
theorem Th11:
  for V being Abelian add-associative vector-distributive scalar-distributive
  scalar-associative scalar-unital non
  empty RLSStruct, M1,M2 being Subset of V, r1,r2 being Real
   st M1 is convex & M2 is convex holds r1*M1 + r2*M2 is convex
proof
  let V be Abelian add-associative vector-distributive scalar-distributive
  scalar-associative scalar-unital non empty RLSStruct;
  let M1,M2 be Subset of V;
  let r1,r2 be Real;
  assume that
A1: M1 is convex and
A2: M2 is convex;
  let u,v be VECTOR of V;
  let p be Real;
  assume that
A3: 0 < p & p < 1 and
A4: u in r1*M1 + r2*M2 and
A5: v in r1*M1 + r2*M2;
  v in {x+y where x,y is VECTOR of V : x in r1*M1 & y in r2*M2} by A5,
RUSUB_4:def 9;
  then consider v1,v2 be VECTOR of V such that
A6: v = v1 + v2 and
A7: v1 in r1*M1 and
A8: v2 in 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 be VECTOR of V such that
A9: u = u1 + u2 and
A10: u1 in r1*M1 and
A11: u2 in r2*M2;
  consider y1 be VECTOR of V such that
A12: v1 = r1*y1 and
A13: y1 in M1 by A7;
  consider x1 be VECTOR of V such that
A14: u1 = r1*x1 and
A15: x1 in M1 by A10;
A16: p*u1 + (1-p)*v1 = r1*p*x1 + (1-p)*(r1*y1) by A14,A12,RLVECT_1:def 7
    .= r1*p*x1 + r1*(1-p)*y1 by RLVECT_1:def 7
    .= r1*(p*x1) + r1*(1-p)*y1 by RLVECT_1:def 7
    .= r1*(p*x1) + r1*((1-p)*y1) by RLVECT_1:def 7
    .= r1*(p*x1 + (1-p)*y1) by RLVECT_1:def 5;
  consider y2 be VECTOR of V such that
A17: v2 = r2*y2 and
A18: y2 in M2 by A8;
  consider x2 be VECTOR of V such that
A19: u2 = r2*x2 and
A20: x2 in M2 by A11;
A21: p*u2 + (1-p)*v2 = r2*p*x2 + (1-p)*(r2*y2) by A19,A17,RLVECT_1:def 7
    .= r2*p*x2 + r2*(1-p)*y2 by RLVECT_1:def 7
    .= r2*(p*x2) + r2*(1-p)*y2 by RLVECT_1:def 7
    .= r2*(p*x2) + r2*((1-p)*y2) by RLVECT_1:def 7
    .= r2*(p*x2 + (1-p)*y2) by RLVECT_1:def 5;
  p*x2 + (1-p)*y2 in M2 by A2,A3,A20,A18;
  then
A22: p*u2 + (1-p)*v2 in r2*M2 by A21;
  p*x1 + (1-p)*y1 in M1 by A1,A3,A15,A13;
  then
A23: p*u1 + (1-p)*v1 in {r1*x where x is VECTOR of V: x in M1} by A16;
  p*(u1+u2) + (1-p)*(v1+v2) = p*u1 + p*u2 + (1-p)*(v1+v2) by RLVECT_1:def 5
    .= p*u1 + p*u2 + ((1-p)*v1 + (1-p)*v2) by RLVECT_1:def 5
    .= p*u1 + p*u2 + (1-p)*v1 + (1-p)*v2 by RLVECT_1:def 3
    .= p*u1 + (1-p)*v1 + p*u2 + (1-p)*v2 by RLVECT_1:def 3
    .= p*u1 + (1-p)*v1 + (p*u2 + (1-p)*v2) by RLVECT_1:def 3;
  then p*(u1+u2) + (1-p)*(v1+v2) in {x+y where x,y is VECTOR of V: x in r1*M1
  & y in r2*M2} by A23,A22;
  hence thesis by A9,A6,RUSUB_4:def 9;
end;
