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 r,p,i st i in Seg n holds (r*p)/.i=r*p/.i
proof
  let r,p,i;
  reconsider w1=p as Element of REAL n by EUCLID:22;
  reconsider w3=w1 as Element of n-tuples_on REAL;
  reconsider w=r*p as Element of REAL n by EUCLID:22;
  assume
A1: i in Seg n;
  then i in Seg len w1 by CARD_1:def 7;
  then
A2: i in dom w1 by FINSEQ_1:def 3;
  len w=n by CARD_1:def 7;
  then
A3: i in dom w by A1,FINSEQ_1:def 3;
A4: p/.i =w3.i by A2,PARTFUN1:def 6;
  (r*p)/.i = w.i by A3,PARTFUN1:def 6
    .= (r*w3).i
    .= r*p/.i by A4,RVSUM_1:45;
  hence thesis;
end;
