reserve i,j,k,n,m for Nat,
        b,b1,b2 for bag of n;

theorem Th11:
  support b = support (b bag_extend 0)
proof
  set E = b bag_extend 0;
A1: b = E|n by HILBASIS:def 1;
  thus support b c= support E
  proof
    let x be object;
    assume x in support b;
    then
A2: b.x <>0 by PRE_POLY:def 7;
    then x in dom b by FUNCT_1:def 2;
    then E.x = b.x by A1,FUNCT_1:49;
    hence thesis by A2,PRE_POLY:def 7;
  end;
  let x be object;
  assume x in support E;
  then
A3:E.x <>0 by PRE_POLY:def 7;
  then
A4:x <> n & x in dom E by HILBASIS:def 1,FUNCT_1:def 2;
  dom E=Segm(n+1) by PARTFUN1:def 2;
  then x in (Segm n)\/{n} by A4,AFINSQ_1:2;
  then E.x = b.x by A1,FUNCT_1:49,A4,ZFMISC_1:136;
  hence thesis by A3,PRE_POLY:def 7;
end;
