let M1, M2 be Matrix of REAL; :: 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
A8: len M1 = len x and
A9: width M1 = len y and
A10: for i, j being Nat st [i,j] in Indices M holds
M1 * (i,j) = ((x . i) * (M * (i,j))) * (y . j) and
A11: ( len M2 = len x & 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) )
A13: Indices M = [:(dom M),(Seg (width M)):] by MATRIX_1:def 4;
dom M1 = dom M by A1, A8, FINSEQ_3:29;
then A14: Indices M1 = [:(dom M),(Seg (width M)):] by A2, A9, MATRIX_1:def 4;
assume A15: [i,j] in Indices M1 ; :: thesis: M1 * (i,j) = M2 * (i,j)
hence M1 * (i,j) = ((x . i) * (M * (i,j))) * (y . j) by A10, A14, A13
.= M2 * (i,j) by A12, A15, A14, A13 ;
:: thesis: verum

end;

hence M1 = M2 by A8, A9, A11, MATRIX_1:21; :: thesis: verum