
theorem Fsubb:
for F being Field, E being FieldExtension of F,
    L be F-monomorphic Field
for f being Monomorphism of F,L
for S being ascending non empty Subset of Ext_Set(f,E)
for p being Element of S
holds p`1 is Subfield of unionField(S,f,E)
proof
let F be Field, E be FieldExtension of F;
let L be F-monomorphic Field;
let f be Monomorphism of F,L;
let S be ascending non empty Subset of Ext_Set(f,E);
let p be Element of S;
   set K = unionField(S,f,E);
H: the carrier of K = unionCarrier(S,f,E) by duf
    .= union(the set of all the carrier of p`1 where p is Element of S);
J: the carrier of p`1 in
      the set of all the carrier of p`1 where p is Element of S; then
A: the carrier of p`1 c= the carrier of K by H,ZFMISC_1:74;
B: the addF of p`1 = (the addF of K) || the carrier of p`1
   proof
   set aF = the addF of p`1, aK = (the addF of K) || the carrier of p`1;
   B1: dom aK
    = dom(the addF of K) /\ [:the carrier of p`1,the carrier of p`1:]
      by RELAT_1:61
   .= [:the carrier of K,the carrier of K:] /\
      [:the carrier of p`1,the carrier of p`1:] by FUNCT_2:def 1
   .= [:the carrier of p`1,the carrier of p`1:] 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 p`1 & b in the carrier of p`1 & x = [a,b]
         by ZFMISC_1:def 2;
     reconsider a,b as Element of p`1 by B3;
     reconsider y = a, z = b as Element of K by A;
     thus aF.x = a + b by B3
              .= y + z by lem4a
              .= aK.x by B3,B2,FUNCT_1:49;
     end;
   hence thesis by B1,FUNCT_2:def 1;
   end;
C: the multF of p`1 = (the multF of K) || the carrier of p`1
   proof
   set mF = the multF of p`1, mK = (the multF of K) || the carrier of p`1;
   B1: dom mK
    = dom(the multF of K) /\ [:the carrier of p`1,the carrier of p`1:]
      by RELAT_1:61
   .= [:the carrier of K,the carrier of K:] /\
      [:the carrier of p`1,the carrier of p`1:] by FUNCT_2:def 1
   .= [:the carrier of p`1,the carrier of p`1:] 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 p`1 & b in the carrier of p`1 & x = [a,b]
         by ZFMISC_1:def 2;
     reconsider a,b as Element of p`1 by B3;
     reconsider y = a, z = b as Element of K by A;
     thus mF.x = a * b by B3
              .= y * z by lem4a
              .= mK.x by B3,B2,FUNCT_1:49;
     end;
   hence thesis by B1,FUNCT_2:def 1;
   end;
1.(p`1) = 1.K & 0.K = 0.(p`1) by lem5a;
hence thesis by H,J,ZFMISC_1:74,B,C,EC_PF_1:def 1;
end;
