
theorem Th16:
  for X being set, L being non empty ZeroStr, s being Series of X,
L, Y being Subset of Bags X holds Support s|Y = (Support s) /\ Y & for b being
  bag of X st b in Support s|Y holds (s|Y).b = s.b
proof
  let X be set, L be non empty ZeroStr, s be Series of X,L, Y be Subset of
  Bags X;
  set m = (Support s \ Y) --> 0.L;
  set r = s +* m;
A1: now
    let u be object;
    assume
A2: u in Support(s|Y);
    then reconsider u9 = u as Element of Bags X;
A3: now
      assume
A4:   u9 in dom m;
      then r.u9 = m.u9 by FUNCT_4:13
        .= 0.L by A4,FUNCOP_1:7;
      hence contradiction by A2,POLYNOM1:def 4;
    end;
    r.u9 <> 0.L by A2,POLYNOM1:def 4;
    then s.u9 <> 0.L by A3,FUNCT_4:11;
    then
A5: u9 in Support s by POLYNOM1:def 4;
    dom m = Support s \ Y by FUNCOP_1:13;
    then u9 in Y by A3,A5,XBOOLE_0:def 5;
    hence u in (Support s) /\ Y by A5,XBOOLE_0:def 4;
  end;
A6: dom m = Support s \ Y by FUNCOP_1:13;
  now
    let u be object;
    assume
A7: u in Support s /\ Y;
    then
A8: u in Support s by XBOOLE_0:def 4;
    then reconsider u9 = u as Element of Bags X;
    u in Y by A7,XBOOLE_0:def 4;
    then not u in Support s \ Y by XBOOLE_0:def 5;
    then r.u9 = s.u9 by A6,FUNCT_4:11;
    then r.u9 <> 0.L by A8,POLYNOM1:def 4;
    hence u in Support s|Y by POLYNOM1:def 4;
  end;
  hence
A9: Support s|Y = Support s /\ Y by A1,TARSKI:2;
  now
    let b be bag of X;
    assume b in Support s|Y;
    then b in Y by A9,XBOOLE_0:def 4;
    then not b in dom m by XBOOLE_0:def 5;
    hence (s|Y).b = s.b by FUNCT_4:11;
  end;
  hence thesis;
end;
