A4: width M = m by A2, MATRIX_0:23;
then A5: len (M @) = m by A3, MATRIX_0:54;
A6: len M = n by A2, MATRIX_0:23;
then width (M @) = n by A3, A4, MATRIX_0:54;
then M @ is Matrix of m,n,K by A3, A5, MATRIX_0:20;
then consider M1 being Matrix of m,n,K such that
A7: M1 = M @ ;
A8: width (ScalarXLine (M1,l,a)) = n by A3, MATRIX_0:23;
then A9: len ((ScalarXLine (M1,l,a)) @) = n by A2, MATRIX_0:54;
len (ScalarXLine (M1,l,a)) = m by A3, MATRIX_0:23;
then width ((ScalarXLine (M1,l,a)) @) = m by A2, A8, MATRIX_0:54;
then (ScalarXLine (M1,l,a)) @ is Matrix of n,m,K by A2, A9, MATRIX_0:20;
then consider M2 being Matrix of n,m,K such that
A10: M2 = (ScalarXLine (M1,l,a)) @ ;
take M2 ; :: thesis: ( len M2 = len M & ( for i, j being Nat st i in dom M & j in Seg (width M) holds
( ( j = l implies M2 * (i,j) = a * (M * (i,l)) ) & ( j <> l implies M2 * (i,j) = M * (i,j) ) ) ) )
for i, j being Nat st i in dom M & j in Seg (width M) holds
( ( j = l implies M2 * (i,j) = a * (M * (i,l)) ) & ( j <> l implies M2 * (i,j) = M * (i,j) ) )

proof

let i, j be Nat; :: thesis: ( i in dom M & j in Seg (width M) implies ( ( j = l implies M2 * (i,j) = a * (M * (i,l)) ) & ( j <> l implies M2 * (i,j) = M * (i,j) ) ) )
assume that
A11: i in dom M and
A12: j in Seg (width M) ; :: thesis: ( ( j = l implies M2 * (i,j) = a * (M * (i,l)) ) & ( j <> l implies M2 * (i,j) = M * (i,j) ) )
A13: [i,j] in Indices M by A11, A12, ZFMISC_1:87;
then A14: [j,i] in Indices M1 by A7, MATRIX_0:def 6;
dom (ScalarXLine (M1,l,a)) = Seg (len (ScalarXLine (M1,l,a))) by FINSEQ_1:def 3
.= Seg (len M1) by Def2
.= dom M1 by FINSEQ_1:def 3 ;
then A15: [j,i] in Indices (ScalarXLine (M1,l,a)) by A14, Th1;
A16: ( j in dom M1 & i in Seg (width M1) ) by A14, ZFMISC_1:87;
thus ( j = l implies M2 * (i,j) = a * (M * (i,l)) ) :: thesis: ( j <> l implies M2 * (i,j) = M * (i,j) )

proof

A17: [i,l] in Indices M by A1, A11, ZFMISC_1:87;
assume A18: j = l ; :: thesis: M2 * (i,j) = a * (M * (i,l))
M2 * (i,j) = (ScalarXLine (M1,l,a)) * (j,i) by A10, A15, MATRIX_0:def 6
.= a * (M1 * (l,i)) by A16, A18, Def2
.= a * (M * (i,l)) by A7, A17, MATRIX_0:def 6 ;
hence M2 * (i,j) = a * (M * (i,l)) ; :: thesis: verum

end;

thus ( j <> l implies M2 * (i,j) = M * (i,j) ) :: thesis: verum

proof

assume A19: j <> l ; :: thesis: M2 * (i,j) = M * (i,j)
M2 * (i,j) = (ScalarXLine (M1,l,a)) * (j,i) by A10, A15, MATRIX_0:def 6
.= M1 * (j,i) by A16, A19, Def2
.= M * (i,j) by A7, A13, MATRIX_0:def 6 ;
hence M2 * (i,j) = M * (i,j) ; :: thesis: verum

end;

end;

hence ( len M2 = len M & ( for i, j being Nat st i in dom M & j in Seg (width M) holds
( ( j = l implies M2 * (i,j) = a * (M * (i,l)) ) & ( j <> l implies M2 * (i,j) = M * (i,j) ) ) ) ) by A2, A6, MATRIX_0:23; :: thesis: verum