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

proof

let x be object ; :: thesis: ( x in the carrier of W1 iff x in the carrier of W2 )
thus ( x in the carrier of W1 implies x in the carrier of W2 ) :: thesis: ( x in the carrier of W2 implies x in the carrier of W1 )

proof

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;

assume A3: x in the carrier of W2 ; :: thesis: x in the carrier of W1
the carrier of W2 c= the carrier of V by Def1;
then reconsider v = x as VECTOR of V by A3;
v in W2 by A3;
then v in W1 by A1;
hence x in the carrier of W1 ; :: thesis: verum

end;

then the carrier of W1 = the carrier of W2 by TARSKI:2;
hence W1 = W2 by Th24; :: thesis: verum