
theorem Th6:
  for S be non empty finite set,
  D be Element of distribution_family(S)
  holds not D is well-distributed iff
  D = {<*>S}
  proof
    let S be non empty finite set,
    D be Element of distribution_family(S);
    thus not D is well-distributed implies D = {<*>S}
    proof
      assume not D is well-distributed; then
      reconsider o = {} as Element of D by SETFAM_1:def 8;
      A1: for s be Element of D holds s in {<*>S}& s=<*>S
      proof
        let s be Element of D;
        for x be set holds FDprobability(x,s) = 0
        proof
          let x be set;
          FDprobability(x,s)=FDprobability(x,o) by Th4,DIST_1:def 4;
          hence thesis;
        end;
        then s is empty by Th5;
        hence thesis by TARSKI:def 1;
      end;
      then A2: for z be object st z in D holds z in {<*>S};
      for z be object st z in {<*>S} holds z in D
      proof
        let z be object;
        assume A3: z in {<*>S};
        z is Element of D
        proof
          assume A4:not z is Element of D;
          set t = the Element of D;
          t=<*>S by A1;
          hence contradiction by A4,A3,TARSKI:def 1;
        end;
        hence thesis;
      end;
      hence thesis by A2,TARSKI:2;
    end;
    assume A5:D = {<*>S};
    assume A6:D is well-distributed;
    set s = the Element of D;
    s={} by A5,TARSKI:def 1;
    hence contradiction by A6;
  end;
