A2: ( Indices (0. K,n) = [:(Seg n),(Seg n):] & len (0. K,n) = n & width (0. K,n) = n ) by MATRIX_1:25;
A3: len (n |-> (0. K)) = n by FINSEQ_1:def 18;
A4: ( 0. K,n = n |-> (n |-> (0. K)) & 0. K,n,n = n |-> (n |-> (0. K)) ) ;
set p = n |-> (0. K);
set M1 = 0. K,n;
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:
assume B1: ( [i,j] in Indices (0. K,n) & i <= j ) ; :: thesis:
then ((j - i) mod n) + 1 in Seg n by A2, Lm2;
then ((Seg n) --> (0. K)) . (((j - i) mod n) + 1) = 0. K by FUNCOP_1:13;
hence (0. K,n) * i,j = (n |-> (0. K)) . (((j - i) mod (len (n |-> (0. K)))) + 1) by A3, B1, A4, MATRIX_3:3; :: thesis:

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:
assume B1: ( [i,j] in Indices (0. K,n) & i >= j ) ; :: thesis:
then B2: ((j - i) mod n) + 1 in Seg n by A2, Lm2;
(- (n |-> (0. K))) . (((j - i) mod (len (n |-> (0. K)))) + 1) = (n |-> (- (0. K))) . (((j - i) mod n) + 1) by A3, FVSUM_1:34
.= - (0. K) by B2, FUNCOP_1:13
.= 0. K by VECTSP_2:34 ;
hence (0. K,n) * i,j = (- (n |-> (0. K))) . (((j - i) mod (len (n |-> (0. K)))) + 1) by B1, A4, MATRIX_3:3; :: thesis:

end;

then A9: 0. K,n is_anti-circular_about n |-> (0. K) by A2, A3, A5, Def10;
consider p being FinSequence of K such that
A11: ( len p = width (0. K,n) & 0. K,n is_anti-circular_about p ) by A2, A3, A9;
take p ; :: according to MATRIX16:def 10 :: thesis:
thus ( len p = width (0. K,n) & 0. K,n is_anti-circular_about p ) by A11; :: thesis: