let K be Field; :: thesis: for A, B being Matrix of K
for P, Q being finite without_zero Subset of NAT 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 K; :: thesis: for P, Q being finite without_zero Subset of NAT 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 NAT; :: 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