
theorem LM03:
  for R1,R2 be FinSequence of REAL,
      n,i be Nat,
      r be Real st i in dom R1
   & R1 = n |-> (1 qua Real)
   & R2 = R1 +* (i,r)
  holds Product R2 = r
  proof
    let R1,R2 be FinSequence of REAL,
        n,i be Nat,
        r be Real;
    assume that
    A1: i in dom R1 and
    A2: R1 = n |-> (1 qua Real) and
    A3: R2 = R1 +* (i,r);
    i in Seg n by A1,A2,FUNCT_2:def 1; then
    A4: R1.i = 1 by A2,FUNCOP_1:7;
    A5: Product R1 = 1 by A2,RVSUM_1:102;
    thus Product R2 = ((Product R1) * r) / (R1 . i) by A1,A2,A3,RVSUM_3:25
    .= r by A4,A5;
  end;
