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

theorem th00:
   ProjMap1(Partial_Sums Rseq,0) = ProjMap1(Partial_Sums_in_cod2 Rseq,0)
 & ProjMap2(Partial_Sums Rseq,0) = ProjMap2(Partial_Sums_in_cod1 Rseq,0)
proof
A1:now let m be Element of NAT;
    ProjMap1(Partial_Sums Rseq,0).m
      = (Partial_Sums Rseq).(0,m) by MESFUNC9:def 6
     .= (Partial_Sums_in_cod1(Partial_Sums_in_cod2 Rseq)).(0,m) by th103a
     .= (Partial_Sums_in_cod2 Rseq).(0,m) by DefRS;
    hence ProjMap1(Partial_Sums Rseq,0).m
     = ProjMap1(Partial_Sums_in_cod2 Rseq,0).m by MESFUNC9:def 6;
   end;
   now let n be Element of NAT;
    ProjMap2(Partial_Sums Rseq,0).n
      = (Partial_Sums Rseq).(n,0) by MESFUNC9:def 7
     .= (Partial_Sums_in_cod1 Rseq).(n,0) by DefCS;
    hence ProjMap2(Partial_Sums Rseq,0).n
     = ProjMap2(Partial_Sums_in_cod1 Rseq,0).n by MESFUNC9:def 7;
   end;
   hence thesis by A1;
end;
