
theorem Fsub:
for f being ascending Field-yielding sequence
for i being Element of NAT holds (f.i) is Subfield of (SeqField f)
proof
let f be ascending Field-yielding sequence, i be Element of NAT;
set F = f.i, K = SeqField f;
H: the carrier of K = Carrier f by dsf
    .= union(the set of all the carrier of f.i where i is Element of NAT);
J: the carrier of F in
      the set of all the carrier of f.i where i is Element of NAT; then
A: the carrier of F c= the carrier of K by H,ZFMISC_1:74;
B: the addF of F = (the addF of K) || the carrier of F
   proof
   set aF = the addF of F, aK = (the addF of K) || the carrier of F;
   B1: dom aK
    = dom(the addF of K) /\ [:the carrier of F,the carrier of F:] by RELAT_1:61
   .= [:the carrier of K,the carrier of K:] /\
      [:the carrier of F,the carrier of F:] by FUNCT_2:def 1
   .= [:the carrier of F,the carrier of F:] by A,ZFMISC_1:96,XBOOLE_1:28;
   now let x be object;
     assume B2: x in dom aF; then
     consider a,b being object such that
     B3: a in the carrier of F & b in the carrier of F & x = [a,b]
         by ZFMISC_1:def 2;
     reconsider a,b as Element of F by B3;
     reconsider y = a, z = b as Element of K by A;
     thus aF.x = a + b by B3
              .= y + z by lem4
              .= aK.x by B3,B2,FUNCT_1:49;
     end;
   hence thesis by B1,FUNCT_2:def 1;
   end;
C: the multF of F = (the multF of K) || the carrier of F
   proof
   set mF = the multF of F, mK = (the multF of K) || the carrier of F;
   B1: dom mK
    = dom(the multF of K) /\ [:the carrier of F,the carrier of F:]
      by RELAT_1:61
   .= [:the carrier of K,the carrier of K:] /\
      [:the carrier of F,the carrier of F:] by FUNCT_2:def 1
   .= [:the carrier of F,the carrier of F:] by A,ZFMISC_1:96,XBOOLE_1:28;
   now let x be object;
     assume B2: x in dom mF; then
     consider a,b being object such that
     B3: a in the carrier of F & b in the carrier of F & x = [a,b]
         by ZFMISC_1:def 2;
     reconsider a,b as Element of F by B3;
     reconsider y = a, z = b as Element of K by A;
     thus mF.x = a * b by B3
              .= y * z by lem4
              .= mK.x by B3,B2,FUNCT_1:49;
     end;
   hence thesis by B1,FUNCT_2:def 1;
   end;
   1.F = 1.K & 0.F = 0.K by lem5;
hence thesis by H,J,ZFMISC_1:74,B,C,EC_PF_1:def 1;
end;
