
theorem
  for S be non empty finite set,
  D be EqSampleSpaces of S, x be sample of D
  holds Prob(x)= (ProbFinS_of D).(index(x))
  proof
    let S be non empty finite set,
    D be EqSampleSpaces of S, x be sample of D;
    reconsider f = chi({x},S) as Element of Funcs(S,BOOLEAN)
      by FUNCT_2:8,MARGREL1:def 11;
    set s = the Element of D;
A1: for a be set holds a=x implies f.a=TRUE
    proof
      let a be set;
      assume a=x;then
      a in {x} by TARSKI:def 1;
      hence thesis by FUNCT_3:def 3;
    end;
    for a be set holds f.a=TRUE implies a=x
    proof
      let a be set;
      assume f.a=TRUE;then
      a in {x} by FUNCT_3:36;
      hence thesis by TARSKI:def 1;
    end; then
    A2: Prob(f,s) = FDprobability (x,s) by Th10,A1;
    A3: Prob(x) = FDprobability (x,s) by A2,Def6;
    consider t be FinSequence of S such that
    A4: t in D & (GenProbSEQ(S)).D=FDprobSEQ (t) by DIST_1:def 7;
    A5:(GenProbSEQ(S)).D=FDprobSEQ (s) by A4,DIST_1:10,Th4;
    reconsider n= x..(canFS(S)) as Nat;
    len canFS(S) = card S by FINSEQ_1:93; then
    A6: dom canFS(S) = Seg (card S) by FINSEQ_1:def 3;
    A7: x in rng canFS(S) by Th3;then
    n in dom canFS(S) by FINSEQ_4:20;then
    A8: n in dom (FDprobSEQ (s)) by A6,DIST_1:def 3;
    (canFS(S)).n =x by A7,FINSEQ_4:19;
    hence thesis by A3,A5,A8,DIST_1:def 3;
  end;
