reserve x,y,y1,y2 for object;
reserve GF for add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr,
  V,X,Y for Abelian add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital non
  empty ModuleStr over GF;
reserve a for Element of GF;
reserve u,u1,u2,v,v1,v2 for Element of V;
reserve W,W1,W2 for Subspace of V;
reserve V1 for Subset of V;
reserve w,w1,w2 for Element of W;

theorem Th20:
  u in W & v in W implies u + v in W
proof
  reconsider VW = the carrier of W as Subset of V by Def2;
  assume u in W & v in W;
  then
A1: u in the carrier of W & v in the carrier of W;
  VW is linearly-closed by Lm2;
  then u + v in the carrier of W by A1;
  hence thesis;
end;
