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
  r(#)(G + H) = r(#)G + r(#)H & r(#)(G - H) = r(#)G - r(#)H
proof
  now
    let n be Element of NAT;
    thus (r(#)(G + H)).n = r(#)(G + H).n by Def1
      .= r(#)(G.n + H.n) by Def5
      .= r(#)(G.n) + r(#)(H.n) by RFUNCT_1:16
      .=(r(#)G).n + r(#)(H.n) by Def1
      .=(r(#)G).n + (r(#)H).n by Def1
      .=(r(#)G + r(#)H).n by Def5;
  end;
  hence r(#)(G + H) = r(#)G + r(#)H by FUNCT_2:63;
  now
    let n be Element of NAT;
    thus (r(#)(G - H)).n = r(#)(G - H).n by Def1
      .= r(#)(G.n - H.n) by Th3
      .= r(#)G.n - r(#)H.n by RFUNCT_1:18
      .= (r(#)G).n - r(#)H.n by Def1
      .= (r(#)G).n - (r(#)H).n by Def1
      .= (r(#)G - r(#)H).n by Th3;
  end;
  hence thesis by FUNCT_2:63;
end;
