let F be real-valued FinSequence; :: thesis: ( ( for i being Nat st i in dom F holds
0 <= F . i ) & ex i being Nat st
( i in dom F & 0 < F . i ) implies 0 < Sum F )
reconsider F1 = F as FinSequence of REAL by Lm2;
set i = len F;
set R1 = (len F) |-> (In (0,REAL));
reconsider R2 = F1 as Element of (len F) -tuples_on REAL by FINSEQ_2:92;
A1: Seg (len F) = dom F by FINSEQ_1:def 3;
assume A2: for i being Nat st i in dom F holds
0 <= F . i ; :: thesis: ( for i being Nat holds
( not i in dom F or not 0 < F . i ) or 0 < Sum F )
A3: for j being Nat st j in Seg (len F) holds
((len F) |-> (In (0,REAL))) . j <= R2 . j

proof

let j be Nat; :: thesis: ( j in Seg (len F) implies ((len F) |-> (In (0,REAL))) . j <= R2 . j )
((len F) |-> (In (0,REAL))) . j = In (0,REAL) ;
hence ( j in Seg (len F) implies ((len F) |-> (In (0,REAL))) . j <= R2 . j ) by A2, A1; :: thesis: verum

end;

given j being Nat such that A4: j in dom F and
A5: 0 < F . j ; :: thesis: 0 < Sum F
((len F) |-> (In (0,REAL))) . j = 0 ;
then Sum ((len F) |-> (In (0,REAL))) < Sum R2 by A1, A3, A4, A5, Th83;
hence 0 < Sum F by Th81; :: thesis: verum