V 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 that
A1: V1 <> {} and
A2: V1 is linearly-closed;
  set x = the Element of V1;
  reconsider x as Element of V by A1,TARSKI:def 3;
  0.GF * x in V1 by A1,A2;
  hence thesis by VECTSP_1:14;
end;
