
theorem up:
for F being Field, E being FieldExtension of F,
    L being F-monomorphic Field,
    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 <= upper_bound S
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 q = upper_bound S, K = unionField(S,f,E), h = unionExt(S,f,E);
C: p`1 is Subfield of K by Fsubb;
now let K1 be FieldExtension of p`1, g1 be Function of K1,L;
  assume K1 = q`1 & g1 = q`2; then
  g1|(the carrier of p`1) = p`2 by dufe;
  hence g1 is (p`2)-extending by FUNCT_1:49;
  end;
hence thesis by C,FIELD_4:7;
end;
