reserve n for Nat;

theorem
  for C be compact connected non vertical non horizontal Subset of
  TOP-REAL 2 holds (Lower_Seq(C,n)/.2)`1 = E-bound L~Cage(C,n)
proof
  let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
  set Ca = Cage(C,n);
  set LS = Lower_Seq(C,n);
  set Emax = E-max L~Cage(C,n);
  set Emin = E-min L~Cage(C,n);
  set Smax = S-max L~Cage(C,n);
  set Smin = S-min L~Cage(C,n);
  set Wmin = W-min L~Cage(C,n);
  set Nmin = N-min L~Cage(C,n);
  Wmin in rng Ca by SPRECT_2:43;
  then
A1: Wmin in rng Rotate(Ca,Emax) by FINSEQ_6:90,SPRECT_2:46;
  len LS >= 3 by JORDAN1E:15;
  then len LS >= 2 by XXREAL_0:2;
  then 2 <= Wmin..LS by Th30;
  then 2 <= Wmin..(Rotate(Ca,Emax)-:Wmin) by Th18;
  then 2 <= Wmin..Rotate(Ca,Emax) by A1,FINSEQ_6:72;
  then
A2: 2 in Seg (Wmin..Rotate(Ca,Emax)) by FINSEQ_1:1;
  (Ca:-Emax)/.1 = Emax by FINSEQ_5:53;
  then
A3: Emax in rng (Ca:-Emax) by FINSEQ_6:42;
  N-max L~Ca in L~Ca & Emax`1 = E-bound L~Ca by EUCLID:52,SPRECT_1:11;
  then (N-max L~Ca)`1 <= Emax`1 by PSCOMP_1:24;
  then Nmin <> Emax by SPRECT_2:51;
  then
A4: card {Nmin,Emax} = 2 by CARD_2:57;
A5: Ca/.1 = Nmin by JORDAN9:32;
  then Emax..Ca < Emin..Ca by SPRECT_2:71;
  then Emax..Ca < Smax..Ca by A5,SPRECT_2:72,XXREAL_0:2;
  then Emax..Ca < Smin..Ca by A5,SPRECT_2:73,XXREAL_0:2;
  then Emax..Ca < Wmin..Ca by A5,SPRECT_2:74,XXREAL_0:2;
  then Emax..Ca < len Ca by A5,SPRECT_2:76,XXREAL_0:2;
  then
A6: Emax..Ca+1 <= len Ca by NAT_1:13;
A7: Emax in rng Ca by SPRECT_2:46;
  then
A8: 1 <= Emax..Ca by FINSEQ_4:21;
  (Ca:-Emax)/.len(Ca:-Emax) = Ca/.len Ca by A7,FINSEQ_5:54
    .= Ca/.1 by FINSEQ_6:def 1
    .= Nmin by JORDAN9:32;
  then
A9: Nmin in rng (Ca:-Emax) by FINSEQ_6:168;
  {Nmin,Emax} c= rng (Ca:-Emax)
  by A9,A3,TARSKI:def 2;
  then
A10: card {Nmin,Emax} c= card rng (Ca:-Emax) by CARD_1:11;
  card rng (Ca:-Emax) c= card dom (Ca:-Emax) by CARD_2:61;
  then card rng (Ca:-Emax) c= len (Ca:-Emax) by CARD_1:62;
  then Segm 2 c= Segm len (Ca:-Emax) by A4,A10;
  then
A11: len (Ca:-Emax) >= 2 by NAT_1:39;
  then
A12: len(Ca:-Emax) >= 1 by XXREAL_0:2;
A13: LS/.1 = (Rotate(Ca,Emax)-:Wmin)/.1 by Th18
    .= Rotate(Ca,Emax)/.1 by A1,FINSEQ_5:44
    .= Ca/.(1-'1+Emax..Ca) by A7,A12,FINSEQ_6:174
    .= Ca/.(0+Emax..Ca) by XREAL_1:232;
  LS/.2 = (Rotate(Ca,Emax)-:Wmin)/.2 by Th18
    .= Rotate(Ca,Emax)/.2 by A1,A2,FINSEQ_5:43
    .= Ca/.(2-'1+Emax..Ca) by A7,A11,FINSEQ_6:174
    .= Ca/.(2-1+Emax..Ca) by XREAL_0:def 2;
  hence thesis by A8,A6,A13,JORDAN1E:20,JORDAN1F:6;
end;
