
theorem
  for Rseq be Function of [:NAT,NAT:],REAL st
   lim_in_cod1 Rseq is convergent holds
    cod1_major_iterated_lim Rseq = lim (lim_in_cod1 Rseq)
proof
   let Rseq be Function of [:NAT,NAT:],REAL;
   assume A1: lim_in_cod1 Rseq is convergent; then
   consider g be Real such that
A2: for p be Real st 0<p
     ex M be Nat st
      for m be Nat st M<=m holds
       |. (lim_in_cod1 Rseq).m - g qua Complex .| < p by SEQ_2:def 6;
   g = lim (lim_in_cod1 Rseq) by A1,A2,SEQ_2:def 7;
   hence thesis by A1,A2,DBLSEQ_1:def 7;
end;
