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
  H is_point_conv_on X implies r(#)H is_point_conv_on X & lim (r(#)H,X)
  = r(#)(lim(H,X))
proof
  assume
A1: H is_point_conv_on X;
  then
A2: X common_on_dom H;
A3: now
    let x;
    assume
A4: x in dom(r(#)(lim(H,X)));
    then
A5: x in dom lim(H,X) by VALUED_1:def 5;
    then
A6: x in X by A1,Def13;
    then
A7: H#x is convergent by A1,Th19;
    thus (r(#)lim(H,X)).x = r*(lim(H,X).x) by A4,VALUED_1:def 5
      .= r*(lim(H#x)) by A1,A5,Def13
      .= lim(r(#)(H#x)) by A7,SEQ_2:8
      .= lim((r(#)H)#x) by A2,A6,Th32;
  end;
A8: now
    let x;
    assume
A9: x in X;
    then H#x is convergent by A1,Th19;
    then r(#)(H#x) is convergent by SEQ_2:7;
    hence (r(#)H)#x is convergent by A2,A9,Th32;
  end;
  X common_on_dom r(#)H by A2,Th38;
  hence
A10: r(#)H is_point_conv_on X by A8,Th19;
  dom(r(#)(lim(H,X))) = dom lim(H,X) by VALUED_1:def 5
    .= X by A1,Def13;
  hence thesis by A10,A3,Def13;
end;
