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+j) |->r ) = (Product (i|->r))*(Product (j|->r))
proof
  reconsider i,j as Nat by TARSKI:1;
  reconsider s=r as Element of REAL by XREAL_0:def 1;
  Product ((i+j) |->s) = multreal.(multreal$$(i|->s),multreal$$(j|->s))
  by SETWOP_2:26
    .= (Product (i|->s))*(Product (j|->s)) by BINOP_2:def 11;
  hence thesis;
end;
