let K be Field; :: thesis: for M being Matrix of K
for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices M & card P = card Q holds
( card P <= len M & card Q <= width M )
let M be Matrix of K; :: thesis: for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices M & card P = card Q holds
( card P <= len M & card Q <= width M )
let P, Q be finite without_zero Subset of NAT; :: thesis: ( [:P,Q:] c= Indices M & card P = card Q implies ( card P <= len M & card Q <= width M ) )
assume that
A1: [:P,Q:] c= Indices M and
A2: card P = card Q ; :: thesis: ( card P <= len M & card Q <= width M )
Q c= Seg (width M) by A1, A2, Th67;
then A3: card Q <= card (Seg (width M)) by NAT_1:43;
P c= Seg (len M) by A1, A2, Th67;
then card P <= card (Seg (len M)) by NAT_1:43;
hence ( card P <= len M & card Q <= width M ) by A3, FINSEQ_1:57; :: thesis: verum