let i, j, n be Nat; :: thesis: for K being non empty doubleLoopStr st [i,j] in Indices (0. (K,n)) holds
(0. (K,n)) * (i,j) = 0. K
let K be non empty doubleLoopStr ; :: thesis: ( [i,j] in Indices (0. (K,n)) implies (0. (K,n)) * (i,j) = 0. K )
reconsider n1 = n as Element of NAT by ORDINAL1:def 12;
set M = 0. (K,n);
assume A1: [i,j] in Indices (0. (K,n)) ; :: thesis: (0. (K,n)) * (i,j) = 0. K
then A2: [i,j] in [:(Seg n),(Seg n):] by MATRIX_0:24;
then j in Seg n by ZFMISC_1:87;
then A3: (n1 |-> (0. K)) . j = 0. K by FUNCOP_1:7;
i in Seg n by A2, ZFMISC_1:87;
then (0. (K,n)) . i = n1 |-> (0. K) by FUNCOP_1:7;
hence (0. (K,n)) * (i,j) = 0. K by A1, A3, MATRIX_0:def 5; :: thesis: verum