
theorem
  for f be Function of [:NAT,NAT:],ExtREAL,
      g be ext-real-valued Function, n be Nat
   st (for k be Nat holds (Partial_Sums_in_cod1 f).(n,k) = g.k) holds
    (for k be Nat holds (Partial_Sums f).(n,k) = (Partial_Sums g).k)
  & (lim_in_cod2(Partial_Sums f)).n = Sum g
proof
   let f be Function of [:NAT,NAT:],ExtREAL,
       g be ext-real-valued Function, n be Nat;
   assume A1: for k be Nat holds (Partial_Sums_in_cod1 f).(n,k) = g.k;
A4:now let k be Nat;
    defpred P[Nat] means (Partial_Sums f).(n,$1) = (Partial_Sums g).$1;
    (Partial_Sums f).(n,0) = (Partial_Sums_in_cod1 f).(n,0) by DefCSM
    .= g.0 by A1; then
A2: P[0] by MESFUNC9:def 1;
A3: for m be Nat st P[m] holds P[m+1]
    proof
     let m be Nat;
     assume A4: P[m];
     (Partial_Sums f).(n,m+1)
      = (Partial_Sums_in_cod2(Partial_Sums_in_cod1 f)).(n,m)
         + (Partial_Sums_in_cod1 f).(n,m+1) by DefCSM
     .= (Partial_Sums g).m + g.(m+1) by A1,A4;
     hence P[m+1] by MESFUNC9:def 1;
    end;
    for m be Nat holds P[m] from NAT_1:sch 2(A2,A3);
    hence (Partial_Sums f).(n,k) = (Partial_Sums g).k;
   end;
   reconsider n1=n as Element of NAT by ORDINAL1:def 12;
   now let k be Element of NAT;
    ProjMap1(Partial_Sums f,n1).k = (Partial_Sums f).(n,k) by MESFUNC9:def 6;
    hence ProjMap1(Partial_Sums f,n1).k = (Partial_Sums g).k by A4;
   end; then
   ProjMap1(Partial_Sums f,n1) = Partial_Sums g by FUNCT_2:def 8; then
   (lim_in_cod2(Partial_Sums f)).n1 = lim (Partial_Sums g) by D1DEF6;
   hence thesis by A4,MESFUNC9:def 3;
end;
