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 ComplexLinearSpace, M being Subset of V, z1,z2 being
  Complex st (ex r1,r2 being Real st z1 = r1 & z2 = r2 & r1 >= 0 & r2 >= 0) & M
  is convex holds z1*M + z2*M = (z1 + z2)*M
proof
  let V be ComplexLinearSpace, M be Subset of V, z1,z2 be Complex;
  assume
  ( ex r1,r2 being Real st z1=r1 & z2=r2 & r1 >= 0 & r2 >= 0)& M is convex;
  hence z1*M + z2*M c= (z1 + z2)*M by Lm2;
  thus thesis by Th60;
end;
