
theorem lem1:
for f being ascending Field-yielding sequence
for i,j being Element of NAT
for a being Element of f.i holds i <= j implies a in the carrier of f.j
proof
let f be ascending Field-yielding sequence, i,j be Element of NAT;
let a be Element of f.i;
assume AS: i <= j;
defpred P[Nat] means
  ex k being Element of NAT st k = i + $1 & a in the carrier of f.k;
IA: P[0];
IS: now let k be Nat;
    assume P[k]; then
    consider n being Element of NAT such that
    IV: n = i + k & a in the carrier of f.n;
    f.(n+1) is FieldExtension of f.n by defasc; then
    f.n is Subring of f.(n+1) by FIELD_4:def 1; then
    the carrier of f.n c= the carrier of f.(n+1) by C0SP1:def 3;
    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 AS,NAT_1:10;
P[n] by I;
hence thesis by H;
end;
