let M1, M2 be Matrix of n,K; :: thesis: ( ( for i, j being Nat st [i,j] in Indices A holds

M1 * (i,j) = (A * (i,j)) + (B * (i,j)) ) & ( for i, j being Nat st [i,j] in Indices A holds

M2 * (i,j) = (A * (i,j)) + (B * (i,j)) ) implies M1 = M2 )

assume that

A22: for i, j being Nat st [i,j] in Indices A holds

M1 * (i,j) = (A * (i,j)) + (B * (i,j)) and

A23: for i, j being Nat st [i,j] in Indices A holds

M2 * (i,j) = (A * (i,j)) + (B * (i,j)) ; :: thesis: M1 = M2

then Indices A = Indices M1 by MATRIX_0:24;

hence M1 = M2 by A24, MATRIX_0:27; :: thesis: verum

M1 * (i,j) = (A * (i,j)) + (B * (i,j)) ) & ( for i, j being Nat st [i,j] in Indices A holds

M2 * (i,j) = (A * (i,j)) + (B * (i,j)) ) implies M1 = M2 )

assume that

A22: for i, j being Nat st [i,j] in Indices A holds

M1 * (i,j) = (A * (i,j)) + (B * (i,j)) and

A23: for i, j being Nat st [i,j] in Indices A holds

M2 * (i,j) = (A * (i,j)) + (B * (i,j)) ; :: thesis: M1 = M2

A24: now :: thesis: for i, j being Nat st [i,j] in Indices A holds

M1 * (i,j) = M2 * (i,j)

Indices M1 = [:(Seg n),(Seg n):]
by MATRIX_0:24;M1 * (i,j) = M2 * (i,j)

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

assume A25: [i,j] in Indices A ; :: thesis: M1 * (i,j) = M2 * (i,j)

then M1 * (i,j) = (A * (i,j)) + (B * (i,j)) by A22;

hence M1 * (i,j) = M2 * (i,j) by A23, A25; :: thesis: verum

end;assume A25: [i,j] in Indices A ; :: thesis: M1 * (i,j) = M2 * (i,j)

then M1 * (i,j) = (A * (i,j)) + (B * (i,j)) by A22;

hence M1 * (i,j) = M2 * (i,j) by A23, A25; :: thesis: verum

then Indices A = Indices M1 by MATRIX_0:24;

hence M1 = M2 by A24, MATRIX_0:27; :: thesis: verum