let V be strict RealLinearSpace; :: thesis: for W being strict Subspace of V st ( for v being VECTOR of V holds
( v in W iff v in V ) ) holds
W = V
let W be strict Subspace of V; :: thesis: ( ( for v being VECTOR of V holds
( v in W iff v in V ) ) implies W = V )
assume A1: for v being VECTOR of V holds
( v in W iff v in V ) ; :: thesis: W = V
V is Subspace of V by Th25;
hence W = V by A1, Th31; :: thesis: verum