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 Th10:
  for r being Real st i in Seg n holds Sum((0*n)+*(i,r))=r
proof
  let r be Real;
A1: len(0*n) = n by CARD_1:def 7;
  reconsider w=0*n as FinSequence of REAL;
  assume
A2: i in Seg n;
  then
A3: i<=n by FINSEQ_1:1;
   reconsider r as Element of REAL by XREAL_0:def 1;
  1<=i by A2,FINSEQ_1:1;
  then i in dom (0*n) by A3,A1,FINSEQ_3:25;
  then Sum( w+*(i,r))=Sum((w | (i-'1))^<*r*>^(w/^i)) by FUNCT_7:98
    .= Sum(((0*n) | (i-'1))^<*r*>)+Sum((0*n)/^i) by RVSUM_1:75
    .= Sum((0*n) | (i-'1))+Sum<*r*>+Sum((0*n)/^i) by RVSUM_1:75
    .= Sum((0*n) | (i-'1))+r+Sum((0*n)/^i) by FINSOP_1:11
    .= Sum(0*(i-'1))+r+Sum((0*n)/^i) by A3,Th7,NAT_D:44
    .= 0+r+Sum((0*n)/^i) by Th9
    .= 0+r+Sum(0*(n-'i)) by Th8
    .= r by Th9;
  hence thesis;
end;
