
theorem Th40:
  for f be Function of [:NAT,NAT:],ExtREAL holds
       ~Partial_Sums_in_cod1 f = Partial_Sums_in_cod2 ~f
     & ~Partial_Sums_in_cod2 f = Partial_Sums_in_cod1 ~f
proof
   let f be Function of [:NAT,NAT:],ExtREAL;
   now let z be Element of [:NAT,NAT:];
    consider n,m be object such that
A1:  n in NAT & m in NAT & z = [n,m] by ZFMISC_1:def 2;
    reconsider n,m as Element of NAT by A1;
    (Partial_Sums_in_cod2 ~f).z = (Partial_Sums_in_cod2 ~f).(n,m) by A1
      .= (Partial_Sums_in_cod1 f).(m,n) by Th39
      .= ~(Partial_Sums_in_cod1 f).(n,m) by FUNCT_4:def 8;
    hence (Partial_Sums_in_cod2 ~f).z = ~(Partial_Sums_in_cod1 f).z by A1;
   end;
   hence ~Partial_Sums_in_cod1 f = Partial_Sums_in_cod2 ~f by FUNCT_2:def 8;
   now let z be Element of [:NAT,NAT:];
    consider n,m be object such that
A2:  n in NAT & m in NAT & z = [n,m] by ZFMISC_1:def 2;
    reconsider n,m as Element of NAT by A2;
    (Partial_Sums_in_cod1 ~f).z = (Partial_Sums_in_cod1 ~f).(n,m) by A2
      .= (Partial_Sums_in_cod2 f).(m,n) by Th39
      .= ~(Partial_Sums_in_cod2 f).(n,m) by FUNCT_4:def 8;
    hence (Partial_Sums_in_cod1 ~f).z = ~(Partial_Sums_in_cod2 f).z by A2;
   end;
   hence ~Partial_Sums_in_cod2 f = Partial_Sums_in_cod1 ~f by FUNCT_2:def 8;
end;
