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;

theorem Th18:
  for V being ComplexLinearSpace, L being C_Linear_Combination of
V, v1, v2 being VECTOR of V holds Carrier L = {v1,v2} & v1 <> v2 implies Sum L
  = L.v1 * v1 + L.v2 * v2
proof
  let V be ComplexLinearSpace;
  let L be C_Linear_Combination of V;
  let v1, v2 be VECTOR of V;
  assume that
A1: Carrier L = {v1,v2} and
A2: v1 <> v2;
  L is C_Linear_Combination of {v1,v2} by A1,Def4;
  hence thesis by A2,Th15;
end;
