 reserve R for domRing;
 reserve p for odd prime Nat, m for positive Nat;

theorem Th3:
  for i be Nat, r be Element of F_Real holds
    Sum(i |-> r) = i*r
   proof
     let i be Nat, r be Element of F_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;
       assume
A2:    Sum(i |-> r) = i*r;
       thus Sum((i+1) |-> r) = Sum((i |-> r)^<*r*>) by FINSEQ_2:60
       .= i*r + r by A2,FVSUM_1:71
       .= i*r + 1*r by BINOM:13
       .= (i+1)*r by BINOM:15;
     end;
     0|-> r = <*> the carrier of F_Real; then
     Sum (0|-> r) = 0.F_Real by RLVECT_1:43 .= 0*r by BINOM:12; then
A3:  P[0];
     for i be Nat holds P[i] from NAT_1:sch 2(A3,A1);
     hence thesis;
   end;
