
theorem Th54:
 for f be Function of [:NAT,NAT:],ExtREAL st
  f is without-infty or f is without+infty holds
   ProjMap1(Partial_Sums f,0) = ProjMap1(Partial_Sums_in_cod2 f,0)
 & ProjMap2(Partial_Sums f,0) = ProjMap2(Partial_Sums_in_cod1 f,0)
proof
   let f be Function of [:NAT,NAT:],ExtREAL;
   assume A0: f is without-infty or f is without+infty;
A1:now let m be Element of NAT;
    ProjMap1(Partial_Sums f,0).m
      = (Partial_Sums f).(0,m) by MESFUNC9:def 6
     .= (Partial_Sums_in_cod1(Partial_Sums_in_cod2 f)).(0,m)
       by A0,Lm8,Lm9
     .= (Partial_Sums_in_cod2 f).(0,m) by DefRSM;
    hence ProjMap1(Partial_Sums f,0).m
     = ProjMap1(Partial_Sums_in_cod2 f,0).m by MESFUNC9:def 6;
   end;
   now let n be Element of NAT;
    ProjMap2(Partial_Sums f,0).n
      = (Partial_Sums f).(n,0) by MESFUNC9:def 7
     .= (Partial_Sums_in_cod1 f).(n,0) by DefCSM;
    hence ProjMap2(Partial_Sums f,0).n
     = ProjMap2(Partial_Sums_in_cod1 f,0).n by MESFUNC9:def 7;
   end;
   hence thesis by A1,FUNCT_2:63;
end;
