reserve V,X,Y for RealLinearSpace;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve a for Real;
reserve V1,V2,V3 for Subset of V;
reserve x for object;
reserve W,W1,W2 for Subspace 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;
  w1 + w2 = ((the addF of V)||the carrier of W).[w1,w2] by Def2;
  hence thesis by A1,FUNCT_1:49;
end;
