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 Th15:
  abs(G(#)H) = abs(G) (#) abs(H)
proof
  now
    let n be Element of NAT;
    thus (abs(G(#)H)).n=abs(((G(#)H).n)) by Def4
      .=abs((G.n)(#)(H.n)) by Def7
      .=abs((G.n))(#)abs((H.n)) by RFUNCT_1:24
      .=((abs(G)).n)(#)abs((H.n)) by Def4
      .=((abs(G)).n)(#)(abs(H)).n by Def4
      .=(abs(G)(#)abs(H)).n by Def7;
  end;
  hence thesis by FUNCT_2:63;
end;
