reserve n for Nat;

theorem Th52:
  for C be compact connected non vertical non horizontal Subset of
  TOP-REAL 2 for n be Nat st n > 0 for k be Nat st 1 <= k & k <
  First_Point(L~Rev Lower_Seq(C,n),W-min L~Cage(C,n),E-max L~Cage(C,n),
Vertical_Line ((W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/2)).. Rev Lower_Seq(C,
n) holds (Rev Lower_Seq(C,n)/.k)`1 < (W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/
  2
proof
  let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
  let n be Nat;
  assume
A1: n > 0;
  set LS = Lower_Seq(C,n);
  set sr = (W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/2;
  set Ebo = E-bound L~Cage(C,n);
  set Wbo = W-bound L~Cage(C,n);
  set Emax = E-max L~Cage(C,n);
  set Wmin = W-min L~Cage(C,n);
  set RLS = Rev LS;
  set FiP = First_Point(L~RLS,Wmin,Emax,Vertical_Line sr);
  set LaP = Last_Point(L~LS,Emax,Wmin,Vertical_Line sr);
A2: L~RLS = L~LS by SPPOL_2:22;
A3: len RLS = len LS by FINSEQ_5:def 3;
  defpred P[Nat] means 1 <= $1 & $1 < FiP..RLS implies (RLS/.$1)`1 < sr;
A4: rng RLS = rng LS by FINSEQ_5:57;
A5: Wbo < Ebo by SPRECT_1:31;
  then
A6: Wbo < sr by XREAL_1:226;
A7: sr < Ebo by A5,XREAL_1:226;
A8: for k be non zero Nat st P[k] holds P[k+1]
  proof
A9: Wbo <= Ebo by SPRECT_1:21;
    then Wbo <= sr by JORDAN6:1;
    then
A10: Wmin`1 <= sr by EUCLID:52;
    sr <= Ebo by A9,JORDAN6:1;
    then
A11: sr <= Emax`1 by EUCLID:52;
A12: RLS/.len RLS = LS/.1 by A3,FINSEQ_5:65
      .= Emax by JORDAN1F:6;
    set GC1 = Gauge(C,n)*(Center Gauge(C,n),1);
    let k be non zero Nat;
    assume
A13: 1 <= k & k < FiP..RLS implies (RLS/.k)`1 < sr;
    4 <= len Gauge(C,n) by JORDAN8:10;
    then 1 <= len Gauge(C,n) by XXREAL_0:2;
    then
A14: 1 <= width Gauge(C,n) by JORDAN8:def 1;
    then
A15: GC1`1 = (W-bound C + E-bound C)/2 by A1,Th35
      .= sr by Th33;
A16: LS/.1 = Emax & LS/.len LS = Wmin by JORDAN1F:6,8;
    then
A17: L~LS is_an_arc_of Emax,Wmin by TOPREAL1:25;
A18: 1 <= Center Gauge(C,n) by JORDAN1B:11;
A19: RLS/.1 = LS/.len LS by FINSEQ_5:65
      .= Wmin by JORDAN1F:8;
    L~LS is_an_arc_of Wmin,Emax by A16,JORDAN5B:14,TOPREAL1:25;
    then
    L~LS meets Vertical_Line(sr) & L~LS /\ Vertical_Line(sr) is closed by A10
,A11,JORDAN6:49;
    then
A20: FiP = LaP by A2,A17,JORDAN5C:18;
    then
A21: FiP..RLS in dom RLS by A1,A4,Th48,FINSEQ_4:20;
    then
A22: 1 <= FiP..RLS by FINSEQ_3:25;
A23: k >= 1 by NAT_1:14;
    reconsider kk=k as Nat;
    assume that
A24: 1 <= k+1 and
A25: k+1 < FiP..RLS;
A26: FiP..RLS <= len RLS by A21,FINSEQ_3:25;
    then
A27: k+1 <= len RLS by A25,XXREAL_0:2;
    LS is_sequence_on Gauge(C,n) by Th5;
    then RLS is_sequence_on Gauge(C,n) by JORDAN9:5;
    then consider i1,j1,i2,j2 be Nat such that
A28: [i1,j1] in Indices Gauge(C,n) and
A29: RLS/.kk = Gauge(C,n)*(i1,j1) and
A30: [i2,j2] in Indices Gauge(C,n) and
A31: RLS/.(kk+1) = Gauge(C,n)*(i2,j2) and
A32: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2
    or i1 = i2 & j1 = j2+1 by A23,A27,JORDAN8:3;
A33: 1 <= i1 by A28,MATRIX_0:32;
A34: 1 <= j1 & j1 <= width Gauge(C,n) by A28,MATRIX_0:32;
A35: i2 <= len Gauge(C,n) by A30,MATRIX_0:32;
A36: i1 <= len Gauge(C,n) by A28,MATRIX_0:32;
A37: j2 <= width Gauge(C,n) by A30,MATRIX_0:32;
A38: 1 <= i2 & 1 <= j2 by A30,MATRIX_0:32;
A39: Center Gauge(C,n) <= len Gauge(C,n) & i1+1 >= 1 by JORDAN1B:13,NAT_1:11;
    now
      per cases by A32;
      suppose
        i1 = i2 & j1+1 = j2;
        then (RLS/.k)`1 = Gauge(C,n)*(i2,1)`1 by A29,A33,A36,A34,GOBOARD5:2
          .= (RLS/.(k+1))`1 by A31,A35,A38,A37,GOBOARD5:2;
        hence thesis by A13,A25,NAT_1:13,14;
      end;
      suppose
A40:    i1+1 = i2 & j1 = j2;
A41:    now
A42:      now
            assume RLS/.1 in Vertical_Line sr;
            then Wmin`1 = sr by A19,JORDAN6:31;
            hence contradiction by A6,EUCLID:52;
          end;
          assume (RLS/.(k+1))`1 = sr;
          then RLS/.(k+1) in {p where p is Point of TOP-REAL 2 : p`1 = sr};
          then
A43:      RLS/.(k+1) in Vertical_Line sr by JORDAN6:def 6;
A44:      sr <= Emax`1 by A7,EUCLID:52;
          L~RLS is_an_arc_of Wmin,Emax & Wmin`1 <= sr by A6,A19,A12,EUCLID:52
,TOPREAL1:25;
          then
A45:      L~RLS meets Vertical_Line sr by A44,JORDAN6:49;
A46:      RLS/.(FiP..RLS) = FiP by A1,A4,A20,Th48,FINSEQ_5:38;
A47:      k+1 >= 1+1 by A23,XREAL_1:7;
          len mid(RLS,1,FiP..RLS) = FiP..RLS-'1+1 by A22,A26,FINSEQ_6:186
            .= FiP..RLS by A22,XREAL_1:235;
          then
A48:      rng mid(RLS,1,FiP..RLS) c= L~mid(RLS,1,FiP..RLS) by A25,A47,
SPPOL_2:18,XXREAL_0:2;
A49:      k+1 in dom RLS by A24,A27,FINSEQ_3:25;
          Vertical_Line sr is closed & RLS is being_S-Seq by JORDAN6:30;
          then
A50:      L~mid(RLS,1,FiP..RLS) /\ Vertical_Line sr = {FiP} by A1,A4,A20,A19
,A12,A45,A42,Th48,Th50;
A51:      mid(RLS,1,FiP..RLS) = RLS|(FiP..RLS) & RLS|Seg(FiP..RLS) = RLS|
          (FiP..RLS) by A22,FINSEQ_1:def 16,FINSEQ_6:116;
          k+1 in Seg (FiP..RLS) by A24,A25,FINSEQ_1:1;
          then RLS/.(k+1) in rng mid(RLS,1,FiP..RLS) by A51,A49,PARTFUN2:18;
          then RLS/.(k+1) in {FiP} by A43,A48,A50,XBOOLE_0:def 4;
          then RLS/.(k+1) = FiP by TARSKI:def 1;
          hence contradiction by A25,A21,A49,A46,PARTFUN2:10;
        end;
        i1 < Center Gauge(C,n) by A13,A25,A29,A36,A34,A18,A14,A15,JORDAN1A:18
,NAT_1:13,14;
        then i1+1 <= Center Gauge(C,n) by NAT_1:13;
        then (RLS/.(k+1))`1 <= sr by A31,A34,A14,A15,A39,A40,JORDAN1A:18;
        hence thesis by A41,XXREAL_0:1;
      end;
      suppose
        i1 = i2+1 & j1 = j2;
        then i2 < i1 by NAT_1:13;
        then (RLS/.(k+1))`1 <= (RLS/.k)`1 by A29,A31,A36,A34,A38,A37,
JORDAN1A:18;
        hence thesis by A13,A25,NAT_1:13,14,XXREAL_0:2;
      end;
      suppose
        i1 = i2 & j1 = j2+1;
        then (RLS/.k)`1 = Gauge(C,n)*(i2,1)`1 by A29,A33,A36,A34,GOBOARD5:2
          .= (RLS/.(k+1))`1 by A31,A35,A38,A37,GOBOARD5:2;
        hence thesis by A13,A25,NAT_1:13,14;
      end;
    end;
    hence thesis;
  end;
A52: P[1]
  proof
    assume that
    1 <= 1 and
    1 < FiP..RLS;
    RLS/.1 = LS/.len LS by FINSEQ_5:65
      .= Wmin by JORDAN1F:8;
    hence thesis by A6,EUCLID:52;
  end;
A53: for k being non zero Nat holds P[k] from NAT_1:sch 10(A52, A8);
  let k be Nat;
  assume 1 <= k & k < First_Point(L~Rev Lower_Seq(C,n),W-min L~Cage(C,n),
E-max L~Cage (C,n),Vertical_Line ((W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/2))
  ..Rev Lower_Seq(C,n);
  hence thesis by A53;
end;
