
theorem
for D be non empty set, Y be with_non-empty_element FinSequenceSet of D,
    s be non-empty sequence of Y, s1 be sequence of D st
  ( for n be Nat holds s1.n = (joined_seq s).(Partial_Sums(Length s).n - 1) )
 holds s1 is subsequence of joined_seq s
proof
   let D be non empty set, Y be with_non-empty_element FinSequenceSet of D;
   let s be non-empty sequence of Y, s1 be sequence of D;
   assume
A1:for n be Nat holds s1.n = (joined_seq s).(Partial_Sums(Length s).n - 1);
   consider N be increasing sequence of NAT such that
A2: for n be Nat holds N.n = (Partial_Sums(Length s)).n - 1 by Th11;
   for n be Element of NAT holds s1.n = ((joined_seq s)*N).n
   proof
    let n be Element of NAT;
    s1.n = (joined_seq s).(Partial_Sums(Length s).n - 1) by A1; then
    s1.n = (joined_seq s).(N.n) by A2;
    hence s1.n = ((joined_seq s)*N).n by FUNCT_2:15;
   end;
   hence s1 is subsequence of joined_seq s by FUNCT_2:def 8;
end;
