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, v being VECTOR of V, L being
C_Linear_Combination of {v} st L is convex holds ( ex r being Real st r = L.v &
  r = 1 ) & Sum L = L.v * v
proof
  let V be ComplexLinearSpace;
  let v be VECTOR of V;
  let L be C_Linear_Combination of {v};
  Carrier L c= {v} by Def4;
  then
A1: Carrier L = {} or Carrier L = {v} by ZFMISC_1:33;
  assume L is convex;
  hence thesis by A1,Th77,Th80;
end;
