theorem Th57:
  for i be Element of NAT holds 1.(K,<*i*>) = <*1.(K,i)*>
proof
  let i be Element of NAT;
A1: dom (1.(K,<*i*>)) = dom <*i*> by Def8
    .= Seg 1 by FINSEQ_1:38;
  then 1 in dom (1.(K,<*i*>));
  then
A2: (1.(K,<*i*>)).1 = 1.(K,<*i*>.1) by Def8
    .= 1.(K,i);
  len (1.(K,<*i*>))=1 by A1,FINSEQ_1:def 3;
  hence thesis by A2,FINSEQ_1:40;
end;
