reserve n for Nat;

theorem Th43:
  for C be compact connected non vertical non horizontal Subset of
  TOP-REAL 2 for i be Nat st 1 <= i & i < len Gauge(C,n) holds not
  Gauge(C,n)*(i,width Gauge(C,n)) in rng Lower_Seq(C,n)
proof
  let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
  let i be Nat;
  set wi = width Gauge(C,n);
  assume that
A1: 1 <= i & i < len Gauge(C,n) and
A2: Gauge(C,n)*(i,wi) in rng Lower_Seq(C,n);
  consider i2 be Nat such that
A3: i2 in dom Lower_Seq(C,n) and
A4: Lower_Seq(C,n).i2 = Gauge(C,n)*(i,wi) by A2,FINSEQ_2:10;
  reconsider i2 as Nat;
A5: 1 <= i2 & i2 <= len Lower_Seq(C,n) by A3,FINSEQ_3:25;
  3 <= len Upper_Seq(C,n) by JORDAN1E:15;
  then
A6: 2 <= len Upper_Seq(C,n) by XXREAL_0:2;
  set f = Rotate(Cage(C,n),E-max L~Cage(C,n));
  set i1 = (S-max L~Cage(C,n))..Lower_Seq(C,n);
A7: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1;
  E-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:46;
  then
A8: f/.1 = E-max L~Cage(C,n) by FINSEQ_6:92;
  L~Cage(C,n) = L~f by REVROT_1:33;
  then
A9: (S-max L~Cage(C,n))..f < (S-min L~Cage(C,n))..f & (S-min L~Cage(C,n))..
  f <= (W-min L~Cage(C,n))..f by A8,SPRECT_5:40,41;
A10: W-min L~Cage(C,n) in rng Cage(C,n) & rng f = rng Cage(C,n) by FINSEQ_6:90
,SPRECT_2:43,46;
  (W-min L~Cage(C,n))..Lower_Seq(C,n) = len Lower_Seq(C,n) by Th30;
  then (S-min L~Cage(C,n))..Lower_Seq(C,n) <= len Lower_Seq(C,n) by Th29;
  then
A11: i1 < len Lower_Seq(C,n) by Th28,XXREAL_0:2;
  (E-max L~Cage(C,n))..Lower_Seq(C,n) = 1 & (E-min L~Cage(C,n))..Lower_Seq
  (C,n ) <= i1 by Th25,Th27;
  then
A12: i1 > 1 by Th26,XXREAL_0:2;
  then
A13: i1 in dom Lower_Seq(C,n) by A11,FINSEQ_3:25;
  Lower_Seq(C,n) = f-:W-min L~Cage(C,n) & S-max L~Cage(C,n) in rng Cage(C
  ,n) by Th18,SPRECT_2:42;
  then
A14: S-max L~Cage(C,n) in rng Lower_Seq(C,n) by A10,A9,FINSEQ_5:46,XXREAL_0:2;
  then
A15: Lower_Seq(C,n)/.i1 = S-max L~Cage(C,n) by FINSEQ_5:38;
A16: i1 in NAT & i2 in NAT by ORDINAL1:def 12;
A17: i1 <> i2
  proof
    assume i1 = i2;
    then Gauge(C,n)*(i,wi) = S-max L~Cage(C,n) by A4,A13,A15,PARTFUN1:def 6;
    then (Gauge(C,n)*(i,wi))`2 = S-bound L~Cage(C,n) by EUCLID:52;
    then N-bound L~Cage(C,n) = S-bound L~Cage(C,n) by A1,A7,JORDAN1A:70;
    hence contradiction by SPRECT_1:16;
  end;
  then mid(Lower_Seq(C,n),i1,i2) is being_S-Seq by A12,A11,A5,JORDAN3:6,A16;
  then reconsider
  h = mid(Lower_Seq(C,n),i1,i2) as one-to-one special FinSequence
  of TOP-REAL 2;
A18: (h/.1)`2 = (Lower_Seq(C,n)/.i1)`2 by A3,A13,SPRECT_2:8
    .= (S-max L~Cage(C,n))`2 by A14,FINSEQ_5:38
    .= S-bound L~Cage(C,n) by EUCLID:52;
  len h >= 1 by A3,A13,SPRECT_2:5;
  then len h > 1 by A3,A13,A17,SPRECT_2:6,XXREAL_0:1;
  then
A19: 1+1 <= len h by NAT_1:13;
A20: h is_in_the_area_of Cage(C,n) by A3,A13,JORDAN1E:18,SPRECT_2:22;
  (h/.len h)`2 = (Lower_Seq(C,n)/.i2)`2 by A3,A13,SPRECT_2:9
    .= (Gauge(C,n)*(i,wi))`2 by A3,A4,PARTFUN1:def 6
    .= N-bound L~Cage(C,n) by A1,A7,JORDAN1A:70;
  then h is_a_v.c._for Cage(C,n) by A20,A18,SPRECT_2:def 3;
  then L~Upper_Seq(C,n) meets L~h by A6,A19,Th40,SPRECT_2:29;
  then consider x be object such that
A21: x in L~Upper_Seq(C,n) and
A22: x in L~h by XBOOLE_0:3;
  L~mid(Lower_Seq(C,n),i1,i2) c= L~Lower_Seq(C,n) by A12,A11,A5,JORDAN4:35;
  then x in L~Lower_Seq(C,n) /\ L~Upper_Seq(C,n) by A21,A22,XBOOLE_0:def 4;
  then
A23: x in {W-min L~Cage(C,n),E-max L~Cage(C,n)} by JORDAN1E:16;
  4 <= len Gauge(C,n) by JORDAN8:10;
  then
A24: 1 <= len Gauge(C,n) by XXREAL_0:2;
  per cases by A23,TARSKI:def 2;
  suppose
    x = E-max L~Cage(C,n);
    then x = Lower_Seq(C,n)/.1 by JORDAN1F:6;
    then i2 = 1 by A12,A11,A5,A22,Th37;
    then Lower_Seq(C,n)/.1 = Gauge(C,n)*(i,wi) by A3,A4,PARTFUN1:def 6;
    then E-max(L~Cage(C,n)) = Gauge(C,n)*(i,wi) by JORDAN1F:6;
    then Gauge(C,n)*(i,wi)`1 = E-bound(L~Cage(C,n)) by EUCLID:52
      .= Gauge(C,n)*(len Gauge(C,n),wi)`1 by A7,A24,JORDAN1A:71;
    hence contradiction by A1,A7,A24,GOBOARD5:3;
  end;
  suppose
    x = W-min L~Cage(C,n);
    then x = Lower_Seq(C,n)/.len Lower_Seq(C,n) by JORDAN1F:8;
    then i2 = len Lower_Seq(C,n) by A12,A11,A5,A22,Th38;
    then Lower_Seq(C,n)/.len Lower_Seq(C,n) = Gauge(C,n)*(i,wi) by A3,A4,
PARTFUN1:def 6;
    then
A25: W-min(L~Cage(C,n)) = Gauge(C,n)*(i,wi) by JORDAN1F:8;
    (NW-corner L~Cage(C,n))`2 >= (W-max L~Cage(C,n))`2 by PSCOMP_1:30;
    then (NW-corner L~Cage(C,n))`2 > (W-min L~Cage(C,n))`2 by SPRECT_2:57
,XXREAL_0:2;
    then N-bound L~Cage(C,n) > (Gauge(C,n)*(i,wi))`2 by A25,EUCLID:52;
    hence contradiction by A1,A7,JORDAN1A:70;
  end;
end;
