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
  lmlt(p,R1 + R2) = lmlt(p,R1) + lmlt(p,R2)
proof
  set L12=lmlt(p,R1+R2);
  set L1=lmlt(p,R1);
  set L2=lmlt(p,R2);
A1: dom (L1+L2)=dom L1/\dom L2 by Lm3;
A2: dom L12=dom p/\dom (R1+R2) by Lm1;
A3: dom (R1+R2)=dom R1/\dom R2 by Lm3;
A4: dom L1=dom p/\dom R1 by Lm1;
A5: dom L2=dom p/\dom R2 by Lm1;
  then
A6: dom(L1+L2) = dom p/\dom R1/\dom p/\dom R2 by A1,A4,XBOOLE_1:16
    .= dom p/\dom p/\dom R1/\dom R2 by XBOOLE_1:16
    .= dom L12 by A3,A2,XBOOLE_1:16;
  now
    let x be object such that
A7: x in dom (L1+L2);
A8: x in dom L2 by A1,A7,XBOOLE_0:def 4;
    then
A9: L2/.x=L2.x by PARTFUN1:def 6;
    x in dom R2 by A5,A8,XBOOLE_0:def 4;
    then
A10: R2/.x=R2.x by PARTFUN1:def 6;
A11: x in dom (R1+R2) by A2,A6,A7,XBOOLE_0:def 4;
    then
A12: (R1+R2).x=(R1+R2)/.x by PARTFUN1:def 6;
A13: x in dom L1 by A1,A7,XBOOLE_0:def 4;
    then x in dom p by A4,XBOOLE_0:def 4;
    then
A14: p/.x=p.x by PARTFUN1:def 6;
    x in dom R1 by A4,A13,XBOOLE_0:def 4;
    then
A15: R1/.x=R1.x by PARTFUN1:def 6;
A16: L1/.x=L1.x by A13,PARTFUN1:def 6;
    hence (L1+L2).x = L1/.x+L2/.x by A7,A9,FVSUM_1:17
      .= (the lmult of V1).(p/.x,R1/.x)+L2/.x by A13,A16,A14,A15,FUNCOP_1:22
      .= ((p/.x)*(R1/.x))+((p/.x)*(R2/.x)) by A8,A9,A14,A10,FUNCOP_1:22
      .= p/.x*(R1/.x+R2/.x) by VECTSP_1:def 14
      .= p/.x*(R1+R2)/.x by A11,A15,A10,A12,FVSUM_1:17
      .= L12.x by A6,A7,A14,A12,FUNCOP_1:22;
  end;
  hence thesis by A6;
end;
