reserve n for Nat;

theorem Th48:
  for C be compact connected non vertical non horizontal Subset of
TOP-REAL 2 for n be Nat st n > 0 holds Last_Point(L~Lower_Seq(C,n),
  E-max L~Cage(C,n),W-min L~Cage(C,n), Vertical_Line ((W-bound L~Cage(C,n)+
  E-bound L~Cage(C,n))/2)) in rng Lower_Seq(C,n)
proof
  let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
  let n be Nat;
  assume
A1: n > 0;
  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 LaP = Last_Point(L~Lower_Seq(C,n),Emax,Wmin,Vertical_Line sr);
A2: Lower_Seq(C,n)/.1 = E-max L~Cage(C,n) & Lower_Seq(C,n)/.len Lower_Seq(C,
  n) = W-min L~Cage(C,n) by JORDAN1F:6,8;
  then
A3: L~Lower_Seq(C,n) is_an_arc_of Emax,Wmin by TOPREAL1:25;
A4: Wbo <= Ebo by SPRECT_1:21;
  then Wbo <= sr by JORDAN6:1;
  then
A5: Wmin`1 <= sr by EUCLID:52;
  sr <= Ebo by A4,JORDAN6:1;
  then
A6: sr <= Emax`1 by EUCLID:52;
A7: L~Lower_Seq(C,n) is_an_arc_of Wmin,Emax by A2,JORDAN5B:14,TOPREAL1:25;
  then
A8: L~Lower_Seq(C,n) meets Vertical_Line(sr) by A5,A6,JORDAN6:49;
  L~Lower_Seq(C,n) /\ Vertical_Line(sr) is closed by A7,A5,A6,JORDAN6:49;
  then
A9: LaP in L~Lower_Seq(C,n) /\ Vertical_Line sr by A3,A8,JORDAN5C:def 2;
  then LaP in L~Lower_Seq(C,n) by XBOOLE_0:def 4;
  then consider t be Nat such that
A10: 1 <= t and
A11: t+1 <= len Lower_Seq(C,n) and
A12: LaP in LSeg(Lower_Seq(C,n),t) by SPPOL_2:13;
A13: LSeg(Lower_Seq(C,n),t) = LSeg(Lower_Seq(C,n)/.t,Lower_Seq(C,n)/.(t+1))
  by A10,A11,TOPREAL1:def 3;
  1 <= t+1 by A10,NAT_1:13;
  then
A14: t+1 in dom Lower_Seq(C,n) by A11,FINSEQ_3:25;
  t < len Lower_Seq(C,n) by A11,NAT_1:13;
  then
A15: t in dom Lower_Seq(C,n) by A10,FINSEQ_3:25;
  LaP in Vertical_Line sr by A9,XBOOLE_0:def 4;
  then
A16: LaP`1 = sr by JORDAN6:31;
A17: LaP = Last_Point (LSeg (Lower_Seq(C,n),t),Lower_Seq(C,n)/.t, Lower_Seq(
  C,n)/.(t+1),Vertical_Line sr) by A2,A8,A10,A11,A12,JORDAN5C:20,JORDAN6:30;
  now
    per cases by SPPOL_1:19;
    suppose
A18:  LSeg(Lower_Seq(C,n),t) is vertical;
      then (Lower_Seq(C,n)/.(t+1))`1 = sr by A12,A13,A16,SPPOL_1:41;
      then Lower_Seq(C,n)/.(t+1) in {p where p is Point of TOP-REAL 2: p`1 =
      sr};
      then
A19:  Lower_Seq(C,n)/.(t+1) in Vertical_Line sr by JORDAN6:def 6;
A20:  LSeg(Lower_Seq(C,n),t) is closed & LSeg(Lower_Seq(C,n),t)
is_an_arc_of Lower_Seq(C,n)/.t, Lower_Seq(C,n)/.(t+1) by A13,A15,A14,
GOBOARD7:29,TOPREAL1:9;
      (Lower_Seq(C,n)/.t)`1 = sr by A12,A13,A16,A18,SPPOL_1:41;
      then Lower_Seq(C,n)/.t in {p where p is Point of TOP-REAL 2: p`1 = sr};
      then Lower_Seq(C,n)/.t in Vertical_Line sr by JORDAN6:def 6;
      then LSeg(Lower_Seq(C,n),t) c= Vertical_Line sr by A13,A19,JORDAN1A:13;
      then
      Last_Point (LSeg(Lower_Seq(C,n),t),Lower_Seq(C,n)/.t, Lower_Seq(C,n
      )/.(t+1),Vertical_Line sr) = Lower_Seq(C,n)/.(t+1) by A20,JORDAN5C:7;
      hence thesis by A17,A14,PARTFUN2:2;
    end;
    suppose
      LSeg(Lower_Seq(C,n),t) is horizontal;
      then
A21:  (Lower_Seq(C,n)/.t)`2 = (Lower_Seq(C,n)/.(t+1))`2 by A13,SPPOL_1:15;
      then
A22:  LaP`2 = (Lower_Seq(C,n)/.t)`2 by A12,A13,GOBOARD7:6;
      Lower_Seq(C,n) is_sequence_on Gauge(C,n) by Th5;
      then consider i1,j1,i2,j2 be Nat such that
A23:  [i1,j1] in Indices Gauge(C,n) and
A24:  Lower_Seq(C,n)/.t = Gauge(C,n)*(i1,j1) and
A25:  [i2,j2] in Indices Gauge(C,n) and
A26:  Lower_Seq(C,n)/.(t+1) = Gauge(C,n)*(i2,j2) and
A27:  i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 =
      j2 or i1 = i2 & j1 = j2+1 by A10,A11,JORDAN8:3;
A28:  1 <= i1 by A23,MATRIX_0:32;
A29:  j1 = j2 by A21,A23,A24,A25,A26,Th6;
A30:  i2 <= len Gauge(C,n) by A25,MATRIX_0:32;
A31:  i1 <= len Gauge(C,n) by A23,MATRIX_0:32;
A32:  1 <= i2 by A25,MATRIX_0:32;
A33:  Center Gauge(C,n) <= len Gauge(C,n) by JORDAN1B:13;
A34:  1 <= j1 & j1 <= width Gauge(C,n) by A23,MATRIX_0:32;
      then
A35:  Gauge(C,n)*(Center Gauge(C,n),j1)`1 = (W-bound C + E-bound C)/2 by A1
,Th35
        .= LaP`1 by A16,Th33;
A36:  1 <= Center Gauge(C,n) by JORDAN1B:11;
      then Gauge(C,n)*(Center Gauge(C,n),j1)`2 = Gauge(C,n)*(1,j1)`2 by A34,A33
,GOBOARD5:1
        .= LaP`2 by A22,A24,A28,A31,A34,GOBOARD5:1;
      then
A37:  LaP = Gauge(C,n)*(Center Gauge(C,n),j1) by A35,TOPREAL3:6;
      now
        per cases by A27,A29;
        suppose
A38:      i1+1 = i2;
          i1 < i1+1 by NAT_1:13;
          then
A39:      Gauge(C,n)*(i1,j1)`1 <= Gauge(C,n)*(i1+1,j1)`1 by A28,A34,A30,A38,
SPRECT_3:13;
          then Gauge(C,n)*(i1,j1)`1 <= LaP`1 by A12,A13,A24,A26,A29,A38,
TOPREAL1:3;
          then i1 <= Center Gauge(C,n) by A31,A34,A36,A35,GOBOARD5:3;
          then i1 = Center Gauge(C,n) or i1 < Center Gauge(C,n) by XXREAL_0:1;
          then
A40:      i1 = Center Gauge(C,n) or i1+1 <= Center Gauge(C,n) by NAT_1:13;
          LaP`1 <= Gauge(C,n)*(i1+1,j1)`1 by A12,A13,A24,A26,A29,A38,A39,
TOPREAL1:3;
          then Center Gauge(C,n) <= i1+1 by A34,A32,A33,A35,A38,GOBOARD5:3;
          then i1 = Center Gauge(C,n) or i1+1 = Center Gauge(C,n) by A40,
XXREAL_0:1;
          hence thesis by A15,A14,A24,A26,A29,A37,A38,PARTFUN2:2;
        end;
        suppose
A41:      i1 = i2+1;
          i2 < i2+1 by NAT_1:13;
          then
A42:      Gauge(C,n)*(i2,j1)`1 <= Gauge(C,n)*(i2+1,j1)`1 by A31,A34,A32,A41,
SPRECT_3:13;
          then Gauge(C,n)*(i2,j1)`1 <= LaP`1 by A12,A13,A24,A26,A29,A41,
TOPREAL1:3;
          then i2 <= Center Gauge(C,n) by A34,A30,A36,A35,GOBOARD5:3;
          then i2 = Center Gauge(C,n) or i2 < Center Gauge(C,n) by XXREAL_0:1;
          then
A43:      i2 = Center Gauge(C,n) or i2+1 <= Center Gauge(C,n) by NAT_1:13;
          LaP`1 <= Gauge(C,n)*(i2+1,j1)`1 by A12,A13,A24,A26,A29,A41,A42,
TOPREAL1:3;
          then Center Gauge(C,n) <= i2+1 by A28,A34,A33,A35,A41,GOBOARD5:3;
          then i2 = Center Gauge(C,n) or i2+1 = Center Gauge(C,n) by A43,
XXREAL_0:1;
          hence thesis by A15,A14,A24,A26,A29,A37,A41,PARTFUN2:2;
        end;
      end;
      hence thesis;
    end;
  end;
  hence thesis;
end;
