reserve s for set,
  i,j for Nat,
  c,c1,c2,c3 for Complex,
  F,F1,F2 for complex-valued FinSequence,
  R,R1,R2 for i-element complex-valued FinSequence;

theorem
  for x,y,z being complex-valued FinSequence st len x = len y & len y = len z
  holds mlt(x+y,z) = mlt(x,z) + mlt(y,z)
proof
  let x,y,z be complex-valued FinSequence;
A1:x is FinSequence of COMPLEX & y is FinSequence of COMPLEX &
  z is FinSequence of COMPLEX by Lm2;
  assume len x=len y & len y=len z; then
  reconsider x2=x, y2=y, z2=z as Element of (len x)-tuples_on COMPLEX
  by A1,FINSEQ_2:92;
A2: dom (mlt(x+y,z))=Seg len(mlt(x2+y2,z2)) by FINSEQ_1:def 3
    .=Seg len x by CARD_1:def 7
    .=Seg len (mlt(x2,z2)+mlt(y2,z2)) by CARD_1:def 7
    .=dom (mlt(x2,z2)+mlt(y2,z2)) by FINSEQ_1:def 3;
A3: dom (mlt(x,z))=Seg len(mlt(x2,z2)) by FINSEQ_1:def 3
    .=Seg len x by CARD_1:def 7
    .=Seg len (mlt(x2,z2)+mlt(y2,z2)) by CARD_1:def 7
    .=dom (mlt(x2,z2)+mlt(y2,z2)) by FINSEQ_1:def 3;
  for i being Nat st i in dom (mlt(x+y,z)) holds mlt(x+y,z).i=(mlt(x,z)+
  mlt(y,z)).i
  proof
    let i be Nat;
    assume A4: i in dom mlt(x+y,z);
    set a=x.i, b=y.i, d=(x+y).i, e=z.i;
    len (x2+y2)=len x by CARD_1:def 7; then
    dom (x2+y2)=Seg len x by FINSEQ_1:def 3
      .=Seg len(mlt(x2,z2)) by CARD_1:def 7
      .=dom (mlt(x,z)) by FINSEQ_1:def 3; then
A5: d=a+b by A2,A3,A4,VALUED_1:def 1;
    thus mlt(x+y,z).i=d*e by VALUED_1:5
      .=a*e+b*e by A5
      .=mlt(x,z).i +b*e by VALUED_1:5
      .=mlt(x,z).i +mlt(y,z).i by VALUED_1:5
      .=(mlt(x,z)+mlt(y,z)).i by A2,A4,VALUED_1:def 1;
  end;
  hence thesis by A2,FINSEQ_1:13;
end;
