let M1, M2 be Matrix of len p,K; :: thesis: ( M1 is_col_circulant_about p & M2 is_col_circulant_about p implies M1 = M2 )
assume that
B1: M1 is_col_circulant_about p and
B2: M2 is_col_circulant_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 [i,j] in Indices M1 ; :: thesis: M1 * i,j = M2 * i,j
then ( M1 * i,j = p . (((i - j) mod (len p)) + 1) & M2 * i,j = p . (((i - j) mod (len p)) + 1) ) by B5, B1, B2, Def4;
hence M1 * i,j = M2 * i,j ; :: thesis: verum

end;

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