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