
theorem lem3:
for f being ascending Field-yielding sequence
for i,j being Element of NAT st i <= j holds f.j is FieldExtension of f.i
proof
let f be ascending Field-yielding sequence, i,j be Element of NAT;
assume AS: i <= j;
defpred P[Nat] means
  ex k being Element of NAT st k = i + $1 & f.k is FieldExtension of f.i;
IA: P[0]
    proof
    take k = i;
    thus thesis by FIELD_4:6;
    end;
IS: now let k be Nat;
    assume P[k]; then
    consider n being Element of NAT such that
    IV: n = i + k & f.n is FieldExtension of f.i;
    f.(n+1) is FieldExtension of f.n by defasc;
    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;
