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 :: thesis: for x being object st x in dom ((lmlt (p1,R)) + (lmlt (p2,R))) holds
((lmlt (p1,R)) + (lmlt (p2,R))) . x = (lmlt ((p1 + p2),R)) . x

let x be object ; :: 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 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 (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 6;
x in dom R by A3, A12, XBOOLE_0:def 4;
then A14: R /. x = R . x by PARTFUN1:def 6;
A15: (lmlt (p1,R)) /. x = (lmlt (p1,R)) . x by A12, PARTFUN1:def 6;
hence ((lmlt (p1,R)) + (lmlt (p2,R))) . x = ((lmlt (p1,R)) /. x) + ((lmlt (p2,R)) /. x) by A6, A8, FVSUM_1:17
.= ( the lmult of V1 . ((p1 /. x),(R /. x))) + ((lmlt (p2,R)) /. 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
.= (lmlt ((p1 + p2),R)) . x by A5, A6, A14, A11, FUNCOP_1:22 ;
:: thesis: verum

end;

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