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;
