reserve V for non empty RLSStruct;
reserve x,y,y1 for set;
reserve v for VECTOR of V;
reserve a,b for Real;
reserve V for non empty addLoopStr;
reserve F for FinSequence-like PartFunc of NAT,V;
reserve f,f9,g for sequence of V;
reserve v,u for Element of V;
reserve j,k,n for Nat;
reserve V for RealLinearSpace;
reserve v for VECTOR of V;
reserve F,G,H,I for FinSequence of V;
reserve V for add-associative right_zeroed right_complementable non empty
  addLoopStr;
reserve F for FinSequence of V;
reserve v,v1,v2,u,w for Element of V;
reserve j,k for Nat;

theorem
  Sum<*0.V,u,v*> = u + v & Sum<*u,v,0.V*> = u + v & Sum<*u,0.V,v*> = u + v
proof
  thus Sum<* 0.V,u,v *> = 0.V + u + v by Th46
    .= u + v;
  thus Sum<* u,v,0.V *> = u + v + 0.V by Th46
    .= u + v;
  thus Sum<* u,0.V,v *> = u + 0.V + v by Th46
    .= u + v;
end;
