let V be RealLinearSpace; :: thesis: for M1, M2 being Subset of V
for r1, r2 being Real st M1 is circled & M2 is circled holds
(r1 * M1) + (r2 * M2) is circled
let M1, M2 be Subset of V; :: thesis: for r1, r2 being Real st M1 is circled & M2 is circled holds
(r1 * M1) + (r2 * M2) is circled
let r1, r2 be Real; :: thesis: ( M1 is circled & M2 is circled implies (r1 * M1) + (r2 * M2) is circled )
assume that
A1: M1 is circled and
A2: M2 is circled ; :: thesis: (r1 * M1) + (r2 * M2) is circled
let u be VECTOR of V; :: according to CIRCLED1:def 1 :: thesis: for r being Real st |.r.| <= 1 & u in (r1 * M1) + (r2 * M2) holds
r * u in (r1 * M1) + (r2 * M2)
let p be Real; :: thesis: ( |.p.| <= 1 & u in (r1 * M1) + (r2 * M2) implies p * u in (r1 * M1) + (r2 * M2) )
assume that
A3: |.p.| <= 1 and
A4: u in (r1 * M1) + (r2 * M2) ; :: thesis: p * u in (r1 * M1) + (r2 * M2)
u in { (x + y) where x, y is VECTOR of V : ( x in r1 * M1 & y in r2 * M2 ) } by A4, RUSUB_4:def 9;
then consider u1, u2 being VECTOR of V such that
A5: u = u1 + u2 and
A6: u1 in r1 * M1 and
A7: u2 in r2 * M2 ;
u1 in { (r1 * x) where x is VECTOR of V : x in M1 } by A6, CONVEX1:def 1;
then consider x1 being VECTOR of V such that
A8: u1 = r1 * x1 and
A9: x1 in M1 ;
A10: p * u1 = (r1 * p) * x1 by A8, RLVECT_1:def 7
.= r1 * (p * x1) by RLVECT_1:def 7 ;
u2 in { (r2 * x) where x is VECTOR of V : x in M2 } by A7, CONVEX1:def 1;
then consider x2 being VECTOR of V such that
A11: u2 = r2 * x2 and
A12: x2 in M2 ;
A13: p * u2 = (r2 * p) * x2 by A11, RLVECT_1:def 7
.= r2 * (p * x2) by RLVECT_1:def 7 ;
reconsider r1 = r1, r2 = r2 as Real ;
p * x2 in M2 by A2, A3, A12;
then A14: p * u2 in r2 * M2 by A13, RLTOPSP1:1;
p * x1 in M1 by A1, A3, A9;
then A15: p * u1 in r1 * M1 by A10, RLTOPSP1:1;
p * (u1 + u2) = (p * u1) + (p * u2) by RLVECT_1:def 5;
then p * (u1 + u2) in { (x + y) where x, y is VECTOR of V : ( x in r1 * M1 & y in r2 * M2 ) } by A15, A14;
hence p * u in (r1 * M1) + (r2 * M2) by A5, RUSUB_4:def 9; :: thesis: verum