
theorem Th21:
  for S be finite set, s,t be FinSequence of S st
  s is uniformly_distributed & t is uniformly_distributed holds
  s,t -are_prob_equivalent
proof
  let S be finite set, s,t be FinSequence of S;
  assume that
A1: s is uniformly_distributed and
A2: t is uniformly_distributed;
A3: dom FDprobSEQ (s)= Seg (card (S)) & dom FDprobSEQ (t)= Seg (card (S)) by
Def3;
  for n be object st n in dom FDprobSEQ (s) holds (FDprobSEQ (s)).n=(
  FDprobSEQ (t)).n
  proof
    let n be object;
    assume
A4: n in dom FDprobSEQ (s);
    then (FDprobSEQ (s)).n= 1/(card S) by A1;
    hence thesis by A2,A3,A4;
  end;
  then FDprobSEQ (s) = FDprobSEQ (t) by A3;
  hence thesis by Th8;
end;
