
theorem Th7:
  for S be non empty finite set,
  D be EqSampleSpaces of S holds
  (GenProbSEQ(S)).D is distProbFinS of S
  proof
    let S be non empty finite set,
    D be well-distributed Element of distribution_family(S);
    set s = the Element of D;
    reconsider p=FDprobSEQ (s) as ProbFinS FinSequence of REAL;
    dom p= Seg (card (S)) by DIST_1:def 3;
    then len p= card(S) by FINSEQ_1:def 3;
    then A1:p is distProbFinS of S by DIST_1:def 10;
    consider t being FinSequence of S
    such that A2: D = Finseq-EQclass(t) by DIST_1:def 6;
    D=Finseq-EQclass(s) by A2,DIST_1:9,7;
    hence thesis by A1,DIST_1:12;
  end;
