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 (i|->(r1*r2)) = (Product (i|->r1))*(Product (i|->r2))
proof
  reconsider i as Nat by TARSKI:1;
  reconsider s1=r1, s2=r2 as Element of REAL by XREAL_0:def 1;
  reconsider ss=s1*s2 as Element of REAL;
  Product (i|->(ss)) = multreal$$(i|->multreal.(s1,s2)) by BINOP_2:def 11
    .= multreal.(multreal$$(i|->s1),multreal$$(i|->s2)) by SETWOP_2:36
    .= (Product (i|->s1))*(Product (i|->s2)) by BINOP_2:def 11;
  hence thesis;
end;
