
theorem Th48:
for f,g be sequence of ExtREAL, j,k be Nat st
 k < j & (for n be Nat st n < j holds f.n = g.n)
holds (Ser f).k = (Ser g).k
proof
    let f,g be sequence of ExtREAL, j,k be Nat;
    assume that
A1:  k < j and
A2:  for n be Nat st n < j holds f.n = g.n;

    defpred P[Nat] means $1 <= k implies (Ser f).$1 = (Ser g).$1;

    now assume 0 <= k;
     f.0 = g.0 by A1,A2; then
     (Ser f).0 = g.0 by SUPINF_2:def 11;
     hence (Ser f).0 = (Ser g).0 by SUPINF_2:def 11;
    end; then
A3: P[0];

A4: for m be Nat st P[m] holds P[m+1]
    proof
     let m be Nat;
     assume A5: P[m];
     assume A6: m+1 <= k; then
A7:  m+1 < j by A1,XXREAL_0:2;

     (Ser f).(m+1) = (Ser f).m + f.(m+1) by SUPINF_2:def 11; then
     (Ser f).(m+1) = (Ser g).m + g.(m+1) by A2,A5,A6,A7,NAT_1:13;
     hence (Ser f).(m+1) = (Ser g).(m+1) by SUPINF_2:def 11;
    end;
    for m be Nat holds P[m] from NAT_1:sch 2(A3,A4);
    hence (Ser f).k = (Ser g).k;
end;
