
theorem
  for X being set, L being non empty ZeroStr, s being Series of X,L
  holds s|(Support s) = s & s|({} Bags X) = 0_(X,L)
proof
  let X be set, L be non empty ZeroStr, s be Series of X,L;
  set r = s|(Support s);
  set e = s|({} Bags X);
  r = s +* ({} --> 0.L) & dom({} --> 0.L) = {} by XBOOLE_1:37;
  hence r = s +* {}
    .= s;
A1: dom((Support s) --> 0.L) = Support s by FUNCOP_1:13;
A2: now
    let u be object;
    assume u in dom e;
    then reconsider u9 = u as Element of Bags X;
    now
      per cases;
      case
A3:     u9 in Support s;
        then e.u9 = ((Support s) --> 0.L).u9 by A1,FUNCT_4:13
          .= 0.L by A3,FUNCOP_1:7;
        hence e.u9 = 0_(X,L).u9 by POLYNOM1:22;
      end;
      case
A4:     not u9 in Support s;
        then e.u9 = s.u9 by A1,FUNCT_4:11;
        then e.u9 = 0.L by A4,POLYNOM1:def 4;
        hence e.u9 = 0_(X,L).u9 by POLYNOM1:22;
      end;
    end;
    hence e.u = 0_(X,L).u;
  end;
  dom e = Bags X by FUNCT_2:def 1
    .= dom(0_(X,L)) by FUNCT_2:def 1;
  hence thesis by A2,FUNCT_1:2;
end;
