let K be Field; :: thesis: for M being Matrix of K
for P, Q being finite without_zero Subset of NAT
for i being Nat st i in Seg (card P) & card P = card Q & Det (EqSegm M,P,Q) <> 0. K holds
ex j being Nat st
( j in Seg (card P) & Det (Delete (EqSegm M,P,Q),i,j) <> 0. K )
let M be Matrix of K; :: thesis: for P, Q being finite without_zero Subset of NAT
for i being Nat st i in Seg (card P) & card P = card Q & Det (EqSegm M,P,Q) <> 0. K holds
ex j being Nat st
( j in Seg (card P) & Det (Delete (EqSegm M,P,Q),i,j) <> 0. K )
let P, Q be finite without_zero Subset of NAT ; :: thesis: for i being Nat st i in Seg (card P) & card P = card Q & Det (EqSegm M,P,Q) <> 0. K holds
ex j being Nat st
( j in Seg (card P) & Det (Delete (EqSegm M,P,Q),i,j) <> 0. K )
let i be Nat; :: thesis: ( i in Seg (card P) & card P = card Q & Det (EqSegm M,P,Q) <> 0. K implies ex j being Nat st
( j in Seg (card P) & Det (Delete (EqSegm M,P,Q),i,j) <> 0. K ) )
assume that
A1: ( i in Seg (card P) & card P = card Q ) and
A2: Det (EqSegm M,P,Q) <> 0. K ; :: thesis: ex j being Nat st
( j in Seg (card P) & Det (Delete (EqSegm M,P,Q),i,j) <> 0. K )
set E = EqSegm M,P,Q;
set C = card P;
set LL = LaplaceExpL (EqSegm M,P,Q),i;
set CC = (card P) |-> (0. K);
( Sum ((card P) |-> (0. K)) = 0. K & len (LaplaceExpL (EqSegm M,P,Q),i) = card P ) by LAPLACE:def 7, MATRIX_3:13;
then A3: ( LaplaceExpL (EqSegm M,P,Q),i <> (card P) |-> (0. K) & dom (LaplaceExpL (EqSegm M,P,Q),i) = Seg (card P) & dom ((card P) |-> (0. K)) = Seg (card P) ) by A1, A2, FINSEQ_1:def 3, FUNCOP_1:19, LAPLACE:25;
then consider j being Nat such that
A4: ( j in dom (LaplaceExpL (EqSegm M,P,Q),i) & (LaplaceExpL (EqSegm M,P,Q),i) . j <> ((card P) |-> (0. K)) . j ) by FINSEQ_1:17;
( ((card P) |-> (0. K)) . j = 0. K & (LaplaceExpL (EqSegm M,P,Q),i) . j = ((EqSegm M,P,Q) * i,j) * (Cofactor (EqSegm M,P,Q),i,j) ) by A3, A4, FINSEQ_2:71, LAPLACE:def 7;
then Cofactor (EqSegm M,P,Q),i,j <> 0. K by A4, VECTSP_1:44;
then Minor (EqSegm M,P,Q),i,j <> 0. K by VECTSP_1:44;
hence ex j being Nat st
( j in Seg (card P) & Det (Delete (EqSegm M,P,Q),i,j) <> 0. K ) by A3, A4; :: thesis: verum