
theorem Th23:
  for K1 be Subfield of F_Rat holds NAT c= the carrier of K1
  proof
    let K1 be Subfield of F_Rat;
    set C1 = the carrier of K1;
    defpred P[Nat] means $1 in C1;
    0.K = 0;
    then 0.K1 = 0 by EC_PF_1:def 1;
    then A1: P[0];
    A2: now let n be Nat;
    assume A3: P[n];
    A4: 1.K1 = 1.K by EC_PF_1:def 1
    .= 1;
    A5: the addF of K1 = (the addF of K) || C1 by EC_PF_1:def 1;
    (the addF of K1).(1,n) = (addrat).(1,n) by A3,A4,A5,FUNCT_1:49,ZFMISC_1:87
    .= 1+n by BINOP_2:def 15;
    hence P[n+1] by A3,A4,BINOP_1:17;
  end;
  for n being Nat holds P[n] from NAT_1:sch 2(A1,A2);
  hence thesis;
end;
