Article JORDAN1G, MML version 4.99.1005

:: JORDAN1G:exreg 1
registration
  cluster Relation-like Function-like finite FinSequence-like trivial set;
end;

:: JORDAN1G:th 1
theorem
for b1 being Relation-like Function-like FinSequence-like trivial set
      st b1 is not empty
   holds ex b2 being set st
      b1 = <*b2*>;

:: JORDAN1G:funcreg 1
registration
  let a1 be Relation-like Function-like FinSequence-like non trivial set;
  cluster Rev a1 -> Relation-like Function-like FinSequence-like non trivial;
end;

:: JORDAN1G:th 2
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3 being tabular FinSequence of b1 *
for b4 being set
      st b2 is_sequence_on b3
   holds b2 -: b4 is_sequence_on b3;

:: JORDAN1G:th 3
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3 being tabular FinSequence of b1 *
for b4 being Element of b1
      st b4 in proj2 b2 & b2 is_sequence_on b3
   holds b2 :- b4 is_sequence_on b3;

:: JORDAN1G:th 4
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   Upper_Seq(b2,b1) is_sequence_on Gauge(b2,b1);

:: JORDAN1G:th 5
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   Lower_Seq(b2,b1) is_sequence_on Gauge(b2,b1);

:: JORDAN1G:funcreg 2
registration
  let a1 be connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2;
  let a2 be Element of NAT;
  cluster Upper_Seq(a1,a2) -> standard;
end;

:: JORDAN1G:funcreg 3
registration
  let a1 be connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2;
  let a2 be Element of NAT;
  cluster Lower_Seq(a1,a2) -> standard;
end;

:: JORDAN1G:th 6
theorem
for b1 being tabular Y_equal-in-column Y_increasing-in-line FinSequence of (the carrier of TOP-REAL 2) *
for b2, b3, b4, b5 being Element of NAT
      st [b2,b4] in Indices b1 &
         [b3,b5] in Indices b1 &
         (b1 *(b2,b4)) `2 = (b1 *(b3,b5)) `2
   holds b4 = b5;

:: JORDAN1G:th 7
theorem
for b1 being tabular X_equal-in-line X_increasing-in-column FinSequence of (the carrier of TOP-REAL 2) *
for b2, b3, b4, b5 being Element of NAT
      st [b2,b4] in Indices b1 &
         [b3,b5] in Indices b1 &
         (b1 *(b2,b4)) `1 = (b1 *(b3,b5)) `1
   holds b2 = b3;

:: JORDAN1G:th 16
theorem
for b1 being non trivial special unfolded standard FinSequence of the carrier of TOP-REAL 2
      st (b1 /. 1 <> N-min L~ b1 & b1 /. len b1 <> N-min L~ b1 or b1 /. 1 <> N-max L~ b1 & b1 /. len b1 <> N-max L~ b1)
   holds (N-min L~ b1) `1 < (N-max L~ b1) `1;

:: JORDAN1G:th 17
theorem
for b1 being non trivial special unfolded standard FinSequence of the carrier of TOP-REAL 2
      st (b1 /. 1 <> N-min L~ b1 & b1 /. len b1 <> N-min L~ b1 or b1 /. 1 <> N-max L~ b1 & b1 /. len b1 <> N-max L~ b1)
   holds N-min L~ b1 <> N-max L~ b1;

:: JORDAN1G:th 18
theorem
for b1 being non trivial special unfolded standard FinSequence of the carrier of TOP-REAL 2
      st (b1 /. 1 <> S-min L~ b1 & b1 /. len b1 <> S-min L~ b1 or b1 /. 1 <> S-max L~ b1 & b1 /. len b1 <> S-max L~ b1)
   holds (S-min L~ b1) `1 < (S-max L~ b1) `1;

:: JORDAN1G:th 19
theorem
for b1 being non trivial special unfolded standard FinSequence of the carrier of TOP-REAL 2
      st (b1 /. 1 <> S-min L~ b1 & b1 /. len b1 <> S-min L~ b1 or b1 /. 1 <> S-max L~ b1 & b1 /. len b1 <> S-max L~ b1)
   holds S-min L~ b1 <> S-max L~ b1;

:: JORDAN1G:th 20
theorem
for b1 being non trivial special unfolded standard FinSequence of the carrier of TOP-REAL 2
      st (b1 /. 1 <> W-min L~ b1 & b1 /. len b1 <> W-min L~ b1 or b1 /. 1 <> W-max L~ b1 & b1 /. len b1 <> W-max L~ b1)
   holds (W-min L~ b1) `2 < (W-max L~ b1) `2;

:: JORDAN1G:th 21
theorem
for b1 being non trivial special unfolded standard FinSequence of the carrier of TOP-REAL 2
      st (b1 /. 1 <> W-min L~ b1 & b1 /. len b1 <> W-min L~ b1 or b1 /. 1 <> W-max L~ b1 & b1 /. len b1 <> W-max L~ b1)
   holds W-min L~ b1 <> W-max L~ b1;

:: JORDAN1G:th 22
theorem
for b1 being non trivial special unfolded standard FinSequence of the carrier of TOP-REAL 2
      st (b1 /. 1 <> E-min L~ b1 & b1 /. len b1 <> E-min L~ b1 or b1 /. 1 <> E-max L~ b1 & b1 /. len b1 <> E-max L~ b1)
   holds (E-min L~ b1) `2 < (E-max L~ b1) `2;

:: JORDAN1G:th 23
theorem
for b1 being non trivial special unfolded standard FinSequence of the carrier of TOP-REAL 2
      st (b1 /. 1 <> E-min L~ b1 & b1 /. len b1 <> E-min L~ b1 or b1 /. 1 <> E-max L~ b1 & b1 /. len b1 <> E-max L~ b1)
   holds E-min L~ b1 <> E-max L~ b1;

:: JORDAN1G:th 24
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4 being Element of b1
      st b3 in proj2 b2 & b4 in proj2 b2 & b4 .. b2 <= b3 .. b2
   holds (b2 -: b3) :- b4 = (b2 :- b4) -: b3;

:: JORDAN1G:th 25
theorem
for b1 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT holds
   (L~ ((Cage(b1,b2)) -: W-min L~ Cage(b1,b2))) /\ L~ ((Cage(b1,b2)) :- W-min L~ Cage(b1,b2)) = {N-min L~ Cage(b1,b2),W-min L~ Cage(b1,b2)};

:: JORDAN1G:th 26
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   Lower_Seq(b2,b1) = (Rotate(Cage(b2,b1),E-max L~ Cage(b2,b1))) -: W-min L~ Cage(b2,b1);

:: JORDAN1G:th 27
theorem
for b1 being Element of NAT
for b2 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   (W-min L~ Cage(b2,b1)) .. Upper_Seq(b2,b1) = 1;

:: JORDAN1G:th 28
theorem
for b1 being Element of NAT
for b2 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   (W-min L~ Cage(b2,b1)) .. Upper_Seq(b2,b1) < (W-max L~ Cage(b2,b1)) .. Upper_Seq(b2,b1);

:: JORDAN1G:th 29
theorem
for b1 being Element of NAT
for b2 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   (W-max L~ Cage(b2,b1)) .. Upper_Seq(b2,b1) <= (N-min L~ Cage(b2,b1)) .. Upper_Seq(b2,b1);

:: JORDAN1G:th 30
theorem
for b1 being Element of NAT
for b2 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   (N-min L~ Cage(b2,b1)) .. Upper_Seq(b2,b1) < (N-max L~ Cage(b2,b1)) .. Upper_Seq(b2,b1);

:: JORDAN1G:th 31
theorem
for b1 being Element of NAT
for b2 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   (N-max L~ Cage(b2,b1)) .. Upper_Seq(b2,b1) <= (E-max L~ Cage(b2,b1)) .. Upper_Seq(b2,b1);

:: JORDAN1G:th 32
theorem
for b1 being Element of NAT
for b2 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   (E-max L~ Cage(b2,b1)) .. Upper_Seq(b2,b1) = len Upper_Seq(b2,b1);

:: JORDAN1G:th 33
theorem
for b1 being Element of NAT
for b2 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   (E-max L~ Cage(b2,b1)) .. Lower_Seq(b2,b1) = 1;

:: JORDAN1G:th 34
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   (E-max L~ Cage(b2,b1)) .. Lower_Seq(b2,b1) < (E-min L~ Cage(b2,b1)) .. Lower_Seq(b2,b1);

:: JORDAN1G:th 35
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   (E-min L~ Cage(b2,b1)) .. Lower_Seq(b2,b1) <= (S-max L~ Cage(b2,b1)) .. Lower_Seq(b2,b1);

:: JORDAN1G:th 36
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   (S-max L~ Cage(b2,b1)) .. Lower_Seq(b2,b1) < (S-min L~ Cage(b2,b1)) .. Lower_Seq(b2,b1);

:: JORDAN1G:th 37
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   (S-min L~ Cage(b2,b1)) .. Lower_Seq(b2,b1) <= (W-min L~ Cage(b2,b1)) .. Lower_Seq(b2,b1);

:: JORDAN1G:th 38
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   (W-min L~ Cage(b2,b1)) .. Lower_Seq(b2,b1) = len Lower_Seq(b2,b1);

:: JORDAN1G:th 39
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   ((Upper_Seq(b2,b1)) /. 2) `1 = W-bound L~ Cage(b2,b1);

:: JORDAN1G:th 40
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   ((Lower_Seq(b2,b1)) /. 2) `1 = E-bound L~ Cage(b2,b1);

:: JORDAN1G:th 41
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   (W-bound L~ Cage(b2,b1)) + E-bound L~ Cage(b2,b1) = (W-bound b2) + E-bound b2;

:: JORDAN1G:th 42
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   (S-bound L~ Cage(b2,b1)) + N-bound L~ Cage(b2,b1) = (S-bound b2) + N-bound b2;

:: JORDAN1G:th 43
theorem
for b1 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
      st 1 <= b3 & b3 <= width Gauge(b1,b2) & 0 < b2
   holds ((Gauge(b1,b2)) *(Center Gauge(b1,b2),b3)) `1 = ((W-bound b1) + E-bound b1) / 2;

:: JORDAN1G:th 44
theorem
for b1 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
      st 1 <= b3 & b3 <= len Gauge(b1,b2) & 0 < b2
   holds ((Gauge(b1,b2)) *(b3,Center Gauge(b1,b2))) `2 = ((S-bound b1) + N-bound b1) / 2;

:: JORDAN1G:th 45
theorem
for b1 being being_S-Seq FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
      st 1 <= b2 & b2 <= len b1 & 1 <= b3 & b3 <= len b1 & b1 /. 1 in L~ mid(b1,b2,b3) & b2 <> 1
   holds b3 = 1;

:: JORDAN1G:th 46
theorem
for b1 being being_S-Seq FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
      st 1 <= b2 &
         b2 <= len b1 &
         1 <= b3 &
         b3 <= len b1 &
         b1 /. len b1 in L~ mid(b1,b2,b3) &
         b2 <> len b1
   holds b3 = len b1;

:: JORDAN1G:th 47
theorem
for b1 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT holds
   proj2 Upper_Seq(b1,b2) c= proj2 Cage(b1,b2) & proj2 Lower_Seq(b1,b2) c= proj2 Cage(b1,b2);

:: JORDAN1G:th 48
theorem
for b1 being Element of NAT
for b2 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   Upper_Seq(b2,b1) is_a_h.c._for Cage(b2,b1);

:: JORDAN1G:th 49
theorem
for b1 being Element of NAT
for b2 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   Rev Lower_Seq(b2,b1) is_a_h.c._for Cage(b2,b1);

:: JORDAN1G:th 50
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b3 being Element of NAT
      st 1 < b3 & b3 <= len Gauge(b2,b1)
   holds not (Gauge(b2,b1)) *(b3,1) in proj2 Upper_Seq(b2,b1);

:: JORDAN1G:th 51
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b3 being Element of NAT
      st 1 <= b3 & b3 < len Gauge(b2,b1)
   holds not (Gauge(b2,b1)) *(b3,width Gauge(b2,b1)) in proj2 Lower_Seq(b2,b1);

:: JORDAN1G:th 52
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b3 being Element of NAT
      st 1 < b3 & b3 <= len Gauge(b2,b1)
   holds not (Gauge(b2,b1)) *(b3,1) in L~ Upper_Seq(b2,b1);

:: JORDAN1G:th 53
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b3 being Element of NAT
      st 1 <= b3 & b3 < len Gauge(b2,b1)
   holds not (Gauge(b2,b1)) *(b3,width Gauge(b2,b1)) in L~ Lower_Seq(b2,b1);

:: JORDAN1G:th 54
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b3, b4 being Element of NAT
      st 1 <= b3 &
         b3 <= len Gauge(b2,b1) &
         1 <= b4 &
         b4 <= width Gauge(b2,b1) &
         (Gauge(b2,b1)) *(b3,b4) in L~ Cage(b2,b1)
   holds LSeg((Gauge(b2,b1)) *(b3,1),(Gauge(b2,b1)) *(b3,b4)) meets L~ Lower_Seq(b2,b1);

:: JORDAN1G:th 55
theorem
for b1 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT
      st 0 < b2
   holds First_Point(L~ Upper_Seq(b1,b2),W-min L~ Cage(b1,b2),E-max L~ Cage(b1,b2),Vertical_Line (((W-bound L~ Cage(b1,b2)) + E-bound L~ Cage(b1,b2)) / 2)) in proj2 Upper_Seq(b1,b2);

:: JORDAN1G:th 56
theorem
for b1 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT
      st 0 < b2
   holds Last_Point(L~ Lower_Seq(b1,b2),E-max L~ Cage(b1,b2),W-min L~ Cage(b1,b2),Vertical_Line (((W-bound L~ Cage(b1,b2)) + E-bound L~ Cage(b1,b2)) / 2)) in proj2 Lower_Seq(b1,b2);

:: JORDAN1G:th 57
theorem
for b1 being being_S-Seq FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in proj2 b1
   holds R_Cut(b1,b2) = mid(b1,1,b2 .. b1);

:: JORDAN1G:th 58
theorem
for b1 being being_S-Seq FinSequence of the carrier of TOP-REAL 2
for b2 being closed Element of bool the carrier of TOP-REAL 2
      st L~ b1 meets b2 &
         not b1 /. 1 in b2 &
         First_Point(L~ b1,b1 /. 1,b1 /. len b1,b2) in proj2 b1
   holds (L~ mid(b1,1,(First_Point(L~ b1,b1 /. 1,b1 /. len b1,b2)) .. b1)) /\ b2 = {First_Point(L~ b1,b1 /. 1,b1 /. len b1,b2)};

:: JORDAN1G:th 59
theorem
for b1 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT
   st 0 < b2
for b3 being Element of NAT
      st 1 <= b3 &
         b3 < (First_Point(L~ Upper_Seq(b1,b2),W-min L~ Cage(b1,b2),E-max L~ Cage(b1,b2),Vertical_Line (((W-bound L~ Cage(b1,b2)) + E-bound L~ Cage(b1,b2)) / 2))) .. Upper_Seq(b1,b2)
   holds ((Upper_Seq(b1,b2)) /. b3) `1 < ((W-bound L~ Cage(b1,b2)) + E-bound L~ Cage(b1,b2)) / 2;

:: JORDAN1G:th 60
theorem
for b1 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT
   st 0 < b2
for b3 being Element of NAT
      st 1 <= b3 &
         b3 < (First_Point(L~ Rev Lower_Seq(b1,b2),W-min L~ Cage(b1,b2),E-max L~ Cage(b1,b2),Vertical_Line (((W-bound L~ Cage(b1,b2)) + E-bound L~ Cage(b1,b2)) / 2))) .. Rev Lower_Seq(b1,b2)
   holds ((Rev Lower_Seq(b1,b2)) /. b3) `1 < ((W-bound L~ Cage(b1,b2)) + E-bound L~ Cage(b1,b2)) / 2;

:: JORDAN1G:th 61
theorem
for b1 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT
   st 0 < b2
for b3 being Element of the carrier of TOP-REAL 2
      st b3 in proj2 mid(Upper_Seq(b1,b2),2,(First_Point(L~ Upper_Seq(b1,b2),W-min L~ Cage(b1,b2),E-max L~ Cage(b1,b2),Vertical_Line (((W-bound L~ Cage(b1,b2)) + E-bound L~ Cage(b1,b2)) / 2))) .. Upper_Seq(b1,b2))
   holds b3 `1 <= ((W-bound L~ Cage(b1,b2)) + E-bound L~ Cage(b1,b2)) / 2;

:: JORDAN1G:th 62
theorem
for b1 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT
      st 0 < b2
   holds (Last_Point(L~ Lower_Seq(b1,b2),E-max L~ Cage(b1,b2),W-min L~ Cage(b1,b2),Vertical_Line (((W-bound L~ Cage(b1,b2)) + E-bound L~ Cage(b1,b2)) / 2))) `2 < (First_Point(L~ Upper_Seq(b1,b2),W-min L~ Cage(b1,b2),E-max L~ Cage(b1,b2),Vertical_Line (((W-bound L~ Cage(b1,b2)) + E-bound L~ Cage(b1,b2)) / 2))) `2;

:: JORDAN1G:th 63
theorem
for b1 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT
      st 0 < b2
   holds L~ Upper_Seq(b1,b2) = Upper_Arc L~ Cage(b1,b2);

:: JORDAN1G:th 64
theorem
for b1 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT
      st 0 < b2
   holds L~ Lower_Seq(b1,b2) = Lower_Arc L~ Cage(b1,b2);

:: JORDAN1G:th 65
theorem
for b1 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT
   st 0 < b2
for b3, b4 being Element of NAT
      st 1 <= b3 &
         b3 <= len Gauge(b1,b2) &
         1 <= b4 &
         b4 <= width Gauge(b1,b2) &
         (Gauge(b1,b2)) *(b3,b4) in L~ Cage(b1,b2)
   holds LSeg((Gauge(b1,b2)) *(b3,1),(Gauge(b1,b2)) *(b3,b4)) meets Lower_Arc L~ Cage(b1,b2);