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 Th37:
  X common_on_dom H implies X common_on_dom abs(H) & X common_on_dom (-H)
proof
  assume
A1: X common_on_dom H;
  then
A2: X <> {};
  now
    let n;
    dom (H.n) = dom |.H.n.| by VALUED_1:def 11
      .= dom (abs(H).n) by Def4;
    hence X c= dom (abs(H).n) by A1;
  end;
  hence X common_on_dom abs(H) by A2;
  now
    let n;
    dom (H.n) = dom (-(H.n)) by VALUED_1:8
      .= dom ((-H).n) by Def3;
    hence X c= dom ((-H).n) by A1;
  end;
  hence thesis by A2;
end;
