reserve x,x1,x2,y,z,z1 for set;
reserve s1,r,r1,r2 for Real;
reserve s,w1,w2 for Real;
reserve n,i for Element of NAT;
reserve X for non empty TopSpace;
reserve p,p1,p2,p3 for Point of TOP-REAL n;
reserve P for Subset of TOP-REAL n;

theorem
  for p,i st i in Seg n holds (-p)/.i=-p/.i
proof
  let p,i;
  assume
A1: i in Seg n;
  reconsider w1=p as Element of REAL n by EUCLID:22;
  len w1=n by CARD_1:def 7;
  then
A2: i in dom w1 by A1,FINSEQ_1:def 3;
  reconsider w3=w1 as Element of n-tuples_on REAL;
A3: p/.i =w3.i by A2,PARTFUN1:def 6;
  reconsider w=-p as Element of REAL n by EUCLID:22;
  len w=n by CARD_1:def 7;
  then
 i in dom w by A1,FINSEQ_1:def 3;
  then (-p)/.i = w.i by PARTFUN1:def 6
    .=(-w3).i
    .= (-1)*p/.i by A3,RVSUM_1:45
    .=-p/.i;
  hence thesis;
end;
