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 Th24:
  V is Submodule of V
proof
A1: the rmult of V = (the rmult of V) |([:the carrier of V, the carrier of R
  :] qua set) by RELSET_1:19;
  0.V = 0.V & the addF of V = (the addF of V) | ([:the carrier of V, the
  carrier of V :] qua set) by RELSET_1:19;
  hence thesis by A1,Def2;
end;
