let V be RealUnitarySpace; :: thesis: for W1, W2 being Subspace of V st ( for v being VECTOR of V st v in W1 holds
v in W2 ) holds
W1 is Subspace of W2
let W1, W2 be Subspace of V; :: thesis: ( ( for v being VECTOR of V st v in W1 holds
v in W2 ) implies W1 is Subspace of W2 )
assume A1: for v being VECTOR of V st v in W1 holds
v in W2 ; :: thesis: W1 is Subspace of W2
the carrier of W1 c= the carrier of W2

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in the carrier of W1 or x in the carrier of W2 )
assume A2: x in the carrier of W1 ; :: thesis: x in the carrier of W2
the carrier of W1 c= the carrier of V by Def1;
then reconsider v = x as VECTOR of V by A2;
v in W1 by A2;
then v in W2 by A1;
hence x in the carrier of W2 ; :: thesis: verum

end;

hence W1 is Subspace of W2 by Th22; :: thesis: verum