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 st v in W1 holds v in W2) implies W1 is Subspace of W2
proof
  assume
A1: for v st v in W1 holds v in W2;
  the carrier of W1 c= the carrier of W2
  proof
    let x be object;
    assume
A2: x in the carrier of W1;
    the carrier of W1 c= the carrier of V by Def8;
    then reconsider v = x as VECTOR of V by A2;
    v in W1 by A2;
    then v in W2 by A1;
    hence thesis;
  end;
  hence thesis by Th47;
end;
