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 Th7:
  lmlt(p1 + p2,R) = lmlt(p1,R) + lmlt(p2,R)
proof
  set L12=lmlt(p1+p2,R);
  set L1=lmlt(p1,R);
  set L2=lmlt(p2,R);
A1: dom (L1+L2)=dom L1/\dom L2 by Lm3;
A2: dom L12=dom (p1+p2)/\dom R by Lm1;
A3: dom L1=dom p1/\dom R by Lm1;
A4: dom L2=dom p2/\dom R by Lm1;
  then
A5: dom(L1+L2) = dom p1/\dom R/\dom p2/\dom R by A1,A3,XBOOLE_1:16
    .= dom p1/\dom p2/\dom R/\dom R by XBOOLE_1:16
    .= (dom p1/\dom p2)/\(dom R/\dom R) by XBOOLE_1:16
    .= dom L12 by A2,Lm2;
  now
    let x be object such that
A6: x in dom (L1+L2);
A7: x in dom L2 by A1,A6,XBOOLE_0:def 4;
    then
A8: L2/.x=L2.x by PARTFUN1:def 6;
    x in dom p2 by A4,A7,XBOOLE_0:def 4;
    then
A9: p2/.x=p2.x by PARTFUN1:def 6;
A10: x in dom (p1+p2) by A2,A5,A6,XBOOLE_0:def 4;
    then
A11: (p1+p2).x=(p1+p2)/.x by PARTFUN1:def 6;
A12: x in dom L1 by A1,A6,XBOOLE_0:def 4;
    then x in dom p1 by A3,XBOOLE_0:def 4;
    then
A13: p1/.x=p1.x by PARTFUN1:def 6;
    x in dom R by A3,A12,XBOOLE_0:def 4;
    then
A14: R/.x=R.x by PARTFUN1:def 6;
A15: L1/.x=L1.x by A12,PARTFUN1:def 6;
    hence (L1+L2).x = L1/.x+L2/.x by A6,A8,FVSUM_1:17
      .= (the lmult of V1).(p1/.x,R/.x)+L2/.x by A12,A15,A13,A14,FUNCOP_1:22
      .= ((p1/.x)*(R/.x))+((p2/.x)*(R/.x)) by A7,A8,A9,A14,FUNCOP_1:22
      .= (p1/.x+p2/.x)*R/.x by VECTSP_1:def 15
      .= (p1+p2)/.x*R/.x by A10,A13,A9,A11,FVSUM_1:17
      .= L12.x by A5,A6,A14,A11,FUNCOP_1:22;
  end;
  hence thesis by A5;
end;
