
theorem Th36:
  for f be Function of [:NAT,NAT:],ExtREAL holds
     (f is convergent_in_cod1_to_+infty
          iff ~f is convergent_in_cod2_to_+infty)
   & (f is convergent_in_cod2_to_+infty
          iff ~f is convergent_in_cod1_to_+infty)
   & (f is convergent_in_cod1_to_-infty
          iff ~f is convergent_in_cod2_to_-infty)
   & (f is convergent_in_cod2_to_-infty
          iff ~f is convergent_in_cod1_to_-infty)
   & (f is convergent_in_cod1_to_finite
          iff ~f is convergent_in_cod2_to_finite)
   & (f is convergent_in_cod2_to_finite
          iff ~f is convergent_in_cod1_to_finite)
proof
   let f be Function of [:NAT,NAT:],ExtREAL;
   now hereby assume
A1:  f is convergent_in_cod1_to_+infty;
     now let m be Element of NAT;
      ProjMap2(f,m) = ProjMap1(~f,m) by Th33;
      hence ProjMap1(~f,m) is convergent_to_+infty by A1;
     end;
     hence ~f is convergent_in_cod2_to_+infty;
    end;
    assume A2: ~f is convergent_in_cod2_to_+infty;
    now let m be Element of NAT;
     ProjMap2(f,m) = ProjMap1(~f,m) by Th33;
     hence ProjMap2(f,m) is convergent_to_+infty by A2;
    end;
    hence f is convergent_in_cod1_to_+infty;
   end;
   hence
    f is convergent_in_cod1_to_+infty iff ~f is convergent_in_cod2_to_+infty;
   now hereby assume
A3:  f is convergent_in_cod2_to_+infty;
     now let m be Element of NAT;
      ProjMap1(f,m) = ProjMap2(~f,m) by Th32;
      hence ProjMap2(~f,m) is convergent_to_+infty by A3;
     end;
     hence ~f is convergent_in_cod1_to_+infty;
    end;
    assume A4: ~f is convergent_in_cod1_to_+infty;
    now let m be Element of NAT;
     ProjMap1(f,m) = ProjMap2(~f,m) by Th32;
     hence ProjMap1(f,m) is convergent_to_+infty by A4;
    end;
    hence f is convergent_in_cod2_to_+infty;
   end;
   hence
    f is convergent_in_cod2_to_+infty iff ~f is convergent_in_cod1_to_+infty;
   now hereby assume
A5:  f is convergent_in_cod1_to_-infty;
     now let m be Element of NAT;
      ProjMap2(f,m) = ProjMap1(~f,m) by Th33;
      hence ProjMap1(~f,m) is convergent_to_-infty by A5;
     end;
     hence ~f is convergent_in_cod2_to_-infty;
    end;
    assume A6: ~f is convergent_in_cod2_to_-infty;
    now let m be Element of NAT;
     ProjMap2(f,m) = ProjMap1(~f,m) by Th33;
     hence ProjMap2(f,m) is convergent_to_-infty by A6;
    end;
    hence f is convergent_in_cod1_to_-infty;
   end;
   hence
    f is convergent_in_cod1_to_-infty iff ~f is convergent_in_cod2_to_-infty;
   now hereby assume
A7:  f is convergent_in_cod2_to_-infty;
     now let m be Element of NAT;
      ProjMap1(f,m) = ProjMap2(~f,m) by Th32;
      hence ProjMap2(~f,m) is convergent_to_-infty by A7;
     end;
     hence ~f is convergent_in_cod1_to_-infty;
    end;
    assume A8: ~f is convergent_in_cod1_to_-infty;
    now let m be Element of NAT;
     ProjMap1(f,m) = ProjMap2(~f,m) by Th32;
     hence ProjMap1(f,m) is convergent_to_-infty by A8;
    end;
    hence f is convergent_in_cod2_to_-infty;
   end;
   hence
    f is convergent_in_cod2_to_-infty iff ~f is convergent_in_cod1_to_-infty;
   now hereby assume
A9:  f is convergent_in_cod1_to_finite;
     now let m be Element of NAT;
      ProjMap2(f,m) = ProjMap1(~f,m) by Th33;
      hence ProjMap1(~f,m) is convergent_to_finite_number by A9;
     end;
     hence ~f is convergent_in_cod2_to_finite;
    end;
    assume A10: ~f is convergent_in_cod2_to_finite;
    now let m be Element of NAT;
     ProjMap2(f,m) = ProjMap1(~f,m) by Th33;
     hence ProjMap2(f,m) is convergent_to_finite_number by A10;
    end;
    hence f is convergent_in_cod1_to_finite;
   end;
   hence
    f is convergent_in_cod1_to_finite iff ~f is convergent_in_cod2_to_finite;
   now hereby assume
A11: f is convergent_in_cod2_to_finite;
     now let m be Element of NAT;
      ProjMap1(f,m) = ProjMap2(~f,m) by Th32;
      hence ProjMap2(~f,m) is convergent_to_finite_number by A11;
     end;
     hence ~f is convergent_in_cod1_to_finite;
    end;
    assume A12: ~f is convergent_in_cod1_to_finite;
    now let m be Element of NAT;
     ProjMap1(f,m) = ProjMap2(~f,m) by Th32;
     hence ProjMap1(f,m) is convergent_to_finite_number by A12;
    end;
    hence f is convergent_in_cod2_to_finite;
   end;
   hence
    f is convergent_in_cod2_to_finite iff ~f is convergent_in_cod1_to_finite;
end;
