let C1, C2 be Matrix of n,K; :: thesis: ( ( for i, j being Nat st [i,j] in Indices C1 holds
C1 * i,j = Cofactor M,i,j ) & ( for i, j being Nat st [i,j] in Indices C2 holds
C2 * i,j = Cofactor M,i,j ) implies C1 = C2 )
assume that
A1: for i, j being Nat st [i,j] in Indices C1 holds
C1 * i,j = Cofactor M,i,j and
A2: for i, j being Nat st [i,j] in Indices C2 holds
C2 * i,j = Cofactor M,i,j ; :: thesis: C1 = C2

now

A3: Indices C1 = Indices C2 by MATRIX_1:27;
let i, j be Nat; :: thesis: ( [i,j] in Indices C1 implies C1 * i,j = C2 * i,j )
assume A4: [i,j] in Indices C1 ; :: thesis: C1 * i,j = C2 * i,j
reconsider i9 = i, j9 = j as Element of NAT by ORDINAL1:def 13;
C1 * i,j = Cofactor M,i9,j9 by A1, A4;
hence C1 * i,j = C2 * i,j by A2, A4, A3; :: thesis: verum

end;

hence C1 = C2 by MATRIX_1:28; :: thesis: verum