reserve a, b for Real,
  r for Real,
  rr for Real,
  i, j, n for Nat,
  M for non empty MetrSpace,
  p, q, s for Point of TOP-REAL 2,
  e for Point of Euclid 2,
  w for Point of Euclid n,
  z for Point of M,
  A, B for Subset of TOP-REAL n,
  P for Subset of TOP-REAL 2,
  D for non empty Subset of TOP-REAL 2;

theorem Th8:
  rr <> 0 & j <= i implies Product((i-'j) |-> rr) = Product(i|->rr) /
  Product(j|->rr)
proof
  assume that
A1: rr <> 0 and
A2: j <= i;
  defpred P[Nat] means j <= $1
   implies Product(($1-'j) |-> rr) =
  Product($1|->rr) / Product(j|->rr);
A3: Product(j|->rr) <> 0 by A1,Th7;
A4: for n being Nat st P[n] holds P[n+1]
  proof
    let n be Nat;
    assume that
A5: P[n] and
A6: j <= n+1;
    per cases by A6,NAT_1:8;
    suppose
A7:   j <= n;
      Product((n-'j+1) |-> rr) = Product((n-'j) |-> rr)*Product(1|->rr)
      by RVSUM_1:104
        .= Product(n|->rr) / Product(j|->rr) * rr by A5,A7,Th4
        .= Product(n|->rr) * rr / Product(j|->rr) by XCMPLX_1:74
        .= Product((n+1) |-> rr) / Product(j|->rr) by Th6;
      hence thesis by A7,NAT_D:38;
    end;
    suppose
A8:   j = n+1;
      hence Product(((n+1)-'j) |-> rr) = Product(0|->rr) by XREAL_1:232
        .= 1 by RVSUM_1:94
        .= Product((n+1) |-> rr) / Product(j|->rr) by A3,A8,XCMPLX_1:60;
    end;
  end;
A9: P[0]
  proof
    assume
A10: j <= 0;
    then j = 0;
    hence Product((0-'j) |-> rr) = Product(0|->rr) / Product<*>REAL by NAT_D:40
,RVSUM_1:94
      .= Product(0|->rr) / Product(j|->rr) by A10;
  end;
  for n being Nat holds P[n] from NAT_1:sch 2(A9,A4);
  hence thesis by A2;
end;
