reserve G,F for RealLinearSpace;

theorem
  for G,F be add-associative right_zeroed
  right_complementable non empty RLSStruct,
  x be Point of [:G,F:], x1 be Point of G, x2 be Point of F
  st x=[x1,x2] holds -x = [-x1,-x2]
  proof
    let G,F be add-associative right_zeroed right_complementable
    non empty RLSStruct;
    let x be Point of [:G,F:], x1 be Point of G, x2 be Point of F;
    assume A1: x=[x1,x2];
    reconsider y = [-x1,-x2 ] as Point of [:G,F:];
    x+y = [x1+-x1,x2+-x2] by A1,Def1
    .= [0.G,x2+-x2] by RLVECT_1:def 10
    .= 0.[:G,F:] by RLVECT_1:def 10;
    hence thesis by RLVECT_1:def 10;
  end;
