let D be non empty set ; :: thesis: for A being Matrix of D
for P, Q being finite without_zero Subset of NAT holds
not ( [:P,Q:] c= Indices A & ( card P = 0 implies card Q = 0 ) & ( card Q = 0 implies card P = 0 ) & not (Segm (A,P,Q)) @ = Segm ((A @),Q,P) )
let A be Matrix of D; :: thesis: for P, Q being finite without_zero Subset of NAT holds
not ( [:P,Q:] c= Indices A & ( card P = 0 implies card Q = 0 ) & ( card Q = 0 implies card P = 0 ) & not (Segm (A,P,Q)) @ = Segm ((A @),Q,P) )
let P, Q be finite without_zero Subset of NAT; :: thesis: not ( [:P,Q:] c= Indices A & ( card P = 0 implies card Q = 0 ) & ( card Q = 0 implies card P = 0 ) & not (Segm (A,P,Q)) @ = Segm ((A @),Q,P) )
assume that
A1: [:P,Q:] c= Indices A and
A2: ( card P = 0 iff card Q = 0 ) ; :: thesis: (Segm (A,P,Q)) @ = Segm ((A @),Q,P)
ex m being Nat st Q c= Seg m by Th43;
then A3: rng (Sgm Q) = Q by FINSEQ_1:def 13;
ex n being Nat st P c= Seg n by Th43;
then rng (Sgm P) = P by FINSEQ_1:def 13;
hence (Segm (A,P,Q)) @ = Segm ((A @),Q,P) by A1, A2, A3, Th18; :: thesis: verum