let K be Field; :: thesis: for A, B being Matrix of
for P, Q being finite without_zero Subset of st [:P,Q:] c= Indices A holds
Segm (A + B),P,Q = (Segm A,P,Q) + (Segm B,P,Q)
let A, B be Matrix of ; :: thesis: for P, Q being finite without_zero Subset of st [:P,Q:] c= Indices A holds
Segm (A + B),P,Q = (Segm A,P,Q) + (Segm B,P,Q)
let P, Q be finite without_zero Subset of ; :: thesis: ( [:P,Q:] c= Indices A implies Segm (A + B),P,Q = (Segm A,P,Q) + (Segm B,P,Q) )
assume A1: [:P,Q:] c= Indices A ; :: thesis: Segm (A + B),P,Q = (Segm A,P,Q) + (Segm B,P,Q)
ex m being Nat st Q c= Seg m by MATRIX13:43;
then A2: rng (Sgm Q) = Q by FINSEQ_1:def 13;
ex n being Nat st P c= Seg n by MATRIX13:43;
then rng (Sgm P) = P by FINSEQ_1:def 13;
hence Segm (A + B),P,Q = (Segm A,P,Q) + (Segm B,P,Q) by A1, A2, Th1; :: thesis: verum