reserve i,j,n,k for Nat,
  a for Element of COMPLEX,
  R1,R2 for Element of i-tuples_on COMPLEX;

theorem Th33:
  for x,y,z being FinSequence of COMPLEX 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 FinSequence of COMPLEX;
  assume that
A1: len x=len y and
A2: len y=len z;
  reconsider x2=x, y2=y, z2=z as Element of (len x)-tuples_on COMPLEX by A1,A2,
FINSEQ_2:92;
A3: 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;
A4: 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;
A5: dom mlt(y,z)=Seg len mlt(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;
  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
A6: 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
A7: d=a+b by A3,A4,A6,COMPLSP2:1;
    thus mlt(x+y,z).i=d*e by A6,Th17
      .=a*e+b*e by A7
      .=mlt(x,z).i +b*e by A3,A4,A6,Th17
      .=mlt(x,z).i +mlt(y,z).i by A3,A5,A6,Th17
      .=(mlt(x,z)+mlt(y,z)).i by A3,A6,COMPLSP2:1;
  end;
  hence thesis by A3,FINSEQ_1:13;
end;
