reserve a,b,r for non unit non zero Real;
reserve X for non empty set,
        x for Tuple of 4,X;
reserve V             for RealLinearSpace,
        A,B,C,P,Q,R,S for Element of V;

theorem Th16:
  for T being RealLinearSpace
  for x being Element of T
  for p being Tuple of 1,REAL st
  T = TOP-REAL 1 & p = x
  holds - p = - x
  proof
    let T be RealLinearSpace;
    let x be Element of T;
    let p be Tuple of 1,REAL;
    assume that
A1: T = TOP-REAL 1 and
A2: p = x;
    consider d be Element of REAL such that
A3: p = <* d *> by FINSEQ_2:97;
    reconsider x9 = <* -d *> as Tuple of 1,REAL;
    reconsider n = 1 as Nat;
    REAL is non empty & REAL c= REAL;
    then reconsider p9 = p as Element of 1-tuples_on REAL
      by FINSEQ_2:109;
A4: the RLSStruct of TOP-REAL 1 = RealVectSpace Seg 1 & 0.REAL 1 = <* 0 *>
      by EUCLID:def 8,FINSEQ_2:59;
A5: p + (-p) = p9 + (-p9)
            .= 0.TOP-REAL 1 by A4,RVSUM_1:22;
    the TopStruct of TOP-REAL 1 = TopSpaceMetr Euclid 1 by EUCLID:def 8;
    then reconsider x99 = x9 as Element of T by A1;
A6: -p = <* -d *> by A3,RVSUM_1:20;
    x + x99 = 0.T by A1,A2,A6,Th14,A5;
    then - x = x99 by RLVECT_1:def 10;
    hence thesis by A3,RVSUM_1:20;
  end;
