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
A2: M1 is_anti-circular_about p and
A3: M2 is_anti-circular_about p ; :: thesis: M1 = M2
A4: Indices M1 = Indices M2 by MATRIX_0:26;
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 A5: [i,j] in Indices M1 ; :: thesis: M1 * (i,j) = M2 * (i,j)

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

suppose A6: i <= j ; :: thesis: M1 * (i,j) = M2 * (i,j)

then M1 * (i,j) = p . (((j - i) mod (len p)) + 1) by A2, A5;
hence M1 * (i,j) = M2 * (i,j) by A3, A4, A5, A6; :: thesis: verum

end;

suppose A7: i > j ; :: thesis: M1 * (i,j) = M2 * (i,j)

then M1 * (i,j) = (- p) . (((j - i) mod (len p)) + 1) by A2, A5;
hence M1 * (i,j) = M2 * (i,j) by A3, A4, A5, A7; :: thesis: verum

end;

end;

end;

hence M1 = M2 by MATRIX_0:27; :: thesis: verum