
theorem lem4a:
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 a,b being Element of unionField(S,f,E)
for p being Element of S, x,y being Element of p`1
st x = a & y = b holds a + b = x + y & a * b = x * y
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 a,b be Element of unionField(S,f,E);
let p be Element of S, x,y being Element of p`1;
assume AS: x = a & y = b;
H: the carrier of unionField(S,f,E) = unionCarrier(S,f,E) by duf; then
consider q being Element of S, x1,y1 being Element of q`1 such that
A: x1 = a & y1 = b & unionAdd(S,f,E).(a,b) = x1 + y1 by dua;
now per cases by dasc;
  suppose p <= q;
    then p`1 is Subfield of q`1 by FIELD_4:7; then
    B: the addF of p`1 = (the addF of q`1) || (the carrier of p`1)
       by EC_PF_1:def 1;
    C: [x1,y1] in [:the carrier of p`1,the carrier of p`1:]
       by AS,A,ZFMISC_1:def 2;
    thus a + b = x1 + y1 by A,duf .= x + y by AS,A,B,C,FUNCT_1:49;
    end;
  suppose q <= p;
    then q`1 is Subfield of p`1 by FIELD_4:7; then
    B: the addF of q`1 = (the addF of p`1) || (the carrier of q`1)
       by EC_PF_1:def 1;
    C: [x1,y1] in [:the carrier of q`1,the carrier of q`1:]
       by ZFMISC_1:def 2;
    thus a + b = x1 + y1 by A,duf .= x + y by AS,A,B,C,FUNCT_1:49;
    end;
  end;
hence a + b = x + y;
consider q being Element of S, x1,y1 being Element of q`1 such that
A: x1 = a & y1 = b & unionMult(S,f,E).(a,b) = x1 * y1 by H,dum;
now per cases by dasc;
  suppose p <= q;
    then p`1 is Subfield of q`1 by FIELD_4:7; then
    B: the multF of p`1 = (the multF of q`1) || (the carrier of p`1)
       by EC_PF_1:def 1;
    C: [x1,y1] in [:the carrier of p`1,the carrier of p`1:]
       by AS,A,ZFMISC_1:def 2;
    thus a * b = x1 * y1 by A,duf .= x * y by AS,A,B,C,FUNCT_1:49;
    end;
  suppose q <= p;
    then q`1 is Subfield of p`1 by FIELD_4:7; then
    B: the multF of q`1 = (the multF of p`1) || (the carrier of q`1)
       by EC_PF_1:def 1;
    C: [x1,y1] in [:the carrier of q`1,the carrier of q`1:]
       by ZFMISC_1:def 2;
    thus a * b = x1 * y1 by A,duf .= x * y by AS,A,B,C,FUNCT_1:49;
    end;
  end;
hence a * b = x * y;
end;
