theorem
  for V being Abelian add-associative vector-distributive scalar-distributive
  scalar-associative scalar-unital non empty
CLSStruct, M1,M2,M3 being Subset of V, z1,z2,z3 being Complex st M1 is convex
  & M2 is convex & M3 is convex holds z1*M1 + z2*M2 + z3*M3 is convex
proof
  let V be Abelian add-associative vector-distributive scalar-distributive
  scalar-associative scalar-unital non empty
  CLSStruct;
  let M1,M2,M3 be Subset of V;
  let z1,z2,z3 be Complex;
  assume that
A1: M1 is convex & M2 is convex and
A2: M3 is convex;
  z1*M1 + z2*M2 is convex by A1,Th59;
  then 1r*(z1*M1 + z2*M2) + z3*M3 is convex by A2,Th59;
  hence thesis by Th51;
end;
