
theorem
  for D be set, Y be FinSequenceSet of D, F be FinSequence of Y
    st (for n be Nat st n in dom F holds F.n = <*>D) holds Sum Length F = 0
proof
   let D be set, Y be FinSequenceSet of D, F be FinSequence of Y;
   assume A1: for n be Nat st n in dom F holds F.n = <*>D;
A2:dom (Length F) = dom F by Def1 .= Seg len F by FINSEQ_1:def 3;
A6:len F |-> (0 qua Real) = Seg len F --> (0 qua Real) by FINSEQ_2:def 2; then
A3:dom(len F |-> (0 qua Real)) = Seg len F by FUNCT_2:def 1;
   now let k be Nat;
    assume A4: k in dom (Length F); then
    k in dom F by Def1; then
    F.k = <*>D by A1; then
    (Length F).k = 0 by A4,Def1;
    hence (Length F).k = (len F |-> (0 qua Real)).k by A2,A4,A6,FUNCOP_1:7;
   end; then
   Length F = (len F |-> (0 qua Real)) by A2,A3,FINSEQ_1:13;
   hence Sum Length F = 0 by RVSUM_1:81;
end;
