reserve s for set,
  i,j for natural Number,
  k for Nat,
  x,x1,x2,x3 for Real,
  r,r1,r2,r3,r4 for Real,
  F,F1,F2,F3 for real-valued FinSequence,
  R,R1,R2 for Element of i-tuples_on REAL;

theorem Th80:
  Sum(i |-> r) = i*r
proof
A0: i is Nat by TARSKI:1;
  reconsider r as Element of REAL by XREAL_0:def 1;
  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;
    thus Sum((i+1) |-> r) = Sum((i |-> r)^<*r*>) by FINSEQ_2:60
      .= i*r + 1*r by A2,Th74
      .= (i+1)*r;
  end;
A3: P[0] by Th72;
  for i be Nat holds P[i] from NAT_1:sch 2(A3,A1);
  hence thesis by A0;
end;
