
theorem inSa:
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)
holds unionField(S,f,E) is Subfield of 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);
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);
A: the carrier of K c= the carrier of E
   proof
   now let o be object;
     assume o in the carrier of K; then
     consider Y being set such that
     A1: o in Y &
         Y in the set of all the carrier of p`1 where p is Element of S
         by H,TARSKI:def 4;
     consider p being Element of S such that
     A2: Y = the carrier of p`1 by A1;
     p in Ext_Set(f,E); then
     consider K be Element of SubFields(E), g be Function of K,L such that
     A3: p = [K,g] &
         ex K1 being FieldExtension of F, g1 being Function of K1,L
         st K1 = K & g1 = g & g1 is monomorphism f-extending;
     p`1 is Subfield of E by A3,subfie; then
     the carrier of p`1 c= the carrier of E by EC_PF_1:def 1;
     hence o in the carrier of E by A1,A2;
     end;
   hence thesis;
   end;
B: the addF of K = (the addF of E) || the carrier of K
   proof
   set aF = the addF of K, aK = (the addF of E) || the carrier of K;
   B1: dom aK
         = dom(the addF of E) /\
           [:the carrier of K,the carrier of K:] by RELAT_1:61
        .= [:the carrier of E,the carrier of E:] /\
           [:the carrier of K,the carrier of K:] by FUNCT_2:def 1
        .= [:the carrier of K,the carrier of K:] 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 K & b in the carrier of K & x = [a,b]
         by ZFMISC_1:def 2;
     reconsider a,b as Element of K by B3;
     consider Y1 being set such that
     C1: a in Y1 &
         Y1 in the set of all the carrier of p`1 where p is Element of S
         by H,TARSKI:def 4;
     consider pa being Element of S such that
     B4: Y1 = the carrier of pa`1 by C1;
     consider Y2 being set such that
     C2: b in Y2 &
         Y2 in the set of all the carrier of p`1 where p is Element of S
         by H,TARSKI:def 4;
     consider pb being Element of S such that
     B5: Y2 = the carrier of pb`1 by C2;
     pa in Ext_Set(f,E); then
     consider U be Element of SubFields(E), g be Function of U,L such that
     A3: pa = [U,g] &
         ex K1 being FieldExtension of F, g1 being Function of K1,L
         st K1 = U & g1 = g & g1 is monomorphism f-extending;
AS1: pa`1 is Subfield of E by A3,subfie;
     pb in Ext_Set(f,E); then
     consider U be Element of SubFields(E), g be Function of U,L such that
     A3: pb = [U,g] &
         ex K1 being FieldExtension of F, g1 being Function of K1,L
         st K1 = U & g1 = g & g1 is monomorphism f-extending;
AS2: pb`1 is Subfield of E by A3,subfie;
     per cases by dasc;
     suppose pa <= pb; then
       reconsider a1 = a, b1 = b as Element of pb`1 by C1,C2,B4,B5,lem1a;
       B4: the addF of pb`1 = (the addF of E) || the carrier of (pb`1)
           by AS2,EC_PF_1:def 1;
       B5: [a1,b1] in [:the carrier of pb`1,the carrier of pb`1:]
           by ZFMISC_1:def 2;
       B6: [a,b] in [:the carrier of K,the carrier of K:] by ZFMISC_1:def 2;
       thus aF.x
          = a + b by B3
         .= a1 + b1 by lem4a
         .= (the addF of E).(a,b) by B4,B5,FUNCT_1:49
         .= aK.x by B3,B6,FUNCT_1:49;
       end;
     suppose pb <= pa; then
       reconsider a1 = a, b1 = b as Element of pa`1 by C1,C2,B4,B5,lem1a;
       B4: the addF of pa`1 = (the addF of E) || the carrier of (pa`1)
           by AS1,EC_PF_1:def 1;
       B5: [a1,b1] in [:the carrier of pa`1,the carrier of pa`1:]
           by ZFMISC_1:def 2;
       B6: [a1,b1] in [:the carrier of K,the carrier of K:]
           by ZFMISC_1:def 2;
       thus aF.x
          = a + b by B3
         .= a1 + b1 by lem4a
         .= (the addF of E).(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 K = (the multF of E) || the carrier of K
   proof
   set aF = the multF of K, aK = (the multF of E) || the carrier of K;
   B1: dom aK
         = dom(the multF of E) /\
           [:the carrier of K,the carrier of K:] by RELAT_1:61
        .= [:the carrier of E,the carrier of E:] /\
           [:the carrier of K,the carrier of K:] by FUNCT_2:def 1
        .= [:the carrier of K,the carrier of K:] 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 K & b in the carrier of K & x = [a,b]
         by ZFMISC_1:def 2;
     reconsider a,b as Element of K by B3;
     consider Y1 being set such that
     C1: a in Y1 &
         Y1 in the set of all the carrier of p`1 where p is Element of S
         by H,TARSKI:def 4;
     consider pa being Element of S such that
     B4: Y1 = the carrier of pa`1 by C1;
     consider Y2 being set such that
     C2: b in Y2 &
         Y2 in the set of all the carrier of p`1 where p is Element of S
         by H,TARSKI:def 4;
     consider pb being Element of S such that
     B5: Y2 = the carrier of pb`1 by C2;
     pa in Ext_Set(f,E); then
     consider U be Element of SubFields(E), g be Function of U,L such that
     A3: pa = [U,g] &
         ex K1 being FieldExtension of F, g1 being Function of K1,L
         st K1 = U & g1 = g & g1 is monomorphism f-extending;
AS1: pa`1 is Subfield of E by A3,subfie;
     pb in Ext_Set(f,E); then
     consider U be Element of SubFields(E), g be Function of U,L such that
     A3: pb = [U,g] &
         ex K1 being FieldExtension of F, g1 being Function of K1,L
         st K1 = U & g1 = g & g1 is monomorphism f-extending;
AS2: pb`1 is Subfield of E by A3,subfie;
     per cases by dasc;
     suppose pa <= pb; then
       reconsider a1 = a, b1 = b as Element of pb`1 by C1,C2,B4,B5,lem1a;
       B4: the multF of pb`1 = (the multF of E) || the carrier of (pb`1)
           by AS2,EC_PF_1:def 1;
       B5: [a1,b1] in [:the carrier of pb`1,the carrier of pb`1:]
           by ZFMISC_1:def 2;
       B6: [a,b] in [:the carrier of K,the carrier of K:]
           by ZFMISC_1:def 2;
       thus aF.x
          = a * b by B3
         .= a1 * b1 by lem4a
         .= (the multF of E).(a,b) by B4,B5,FUNCT_1:49
         .= aK.x by B3,B6,FUNCT_1:49;
       end;
     suppose pb <= pa; then
       reconsider a1 = a, b1 = b as Element of pa`1 by C1,C2,B4,B5,lem1a;
       B4: the multF of pa`1 = (the multF of E) || the carrier of (pa`1)
           by AS1,EC_PF_1:def 1;
       B5: [a1,b1] in [:the carrier of pa`1,the carrier of pa`1:]
           by ZFMISC_1:def 2;
       B6: [a1,b1] in [:the carrier of K,the carrier of K:]
           by ZFMISC_1:def 2;
       thus aF.x
          = a * b by B3
         .= a1 * b1 by lem4a
         .= (the multF of E).(a1,b1) by B4,B5,FUNCT_1:49
         .= aK.x by B3,B6,FUNCT_1:49;
       end;
     end;
   hence thesis by B1;
   end;
set p = the Element of S;
p in Ext_Set(f,E); then
consider U be Element of SubFields(E), g be Function of U,L such that
A3: p = [U,g] &
    ex K1 being FieldExtension of F, g1 being Function of K1,L
    st K1 = U & g1 = g & g1 is monomorphism f-extending;
AS3: p`1 is Subfield of E by A3,subfie;
D: 1.K = 1.(p`1) by lem5a .= 1.E by AS3,EC_PF_1:def 1;
   0.K = 0.(p`1) by lem5a .= 0.E by AS3,EC_PF_1:def 1;
hence thesis by A,B,C,D,EC_PF_1:def 1;
end;
