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;
