
theorem
  for V being Abelian add-associative vector-distributive scalar-distributive
  scalar-associative scalar-unital non empty
  RLSStruct, M1,M2,M3 being Subset of V,
   r1,r2,r3 being Real st M1 is convex &
  M2 is convex & M3 is convex holds r1*M1 + r2*M2 + r3*M3 is convex
proof
  let V be Abelian add-associative vector-distributive scalar-distributive
  scalar-associative scalar-unital non empty RLSStruct;
  let M1,M2,M3 be Subset of V;
  let r1,r2,r3 be Real;
  assume that
A1: M1 is convex & M2 is convex and
A2: M3 is convex;
  r1*M1 + r2*M2 is convex by A1,Th11;
  then 1*(r1*M1 + r2*M2) + r3*M3 is convex by A2,Th11;
  hence thesis by Lm1;
end;
