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

theorem Th31:
  for x,y,z being FinSequence of COMPLEX st len x=len y & len y=
  len z holds mlt(x,y-z) = mlt(x,y)-mlt(x,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,y2)-mlt(x2,z2)) by CARD_1:def 7
    .=dom (mlt(x2,y2)-mlt(x2,z2)) by FINSEQ_1:def 3;
A4: dom mlt(x,y)=Seg len(mlt(x2,y2)) by FINSEQ_1:def 3
    .=Seg len x by CARD_1:def 7
    .=Seg len (mlt(x2,y2)-mlt(x2,z2)) by CARD_1:def 7
    .=dom (mlt(x2,y2)-mlt(x2,z2)) by FINSEQ_1:def 3;
A5: 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,y2)-mlt(x2,z2)) by CARD_1:def 7
    .= dom (mlt(x2,y2)-mlt(x2,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,y)-
  mlt(x,z)).i
  proof
    let i be Nat;
    assume
A6: i in dom mlt(x,y-z);
    set a=y.i, b=z.i, d=(y-z).i, e=x.i;
    len (y2-z2)=len x by CARD_1:def 7;
    then dom (y2-z2)=Seg len x by FINSEQ_1:def 3
      .=Seg len mlt(x2,y2) by CARD_1:def 7
      .=dom mlt(x,y) by FINSEQ_1:def 3;
    then
A7: d=a-b by A3,A4,A6,COMPLSP2:2;
    thus mlt(x,y-z).i=e*d by A6,Th17
      .=e*a-e*b by A7
      .=mlt(x,y).i -e*b by A3,A4,A6,Th17
      .=mlt(x,y).i -mlt(x,z).i by A3,A5,A6,Th17
      .=(mlt(x,y)-mlt(x,z)).i by A3,A6,COMPLSP2:2;
  end;
  hence thesis by A3,FINSEQ_1:13;
end;
