
theorem Th19:
for r be R_eal, F be FinSequence of ExtREAL holds
  Sum(F^<*r*>) = Sum F + r
proof
   let r be R_eal, F be FinSequence of ExtREAL;
   consider f be Function of NAT,ExtREAL such that
A1: Sum(F^<*r*>) = f.(len (F^<*r*>)) &
    f.0 = 0 &
    for i be Nat st i < len (F^<*r*>) holds
     f.(i+1) = f.i + (F^<*r*>).(i+1) by EXTREAL1:def 2;
   consider g be Function of NAT,ExtREAL such that
A2: Sum F = g.(len F) &
    g.0 = 0 &
    for i be Nat st i < len F holds
     g.(i+1) = g.i + F.(i+1) by EXTREAL1:def 2;
   len (F^<*r*>) = len F + len <*r*> by FINSEQ_1:22; then
B1:len (F^<*r*>) = len F + 1 by FINSEQ_1:39; then
B2:len F < len (F^<*r*>) by NAT_1:13;
   defpred P[Nat] means $1 <= len F implies f.$1 = g.$1;
A3:P[0] by A1,A2;
A4:for k be Nat st P[k] holds P[k+1]
   proof
    let k be Nat;
    assume A5: P[k];
    assume A6: k+1 <= len F; then
A7: k < len F by NAT_1:13;
A9: (F^<*r*>).(k+1) = F.(k+1) by A6,FINSEQ_1:64,NAT_1:11;
    k < len (F^<*r*>) by A7,B1,NAT_1:13; then
    f.(k+1) = f.k + (F^<*r*>).(k+1) by A1;
    hence f.(k+1) = g.(k+1) by A2,A6,A5,A9,NAT_1:13;
   end;
   for i be Nat holds P[i] from NAT_1:sch 2(A3,A4); then
   f.(len F) = g.(len F); then
   f.(len F + 1) = g.(len F) + (F^<*r*>).(len F+1) by A1,B2;
   hence Sum(F^<*r*>) = Sum F + r by A1,A2,B1,FINSEQ_1:42;
end;
