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

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 :: thesis: for i, j being Nat st [i,j] in Indices C1 holds

C1 * (i,j) = C2 * (i,j)

hence
C1 = C2
by MATRIX_0:27; :: thesis: verumC1 * (i,j) = C2 * (i,j)

A3:
Indices C1 = Indices C2
by MATRIX_0:26;

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 12;

C1 * (i,j) = Cofactor (M,i9,j9) by A1, A4;

hence C1 * (i,j) = C2 * (i,j) by A2, A4, A3; :: thesis: verum

end;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 12;

C1 * (i,j) = Cofactor (M,i9,j9) by A1, A4;

hence C1 * (i,j) = C2 * (i,j) by A2, A4, A3; :: thesis: verum