
theorem Th20:
for r be R_eal, i be Nat st r is real holds Sum(i |-> r) = i*r
proof
   let r be R_eal, i be Nat;
   assume A0: r is real;
   defpred P[Nat] means Sum($1 |->r) = $1*r;
A1:for i be Nat st P[i] holds P[i+1]
   proof
    let i be Nat such that
A2: Sum(i |-> r) = i*r;
    reconsider i1 = i, One = 1 as ext-real number;
    thus Sum((i+1) |-> r) = Sum((i |-> r)^<*r*>) by FINSEQ_2:60
      .= i*r + r by A2,Th19
      .= i*r + 1*r by XXREAL_3:81
      .= (i1+One)*r by A0,XXREAL_3:95
      .= (i+1)*r by XXREAL_3:def 2;
   end;
A3:P[0] by EXTREAL1:7,FINSEQ_2:58;
   for i be Nat holds P[i] from NAT_1:sch 2(A3,A1);
   hence thesis;
end;
