reserve V,X,Y for RealLinearSpace;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve a for Real;
reserve V1,V2,V3 for Subset of V;
reserve x for object;
reserve W,W1,W2 for Subspace of V;
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 [:REAL,D:],D;

theorem
  for V being strict RealLinearSpace, 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 RealLinearSpace, 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 Th25;
  hence thesis by A1,Th31;
end;
