
theorem
for F be ExtREAL_sequence, n be Nat, a be R_eal st
 (for k be Nat holds F.k = a) holds (Partial_Sums F).n = a*(n+1)
proof
   let F be ExtREAL_sequence, n be Nat, a be R_eal;
   assume A1: for k be Nat holds F.k = a;
   defpred P[Nat] means (Partial_Sums F).$1 = a*($1 + 1);
   (Partial_Sums F).0 = F.0 by MESFUNC9:def 1; then
   (Partial_Sums F).0 = a by A1; then
A2:P[0] by XXREAL_3:81;
A3:for i be Nat st P[i] holds P[i+1]
   proof
    let i be Nat;
    assume A4: P[i];
    i+1 in REAL & 1 in REAL by XREAL_0:def 1; then
    reconsider i1= i+1, One=1 as R_eal by XBOOLE_0:def 3,XXREAL_0:def 4;
    (Partial_Sums F).(i+1) = (Partial_Sums F).i + F.(i+1)
       by MESFUNC9:def 1; then
    (Partial_Sums F).(i+1) = a*(i+1) + a by A1,A4; then
    (Partial_Sums F).(i+1) = a*(i+1) + a*1 by XXREAL_3:81; then
    (Partial_Sums F).(i+1) = a*((i1)+One) by XXREAL_3:96;
    hence P[i+1] by XXREAL_3:def 2;
   end;
   for i be Nat holds P[i] from NAT_1:sch 2(A2,A3);
   hence thesis;
end;
