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
A9: len M1 = len x and
A10: width M1 = len y and
A11: for i, j being Nat st [i,j] in Indices M holds
M1 * (i,j) = ((x . i) * (M * (i,j))) * (y . j) and
A12: ( len M2 = len x & width M2 = len y ) and
A13: 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 :: thesis: for i, j being Nat st [i,j] in Indices M1 holds
M1 * (i,j) = M2 * (i,j)

let i, j be Nat; :: thesis: ( [i,j] in Indices M1 implies M1 * (i,j) = M2 * (i,j) )
A14: Indices M = [:(dom M),(Seg (width M)):] by MATRIX_0:def 4;
dom M1 = dom M by A1, A9, FINSEQ_3:29;
then A15: Indices M1 = [:(dom M),(Seg (width M)):] by A2, A10, MATRIX_0:def 4;
assume A16: [i,j] in Indices M1 ; :: thesis: M1 * (i,j) = M2 * (i,j)
hence M1 * (i,j) = ((x . i) * (M * (i,j))) * (y . j) by A11, A15, A14
.= M2 * (i,j) by A13, A16, A15, A14 ;
:: thesis: verum

end;

hence M1 = M2 by A9, A10, A12, MATRIX_0:21; :: thesis: verum