reserve n for Nat;

theorem Th4:
  for C be compact connected non vertical non horizontal Subset of
TOP-REAL 2 holds Upper_Seq(C,n) = Rotate(Cage(C,n),E-max L~Cage(C,n)):-W-min L~
  Cage(C,n)
proof
  let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
  set Nmi = N-min L~Cage(C,n);
  set Nma = N-max L~Cage(C,n);
  set Wmi = W-min L~Cage(C,n);
  set Wma = W-max L~Cage(C,n);
  set Ema = E-max L~Cage(C,n);
  set Emi = E-min L~Cage(C,n);
  set Sma = S-max L~Cage(C,n);
  set Smi = S-min L~Cage(C,n);
  set RotWmi = Rotate(Cage(C,n),Wmi);
  set RotEma = Rotate(Cage(C,n),Ema);
A1: Nmi in rng Cage(C,n) by SPRECT_2:39;
A2: Cage(C,n)/.1 = Nmi by JORDAN9:32;
  then Emi..Cage(C,n) <= Sma..Cage(C,n) by SPRECT_2:72;
  then Ema..Cage(C,n) < Sma..Cage(C,n) by A2,SPRECT_2:71,XXREAL_0:2;
  then
A3: Ema..Cage(C,n) < Smi..Cage(C,n) by A2,SPRECT_2:73,XXREAL_0:2;
  then
A4: Ema..Cage(C,n) < Wmi..Cage(C,n) by A2,SPRECT_2:74,XXREAL_0:2;
  (Cage(C,n):-Ema)/.1 = Ema by FINSEQ_5:53;
  then
A5: Ema in rng (Cage(C,n):-Ema) by FINSEQ_6:42;
A6: Ema in rng Cage(C,n) by SPRECT_2:46;
  then (Cage(C,n):-Ema)/.len(Cage(C,n):-Ema) = Cage(C,n)/.len Cage(C,n) by
FINSEQ_5:54
    .= Cage(C,n)/.1 by FINSEQ_6:def 1
    .= Nmi by JORDAN9:32;
  then
A7: Nmi in rng (Cage(C,n):-Ema) by FINSEQ_6:168;
  {Nmi,Ema} c= rng (Cage(C,n):-Ema)
  by A7,A5,TARSKI:def 2;
  then
A8: card {Nmi,Ema} c= card rng (Cage(C,n):-Ema) by CARD_1:11;
A9: Wmi in rng Cage(C,n) by SPRECT_2:43;
  then
A10: Cage(C,n)-:Wmi <> {} by FINSEQ_5:47;
  len(Cage(C,n)-:Wmi) = Wmi..Cage(C,n) by A9,FINSEQ_5:42;
  then (Cage(C,n)-:Wmi)/.len (Cage(C,n)-:Wmi) = Wmi by A9,FINSEQ_5:45;
  then
A11: Wmi in rng (Cage(C,n)-:Wmi) by A10,FINSEQ_6:168;
  (Cage(C,n)-:Wmi)/.1 = Cage(C,n)/.1 by A9,FINSEQ_5:44
    .= Nmi by JORDAN9:32;
  then
A12: Nmi in rng (Cage(C,n)-:Wmi) by A10,FINSEQ_6:42;
  {Nmi,Wmi} c= rng (Cage(C,n)-:Wmi)
  by A12,A11,TARSKI:def 2;
  then
A13: card {Nmi,Wmi} c= card rng (Cage(C,n)-:Wmi) by CARD_1:11;
  card rng (Cage(C,n)-:Wmi) c= card dom (Cage(C,n)-:Wmi) by CARD_2:61;
  then
A14: card rng (Cage(C,n)-:Wmi) c= len (Cage(C,n)-:Wmi) by CARD_1:62;
A15: Nmi`2 = N-bound L~Cage(C,n) by EUCLID:52;
  Nma`1 <= (NE-corner L~Cage(C,n))`1 by PSCOMP_1:38;
  then
A16: Nma`1 <= E-bound L~Cage(C,n) by EUCLID:52;
  Nmi`1 < Nma`1 by SPRECT_2:51;
  then
A17: Nmi <> Ema by A16,EUCLID:52;
  then
A18: card {Nmi,Ema} = 2 by CARD_2:57;
A19: Smi..Cage(C,n) <= Wmi..Cage(C,n) by A2,SPRECT_2:74;
  then
A20: Ema in rng (Cage(C,n)-:Wmi) by A9,A6,A3,FINSEQ_5:46,XXREAL_0:2;
A21: Wmi in rng (Cage(C,n):-Ema) by A9,A6,A19,A3,FINSEQ_6:62,XXREAL_0:2;
  Wma in L~Cage(C,n) by SPRECT_1:13;
  then Wma`2 <= Nmi`2 by A15,PSCOMP_1:24;
  then
A22: Nmi <> Wmi by SPRECT_2:57;
  then card {Nmi,Wmi} = 2 by CARD_2:57;
  then Segm 2 c= Segm len (Cage(C,n)-:Wmi) by A13,A14;
  then len (Cage(C,n)-:Wmi) >= 2 by NAT_1:39;
  then
A23: rng (Cage(C,n)-:Wmi) c= L~(Cage(C,n)-:Wmi) by SPPOL_2:18;
A24: not Ema in rng (Cage(C,n):-Wmi)
  proof
    (Cage(C,n):-Wmi)/.1 = Wmi by FINSEQ_5:53;
    then
A25: Wmi in rng (Cage(C,n):-Wmi) by FINSEQ_6:42;
    (Cage(C,n):-Wmi)/.len(Cage(C,n):-Wmi) = Cage(C,n)/.len Cage(C,n) by A9,
FINSEQ_5:54
      .= Cage(C,n)/.1 by FINSEQ_6:def 1
      .= Nmi by JORDAN9:32;
    then
A26: Nmi in rng (Cage(C,n):-Wmi) by FINSEQ_6:168;
    {Nmi,Wmi} c= rng (Cage(C,n):-Wmi)
    by A26,A25,TARSKI:def 2;
    then
A27: card {Nmi,Wmi} c= card rng (Cage(C,n):-Wmi) by CARD_1:11;
A28: Nmi`2 = N-bound L~Cage(C,n) by EUCLID:52;
    Wma in L~Cage(C,n) by SPRECT_1:13;
    then Wma`2 <= Nmi`2 by A28,PSCOMP_1:24;
    then Nmi <> Wmi by SPRECT_2:57;
    then
A29: card {Nmi,Wmi} = 2 by CARD_2:57;
    card rng (Cage(C,n):-Wmi) c= card dom (Cage(C,n):-Wmi) by CARD_2:61;
    then card rng (Cage(C,n):-Wmi) c= len (Cage(C,n):-Wmi) by CARD_1:62;
    then Segm 2 c= Segm len (Cage(C,n):-Wmi) by A29,A27;
    then len (Cage(C,n):-Wmi) >= 2 by NAT_1:39;
    then
A30: rng (Cage(C,n):-Wmi) c= L~(Cage(C,n):-Wmi) by SPPOL_2:18;
    assume Ema in rng (Cage(C,n):-Wmi);
    then Ema in L~(Cage(C,n)-:Wmi) /\ L~(Cage(C,n):-Wmi) by A20,A23,A30,
XBOOLE_0:def 4;
    then Ema in {Nmi,Wmi} by JORDAN1G:17;
    then Ema = Wmi by A17,TARSKI:def 2;
    hence contradiction by TOPREAL5:19;
  end;
A31: Nma..Cage(C,n) <= Ema..Cage(C,n) by A2,SPRECT_2:70;
A32: Nmi..Cage(C,n) < Nma..Cage(C,n) by A2,SPRECT_2:68;
  then
A33: Nmi..Cage(C,n) < Ema..Cage(C,n) by A2,SPRECT_2:70,XXREAL_0:2;
  then
A34: Nmi in rng (Cage(C,n)-:Wmi) by A1,A9,A4,FINSEQ_5:46,XXREAL_0:2;
A35: Ema..(Cage(C,n)-:Wmi) <> 1
  proof
    assume
A36: Ema..(Cage(C,n)-:Wmi) = 1;
    Nmi..(Cage(C,n)-:Wmi) = Nmi..Cage(C,n) by A1,A9,A33,A4,SPRECT_5:3
,XXREAL_0:2
      .= 1 by A2,FINSEQ_6:43;
    hence contradiction by A32,A31,A20,A34,A36,FINSEQ_5:9;
  end;
  then Ema in rng (Cage(C,n)-:Wmi/^1) by A20,FINSEQ_6:78;
  then
A37: Ema in rng (Cage(C,n)-:Wmi/^1) \ rng (Cage(C,n):-Wmi) by A24,
XBOOLE_0:def 5;
  card rng (Cage(C,n):-Ema) c= card dom (Cage(C,n):-Ema) by CARD_2:61;
  then card rng (Cage(C,n):-Ema) c= len (Cage(C,n):-Ema) by CARD_1:62;
  then Segm 2 c= Segm len (Cage(C,n):-Ema) by A18,A8;
  then len (Cage(C,n):-Ema) >= 2 by NAT_1:39;
  then
A38: rng (Cage(C,n):-Ema) c= L~(Cage(C,n):-Ema) by SPPOL_2:18;
  not Wmi in rng (Cage(C,n)-:Ema)
  proof
    assume
A39: Wmi in rng (Cage(C,n)-:Ema);
    (Cage(C,n)-:Ema)/.len (Cage(C,n)-:Ema) = (Cage(C,n)-:Ema)/.(Ema..Cage
    (C,n)) by A6,FINSEQ_5:42
      .= Ema by A6,FINSEQ_5:45;
    then
A40: Ema in rng (Cage(C,n)-:Ema) by A39,RELAT_1:38,FINSEQ_6:168;
    (Cage(C,n)-:Ema)/.1 = Cage(C,n)/.1 by A6,FINSEQ_5:44
      .= Nmi by JORDAN9:32;
    then
A41: Nmi in rng (Cage(C,n)-:Ema) by A39,FINSEQ_6:42,RELAT_1:38;
    {Nmi,Ema} c= rng (Cage(C,n)-:Ema)
    by A41,A40,TARSKI:def 2;
    then
A42: card {Nmi,Ema} c= card rng (Cage(C,n)-:Ema) by CARD_1:11;
    card rng (Cage(C,n)-:Ema) c= card dom (Cage(C,n)-:Ema) by CARD_2:61;
    then card rng (Cage(C,n)-:Ema) c= len (Cage(C,n)-:Ema) by CARD_1:62;
    then Segm 2 c= Segm len (Cage(C,n)-:Ema) by A18,A42;
    then len (Cage(C,n)-:Ema) >= 2 by NAT_1:39;
    then rng (Cage(C,n)-:Ema) c= L~(Cage(C,n)-:Ema) by SPPOL_2:18;
    then Wmi in L~(Cage(C,n)-:Ema) /\ L~(Cage(C,n):-Ema) by A21,A38,A39,
XBOOLE_0:def 4;
    then Wmi in {Nmi,Ema} by Th3;
    then Wmi = Ema by A22,TARSKI:def 2;
    hence contradiction by TOPREAL5:19;
  end;
  then
A43: Wmi in rng Cage(C,n) \ rng(Cage(C,n)-:Ema) by A9,XBOOLE_0:def 5;
  RotWmi-:Ema = ((Cage(C,n):-Wmi)^((Cage(C,n)-:Wmi)/^1))-:Ema by A9,
FINSEQ_6:def 2
    .= (Cage(C,n):-Wmi)^(((Cage(C,n)-:Wmi)/^1)-:Ema) by A37,FINSEQ_6:67
    .= (Cage(C,n):-Wmi)^(((Cage(C,n)-:Wmi)-:Ema)/^1) by A20,A35,FINSEQ_6:60
    .= ((Cage(C,n):-Ema):-Wmi)^(((Cage(C,n)-:Wmi)-:Ema)/^1) by A43,FINSEQ_6:71
,SPRECT_2:46
    .= ((Cage(C,n):-Ema):-Wmi)^((Cage(C,n)-:Ema)/^1) by A9,A20,FINSEQ_6:75
    .= ((Cage(C,n):-Ema)^((Cage(C,n)-:Ema)/^1)):-Wmi by A21,FINSEQ_6:64
    .= RotEma:-Wmi by A6,FINSEQ_6:def 2;
  hence thesis by JORDAN1E:def 1;
end;
