
theorem Th38:
  for f be Function of [:NAT,NAT:],ExtREAL holds
    lim_in_cod1 f = lim_in_cod2 (~f) & lim_in_cod2 f = lim_in_cod1 (~f)
proof
   let f be Function of [:NAT,NAT:],ExtREAL;
   now let n be Element of NAT;
    (lim_in_cod1 f).n = lim ProjMap2(f,n) by D1DEF5
     .= lim ProjMap1(~f,n) by Th33;
    hence (lim_in_cod1 f).n = (lim_in_cod2 (~f)).n by D1DEF6;
   end;
   hence lim_in_cod1 f = lim_in_cod2 (~f) by FUNCT_2:def 8;
   now let n be Element of NAT;
    (lim_in_cod2 f).n = lim ProjMap1(f,n) by D1DEF6
     .= lim ProjMap2(~f,n) by Th32;
    hence (lim_in_cod2 f).n = (lim_in_cod1 (~f)).n by D1DEF5;
   end;
   hence lim_in_cod2 f = lim_in_cod1 (~f) by FUNCT_2:def 8;
end;
