
theorem lem5a:
for F being Field, E being FieldExtension of F,
    L being F-monomorphic Field
for f being Monomorphism of F,L,
    S being ascending non empty Subset of Ext_Set(f,E)
for p being Element of S
holds 1.unionField(S,f,E) = 1.(p`1) & 0.unionField(S,f,E) = 0.(p`1)
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;
1.unionField(S,f,E) = unionOne(S,f,E) by duf; then
consider q being Element of S such that
A: 1.unionField(S,f,E) = 1.(q`1) by defone;
now per cases by dasc;
  suppose p <= q;
    then p`1 is Subfield of q`1 by FIELD_4:7;
    hence 1.unionField(S,f,E) = 1.(p`1) by A,EC_PF_1:def 1;
    end;
  suppose q <= p;
    then q`1 is Subfield of p`1 by FIELD_4:7;
    hence 1.unionField(S,f,E) = 1.(p`1) by A,EC_PF_1:def 1;
    end;
  end;
hence 1.unionField(S,f,E) = 1.(p`1);
0.unionField(S,f,E) = unionZero(S,f,E) by duf; then
consider q being Element of S such that
A: 0.unionField(S,f,E) = 0.(q`1) by defzero;
now per cases by dasc;
  suppose p <= q;
    then p`1 is Subfield of q`1 by FIELD_4:7;
    hence 0.unionField(S,f,E) = 0.(p`1) by A,EC_PF_1:def 1;
    end;
  suppose q <= p;
    then q`1 is Subfield of p`1 by FIELD_4:7;
    hence 0.unionField(S,f,E) = 0.(p`1) by A,EC_PF_1:def 1;
    end;
  end;
hence 0.unionField(S,f,E) = 0.(p`1);
end;
