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 i st i in Seg n holds (0.TOP-REAL n)/.i=0
proof
  let i;
  assume
A1: i in Seg n;
  len (0*n)=n by CARD_1:def 7;
  then
A2: i in dom (0*n) by A1,FINSEQ_1:def 3;
  (0.TOP-REAL n)/.i=(0*n)/.i by EUCLID:70
    .=(0*n).i by A2,PARTFUN1:def 6
    .=0 by A1,FUNCOP_1:7;
  hence thesis;
end;
