
theorem Th13h:
for n,m being Ordinal
for b1,b2 being bag of n st support b1 = {m}
holds b2 divides b1 iff (b2 = EmptyBag n or (support b2 = {m} & b2.m <= b1.m))
proof
let n,m be Ordinal; let b1,b2 be bag of n;
assume AS: support b1 = {m};
A: now assume b2 = EmptyBag n or (support b2 = {m} & b2.m <= b1.m); then
   per cases;
   suppose b2 = EmptyBag n;
     hence b2 divides b1 by PRE_POLY:59;
     end;
   suppose A2: support b2 = {m} & b2.m <= b1.m;
       now let k be object;
         assume k in n;
         per cases;
         suppose k = m;
           hence b2.k <= b1.k by A2;
           end;
         suppose k <> m; then
           A3: not k in {m} by TARSKI:def 1;
           then b2.k = 0 by A2,PRE_POLY:def 7
                    .= b1.k by A3,AS,PRE_POLY:def 7;
           hence b2.k <= b1.k;
           end;
         end;
       hence b2 divides b1 by PRE_POLY:46;
     end;
  end;
H: now let l be object;
   assume l <> m;
   then not l in support b1 by AS,TARSKI:def 1;
   hence b1.l = 0 by PRE_POLY:def 7;
   end;
now assume A4: b2 divides b1;
  then A5: b2 <=' b1 by PRE_POLY:49;
  assume A6: b2 <> EmptyBag n;
  support b2 = {m}
    proof
    B: now let o be object;
       assume B1: o in support b2;
       now assume o <> m;
         then b1.o = 0 by H;
         then b2.o = 0 by A4,PRE_POLY:def 11;
         hence contradiction by B1,PRE_POLY:def 7;
         end;
       hence o in {m} by TARSKI:def 1;
       end;
    now let o be object;
      assume o in {m};
      then B2: o = m by TARSKI:def 1;
      now assume B3: b2.m = 0;
        now let u be object;
          assume u in n;
          per cases;
          suppose u = m;
            hence b2.u = {} by B3;
            end;
          suppose u <> m;
            then b1.u = 0 by H;
            hence b2.u = {} by A4,PRE_POLY:def 11;
            end;
          end;
        hence contradiction by A6,PBOOLE:6;
        end;
      hence o in support b2 by B2,PRE_POLY:def 7;
      end;
    hence thesis by B,TARSKI:2;
    end;
  hence support b2 = {m} & b2.m <= b1.m by A5,AS,Th13e;
  end;
hence thesis by A;
end;
