reserve a,b for Complex;
reserve V,X,Y for ComplexLinearSpace;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve z,z1,z2 for Complex;
reserve V1,V2,V3 for Subset of V;
reserve W,W1,W2 for Subspace of V;
reserve x for set;
reserve w,w1,w2 for VECTOR of W;
reserve D for non empty set;
reserve d1 for Element of D;
reserve A for BinOp of D;
reserve M for Function of [:COMPLEX,D:],D;

theorem
  for V being strict ComplexLinearSpace, W being strict Subspace of V
  holds (for v being VECTOR of V holds v in W iff v in V) implies W = V
proof
  let V be strict ComplexLinearSpace, W be strict Subspace of V;
  assume
A1: for v being VECTOR of V holds v in W iff v in V;
  V is Subspace of V by Th44;
  hence thesis by A1,Th50;
end;
