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 Th23:
  u in W & v in W implies u - v in W
proof
  assume that
A1: u in W and
A2: v in W;
  - v in W by A2,Th22;
  hence thesis by A1,Th20;
end;
