
theorem
for f be Function of [:NAT,NAT:],ExtREAL,
    k be Element of NAT holds
 ProjMap2(Partial_Sums_in_cod1 f,k) = Partial_Sums(ProjMap2(f,k)) &
 ProjMap1(Partial_Sums_in_cod2 f,k) = Partial_Sums(ProjMap1(f,k))
proof
   let f be Function of [:NAT,NAT:],ExtREAL, k be Element of NAT;
   now let n be Element of NAT;
    ProjMap2(Partial_Sums_in_cod1 f,k).n
     = (Partial_Sums_in_cod1 f).(n,k) by MESFUNC9:def 7;
    hence ProjMap2(Partial_Sums_in_cod1 f,k).n
     = Partial_Sums(ProjMap2(f,k)).n by Th43;
   end;
   hence ProjMap2(Partial_Sums_in_cod1 f,k) = Partial_Sums(ProjMap2(f,k))
     by FUNCT_2:def 8;
   now let n be Element of NAT;
    ProjMap1(Partial_Sums_in_cod2 f,k).n
     = (Partial_Sums_in_cod2 f).(k,n) by MESFUNC9:def 6;
    hence ProjMap1(Partial_Sums_in_cod2 f,k).n
     = Partial_Sums(ProjMap1(f,k)).n by Th43;
   end;
   hence ProjMap1(Partial_Sums_in_cod2 f,k) = Partial_Sums(ProjMap1(f,k))
     by FUNCT_2:def 8;
end;
