let k be Nat; :: thesis: for K being Field holds LineVec2Mx (k |-> (0. K)) = 0. (K,1,k)
let K be Field; :: thesis: LineVec2Mx (k |-> (0. K)) = 0. (K,1,k)
card (k |-> (0. K)) = k by CARD_1:def 7;
then reconsider L = LineVec2Mx (k |-> (0. K)) as Matrix of 1,k,K ;
set Z = 0. (K,1,k);

now :: thesis: for i, j being Nat st [i,j] in Indices L holds
(0. (K,1,k)) * (i,j) = L * (i,j)

A1: width L = k by MATRIX_0:23;
A2: ( dom L = Seg (len L) & len L = 1 ) by FINSEQ_1:def 3, MATRIX_0:def 2;
let i, j be Nat; :: thesis: ( [i,j] in Indices L implies (0. (K,1,k)) * (i,j) = L * (i,j) )
assume A3: [i,j] in Indices L ; :: thesis: (0. (K,1,k)) * (i,j) = L * (i,j)
A4: j in Seg (width L) by A3, ZFMISC_1:87;
i in dom L by A3, ZFMISC_1:87;
then A5: i = 1 by A2, FINSEQ_1:2, TARSKI:def 1;
Indices (0. (K,1,k)) = Indices L by MATRIX_0:26;
hence (0. (K,1,k)) * (i,j) = 0. K by A3, MATRIX_3:1
.= (k |-> (0. K)) . j by A4, A1, FINSEQ_2:57
.= (Line (L,i)) . j by A5, Th25
.= L * (i,j) by A4, MATRIX_0:def 7 ;
:: thesis: verum

end;

hence LineVec2Mx (k |-> (0. K)) = 0. (K,1,k) by MATRIX_0:27; :: thesis: verum