reserve n for Nat;

theorem Th15:
  for C be compact non vertical non horizontal Subset of TOP-REAL
2 for n be Nat holds len Upper_Seq(C,n) >= 3 & len Lower_Seq(C,n) >=
  3
proof
  let C be compact non vertical non horizontal Subset of TOP-REAL 2;
  let n be Nat;
  set pWi = W-min L~Cage(C,n);
  set pWa = W-max L~Cage(C,n);
  set pEi = E-min L~Cage(C,n);
  set pEa = E-max L~Cage(C,n);
A1: pWi <> pEa by TOPREAL5:19;
  set f=Rotate(Cage(C,n),W-min L~Cage(C,n));
A2: W-min L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:43;
  E-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:46;
  then
A3: E-max L~Cage(C,n) in rng Rotate(Cage(C,n),W-min L~Cage(C,n)) by FINSEQ_6:90
,SPRECT_2:43;
  then
  Lower_Seq(C,n)/.(len Lower_Seq(C,n)) = Rotate(Cage(C,n),W-min L~Cage(C,
  n))/. (len Rotate(Cage(C,n),W-min L~Cage(C,n))) by FINSEQ_5:54
    .= Rotate(Cage(C,n),W-min L~Cage(C,n))/.1 by FINSEQ_6:def 1
    .= W-min L~Cage(C,n) by A2,FINSEQ_6:92;
  then
A4: pEa in rng Lower_Seq(C,n) & pWi in rng Lower_Seq(C,n) by FINSEQ_6:61
,FINSEQ_6:168;
  (E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n)) <= (E-max L~
  Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n));
  then
A5: pEa in rng Upper_Seq(C,n) by A3,FINSEQ_5:46;
  pWa in rng Cage(C,n) by SPRECT_2:44;
  then
A6: pWa in rng Rotate(Cage(C,n),W-min L~Cage(C,n)) by FINSEQ_6:90,SPRECT_2:43;
A7: Upper_Seq(C,n)/.1 = Rotate(Cage(C,n),W-min L~Cage(C,n))/.1 by A3,
FINSEQ_5:44;
  then
A8: Upper_Seq(C,n)/.1 = W-min L~Cage(C,n) by A2,FINSEQ_6:92;
  then
A9: pWi in rng Upper_Seq(C,n) by FINSEQ_6:42;
A10: L~Cage(C,n) = L~f by REVROT_1:33;
  then (W-max L~f)..f <= (N-min L~f)..f by A2,A7,FINSEQ_6:92,SPRECT_5:23;
  then
A11: (W-max L~f)..f < (N-max L~f)..f by A7,A8,A10,SPRECT_5:24,XXREAL_0:2;
  (N-max L~f)..f <= (E-max L~f)..f by A2,A7,A10,FINSEQ_6:92,SPRECT_5:25;
  then
A12: pWa in rng Upper_Seq(C,n) by A3,A6,A10,A11,FINSEQ_5:46,XXREAL_0:2;
  {pWi,pWa,pEa} c= rng Upper_Seq(C,n)
  by A5,A9,A12,ENUMSET1:def 1;
  then
A13: card {pWi,pWa,pEa} c= card rng Upper_Seq(C,n) by CARD_1:11;
  card rng Upper_Seq(C,n) c= card dom Upper_Seq(C,n) by CARD_2:61;
  then
A14: card rng Upper_Seq(C,n) c= len Upper_Seq(C,n) by CARD_1:62;
  pWi <> pWa & pWa <> pEa by JORDAN1B:5,SPRECT_2:58;
  then card {pWi,pWa,pEa} = 3 by A1,CARD_2:58;
  then Segm 3 c= Segm len Upper_Seq(C,n) by A13,A14;
  hence len Upper_Seq(C,n) >= 3 by NAT_1:39;
A15: pWi <> pEa by TOPREAL5:19;
  pEi in rng Cage(C,n) by SPRECT_2:45;
  then
A16: pEi in rng Rotate(Cage(C,n),W-min L~Cage(C,n)) by FINSEQ_6:90,SPRECT_2:43;
  pEi..Rotate(Cage(C,n),W-min L~Cage(C,n)) > pEa..Rotate(Cage(C,n),W-min
  L~Cage(C,n)) by A2,A7,A10,FINSEQ_6:92,SPRECT_5:26;
  then
A17: pEi in rng Lower_Seq(C,n) by A3,A16,FINSEQ_6:62;
  {pWi,pEi,pEa} c= rng Lower_Seq(C,n)
  by A4,A17,ENUMSET1:def 1;
  then
A18: card {pWi,pEi,pEa} c= card rng Lower_Seq(C,n) by CARD_1:11;
  card rng Lower_Seq(C,n) c= card dom Lower_Seq(C,n) by CARD_2:61;
  then
A19: card rng Lower_Seq(C,n) c= len Lower_Seq(C,n) by CARD_1:62;
  pWi <> pEi & pEi <> pEa by Th14,SPRECT_2:54;
  then card {pWi,pEi,pEa} = 3 by A15,CARD_2:58;
  then Segm 3 c= Segm len Lower_Seq(C,n) by A18,A19;
  hence thesis by NAT_1:39;
end;
