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 Th28:
  (abs(H))#x = abs(H#x) & (-H)#x = -(H#x)
proof
  now
    let n be Element of NAT;
    thus ((abs(H))#x).n = (abs(H).n).x by Def10
      .= |.(H.n).|.x by Def4
      .= |.(H.n).x.| by VALUED_1:18
      .= |.(H#x).n.| by Def10
      .= abs(H#x).n by SEQ_1:12;
  end;
  hence (abs(H))#x = abs(H#x) by FUNCT_2:63;
  now
    let n be Element of NAT;
    thus ((-H)#x).n = ((-H).n).x by Def10
      .= (-H.n).x by Def3
      .= -((H.n).x) by VALUED_1:8
      .= -((H#x).n) by Def10
      .= (-(H#x)).n by SEQ_1:10;
  end;
  hence thesis by FUNCT_2:63;
end;
