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;

theorem
  abs(r(#)H) = |.r.| (#) abs(H)
proof
  now
    let n be Element of NAT;
    thus abs(r(#)H).n=abs((r(#)H).n) by Def4
      .=abs(r(#)(H.n)) by Def1
      .=|.r.|(#)abs(H.n) by RFUNCT_1:25
      .=|.r.|(#)(abs(H)).n by Def4
      .=(|.r.|(#)abs(H)).n by Def1;
  end;
  hence thesis by FUNCT_2:63;
end;
