
theorem
  for S be non empty finite set holds
  uniform_distribution(S) = distribution(Uniform_FDprobSEQ(S),S)
proof
  let S be non empty finite set;
  set p=Uniform_FDprobSEQ(S),s=canFS(S);
A1: for t be FinSequence of S st t is uniformly_distributed holds t in
  Finseq-EQclass(s)
  proof
    let t be FinSequence of S;
    assume t is uniformly_distributed;
    then s,t -are_prob_equivalent by Th21;
    hence thesis;
  end;
  (for t be FinSequence of S st t in Finseq-EQclass(s) holds t
  is uniformly_distributed)&
  Finseq-EQclass(s) is Element of distribution_family S by Def6,Lm3; then
A2: Finseq-EQclass(s)=uniform_distribution(S) by A1,Def12;
  (GenProbSEQ S).(Finseq-EQclass s) = p by Th10;
  then
A3: (GenProbSEQ S).distribution(p,S) = (GenProbSEQ S).(Finseq-EQclass s)
    by Th18;
  dom GenProbSEQ S = distribution_family(S) by FUNCT_2:def 1;
  hence thesis by A2,A3,FUNCT_1:def 4;
end;
