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;
