let M1, M2 be Matrix of COMPLEX; :: thesis: ( len M1 = len x & width M1 = len y & ( for i, j being Nat st [i,j] in Indices M holds
M1 * (i,j) = ((x . i) * (M * (i,j))) * ((y . j) *') ) & len M2 = len x & width M2 = len y & ( for i, j being Nat st [i,j] in Indices M holds
M2 * (i,j) = ((x . i) * (M * (i,j))) * ((y . j) *') ) implies M1 = M2 )
assume that
A7: len M1 = len x and
A8: width M1 = len y and
A9: for i, j being Nat st [i,j] in Indices M holds
M1 * (i,j) = ((x . i) * (M * (i,j))) * ((y . j) *') and
A10: len M2 = len x and
A11: width M2 = len y and
A12: for i, j being Nat st [i,j] in Indices M holds
M2 * (i,j) = ((x . i) * (M * (i,j))) * ((y . j) *') ; :: thesis: M1 = M2

now

let i, j be Nat; :: thesis: ( [i,j] in Indices M1 implies M1 * (i,j) = M2 * (i,j) )
assume A13: [i,j] in Indices M1 ; :: thesis: M1 * (i,j) = M2 * (i,j)
A14: dom M1 = dom M by A1, A7, FINSEQ_3:29;
hence M1 * (i,j) = ((x . i) * (M * (i,j))) * ((y . j) *') by A2, A8, A9, A13
.= M2 * (i,j) by A2, A8, A12, A13, A14 ;
:: thesis: verum

end;

hence M1 = M2 by A7, A8, A10, A11, MATRIX_1:21; :: thesis: verum