reserve n for Nat;

theorem Th53:
  for C be compact connected non vertical non horizontal Subset of
TOP-REAL 2 for n be Nat st n > 0 for q be Point of TOP-REAL 2 holds
q in rng mid(Upper_Seq(C,n),2,First_Point(L~Upper_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))
  ..Upper_Seq(C,n)) implies q`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;
  set Wmin = W-min L~Cage(C,n);
  set Emax = E-max L~Cage(C,n);
  set Wbo = W-bound L~Cage(C,n);
  set Ebo = E-bound L~Cage(C,n);
  set sr = (W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/2;
  set US = Upper_Seq(C,n);
  set FiP = First_Point(L~US,Wmin,Emax,Vertical_Line sr);
A1: US/.1 = Wmin by JORDAN1F:5;
  US/.len US = Emax by JORDAN1F:7;
  then
A2: L~US is_an_arc_of Wmin,Emax by A1,TOPREAL1:25;
  assume
A3: n > 0;
  then
A4: FiP in rng US by Th47;
  then
A5: FiP..US in dom US by FINSEQ_4:20;
  then
A6: FiP..US <= len US by FINSEQ_3:25;
A7: Wbo < Ebo by SPRECT_1:31;
  then
A8: Wbo < sr by XREAL_1:226;
  sr < Ebo by A7,XREAL_1:226;
  then
A9: sr <= Emax`1 by EUCLID:52;
  Wmin`1 <= sr by A8,EUCLID:52;
  then
  L~US meets Vertical_Line sr & L~US /\ Vertical_Line sr is closed by A2,A9,
JORDAN6:49;
  then FiP in L~US /\ Vertical_Line sr by A2,JORDAN5C:def 1;
  then FiP in Vertical_Line sr by XBOOLE_0:def 4;
  then
A10: FiP`1 = sr by JORDAN6:31;
A11: Wmin in rng US by A1,FINSEQ_6:42;
A12: now
    assume FiP..US = 1;
    then FiP..US = (US/.1)..US by FINSEQ_6:43
      .= Wmin..US by JORDAN1F:5;
    then FiP = Wmin by A4,A11,FINSEQ_5:9;
    hence contradiction by A8,A10,EUCLID:52;
  end;
  1 <= FiP..US by A5,FINSEQ_3:25;
  then FiP..US > 1 by A12,XXREAL_0:1;
  then
A13: 1+1+0 <= FiP..US by NAT_1:13;
  then FiP..US-2 >= 0 by XREAL_1:19;
  then FiP..US-'2 = FiP..US-2 by XREAL_0:def 2;
  then
A14: len mid(US,2,FiP..US) = FiP..US-2+1 by A6,A13,FINSEQ_6:186;
  let q be Point of TOP-REAL 2;
  assume q in rng mid(US,2,FiP..US);
  then consider k be Element of NAT such that
A15: k in dom mid(US,2,FiP..US) and
A16: q = mid(US,2,FiP..US)/.k by PARTFUN2:2;
  k+2 >= 1+1 by NAT_1:11;
  then
A17: k+2-1 >= 1+1-1 by XREAL_1:9;
  len US >= 3 by JORDAN1E:15;
  then len US >= 2 by XXREAL_0:2;
  then 2 in dom US by FINSEQ_3:25;
  then
A18: mid(US,2,FiP..US)/.k = US/.(k+2-'1) by A15,A5,A13,SPRECT_2:3
    .= US/.(k+(2-1)) by A17,XREAL_0:def 2;
  k <= len mid(US,2,FiP..US) by A15,FINSEQ_3:25;
  then k < FiP..US-2+1+1 by A14,NAT_1:13;
  then
A19: k+1 <= FiP..US by NAT_1:13;
  per cases by A19,XXREAL_0:1;
  suppose
    k+1 < FiP..US;
    hence thesis by A3,A16,A18,Th51,NAT_1:11;
  end;
  suppose
    k+1 = FiP..US;
    hence thesis by A16,A4,A10,A18,FINSEQ_5:38;
  end;
end;
