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
  Product (r*R) = Product (i|->r) * Product R
proof
  reconsider s=r as Element of REAL by XREAL_0:def 1;
  thus Product (r*R) = Product mlt(i|->s,R) by Th63
    .= Product (i|->r) * Product R by Th107;
end;
