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

theorem th103a:
  Partial_Sums Rseq = Partial_Sums_in_cod1(Partial_Sums_in_cod2 Rseq)
proof
   now let x be Element of [:NAT,NAT:];
    consider n,m be object such that
A1:  n in NAT & m in NAT & x = [n,m] by ZFMISC_1:def 2;
    reconsider n1=n,m1=m as Nat by A1;
    (Partial_Sums Rseq).(n1,m1)
     = (Partial_Sums_in_cod1(Partial_Sums_in_cod2 Rseq)).(n1,m1) by th103;
    hence (Partial_Sums Rseq).x
      = (Partial_Sums_in_cod1(Partial_Sums_in_cod2 Rseq)).x by A1;
   end;
   hence thesis;
end;
