let n, m, l, k be Nat; :: thesis: for K being Field
for M being Matrix of n,m,K holds ILine M,l,k = ILine M,k,l
let K be Field; :: thesis: for M being Matrix of n,m,K holds ILine M,l,k = ILine M,k,l
let M be Matrix of n,m,K; :: thesis: ILine M,l,k = ILine M,k,l
A1: ( len (ILine M,l,k) = len M & width (ILine M,l,k) = width M & len (ILine M,k,l) = len M & width (ILine M,k,l) = width M ) by Def1, Th1;
for i, j being Nat st [i,j] in Indices (ILine M,l,k) holds
(ILine M,l,k) * i,j = (ILine M,k,l) * i,j

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (ILine M,l,k) implies (ILine M,l,k) * i,j = (ILine M,k,l) * i,j )
assume A2: [i,j] in Indices (ILine M,l,k) ; :: thesis: (ILine M,l,k) * i,j = (ILine M,k,l) * i,j
Indices (ILine M,l,k) = Indices M by MATRIX_1:27;
then A3: ( i in dom M & j in Seg (width M) ) by A2, ZFMISC_1:106;
then A4: ( ( i = l implies (ILine M,l,k) * i,j = M * k,j ) & ( i = k implies (ILine M,l,k) * i,j = M * l,j ) & ( i <> l & i <> k implies (ILine M,l,k) * i,j = M * i,j ) ) by Def1;
( ( i = l implies (ILine M,k,l) * i,j = M * k,j ) & ( i = k implies (ILine M,k,l) * i,j = M * l,j ) & ( i <> l & i <> k implies (ILine M,k,l) * i,j = M * i,j ) ) by A3, Def1;
hence (ILine M,l,k) * i,j = (ILine M,k,l) * i,j by A4; :: thesis: verum

end;

hence ILine M,l,k = ILine M,k,l by A1, MATRIX_1:21; :: thesis: verum