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 Th11:
  for V being Abelian add-associative right_zeroed
  right_complementable non empty CLSStruct holds Sum(ZeroCLC V) = 0.V
proof
  let V be Abelian add-associative right_zeroed right_complementable non empty
  CLSStruct;
  consider F being FinSequence of the carrier of V such that
  F is one-to-one and
A1: rng F = Carrier (ZeroCLC V) and
A2: Sum(ZeroCLC V) = Sum(ZeroCLC V (#) F) by Def6;
  F = {} by A1,RELAT_1:41;
  then len(ZeroCLC V (#) F) = 0 by Def5,CARD_1:27;
  hence thesis by A2,RLVECT_1:75;
end;
