let V be RealLinearSpace; :: thesis: for M being Subset of V
for r being Real st M is circled holds
r * M is circled
let M be Subset of V; :: thesis: for r being Real st M is circled holds
r * M is circled
let r be Real; :: thesis: ( M is circled implies r * M is circled )
assume A1: M is circled ; :: thesis: r * M is circled
for u being VECTOR of V
for p being Real st abs p <= 1 & u in r * M holds
p * u in r * M

proof

let u be VECTOR of V; :: thesis: for p being Real st abs p <= 1 & u in r * M holds
p * u in r * M
let p be Real; :: thesis: ( abs p <= 1 & u in r * M implies p * u in r * M )
assume A2: ( abs p <= 1 & u in r * M ) ; :: thesis: p * u in r * M
then u in { (r * w) where w is Element of V : w in M } by CONVEX1:def 1;
then consider u' being Element of V such that
A3: ( u = r * u' & u' in M ) ;
A4: p * u = (r * p) * u' by A3, RLVECT_1:def 9
.= r * (p * u') by RLVECT_1:def 9 ;
p * u' in M by A1, A2, A3, Def1;
hence p * u in r * M by A4, RLTOPSP1:1; :: thesis: verum

end;

hence r * M is circled by Def1; :: thesis: verum