let K be Field; :: thesis: for V1 being finite-dimensional VectSp of K
for M1, M2 being Matrix of the carrier of V1 holds (Sum M1) + (Sum M2) = Sum (M1 ^^ M2)
let V1 be finite-dimensional VectSp of K; :: thesis: for M1, M2 being Matrix of the carrier of V1 holds (Sum M1) + (Sum M2) = Sum (M1 ^^ M2)
let M1, M2 be Matrix of the carrier of V1; :: thesis: (Sum M1) + (Sum M2) = Sum (M1 ^^ M2)
reconsider m = min (len M1),(len M2) as Element of NAT by FINSEQ_2:1;
A1: Seg m = (Seg (len M1)) /\ (Seg (len M2)) by FINSEQ_2:2
.= (Seg (len M1)) /\ (dom M2) by FINSEQ_1:def 3
.= (dom M1) /\ (dom M2) by FINSEQ_1:def 3
.= dom (M1 ^^ M2) by Def2
.= Seg (len (M1 ^^ M2)) by FINSEQ_1:def 3 ;
A2: len ((Sum M1) + (Sum M2)) = min (len (Sum M1)),(len (Sum M2)) by FINSEQ_2:85
.= min (len M1),(len (Sum M2)) by Def8
.= min (len M1),(len M2) by Def8
.= len (M1 ^^ M2) by A1, FINSEQ_1:8
.= len (Sum (M1 ^^ M2)) by Def8 ;
X: dom ((Sum M1) + (Sum M2)) = Seg (len ((Sum M1) + (Sum M2))) by FINSEQ_1:def 3;

now

let i be Nat; :: thesis: ( i in dom ((Sum M1) + (Sum M2)) implies ((Sum M1) + (Sum M2)) . i = (Sum (M1 ^^ M2)) . i )
assume A3: i in dom ((Sum M1) + (Sum M2)) ; :: thesis: ((Sum M1) + (Sum M2)) . i = (Sum (M1 ^^ M2)) . i
then A4: i in dom ((Sum M1) + (Sum M2)) ;
A5: i in dom (Sum (M1 ^^ M2)) by A2, A3, X, FINSEQ_1:def 3;
i in Seg (len (M1 ^^ M2)) by A2, A3, Def8, X;
then A6: i in dom (M1 ^^ M2) by FINSEQ_1:def 3;
then i in (dom M1) /\ (dom M2) by Def2;
then A7: ( i in dom M1 & i in dom M2 ) by XBOOLE_0:def 4;
then ( i in Seg (len M1) & i in Seg (len M2) ) by FINSEQ_1:def 3;
then ( i in Seg (len (Sum M1)) & i in Seg (len (Sum M2)) ) by Def8;
then A8: ( i in dom (Sum M1) & i in dom (Sum M2) ) by FINSEQ_1:def 3;
reconsider m1 = M1 . i, m2 = M2 . i as FinSequence by A7, Def1;
A9: (M1 /. i) ^ (M2 /. i) = m1 ^ (M2 /. i) by A7, PARTFUN1:def 8
.= m1 ^ m2 by A7, PARTFUN1:def 8
.= (M1 ^^ M2) . i by A6, Def2
.= (M1 ^^ M2) /. i by A6, PARTFUN1:def 8 ;
( (Sum M1) /. i = (Sum M1) . i & (Sum M2) /. i = (Sum M2) . i ) by A8, PARTFUN1:def 8;
hence ((Sum M1) + (Sum M2)) . i = ((Sum M1) /. i) + ((Sum M2) /. i) by A4, FUNCOP_1:28
.= (Sum (M1 /. i)) + ((Sum M2) /. i) by A8, Def8
.= (Sum (M1 /. i)) + (Sum (M2 /. i)) by A8, Def8
.= Sum ((M1 ^^ M2) /. i) by A9, RLVECT_1:58
.= (Sum (M1 ^^ M2)) /. i by A5, Def8
.= (Sum (M1 ^^ M2)) . i by A5, PARTFUN1:def 8 ;
:: thesis: verum

end;

hence (Sum M1) + (Sum M2) = Sum (M1 ^^ M2) by A2, FINSEQ_2:10; :: thesis: verum