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;
reserve z,z1,z2 for Element of COMPLEX;
reserve n for Nat,
  x, y, a for Real,
  p, p1, p2, p3, q, q1, q2 for Element of n-tuples_on REAL;

theorem Th118:
  for x,y,z being real-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 real-valued FinSequence;
A1:x is FinSequence of REAL & y is FinSequence of REAL &
  z is FinSequence of REAL 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 REAL 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;
