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 Th13:
  w1 = v & w2 = u implies w1 + w2 = v + u
proof
  assume
A1: v = w1 & u = w2;
  set IW = [:the carrier of W, the carrier of W:];
  w1 + w2 = ((the addF of V)|(IW qua set)).[w1,w2] by Def2;
  hence thesis by A1,FUNCT_1:49;
end;
