reserve x,y,y1,y2 for object;
reserve R for Ring;
reserve a for Scalar of R;
reserve V,X,Y for RightMod of R;
reserve u,u1,u2,v,v1,v2 for Vector of V;
reserve V1,V2,V3 for Subset of V;
reserve W,W1,W2 for Submodule of V;
reserve w,w1,w2 for Vector of W;

theorem
  (for v st v in W1 holds v in W2) implies W1 is Submodule 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 Def2;
    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 Th27;
end;
