let K be Field; :: thesis: for V1 being finite-dimensional VectSp of K
for R being FinSequence of V1
for p1, p2 being FinSequence of K holds lmlt (p1 + p2),R = (lmlt p1,R) + (lmlt p2,R)
let V1 be finite-dimensional VectSp of K; :: thesis: for R being FinSequence of V1
for p1, p2 being FinSequence of K holds lmlt (p1 + p2),R = (lmlt p1,R) + (lmlt p2,R)
let R be FinSequence of V1; :: thesis: for p1, p2 being FinSequence of K holds lmlt (p1 + p2),R = (lmlt p1,R) + (lmlt p2,R)
let p1, p2 be FinSequence of K; :: thesis: lmlt (p1 + p2),R = (lmlt p1,R) + (lmlt p2,R)
set L12 = lmlt (p1 + p2),R;
set L1 = lmlt p1,R;
set L2 = lmlt p2,R;
A1: dom ((lmlt p1,R) + (lmlt p2,R)) = (dom (lmlt p1,R)) /\ (dom (lmlt p2,R)) by Lm3;
A2: dom (lmlt (p1 + p2),R) = (dom (p1 + p2)) /\ (dom R) by Lm1;
A3: dom (lmlt p1,R) = (dom p1) /\ (dom R) by Lm1;
A4: dom (lmlt p2,R) = (dom p2) /\ (dom R) by Lm1;
then A5: dom ((lmlt p1,R) + (lmlt p2,R)) = (((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 (lmlt (p1 + p2),R) by A2, Lm2 ;

now

let x be set ; :: thesis: ( x in dom ((lmlt p1,R) + (lmlt p2,R)) implies ((lmlt p1,R) + (lmlt p2,R)) . x = (lmlt (p1 + p2),R) . x )
assume A6: x in dom ((lmlt p1,R) + (lmlt p2,R)) ; :: thesis: ((lmlt p1,R) + (lmlt p2,R)) . x = (lmlt (p1 + p2),R) . x
A7: x in dom (lmlt p2,R) by A1, A6, XBOOLE_0:def 4;
then A8: (lmlt p2,R) /. x = (lmlt p2,R) . x by PARTFUN1:def 8;
x in dom p2 by A4, A7, XBOOLE_0:def 4;
then A9: p2 /. x = p2 . x by PARTFUN1:def 8;
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 8;
A12: x in dom (lmlt p1,R) 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 8;
x in dom R by A3, A12, XBOOLE_0:def 4;
then A14: R /. x = R . x by PARTFUN1:def 8;
A15: (lmlt p1,R) /. x = (lmlt p1,R) . x by A12, PARTFUN1:def 8;
hence ((lmlt p1,R) + (lmlt p2,R)) . x = ((lmlt p1,R) /. x) + ((lmlt p2,R) /. x) by A6, A8, FVSUM_1:21
.= (the lmult of V1 . (p1 /. x),(R /. x)) + ((lmlt p2,R) /. x) by A12, A15, A13, A14, FUNCOP_1:28
.= ((p1 /. x) * (R /. x)) + ((p2 /. x) * (R /. x)) by A7, A8, A9, A14, FUNCOP_1:28
.= ((p1 /. x) + (p2 /. x)) * (R /. x) by VECTSP_1:def 27
.= ((p1 + p2) /. x) * (R /. x) by A10, A13, A9, A11, FVSUM_1:21
.= (lmlt (p1 + p2),R) . x by A5, A6, A14, A11, FUNCOP_1:28 ;
:: thesis: verum

end;

hence lmlt (p1 + p2),R = (lmlt p1,R) + (lmlt p2,R) by A5, FUNCT_1:9; :: thesis: verum