reserve a,b for object, I,J for set;

theorem Lem13:
  for p,q,r being FinSequence holds r^p c< r^q iff p c< q
  proof let p,q,r be FinSequence;
    thus r^p c< r^q implies p c< q
    proof
      assume
A0:   r^p c< r^q;
      len(r^p) = len r+len p & len(r^q) = len r+len q by FINSEQ_1:22;
      then
A2:   len p < len q by A0,Lem12,XREAL_1:6;
      then
A3:   dom p c= dom q by FINSEQ_3:30;
      for i being Nat st i in dom p holds p.i = q.i
      proof let i be Nat;
        assume i in dom p;
        then p.i = (r^p).(len r+i) & q.i = (r^q).(len r+i) &
        len r+i in dom(r^p) by A3,FINSEQ_1:def 7,28;
        hence thesis by A0,Lem12;
      end;
      hence thesis by A2,Lem12;
    end;
    assume p c< q;
    then r^p c= r^q & r^p <> r^q by FINSEQ_6:13,FINSEQ_1:33,XBOOLE_0:def 8;
    hence thesis by XBOOLE_0:def 8;
  end;
