let M1, M2 be Matrix of len p,K; :: thesis: ( M1 is_anti-circular_about p & M2 is_anti-circular_about p implies M1 = M2 )
assume that
B1: M1 is_anti-circular_about p and
B2: M2 is_anti-circular_about p ; :: thesis: M1 = M2
B5: Indices M1 = Indices M2 by MATRIX_1:27;
for i, j being Nat st [i,j] in Indices M1 holds
M1 * i,j = M2 * i,j

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices M1 implies M1 * i,j = M2 * i,j )
assume C1: [i,j] in Indices M1 ; :: thesis: M1 * i,j = M2 * i,j

per cases ( i <= j or i > j ) ;

suppose i <= j ; :: thesis: M1 * i,j = M2 * i,j

then ( M1 * i,j = p . (((j - i) mod (len p)) + 1) & M2 * i,j = p . (((j - i) mod (len p)) + 1) ) by B5, B1, Def10, B2, C1;
hence M1 * i,j = M2 * i,j ; :: thesis: verum

end;

suppose i > j ; :: thesis: M1 * i,j = M2 * i,j

then ( M1 * i,j = (- p) . (((j - i) mod (len p)) + 1) & M2 * i,j = (- p) . (((j - i) mod (len p)) + 1) ) by B5, B1, Def10, B2, C1;
hence M1 * i,j = M2 * i,j ; :: thesis: verum

end;

end;

end;

hence M1 = M2 by MATRIX_1:28; :: thesis: verum