let K be Field; :: thesis: for a being Element of
for M being Matrix of
for P, Q being finite without_zero Subset of st [:P,Q:] c= Indices M holds
a * (Segm M,P,Q) = Segm (a * M),P,Q
let a be Element of ; :: thesis: for M being Matrix of
for P, Q being finite without_zero Subset of st [:P,Q:] c= Indices M holds
a * (Segm M,P,Q) = Segm (a * M),P,Q
let M be Matrix of ; :: thesis: for P, Q being finite without_zero Subset of st [:P,Q:] c= Indices M holds
a * (Segm M,P,Q) = Segm (a * M),P,Q
let P, Q be finite without_zero Subset of ; :: thesis: ( [:P,Q:] c= Indices M implies a * (Segm M,P,Q) = Segm (a * M),P,Q )
ex n being Nat st P c= Seg n by Th43;
then A1: rng (Sgm P) = P by FINSEQ_1:def 13;
ex k being Nat st Q c= Seg k by Th43;
then A2: rng (Sgm Q) = Q by FINSEQ_1:def 13;
assume [:P,Q:] c= Indices M ; :: thesis: a * (Segm M,P,Q) = Segm (a * M),P,Q
hence a * (Segm M,P,Q) = Segm (a * M),P,Q by A1, A2, Th41; :: thesis: verum