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
  (r1 + r2)*F = r1*F + r2*F
proof
A1: dom ((r1 + r2)*F) = dom F by VALUED_1:def 5;
A2: dom (r1*F + r2*F) = dom (r1*F) /\ dom (r2*F) by VALUED_1:def 1;
A3: dom (r1*F) = dom F by VALUED_1:def 5;
A4: dom (r2*F) = dom F by VALUED_1:def 5;
    now
      let i be Nat;
      assume
A5:   i in dom ((r1+r2)*F);
      thus ((r1+r2)*F).i = (r1+r2)*(F.i) by VALUED_1:6
      .= r1*(F.i) + r2*(F.i)
      .= r1*(F.i) + (r2*F).i by VALUED_1:6
      .= (r1*F).i + (r2*F).i by VALUED_1:6
      .= (r1*F+r2*F).i by A1,A2,A3,A4,A5,VALUED_1:def 1;
    end;
    hence thesis by A1,A2,A3,A4,FINSEQ_1:13;
end;
