let i be Nat; :: thesis: for K being Field
for a being Element of K
for M being Matrix of K st 1 <= i & i <= width M holds
Col ((a * M),i) = a * (Col (M,i))

let K be Field; :: thesis: for a being Element of K
for M being Matrix of K st 1 <= i & i <= width M holds
Col ((a * M),i) = a * (Col (M,i))

let a be Element of K; :: thesis: for M being Matrix of K st 1 <= i & i <= width M holds
Col ((a * M),i) = a * (Col (M,i))

let M be Matrix of K; :: thesis: ( 1 <= i & i <= width M implies Col ((a * M),i) = a * (Col (M,i)) )
assume A1: ( 1 <= i & i <= width M ) ; :: thesis: Col ((a * M),i) = a * (Col (M,i))
A2: Seg (len (a * M)) = dom (a * M) by FINSEQ_1:def 3;
A3: len (a * M) = len M by MATRIX_3:def 5;
then A4: dom M = dom (a * M) by FINSEQ_3:29;
A5: ( len (a * (Col (M,i))) = len (Col (M,i)) & len (Col (M,i)) = len M ) by ;
then A6: dom (a * (Col (M,i))) = Seg (len (a * M)) by ;
for j being Nat st j in dom (a * M) holds
(a * (Col (M,i))) . j = (a * M) * (j,i)
proof
let j be Nat; :: thesis: ( j in dom (a * M) implies (a * (Col (M,i))) . j = (a * M) * (j,i) )
assume A7: j in dom (a * M) ; :: thesis: (a * (Col (M,i))) . j = (a * M) * (j,i)
i in Seg () by ;
then [j,i] in Indices M by ;
then A8: (a * M) * (j,i) = a * (M * (j,i)) by MATRIX_3:def 5;
(Col (M,i)) . j = M * (j,i) by ;
hence (a * (Col (M,i))) . j = (a * M) * (j,i) by ; :: thesis: verum
end;
hence Col ((a * M),i) = a * (Col (M,i)) by ; :: thesis: verum