
theorem lem2:
for f being ascending Field-yielding sequence
for i,j being Element of NAT
for xi,yi being Element of f.i
for xj,yj being Element of f.j
st xi = xj & yi = yj holds xi + yi = xj + yj & xi * yi = xj * yj
proof
let f be ascending Field-yielding sequence, i,j be Element of NAT;
let xi,yi be Element of f.i, xj,yj be Element of f.j;
assume AS2: xi = xj & yi = yj;
per cases;
suppose AS1: i <= j;
defpred P[Nat] means
  ex k being Element of NAT st k = i + $1
  & for xi,yi being Element of f.i
    for xj,yj being Element of f.k
    st xi = xj & yi = yj holds xi + yi = xj + yj & xi * yi = xj * yj;
IA: P[0]
    proof
    take k = i;
    thus k = i + 0;
    thus thesis;
    end;
IS: now let k be Nat;
    assume P[k]; then
    consider n being Element of NAT such that
    IV: n = i + k &
        for xi,yi being Element of f.i
        for xj,yj being Element of f.n
        st xi = xj & yi = yj holds xi + yi = xj + yj & xi * yi = xj * yj;
    now let xi,yi being Element of f.i;
        let xj,yj being Element of f.(n+1);
      assume B: xi = xj & yi = yj;
      i <= i + k & i + k <= n + 1 by IV,NAT_1:11; then
      f.(n+1) is FieldExtension of f.i by lem3,XXREAL_0:2; then
      f.i is Subfield of f.(n+1) by FIELD_4:7; then
      C: f.i is Subring of f.(n+1) by FIELD_5:12;
      hence xi + yi = xj + yj by B,FIELD_6:15;
      thus xi * yi = xj * yj by B,C,FIELD_6:16;
      end;
    hence P[k+1] by IV;
    end;
I: for k being Nat holds P[k] from NAT_1:sch 2(IA,IS);
consider n being Nat such that H: i + n = j by AS1,NAT_1:10;
P[n] by I;
hence thesis by H,AS2;
end;
suppose AS1: j <= i;
defpred P[Nat] means
  ex k being Element of NAT st k = j + $1
  & for xi,yi being Element of f.j
    for xj,yj being Element of f.k
    st xi = xj & yi = yj holds xi + yi = xj + yj & xi * yi = xj * yj;
IA: P[0]
    proof
    take k = j;
    thus k = j + 0;
    thus thesis;
    end;
IS: now let k be Nat;
    assume P[k]; then
    consider n being Element of NAT such that
    IV: n = j + k &
        for xi,yi being Element of f.j
        for xj,yj being Element of f.n
        st xi = xj & yi = yj holds xi + yi = xj + yj & xi * yi = xj * yj;
    now let xi,yi being Element of f.j;
        let xj,yj being Element of f.(n+1);
      assume B: xi = xj & yi = yj;
      j <= j + k & j + k <= n + 1 by IV,NAT_1:11; then
      f.(n+1) is FieldExtension of f.j by lem3,XXREAL_0:2; then
      f.j is Subfield of f.(n+1) by FIELD_4:7; then
      C: f.j is Subring of f.(n+1) by FIELD_5:12;
      hence xi + yi = xj + yj by B,FIELD_6:15;
      thus xi * yi = xj * yj by B,C,FIELD_6:16;
      end;
    hence P[k+1] by IV;
    end;
I: for k being Nat holds P[k] from NAT_1:sch 2(IA,IS);
consider n being Nat such that H: j + n = i by AS1,NAT_1:10;
P[n] by I;
hence thesis by H,AS2;
end;
end;
