
theorem inSb:
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 unionExt(S,f,E) is monomorphism f-extending
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 = unionExt(S,f,E);
set p = the Element of S;
H: the carrier of K = unionCarrier(S,f,E) by duf;
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;
consider K1 being FieldExtension of F, g1 being Function of K1,L such that
A4: K1 = U & g1 = g & g1 is monomorphism f-extending by A3;
AS3: p`1 = U by A3;
AS4: p`2 = g by A3;
D: 1.K = 1.K1 by A4,AS3,lem5a;
F: h|(the carrier of K1) = g by A4,AS4,dufe;
E: g is additive multiplicative unity-preserving by A4;
A: now let a,b be Element of K;
   consider p being Element of S, x,y being Element of p`1 such that
   A1: x = a & y = b & unionAdd(S,f,E).(a,b) = x + y by H,dua;
   A2: h|(the carrier of p`1) = p`2 by dufe; then
   A3: h.a = (p`2).x & h.b = (p`2).y by A1,FUNCT_1:49;
   p in Ext_Set(f,E); then
   consider U be Element of SubFields(E), g2 be Function of U,L such that
   A7: p = [U,g2] &
       ex K1 being FieldExtension of F, g1 being Function of K1,L
       st K1 = U & g1 = g2 & g1 is monomorphism f-extending;
   A4: p`2 is additive by A7;
   A5: x + y = a + b by A1,duf;
   thus h.a + h.b
      = (p`2).(x + y) by A4,A3
     .= h.(a + b) by A2,A5,FUNCT_1:49;
   end;
B: now let a,b be Element of K;
   consider p being Element of S, x,y being Element of p`1 such that
   A1: x = a & y = b & unionMult(S,f,E).(a,b) = x * y by H,dum;
   A2: h|(the carrier of p`1) = p`2 by dufe; then
   A3: h.a = (p`2).x & h.b = (p`2).y by A1,FUNCT_1:49;
   p in Ext_Set(f,E); then
   consider U be Element of SubFields(E), g2 be Function of U,L such that
   A7: p = [U,g2] &
       ex K1 being FieldExtension of F, g1 being Function of K1,L
       st K1 = U & g1 = g2 & g1 is monomorphism f-extending;
   A4: p`2 is multiplicative by A7;
   A5: x * y = a * b by A1,duf;
   thus h.a * h.b
      = (p`2).(x * y) by A4,A3
     .= h.(a * b) by A2,A5,FUNCT_1:49;
   end;
h is additive multiplicative unity-preserving by A,B,E,A4,D,F,FUNCT_1:49;
hence h is monomorphism;
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;
consider K1 being FieldExtension of F, g1 being Function of K1,L such that
A4: K1 = U & g1 = g & g1 is monomorphism f-extending by A3;
reconsider L1 = L as L-extending FieldExtension of L by FIELD_4:6;
p`1 is Subfield of K by Fsubb; then
reconsider K as K1-extending FieldExtension of F by A3,A4,FIELD_4:7;
reconsider h as Function of K,L1;
A7: p`2 = g1 by A3,A4;
now let a be Element of K1;
  h|(the carrier of K1) = g1 by A7,dufe;
  hence g1.a = h.a by FUNCT_1:49;
  end;
then h is g1-extending;
hence thesis by A4,FIELD_8:41;
end;
