theorem
  Sum lmlt(a*p,R) = a * Sum lmlt(p,R)
proof
  set Ma=lmlt(a*p,R);
  set M=lmlt(p,R);
  len (a*p)=len p by MATRIXR1:16;
  then
A1: dom (a*p)=dom p by FINSEQ_3:29;
A2: dom Ma=dom (a*p)/\dom R by Lm1;
A3: dom M=dom p /\dom R by Lm1;
A4: for k be Nat for v1 st k in dom Ma & v1 = M.k holds Ma.k = a * v1
  proof
    let k be Nat;
    let v1 such that
A5: k in dom Ma and
A6: v1=M.k;
    k in dom R by A2,A5,XBOOLE_0:def 4;
    then
A7: R/.k=R.k by PARTFUN1:def 6;
    k in dom p by A1,A2,A5,XBOOLE_0:def 4;
    then
A8: p/.k=p.k by PARTFUN1:def 6;
    k in dom (a*p) by A2,A5,XBOOLE_0:def 4;
    then (a*p).k=a*(p/.k) by A8,FVSUM_1:50;
    hence Ma.k = (a*(p/.k))*R/.k by A5,A7,FUNCOP_1:22
      .= a*((p/.k)*R/.k) by VECTSP_1:def 16
      .= a*v1 by A1,A3,A2,A5,A6,A8,A7,FUNCOP_1:22;
  end;
  len M=len Ma by A1,A3,A2,FINSEQ_3:29;
  hence thesis by A4,RLVECT_2:66;
end;
