let K be Field; :: thesis: for a being Element of K
for M being Matrix of K
for i being Nat st 1 <= i & i <= len M holds
Line (a * M),i = a * (Line M,i)
let a be Element of K; :: thesis: for M being Matrix of K
for i being Nat st 1 <= i & i <= len M holds
Line (a * M),i = a * (Line M,i)
let M be Matrix of K; :: thesis: for i being Nat st 1 <= i & i <= len M holds
Line (a * M),i = a * (Line M,i)
let i be Nat; :: thesis: ( 1 <= i & i <= len M implies Line (a * M),i = a * (Line M,i) )
assume A1: ( 1 <= i & i <= len M ) ; :: thesis: Line (a * M),i = a * (Line M,i)
A2: width (a * M) = width M by MATRIX_3:def 5;
A3: Seg (width M) = Seg (width (a * M)) by MATRIX_3:def 5;
A4: ( len (a * (Line M,i)) = len (Line M,i) & len (Line M,i) = width M ) by Th16, MATRIX_1:def 8;
then A5: dom (a * (Line M,i)) = Seg (width (a * M)) by A2, FINSEQ_1:def 3;
for j being Nat st j in Seg (width (a * M)) holds
(a * (Line M,i)) . j = (a * M) * i,j

proof

let j be Nat; :: thesis: ( j in Seg (width (a * M)) implies (a * (Line M,i)) . j = (a * M) * i,j )
assume A6: j in Seg (width (a * M)) ; :: thesis: (a * (Line M,i)) . j = (a * M) * i,j
i in dom M by A1, FINSEQ_3:27;
then [i,j] in Indices M by A3, A6, ZFMISC_1:106;
then A7: (a * M) * i,j = a * (M * i,j) by MATRIX_3:def 5;
(Line M,i) . j = M * i,j by A3, A6, MATRIX_1:def 8;
hence (a * (Line M,i)) . j = (a * M) * i,j by A5, A6, A7, FVSUM_1:62; :: thesis: verum

end;

hence Line (a * M),i = a * (Line M,i) by A4, A2, MATRIX_1:def 8; :: thesis: verum