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

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 & (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;
