A5: width M = m by A3, MATRIX_1:24;
then A6: len (M @ ) = m by A4, MATRIX_2:12;
A7: len M = n by A3, MATRIX_1:24;
then width (M @ ) = n by A4, A5, MATRIX_2:12;
then M @ is Matrix of m,n,K by A4, A6, MATRIX_1:20;
then consider M1 being Matrix of m,n,K such that
A8: M1 = M @ ;
A9: width (RlineXScalar M1,l,k,a) = n by A4, MATRIX_1:24;
then A10: len ((RlineXScalar M1,l,k,a) @ ) = n by A3, MATRIX_2:12;
len (RlineXScalar M1,l,k,a) = m by A4, MATRIX_1:24;
then width ((RlineXScalar M1,l,k,a) @ ) = m by A3, A9, MATRIX_2:12;
then (RlineXScalar M1,l,k,a) @ is Matrix of n,m,K by A3, A10, MATRIX_1:20;
then consider M2 being Matrix of n,m,K such that
A11: M2 = (RlineXScalar M1,l,k,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,k)) + (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,k)) + (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,k)) + (M * i,l) ) & ( j <> l implies M2 * i,j = M * i,j ) ) )
assume that
A12: i in dom M and
A13: j in Seg (width M) ; :: thesis: ( ( j = l implies M2 * i,j = (a * (M * i,k)) + (M * i,l) ) & ( j <> l implies M2 * i,j = M * i,j ) )
A14: [i,j] in Indices M by A12, A13, ZFMISC_1:106;
then A15: [j,i] in Indices M1 by A8, MATRIX_1:def 7;
then A16: i in Seg (width M1) by ZFMISC_1:106;
A17: len M1 = width M by A8, MATRIX_1:def 7;
then A18: k in dom M1 by A2, FINSEQ_1:def 3;
dom (RlineXScalar M1,l,k,a) = Seg (len (RlineXScalar M1,l,k,a)) by FINSEQ_1:def 3
.= Seg (len M1) by A18, Def3
.= dom M1 by FINSEQ_1:def 3 ;
then A19: [j,i] in Indices (RlineXScalar M1,l,k,a) by A15, Th1;
A20: l in dom M1 by A1, A17, FINSEQ_1:def 3;
thus ( j = l implies M2 * i,j = (a * (M * i,k)) + (M * i,l) ) :: thesis: ( j <> l implies M2 * i,j = M * i,j )

proof

A21: [i,k] in Indices M by A2, A12, ZFMISC_1:106;
A22: [i,l] in Indices M by A1, A12, ZFMISC_1:106;
assume A23: j = l ; :: thesis: M2 * i,j = (a * (M * i,k)) + (M * i,l)
M2 * i,j = (RlineXScalar M1,l,k,a) * j,i by A11, A19, MATRIX_1:def 7
.= (a * (M1 * k,i)) + (M1 * l,i) by A20, A18, A16, A23, Def3
.= (a * (M * i,k)) + (M1 * l,i) by A8, A21, MATRIX_1:def 7
.= (a * (M * i,k)) + (M * i,l) by A8, A22, MATRIX_1:def 7 ;
hence M2 * i,j = (a * (M * i,k)) + (M * i,l) ; :: thesis: verum

end;

A24: j in dom M1 by A15, ZFMISC_1:106;
thus ( j <> l implies M2 * i,j = M * i,j ) :: thesis: verum

proof

assume A25: j <> l ; :: thesis: M2 * i,j = M * i,j
M2 * i,j = (RlineXScalar M1,l,k,a) * j,i by A11, A19, MATRIX_1:def 7
.= M1 * j,i by A18, A24, A16, A25, Def3
.= M * i,j by A8, A14, MATRIX_1:def 7 ;
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,k)) + (M * i,l) ) & ( j <> l implies M2 * i,j = M * i,j ) ) ) ) by A3, A7, MATRIX_1:24; :: thesis: verum