set p = n |-> (0. K);
A1: width (0. (K,n)) = n by MATRIX_0:24;
A2: 0. (K,n,n) = n |-> (n |-> (0. K)) ;
set M1 = 0. (K,n);
A3: len (n |-> (0. K)) = n by CARD_1:def 7;
A4: Indices (0. (K,n)) = [:(Seg n),(Seg n):] by MATRIX_0:24;
A5: for i, j being Nat st [i,j] in Indices (0. (K,n)) & i >= j holds
(0. (K,n)) * (i,j) = (- (n |-> (0. K))) . (((j - i) mod (len (n |-> (0. K)))) + 1)

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (0. (K,n)) & i >= j implies (0. (K,n)) * (i,j) = (- (n |-> (0. K))) . (((j - i) mod (len (n |-> (0. K)))) + 1) )
assume that
A6: [i,j] in Indices (0. (K,n)) and
i >= j ; :: thesis: (0. (K,n)) * (i,j) = (- (n |-> (0. K))) . (((j - i) mod (len (n |-> (0. K)))) + 1)
A7: ((j - i) mod n) + 1 in Seg n by A4, A6, Lm3;
(- (n |-> (0. K))) . (((j - i) mod (len (n |-> (0. K)))) + 1) = (n |-> (- (0. K))) . (((j - i) mod n) + 1) by A3, FVSUM_1:25
.= - (0. K) by A7, FUNCOP_1:7
.= 0. K by VECTSP_2:3 ;
hence (0. (K,n)) * (i,j) = (- (n |-> (0. K))) . (((j - i) mod (len (n |-> (0. K)))) + 1) by A2, A6, MATRIX_3:1; :: thesis: verum

end;

for i, j being Nat st [i,j] in Indices (0. (K,n)) & i <= j holds
(0. (K,n)) * (i,j) = (n |-> (0. K)) . (((j - i) mod (len (n |-> (0. K)))) + 1)

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (0. (K,n)) & i <= j implies (0. (K,n)) * (i,j) = (n |-> (0. K)) . (((j - i) mod (len (n |-> (0. K)))) + 1) )
assume that
A8: [i,j] in Indices (0. (K,n)) and
i <= j ; :: thesis: (0. (K,n)) * (i,j) = (n |-> (0. K)) . (((j - i) mod (len (n |-> (0. K)))) + 1)
((j - i) mod n) + 1 in Seg n by A4, A8, Lm3;
then ((Seg n) --> (0. K)) . (((j - i) mod n) + 1) = 0. K by FUNCOP_1:7;
hence (0. (K,n)) * (i,j) = (n |-> (0. K)) . (((j - i) mod (len (n |-> (0. K)))) + 1) by A3, A2, A8, MATRIX_3:1; :: thesis: verum

end;

then 0. (K,n) is_anti-circular_about n |-> (0. K) by A1, A3, A5;
then consider p being FinSequence of K such that
A9: ( len p = width (0. (K,n)) & 0. (K,n) is_anti-circular_about p ) ;
take p ; :: according to MATRIX16:def 10 :: thesis: ( len p = width (0. (K,n)) & 0. (K,n) is_anti-circular_about p )
thus ( len p = width (0. (K,n)) & 0. (K,n) is_anti-circular_about p ) by A9; :: thesis: verum