
theorem RSF:
  for f be complex-valued XFinSequence holds
   Re (Sequel f) = Sequel (Re f)
   proof
     let f be complex-valued XFinSequence;
     dom Sequel f = NAT by COMSEQ_1:1; then
     A2: dom Re Sequel f = NAT by COMSEQ_3:def 3;
     for x be object st x in dom Re Sequel f holds
       (Re(Sequel f)).x = (Sequel(Re f)).x
     proof
       let x be object; assume
       B1: x in dom Re Sequel f; then
       reconsider x as Nat;
       B2: (Re(Sequel f)).x = Re ((Sequel f).x) by B1,COMSEQ_3:def 3
       .= Re (f.x) by SFX;
       B3: (Re f).x = (Sequel (Re f)).x by SFX;
       per cases;
       suppose
         x in dom Re f;
         hence thesis by B2,B3,COMSEQ_3:def 3;
       end;
       suppose
         not x in dom Re f;
         then C1: (Re f).x = 0 & not x in dom f
           by COMSEQ_3:def 3,FUNCT_1:def 2; then
         f.x = 0 by FUNCT_1:def 2;
         hence thesis by B2,SFX,C1;
       end;
     end;
     hence thesis by A2,COMSEQ_1:1;
   end;
