
theorem Th8:
for X be non empty set, F be summable FinSequence of Funcs(X,ExtREAL)
 st len F >= 2 holds (Partial_Sums F)/.2 = F/.1 + F/.2
proof
   let X be non empty set, F be summable FinSequence of Funcs(X,ExtREAL);
   assume A1: len F >= 2; then
   1+1 <= len F; then
A3:1 < len F by NAT_1:13; then
A6:1 in dom F & 2 in dom F by A1,FINSEQ_3:25;
   len F = len (Partial_Sums F) by MEASUR11:def 11; then
A5:1 in dom (Partial_Sums F) & 2 in dom(Partial_Sums F)
     by A1,A3,FINSEQ_3:25; then
A4:(Partial_Sums F)/.1 = (Partial_Sums F).1 by PARTFUN1:def 6
     .= F.1 by MEASUR11:def 11 .= F/.1 by A6,PARTFUN1:def 6;
   (Partial_Sums F).(1+1) = (Partial_Sums F)/.1 + F/.(1+1)
     by A1,NAT_1:13,MEASUR11:def 11;
   hence (Partial_Sums F)/.2 = F/.1 + F/.2 by A4,A5,PARTFUN1:def 6;
end;
