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
A6: ( 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) ) ) and
A7: ( 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) ) ) ; :: thesis: M1 = M2

now

let i, j be Nat; :: thesis: ( [i,j] in Indices M1 implies M1 * i,j = M2 * i,j )
assume A8: [i,j] in Indices M1 ; :: thesis: M1 * i,j = M2 * i,j
dom M1 = dom M by A1, A6, FINSEQ_3:31;
then A9: ( Indices M1 = [:(dom M),(Seg (width M)):] & Indices M = [:(dom M),(Seg (width M)):] ) by A1, A6, MATRIX_1:def 5;
hence M1 * i,j = ((x . i) * (M * i,j)) * (y . j) by A6, A8
.= M2 * i,j by A7, A8, A9 ;
:: thesis: verum

end;

hence M1 = M2 by A6, A7, MATRIX_1:21; :: thesis: verum