
theorem
for E being Field
for f being ascending Field-yielding sequence
st for i being Element of NAT holds (f.i) is Subfield of E
holds (SeqField f) is Subfield of E
proof
let K be Field, f be ascending Field-yielding sequence;
assume AS: for i being Element of NAT holds (f.i) is Subfield of K;
then X: f.0 is Subfield of K;
set F = SeqField f;
H: the carrier of F = Carrier f by dsf
    .= union(the set of all the carrier of f.i where i is Element of NAT);
A: the carrier of F c= the carrier of K
   proof
   now let o be object;
     assume o in the carrier of F; then
     consider Y being set such that
     A1: o in Y &
         Y in the set of all the carrier of f.i where i is Element of NAT
         by H,TARSKI:def 4;
     consider i being Element of NAT such that
     A2: Y = the carrier of f.i by A1;
     f.i is Subfield of K by AS; then
     the carrier of f.i c= the carrier of K by EC_PF_1:def 1;
     hence o in the carrier of K by A1,A2;
     end;
   hence thesis;
   end;
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,XBOOLE_1:28,ZFMISC_1:96
        .= dom aF by FUNCT_2:def 1;
   now let x be object;
     assume 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;
     consider Y1 being set such that
     C1: a in Y1 &
         Y1 in the set of all the carrier of f.i where i is Element of NAT
         by H,TARSKI:def 4;
     consider j1 being Element of NAT such that
     B4: Y1 = the carrier of f.j1 by C1;
     consider Y2 being set such that
     C2: b in Y2 &
         Y2 in the set of all the carrier of f.i where i is Element of NAT
         by H,TARSKI:def 4;
     consider j2 being Element of NAT such that
     B5: Y2 = the carrier of f.j2 by C2;
     per cases;
     suppose j1 <= j2; then
       reconsider a1 = a, b1 = b as Element of f.j2 by C1,C2,B4,B5,lem1;
       f.j2 is Subfield of K by AS; then
       B4: the addF of (f.j2) = (the addF of K) || the carrier of (f.j2)
           by EC_PF_1:def 1;
       B5: [a1,b1] in [:the carrier of (f.j2),the carrier of (f.j2):]
           by ZFMISC_1:def 2;
       B6: [a1,b1] in [:the carrier of F,the carrier of F:]
           by ZFMISC_1:def 2;
       thus aF.x
          = a + b by B3
         .= a1 + b1 by lem4
         .= (the addF of K).(a1,b1) by B4,B5,FUNCT_1:49
         .= aK.x by B3,B6,FUNCT_1:49;
       end;
     suppose j2 <= j1;then
       reconsider a1 = a, b1 = b as Element of f.j1 by C1,C2,B4,B5,lem1;
       f.j1 is Subfield of K by AS; then
       B4: the addF of (f.j1) = (the addF of K) || the carrier of (f.j1)
           by EC_PF_1:def 1;
       B5: [a1,b1] in [:the carrier of (f.j1),the carrier of (f.j1):]
           by ZFMISC_1:def 2;
       B6: [a1,b1] in [:the carrier of F,the carrier of F:]
           by ZFMISC_1:def 2;
       thus aF.x
          = a + b by B3
         .= a1 + b1 by lem4
         .= (the addF of K).(a1,b1) by B4,B5,FUNCT_1:49
         .= aK.x by B3,B6,FUNCT_1:49;
       end;
     end;
   hence thesis by B1;
   end;
C: the multF of F = (the multF of K) || the carrier of F
   proof
   set aF = the multF of F, aK = (the multF of K) || the carrier of F;
   B1: dom aK
         = 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,XBOOLE_1:28,ZFMISC_1:96
        .= dom aF by FUNCT_2:def 1;
   now let x be object;
     assume 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;
     consider Y1 being set such that
     C1: a in Y1 &
         Y1 in the set of all the carrier of f.i where i is Element of NAT
         by H,TARSKI:def 4;
     consider j1 being Element of NAT such that
     B4: Y1 = the carrier of f.j1 by C1;
     consider Y2 being set such that
     C2: b in Y2 &
         Y2 in the set of all the carrier of f.i where i is Element of NAT
         by H,TARSKI:def 4;
     consider j2 being Element of NAT such that
     B5: Y2 = the carrier of f.j2 by C2;
     per cases;
     suppose j1 <= j2; then
       reconsider a1 = a, b1 = b as Element of f.j2 by C1,C2,B4,B5,lem1;
       f.j2 is Subfield of K by AS; then
       B4: the multF of (f.j2) = (the multF of K) || the carrier of (f.j2)
           by EC_PF_1:def 1;
       B5: [a1,b1] in [:the carrier of (f.j2),the carrier of (f.j2):]
           by ZFMISC_1:def 2;
       B6: [a1,b1] in [:the carrier of F,the carrier of F:] by ZFMISC_1:def 2;
       thus aF.x
          = a * b by B3
         .= a1 * b1 by lem4
         .= (the multF of K).(a1,b1) by B4,B5,FUNCT_1:49
         .= aK.x by B3,B6,FUNCT_1:49;
       end;
     suppose j2 <= j1;then
       reconsider a1 = a, b1 = b as Element of f.j1 by C1,C2,B4,B5,lem1;
       f.j1 is Subfield of K by AS; then
       B4: the multF of (f.j1) = (the multF of K) || the carrier of (f.j1)
           by EC_PF_1:def 1;
       B5: [a1,b1] in [:the carrier of (f.j1),the carrier of (f.j1):]
           by ZFMISC_1:def 2;
       B6: [a1,b1] in [:the carrier of F,the carrier of F:] by ZFMISC_1:def 2;
       thus aF.x
          = a * b by B3
         .= a1 * b1 by lem4
         .= (the multF of K).(a1,b1) by B4,B5,FUNCT_1:49
         .= aK.x by B3,B6,FUNCT_1:49;
       end;
     end;
   hence thesis by B1;
   end;
D: 1.F = 1.(f.0) by dsf .= 1.K by X,EC_PF_1:def 1;
   0.F = 0.(f.0) by dsf .= 0.K by X,EC_PF_1:def 1;
hence thesis by A,B,C,D,EC_PF_1:def 1;
end;
