reserve K,F for Ring;
reserve V,W for VectSp of K;
reserve l for Linear_Combination of V;
reserve T for linear-transformation of V,W;
reserve K for Ring;

theorem
  for G,F being add-associative right_zeroed
      right_complementable non empty ModuleStr over K,
      x being Vector of [:G,F:], x1 being Vector of G, x2 be Vector of F
  st x = [x1,x2] holds -x = [-x1,-x2]
  proof
    let G,F be add-associative right_zeroed right_complementable
    non empty ModuleStr over K;
    let x be Vector of [:G,F:], x1 be Vector of G, x2 be Vector of F;
    assume A1: x=[x1,x2];
    reconsider y = [-x1,-x2 ] as Vector of [:G,F:];
    x+y = [x1+-x1,x2+-x2] by A1,PRVECT_3:def 1
    .= [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;
