 reserve Rseq, Rseq1, Rseq2 for Function of [:NAT,NAT:],REAL;

theorem th100:
for m,n being Element of NAT holds
 (Partial_Sums_in_cod1(Rseq)).(m,n) = Partial_Sums(ProjMap2(Rseq,n)).m &
 (Partial_Sums_in_cod2(Rseq)).(m,n) = Partial_Sums(ProjMap1(Rseq,m)).n
proof
   let m,n be Element of NAT;
   defpred P[Nat] means
    (Partial_Sums_in_cod1(Rseq)).($1,n) = Partial_Sums(ProjMap2(Rseq,n)).$1;
   Partial_Sums(ProjMap2(Rseq,n)).0 = (ProjMap2(Rseq,n)).0 by SERIES_1:def 1
     .= Rseq.(0,n) by MESFUNC9:def 7; then
a1:P[0] by DefRS;
a2:for k be Nat st P[k] holds P[k+1]
   proof
    let k be Nat;
    assume P[k]; then
    (Partial_Sums_in_cod1(Rseq)).(k+1,n)
      = Partial_Sums(ProjMap2(Rseq,n)).k + Rseq.(k+1,n) by DefRS
     .= Partial_Sums(ProjMap2(Rseq,n)).k + (ProjMap2(Rseq,n)).(k+1)
       by MESFUNC9:def 7;
    hence P[k+1] by SERIES_1:def 1;
   end;
   for k be Nat holds P[k] from NAT_1:sch 2(a1,a2);
   hence (Partial_Sums_in_cod1(Rseq)).(m,n) = Partial_Sums(ProjMap2(Rseq,n)).m;
   defpred Q[Nat] means
    (Partial_Sums_in_cod2(Rseq)).(m,$1) = Partial_Sums(ProjMap1(Rseq,m)).$1;
   Partial_Sums(ProjMap1(Rseq,m)).0 = (ProjMap1(Rseq,m)).0 by SERIES_1:def 1
    .= Rseq.(m,0) by MESFUNC9:def 6; then
a3:Q[0] by DefCS;
a4:for k be Nat st Q[k] holds Q[k+1]
   proof
    let k be Nat;
    assume Q[k]; then
    (Partial_Sums_in_cod2(Rseq)).(m,k+1)
      = Partial_Sums(ProjMap1(Rseq,m)).k + Rseq.(m,k+1) by DefCS
     .= Partial_Sums(ProjMap1(Rseq,m)).k + (ProjMap1(Rseq,m)).(k+1)
       by MESFUNC9:def 6;
    hence Q[k+1] by SERIES_1:def 1;
   end;
   for k be Nat holds Q[k] from NAT_1:sch 2(a3,a4);
   hence thesis;
end;
