reserve i, j, m, n, k for Nat,
  x, y for set,
  K for Field,
  a,a1 for Element of K;
reserve V1,V2,V3 for finite-dimensional VectSp of K,
  f for Function of V1,V2,

  b1,b19 for OrdBasis of V1,
  B1 for FinSequence of V1,
  b2 for OrdBasis of V2,
  B2 for FinSequence of V2,

  B3 for FinSequence of V3,
  v1,w1 for Element of V1,
  R,R1,R2 for FinSequence of V1,
  p,p1,p2 for FinSequence of K;

theorem
  Sum lmlt(p,len p|->v1) = (Sum p) * v1
proof
  set L=len p|->v1;
  set M=lmlt(p,L);
  len L = len p by CARD_1:def 7;
  then dom L=dom p by FINSEQ_3:29;
  then
A1: dom M=dom p by MATRLIN:12;
A2: now
    let k, a1 such that
A3: k in dom M and
A4: a1=p.k;
    k in Seg len p by A1,A3,FINSEQ_1:def 3;
    then L.k=v1 by FINSEQ_2:57;
    hence M.k = a1*v1 by A3,A4,FUNCOP_1:22;
  end;
  len p=len M by A1,FINSEQ_3:29;
  hence thesis by A2,MATRLIN:9;
end;
