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(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;
