
theorem
  for f be Function of [:NAT,NAT:],ExtREAL holds
    (f is convergent_in_cod1_to_+infty
       iff -f is convergent_in_cod1_to_-infty)
  & (f is convergent_in_cod1_to_-infty
       iff -f is convergent_in_cod1_to_+infty)
  & (f is convergent_in_cod1_to_finite
       iff -f is convergent_in_cod1_to_finite)
  & (f is convergent_in_cod2_to_+infty
       iff -f is convergent_in_cod2_to_-infty)
  & (f is convergent_in_cod2_to_-infty
       iff -f is convergent_in_cod2_to_+infty)
  & (f is convergent_in_cod2_to_finite
       iff -f is convergent_in_cod2_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;
A2:   ProjMap2(f,m) is convergent_to_+infty by A1;
      -ProjMap2(f,m) = ProjMap2(-f,m) by Th35;
      hence ProjMap2(-f,m) is convergent_to_-infty by A2,Th17;
     end;
     hence -f is convergent_in_cod1_to_-infty;
    end;
    assume A3: -f is convergent_in_cod1_to_-infty;
    now let m be Element of NAT;
     -ProjMap2(f,m) = ProjMap2(-f,m) by Th35; then
     -ProjMap2(f,m) is convergent_to_-infty by A3; then
     -(-ProjMap2(f,m)) is convergent_to_+infty by Th17;
     hence ProjMap2(f,m) is convergent_to_+infty by Th2;
    end;
    hence f is convergent_in_cod1_to_+infty;
   end;
   hence
    f is convergent_in_cod1_to_+infty iff -f is convergent_in_cod1_to_-infty;
   now hereby assume
A1:  f is convergent_in_cod1_to_-infty;
     now let m be Element of NAT;
A2:   ProjMap2(f,m) is convergent_to_-infty by A1;
      -ProjMap2(f,m) = ProjMap2(-f,m) by Th35;
      hence ProjMap2(-f,m) is convergent_to_+infty by A2,Th17;
     end;
     hence -f is convergent_in_cod1_to_+infty;
    end;
    assume A3: -f is convergent_in_cod1_to_+infty;
    now let m be Element of NAT;
     -ProjMap2(f,m) = ProjMap2(-f,m) by Th35; then
     -ProjMap2(f,m) is convergent_to_+infty by A3; then
     -(-ProjMap2(f,m)) is convergent_to_-infty by Th17;
     hence ProjMap2(f,m) is convergent_to_-infty by Th2;
    end;
    hence f is convergent_in_cod1_to_-infty;
   end;
   hence
    f is convergent_in_cod1_to_-infty iff -f is convergent_in_cod1_to_+infty;
   now hereby assume
A1:  f is convergent_in_cod1_to_finite;
     now let m be Element of NAT;
A2:   ProjMap2(f,m) is convergent_to_finite_number by A1;
      -ProjMap2(f,m) = ProjMap2(-f,m) by Th35;
      hence ProjMap2(-f,m) is convergent_to_finite_number by A2,Th17;
     end;
     hence -f is convergent_in_cod1_to_finite;
    end;
    assume A3: -f is convergent_in_cod1_to_finite;
    now let m be Element of NAT;
     -ProjMap2(f,m) = ProjMap2(-f,m) by Th35; then
     -ProjMap2(f,m) is convergent_to_finite_number by A3; then
     -(-ProjMap2(f,m)) is convergent_to_finite_number by Th17;
     hence ProjMap2(f,m) is convergent_to_finite_number by Th2;
    end;
    hence f is convergent_in_cod1_to_finite;
   end;
   hence
    f is convergent_in_cod1_to_finite iff -f is convergent_in_cod1_to_finite;
   now hereby assume
A1:  f is convergent_in_cod2_to_+infty;
     now let m be Element of NAT;
A2:   ProjMap1(f,m) is convergent_to_+infty by A1;
      -ProjMap1(f,m) = ProjMap1(-f,m) by Th34;
      hence ProjMap1(-f,m) is convergent_to_-infty by A2,Th17;
     end;
     hence -f is convergent_in_cod2_to_-infty;
    end;
    assume A3: -f is convergent_in_cod2_to_-infty;
    now let m be Element of NAT;
     -ProjMap1(f,m) = ProjMap1(-f,m) by Th34; then
     -ProjMap1(f,m) is convergent_to_-infty by A3; then
     -(-ProjMap1(f,m)) is convergent_to_+infty by Th17;
     hence ProjMap1(f,m) is convergent_to_+infty by Th2;
    end;
    hence f is convergent_in_cod2_to_+infty;
   end;
   hence
    f is convergent_in_cod2_to_+infty iff -f is convergent_in_cod2_to_-infty;
   now hereby assume
A1:  f is convergent_in_cod2_to_-infty;
     now let m be Element of NAT;
A2:   ProjMap1(f,m) is convergent_to_-infty by A1;
      -ProjMap1(f,m) = ProjMap1(-f,m) by Th34;
      hence ProjMap1(-f,m) is convergent_to_+infty by A2,Th17;
     end;
     hence -f is convergent_in_cod2_to_+infty;
    end;
    assume A3: -f is convergent_in_cod2_to_+infty;
    now let m be Element of NAT;
     -ProjMap1(f,m) = ProjMap1(-f,m) by Th34; then
     -ProjMap1(f,m) is convergent_to_+infty by A3; then
     -(-ProjMap1(f,m)) is convergent_to_-infty by Th17;
     hence ProjMap1(f,m) is convergent_to_-infty by Th2;
    end;
    hence f is convergent_in_cod2_to_-infty;
   end;
   hence
    f is convergent_in_cod2_to_-infty iff -f is convergent_in_cod2_to_+infty;
   now hereby assume
A1:  f is convergent_in_cod2_to_finite;
     now let m be Element of NAT;
A2:   ProjMap1(f,m) is convergent_to_finite_number by A1;
      -ProjMap1(f,m) = ProjMap1(-f,m) by Th34;
      hence ProjMap1(-f,m) is convergent_to_finite_number by A2,Th17;
     end;
     hence -f is convergent_in_cod2_to_finite;
    end;
    assume A3: -f is convergent_in_cod2_to_finite;
    now let m be Element of NAT;
     -ProjMap1(f,m) = ProjMap1(-f,m) by Th34; then
     -ProjMap1(f,m) is convergent_to_finite_number by A3; then
     -(-ProjMap1(f,m)) is convergent_to_finite_number by Th17;
     hence ProjMap1(f,m) is convergent_to_finite_number by Th2;
    end;
    hence f is convergent_in_cod2_to_finite;
   end;
   hence
    f is convergent_in_cod2_to_finite iff -f is convergent_in_cod2_to_finite;
end;
