
theorem
  for f be Function of [:NAT,NAT:],ExtREAL holds
   ( f is convergent_in_cod1_to_+infty or f is convergent_in_cod1_to_-infty
     or f is convergent_in_cod1_to_finite
       implies f is convergent_in_cod1 )
 & ( f is convergent_in_cod2_to_+infty or f is convergent_in_cod2_to_-infty
     or f is convergent_in_cod2_to_finite
       implies f is convergent_in_cod2 )
proof
   let f be Function of [:NAT,NAT:],ExtREAL;
   hereby assume
A1: f is convergent_in_cod1_to_+infty or f is convergent_in_cod1_to_-infty or
    f is convergent_in_cod1_to_finite;
    per cases by A1;
    suppose A2: f is convergent_in_cod1_to_+infty;
     now let m be Element of NAT;
      ProjMap2(f,m) is convergent_to_+infty by A2;
      hence ProjMap2(f,m) is convergent;
     end;
     hence f is convergent_in_cod1;
    end;
    suppose A3: f is convergent_in_cod1_to_-infty;
     now let m be Element of NAT;
      ProjMap2(f,m) is convergent_to_-infty by A3;
      hence ProjMap2(f,m) is convergent;
     end;
     hence f is convergent_in_cod1;
    end;
    suppose A4: f is convergent_in_cod1_to_finite;
     now let m be Element of NAT;
      ProjMap2(f,m) is convergent_to_finite_number by A4;
      hence ProjMap2(f,m) is convergent;
     end;
     hence f is convergent_in_cod1;
    end;
   end;
   assume
A5: f is convergent_in_cod2_to_+infty or f is convergent_in_cod2_to_-infty or
    f is convergent_in_cod2_to_finite;
   per cases by A5;
   suppose A6: f is convergent_in_cod2_to_+infty;
    now let m be Element of NAT;
     ProjMap1(f,m) is convergent_to_+infty by A6;
     hence ProjMap1(f,m) is convergent;
    end;
    hence f is convergent_in_cod2;
   end;
   suppose A7: f is convergent_in_cod2_to_-infty;
    now let m be Element of NAT;
     ProjMap1(f,m) is convergent_to_-infty by A7;
     hence ProjMap1(f,m) is convergent;
    end;
    hence f is convergent_in_cod2;
   end;
   suppose A8: f is convergent_in_cod2_to_finite;
    now let m be Element of NAT;
     ProjMap1(f,m) is convergent_to_finite_number by A8;
     hence ProjMap1(f,m) is convergent;
    end;
    hence f is convergent_in_cod2;
   end;
end;
