reserve D for non empty set,
  D1, D2, x, y, Z for set,
  n, k for Nat,
  p, x1, r for Real,
  f for Function,
  Y for RealNormSpace,
  G, H, H1, H2, J for Functional_Sequence of D,the carrier of Y;
reserve
  x for Element of D,
  X for set,
  S1, S2 for sequence of Y,
  f for PartFunc of D,the carrier of Y;
reserve x for Element of D;

theorem Th30:
  X common_on_dom H1 & X common_on_dom H2 implies for x st x in X
  holds H1#x + H2#x = (H1+H2)#x & H1#x - H2#x = (H1-H2)#x 
  proof
    assume
    A1: X common_on_dom H1 & X common_on_dom H2;
    let x;
    assume x in X;
    then {x} common_on_dom H1 & {x} common_on_dom H2 by A1, Th25;
    hence thesis by Th27;
  end;
