let i be natural Number ; :: thesis: for r being Real holds Sum (i |-> r) = i * r
let r be Real; :: thesis: Sum (i |-> r) = i * r
A0: i is Nat by TARSKI:1;
reconsider r = r as Element of REAL by XREAL_0:def 1;
defpred S₁[ Nat] means Sum ($1 |-> r) = $1 * r;
A1: for i being Nat st S₁[i] holds
S₁[i + 1]

proof

let i be Nat; :: thesis: ( S₁[i] implies S₁[i + 1] )
assume A2: Sum (i |-> r) = i * r ; :: thesis: S₁[i + 1]
thus Sum ((i + 1) |-> r) = Sum ((i |-> r) ^ <*r*>) by FINSEQ_2:60
.= (i * r) + (1 * r) by A2, Th74
.= (i + 1) * r ; :: thesis: verum

end;

A3: S₁[ 0 ] by Th72;
for i being Nat holds S₁[i] from NAT_1:sch 2(A3, A1);
hence Sum (i |-> r) = i * r by A0; :: thesis: verum