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 Th27:
  {x} common_on_dom H1 & {x} common_on_dom H2 implies H1#x + H2#x
  = (H1+H2)#x & H1#x - H2#x = (H1-H2)#x & (H1#x) (#) (H2#x) = (H1(#)H2)#x
proof
  assume that
A1: {x} common_on_dom H1 and
A2: {x} common_on_dom H2;
  now
    let n be Element of NAT;
A3: {x} c= dom (H2.n) by A2;
    x in {x} & {x} c= dom(H1.n) by A1,TARSKI:def 1;
    then x in (dom(H1.n) /\ dom(H2.n)) by A3,XBOOLE_0:def 4;
    then
A4: x in dom(H1.n + H2.n) by VALUED_1:def 1;
    thus (H1#x + H2#x).n = (H1#x).n + (H2#x).n by SEQ_1:7
      .= (H1.n).x + (H2#x).n by Def10
      .= (H1.n).x + (H2.n).x by Def10
      .= ((H1.n) + (H2.n)).x by A4,VALUED_1:def 1
      .= ((H1+H2).n).x by Def5
      .= ((H1+H2)#x).n by Def10;
  end;
  hence H1#x + H2#x = (H1+H2)#x by FUNCT_2:63;
  now
    let n be Element of NAT;
A5: {x} c= dom (H2.n) by A2;
    x in {x} & {x} c= dom(H1.n) by A1,TARSKI:def 1;
    then x in (dom(H1.n) /\ dom(H2.n)) by A5,XBOOLE_0:def 4;
    then
A6: x in dom(H1.n - H2.n) by VALUED_1:12;
    thus (H1#x - H2#x).n = (H1#x).n - (H2#x).n by RFUNCT_2:1
      .= (H1.n).x - (H2#x).n by Def10
      .= (H1.n).x - (H2.n).x by Def10
      .= ((H1.n) - (H2.n)).x by A6,VALUED_1:13
      .= ((H1-H2).n).x by Th3
      .= ((H1-H2)#x).n by Def10;
  end;
  hence H1#x - H2#x = (H1-H2)#x by FUNCT_2:63;
  now
    let n be Element of NAT;
    thus ((H1#x) (#) (H2#x)).n = (H1#x).n * (H2#x).n by SEQ_1:8
      .= (H1.n).x * (H2#x).n by Def10
      .= (H1.n).x * (H2.n).x by Def10
      .= ((H1.n) (#) (H2.n)).x by VALUED_1:5
      .= ((H1(#)H2).n).x by Def7
      .= ((H1(#)H2)#x).n by Def10;
  end;
  hence thesis by FUNCT_2:63;
end;
