let M1, M2 be Matrix of COMPLEX ; :: thesis: ( len M1 = len x & width M1 = 1 & ( for j being Nat st j in Seg (len x) holds
M1 . j = <*(x . j)*> ) & len M2 = len x & width M2 = 1 & ( for j being Nat st j in Seg (len x) holds
M2 . j = <*(x . j)*> ) implies M1 = M2 )
assume that
A9: ( len M1 = len x & width M1 = 1 ) and
A10: for k being Nat st k in Seg (len x) holds
M1 . k = <*(x . k)*> and
A11: ( len M2 = len x & width M2 = 1 ) and
A12: for k being Nat st k in Seg (len x) holds
M2 . k = <*(x . k)*> ; :: thesis: M1 = M2
X: dom M1 = Seg (len x) by A9, FINSEQ_1:def 3;

now

let k be Nat; :: thesis: ( k in dom M1 implies M1 . k = M2 . k )
assume A13: k in dom M1 ; :: thesis: M1 . k = M2 . k
hence M1 . k = <*(x . k)*> by A10, X
.= M2 . k by A12, A13, X ;
:: thesis: verum

end;

hence M1 = M2 by A9, A11, FINSEQ_2:10; :: thesis: verum