reserve x,y for set,
  i for Nat;
reserve V for non empty CLSStruct,
  u,v,v1,v2,v3 for VECTOR of V,
  A for Subset of V,
  l, l1, l2 for C_Linear_Combination of A,
  x,y,y1,y2 for set,
  a,b for Complex,
  F for FinSequence of the carrier of V,
  f for Function of the carrier of V, COMPLEX;
reserve K,L,L1,L2,L3 for C_Linear_Combination of V;
reserve e,e1,e2 for Element of C_LinComb V;

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;
