
theorem Th7:
  for S be finite set, s,t be FinSequence of S holds
  s,t -are_prob_equivalent iff Finseq-EQclass(s) = Finseq-EQclass(t)
proof
  let S be finite set, t,s be FinSequence of S;
  hereby assume
A1: t,s -are_prob_equivalent;
    thus Finseq-EQclass(t) = Finseq-EQclass(s)
    proof
    thus Finseq-EQclass(t) c= Finseq-EQclass(s)
    proof
      let x be object;
      assume x in Finseq-EQclass(t);
      then consider u be FinSequence of S such that
A2:   x=u and
A3:   t,u -are_prob_equivalent;
      s,u -are_prob_equivalent by A1,A3,Th4;
      hence x in Finseq-EQclass(s) by A2;
    end;
      let x be object;
      assume x in Finseq-EQclass(s);
      then consider u be FinSequence of S such that
A4:   x=u and
A5:   s,u -are_prob_equivalent;
      t,u -are_prob_equivalent by A1,A5,Th4;
      hence x in Finseq-EQclass(t) by A4;
  end;
  end;
    assume Finseq-EQclass(t) = Finseq-EQclass(s);
    then t in Finseq-EQclass(s);
    hence t,s -are_prob_equivalent by Th5;
end;
