
theorem ISF:
  for f be complex-valued XFinSequence holds
   Im (Sequel f) = Sequel (Im f)
   proof
     let f be complex-valued XFinSequence;
     dom Sequel f = NAT by COMSEQ_1:1; then
     A2: dom Im (Sequel f) = NAT by COMSEQ_3:def 4;
     for x be object st x in dom Im Sequel f holds
       (Im(Sequel f)).x = (Sequel Im f).x
     proof
       let x be object; assume
       B1: x in dom Im Sequel f; then
       reconsider x as Nat;
       B2: (Im(Sequel f)).x = Im ((Sequel f).x) by B1,COMSEQ_3:def 4
       .= Im (f.x) by SFX;
       B3: (Im f).x = (Sequel Im f).x by SFX;
       per cases;
       suppose
         x in dom Im f;
         hence thesis by B2,B3,COMSEQ_3:def 4;
       end;
       suppose
         not x in dom Im f;
         then C1: (Im f).x = 0 & not x in dom f
           by COMSEQ_3:def 4,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;
