Article SPRECT_1, MML version 4.99.1005

:: SPRECT_1:th 1
theorem
for b1 being trivial set
for b2 being set
      st b2 c= b1
   holds b2 is trivial;

:: SPRECT_1:condreg 1
registration
  cluster Relation-like Function-like non constant -> non trivial (set);
end;

:: SPRECT_1:condreg 2
registration
  cluster Relation-like Function-like trivial -> constant (set);
end;

:: SPRECT_1:th 2
theorem
for b1 being Relation-like Function-like set
      st proj2 b1 is trivial
   holds b1 is constant;

:: SPRECT_1:funcreg 1
registration
  let a1 be Relation-like Function-like constant set;
  cluster proj2 a1 -> trivial;
end;

:: SPRECT_1:exreg 1
registration
  cluster Relation-like Function-like constant non empty finite FinSequence-like set;
end;

:: SPRECT_1:th 3
theorem
for b1, b2 being Relation-like Function-like FinSequence-like set
      st b1 ^ b2 is constant
   holds b1 is constant & b2 is constant;

:: SPRECT_1:th 4
theorem
for b1, b2 being set
      st <*b1,b2*> is constant
   holds b1 = b2;

:: SPRECT_1:th 5
theorem
for b1, b2, b3 being set
      st <*b1,b2,b3*> is constant
   holds b1 = b2 & b2 = b3 & b3 = b1;

:: SPRECT_1:th 6
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being non empty Element of bool the carrier of b1
      st b2 is_a_component_of b3
   holds b2 <> {};

:: SPRECT_1:th 7
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is_a_component_of b3
   holds b2 c= b3;

:: SPRECT_1:th 8
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty Element of bool the carrier of b1
for b3, b4, b5 being Element of bool the carrier of b1
      st b3 is_a_component_of b2 & b4 is_a_component_of b2 & b5 is_a_component_of b2 & b3 \/ b4 = b2 & b5 <> b3
   holds b5 = b4;

:: SPRECT_1:th 9
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty Element of bool the carrier of b1
for b3, b4, b5, b6 being Element of bool the carrier of b1
      st b3 is_a_component_of b2 & b4 is_a_component_of b2 & b5 is_a_component_of b2 & b6 is_a_component_of b2 & b3 \/ b4 = b2 & b5 \/ b6 = b2
   holds {b3,b4} = {b5,b6};

:: SPRECT_1:th 10
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2 holds
L~ <*b1,b2,b3*> = (LSeg(b1,b2)) \/ LSeg(b2,b3);

:: SPRECT_1:funcreg 2
registration
  let a1 be Element of NAT;
  let a2 be non trivial FinSequence of the carrier of TOP-REAL a1;
  cluster L~ a2 -> non empty;
end;

:: SPRECT_1:funcreg 3
registration
  let a1 be FinSequence of the carrier of TOP-REAL 2;
  cluster L~ a1 -> compact;
end;

:: SPRECT_1:th 11
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
      st b1 c= b2 & b2 is horizontal
   holds b1 is horizontal;

:: SPRECT_1:th 12
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
      st b1 c= b2 & b2 is vertical
   holds b1 is vertical;

:: SPRECT_1:funcreg 4
registration
  cluster R^2-unit_square -> being_special_polygon non horizontal non vertical;
end;

:: SPRECT_1:exreg 2
registration
  cluster non empty compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2;
end;

:: SPRECT_1:th 13
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   N-min b1 in b1 & N-max b1 in b1;

:: SPRECT_1:th 14
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   S-min b1 in b1 & S-max b1 in b1;

:: SPRECT_1:th 15
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   W-min b1 in b1 & W-max b1 in b1;

:: SPRECT_1:th 16
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   E-min b1 in b1 & E-max b1 in b1;

:: SPRECT_1:th 17
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
      b1 is vertical
   iff
      W-bound b1 = E-bound b1;

:: SPRECT_1:th 18
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
      b1 is horizontal
   iff
      S-bound b1 = N-bound b1;

:: SPRECT_1:th 19
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
      st NW-corner b1 = NE-corner b1
   holds b1 is vertical;

:: SPRECT_1:th 20
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
      st SW-corner b1 = SE-corner b1
   holds b1 is vertical;

:: SPRECT_1:th 21
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
      st NW-corner b1 = SW-corner b1
   holds b1 is horizontal;

:: SPRECT_1:th 22
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
      st NE-corner b1 = SE-corner b1
   holds b1 is horizontal;

:: SPRECT_1:th 23
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   W-bound b1 <= E-bound b1;

:: SPRECT_1:th 24
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   S-bound b1 <= N-bound b1;

:: SPRECT_1:th 25
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   LSeg(SE-corner b1,NE-corner b1) = {b2 where b2 is Element of the carrier of TOP-REAL 2: b2 `1 = E-bound b1 & b2 `2 <= N-bound b1 & S-bound b1 <= b2 `2};

:: SPRECT_1:th 26
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   LSeg(SW-corner b1,SE-corner b1) = {b2 where b2 is Element of the carrier of TOP-REAL 2: b2 `1 <= E-bound b1 & W-bound b1 <= b2 `1 & b2 `2 = S-bound b1};

:: SPRECT_1:th 27
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   LSeg(NW-corner b1,NE-corner b1) = {b2 where b2 is Element of the carrier of TOP-REAL 2: b2 `1 <= E-bound b1 & W-bound b1 <= b2 `1 & b2 `2 = N-bound b1};

:: SPRECT_1:th 28
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   LSeg(SW-corner b1,NW-corner b1) = {b2 where b2 is Element of the carrier of TOP-REAL 2: b2 `1 = W-bound b1 & b2 `2 <= N-bound b1 & S-bound b1 <= b2 `2};

:: SPRECT_1:th 29
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   (LSeg(SW-corner b1,NW-corner b1)) /\ LSeg(NW-corner b1,NE-corner b1) = {NW-corner b1};

:: SPRECT_1:th 30
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   (LSeg(NW-corner b1,NE-corner b1)) /\ LSeg(NE-corner b1,SE-corner b1) = {NE-corner b1};

:: SPRECT_1:th 31
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   (LSeg(SE-corner b1,NE-corner b1)) /\ LSeg(SW-corner b1,SE-corner b1) = {SE-corner b1};

:: SPRECT_1:th 32
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   (LSeg(NW-corner b1,SW-corner b1)) /\ LSeg(SW-corner b1,SE-corner b1) = {SW-corner b1};

:: SPRECT_1:th 33
theorem
for b1 being non empty compact non vertical Element of bool the carrier of TOP-REAL 2 holds
   W-bound b1 < E-bound b1;

:: SPRECT_1:th 34
theorem
for b1 being non empty compact non horizontal Element of bool the carrier of TOP-REAL 2 holds
   S-bound b1 < N-bound b1;

:: SPRECT_1:th 35
theorem
for b1 being non empty compact non vertical Element of bool the carrier of TOP-REAL 2 holds
   LSeg(SW-corner b1,NW-corner b1) misses LSeg(SE-corner b1,NE-corner b1);

:: SPRECT_1:th 36
theorem
for b1 being non empty compact non horizontal Element of bool the carrier of TOP-REAL 2 holds
   LSeg(SW-corner b1,SE-corner b1) misses LSeg(NW-corner b1,NE-corner b1);

:: SPRECT_1:funcnot 1 => SPRECT_1:func 1
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  func SpStSeq A1 -> FinSequence of the carrier of TOP-REAL 2 equals
    <*NW-corner a1,NE-corner a1,SE-corner a1*> ^ <*SW-corner a1,NW-corner a1*>;
end;

:: SPRECT_1:def 1
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   SpStSeq b1 = <*NW-corner b1,NE-corner b1,SE-corner b1*> ^ <*SW-corner b1,NW-corner b1*>;

:: SPRECT_1:th 37
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   (SpStSeq b1) /. 1 = NW-corner b1;

:: SPRECT_1:th 38
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   (SpStSeq b1) /. 2 = NE-corner b1;

:: SPRECT_1:th 39
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   (SpStSeq b1) /. 3 = SE-corner b1;

:: SPRECT_1:th 40
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   (SpStSeq b1) /. 4 = SW-corner b1;

:: SPRECT_1:th 41
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   (SpStSeq b1) /. 5 = NW-corner b1;

:: SPRECT_1:th 42
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   len SpStSeq b1 = 5;

:: SPRECT_1:th 43
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   L~ SpStSeq b1 = ((LSeg(NW-corner b1,NE-corner b1)) \/ LSeg(NE-corner b1,SE-corner b1)) \/ ((LSeg(SE-corner b1,SW-corner b1)) \/ LSeg(SW-corner b1,NW-corner b1));

:: SPRECT_1:funcreg 5
registration
  let a1 be non empty compact non vertical Element of bool the carrier of TOP-REAL 2;
  cluster SpStSeq a1 -> non constant;
end;

:: SPRECT_1:funcreg 6
registration
  let a1 be non empty compact non horizontal Element of bool the carrier of TOP-REAL 2;
  cluster SpStSeq a1 -> non constant;
end;

:: SPRECT_1:funcreg 7
registration
  let a1 be non empty compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2;
  cluster SpStSeq a1 -> circular special unfolded s.c.c. standard;
end;

:: SPRECT_1:th 44
theorem
for b1 being non empty compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   L~ SpStSeq b1 = [.W-bound b1,E-bound b1,S-bound b1,N-bound b1.];

:: SPRECT_1:th 46
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty compact Element of bool the carrier of b1
for b3 being Function-like quasi_total continuous Relation of the carrier of b1,REAL holds
   b3 .: b2 is bounded_below;

:: SPRECT_1:th 47
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty compact Element of bool the carrier of b1
for b3 being Function-like quasi_total continuous Relation of the carrier of b1,REAL holds
   b3 .: b2 is bounded_above;

:: SPRECT_1:exreg 3
registration
  cluster non empty complex-membered ext-real-membered real-membered bounded_above bounded_below Element of bool REAL;
end;

:: SPRECT_1:th 48
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   W-bound b1 = inf (proj1 .: b1);

:: SPRECT_1:th 49
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   S-bound b1 = inf (proj2 .: b1);

:: SPRECT_1:th 50
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   N-bound b1 = sup (proj2 .: b1);

:: SPRECT_1:th 51
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   E-bound b1 = sup (proj1 .: b1);

:: SPRECT_1:th 52
theorem
for b1, b2 being non empty bounded_below Element of bool REAL holds
inf (b1 \/ b2) = min(inf b1,inf b2);

:: SPRECT_1:th 53
theorem
for b1, b2 being non empty bounded_above Element of bool REAL holds
sup (b1 \/ b2) = max(sup b1,sup b2);

:: SPRECT_1:th 54
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 = b2 \/ b3
   holds W-bound b1 = min(W-bound b2,W-bound b3);

:: SPRECT_1:th 55
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 = b2 \/ b3
   holds S-bound b1 = min(S-bound b2,S-bound b3);

:: SPRECT_1:th 56
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 = b2 \/ b3
   holds N-bound b1 = max(N-bound b2,N-bound b3);

:: SPRECT_1:th 57
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 = b2 \/ b3
   holds E-bound b1 = max(E-bound b2,E-bound b3);

:: SPRECT_1:funcreg 8
registration
  cluster {} REAL -> bounded;
end;

:: SPRECT_1:funcreg 9
registration
  let a1, a2 be real set;
  cluster [.a1,a2.] -> bounded;
end;

:: SPRECT_1:condreg 3
registration
  cluster bounded -> bounded_above bounded_below (Element of bool REAL);
end;

:: SPRECT_1:condreg 4
registration
  cluster bounded_above bounded_below -> bounded (Element of bool REAL);
end;

:: SPRECT_1:th 59
theorem
for b1, b2, b3 being Element of REAL
      st b1 <= b2
   holds    b3 in [.b1,b2.]
   iff
      ex b4 being Element of REAL st
         0 <= b4 &
          b4 <= 1 &
          b3 = (b4 * b1) + ((1 - b4) * b2);

:: SPRECT_1:th 60
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
      st b1 `1 <= b2 `1
   holds proj1 .: LSeg(b1,b2) = [.b1 `1,b2 `1.];

:: SPRECT_1:th 61
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
      st b1 `2 <= b2 `2
   holds proj2 .: LSeg(b1,b2) = [.b1 `2,b2 `2.];

:: SPRECT_1:th 62
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
      st b1 `1 <= b2 `1
   holds W-bound LSeg(b1,b2) = b1 `1;

:: SPRECT_1:th 63
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
      st b1 `2 <= b2 `2
   holds S-bound LSeg(b1,b2) = b1 `2;

:: SPRECT_1:th 64
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
      st b1 `2 <= b2 `2
   holds N-bound LSeg(b1,b2) = b2 `2;

:: SPRECT_1:th 65
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
      st b1 `1 <= b2 `1
   holds E-bound LSeg(b1,b2) = b2 `1;

:: SPRECT_1:th 66
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   W-bound L~ SpStSeq b1 = W-bound b1;

:: SPRECT_1:th 67
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   S-bound L~ SpStSeq b1 = S-bound b1;

:: SPRECT_1:th 68
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   N-bound L~ SpStSeq b1 = N-bound b1;

:: SPRECT_1:th 69
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   E-bound L~ SpStSeq b1 = E-bound b1;

:: SPRECT_1:th 70
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   NW-corner L~ SpStSeq b1 = NW-corner b1;

:: SPRECT_1:th 71
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   NE-corner L~ SpStSeq b1 = NE-corner b1;

:: SPRECT_1:th 72
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   SW-corner L~ SpStSeq b1 = SW-corner b1;

:: SPRECT_1:th 73
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   SE-corner L~ SpStSeq b1 = SE-corner b1;

:: SPRECT_1:th 74
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   W-most L~ SpStSeq b1 = LSeg(SW-corner b1,NW-corner b1);

:: SPRECT_1:th 75
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   N-most L~ SpStSeq b1 = LSeg(NW-corner b1,NE-corner b1);

:: SPRECT_1:th 76
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   S-most L~ SpStSeq b1 = LSeg(SW-corner b1,SE-corner b1);

:: SPRECT_1:th 77
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   E-most L~ SpStSeq b1 = LSeg(SE-corner b1,NE-corner b1);

:: SPRECT_1:th 78
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   proj2 .: LSeg(SW-corner b1,NW-corner b1) = [.S-bound b1,N-bound b1.];

:: SPRECT_1:th 79
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   proj1 .: LSeg(NW-corner b1,NE-corner b1) = [.W-bound b1,E-bound b1.];

:: SPRECT_1:th 80
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   proj2 .: LSeg(NE-corner b1,SE-corner b1) = [.S-bound b1,N-bound b1.];

:: SPRECT_1:th 81
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   proj1 .: LSeg(SE-corner b1,SW-corner b1) = [.W-bound b1,E-bound b1.];

:: SPRECT_1:th 82
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   W-min L~ SpStSeq b1 = SW-corner b1;

:: SPRECT_1:th 83
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   W-max L~ SpStSeq b1 = NW-corner b1;

:: SPRECT_1:th 84
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   N-min L~ SpStSeq b1 = NW-corner b1;

:: SPRECT_1:th 85
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   N-max L~ SpStSeq b1 = NE-corner b1;

:: SPRECT_1:th 86
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   E-min L~ SpStSeq b1 = SE-corner b1;

:: SPRECT_1:th 87
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   E-max L~ SpStSeq b1 = NE-corner b1;

:: SPRECT_1:th 88
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   S-min L~ SpStSeq b1 = SW-corner b1;

:: SPRECT_1:th 89
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
   S-max L~ SpStSeq b1 = SE-corner b1;

:: SPRECT_1:attrnot 1 => SPRECT_1:attr 1
definition
  let a1 be FinSequence of the carrier of TOP-REAL 2;
  attr a1 is rectangular means
    ex b1 being non empty compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 st
       a1 = SpStSeq b1;
end;

:: SPRECT_1:dfs 2
definiens
  let a1 be FinSequence of the carrier of TOP-REAL 2;
To prove
     a1 is rectangular
it is sufficient to prove
  thus ex b1 being non empty compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 st
       a1 = SpStSeq b1;

:: SPRECT_1:def 2
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2 holds
      b1 is rectangular
   iff
      ex b2 being non empty compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 st
         b1 = SpStSeq b2;

:: SPRECT_1:funcreg 10
registration
  let a1 be non empty compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2;
  cluster SpStSeq a1 -> rectangular;
end;

:: SPRECT_1:exreg 4
registration
  cluster Relation-like Function-like finite FinSequence-like rectangular FinSequence of the carrier of TOP-REAL 2;
end;

:: SPRECT_1:th 90
theorem
for b1 being rectangular FinSequence of the carrier of TOP-REAL 2 holds
   len b1 = 5;

:: SPRECT_1:condreg 5
registration
  cluster rectangular -> non constant (FinSequence of the carrier of TOP-REAL 2);
end;

:: SPRECT_1:condreg 6
registration
  cluster non empty rectangular -> circular special unfolded s.c.c. standard (FinSequence of the carrier of TOP-REAL 2);
end;

:: SPRECT_1:th 91
theorem
for b1 being rectangular FinSequence of the carrier of TOP-REAL 2 holds
   b1 /. 1 = N-min L~ b1 & b1 /. 1 = W-max L~ b1;

:: SPRECT_1:th 92
theorem
for b1 being rectangular FinSequence of the carrier of TOP-REAL 2 holds
   b1 /. 2 = N-max L~ b1 & b1 /. 2 = E-max L~ b1;

:: SPRECT_1:th 93
theorem
for b1 being rectangular FinSequence of the carrier of TOP-REAL 2 holds
   b1 /. 3 = S-max L~ b1 & b1 /. 3 = E-min L~ b1;

:: SPRECT_1:th 94
theorem
for b1 being rectangular FinSequence of the carrier of TOP-REAL 2 holds
   b1 /. 4 = S-min L~ b1 & b1 /. 4 = W-min L~ b1;

:: SPRECT_1:th 95
theorem
for b1, b2, b3, b4 being Element of REAL
      st b1 < b2 & b3 < b4
   holds [.b1,b2,b3,b4.] is Jordan;

:: SPRECT_1:funcreg 11
registration
  let a1 be rectangular FinSequence of the carrier of TOP-REAL 2;
  cluster L~ a1 -> Jordan;
end;

:: SPRECT_1:attrnot 2 => JORDAN1:attr 2
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  attr a1 is Jordan means
    a1 ` <> {} &
     (ex b1, b2 being Element of bool the carrier of TOP-REAL 2 st
        a1 ` = b1 \/ b2 & b1 misses b2 & (Cl b1) \ b1 = (Cl b2) \ b2 & b1 is_a_component_of a1 ` & b2 is_a_component_of a1 `);
end;

:: SPRECT_1:dfs 3
definiens
  let a1 be Element of bool the carrier of TOP-REAL 2;
To prove
     a1 is Jordan
it is sufficient to prove
  thus a1 ` <> {} &
     (ex b1, b2 being Element of bool the carrier of TOP-REAL 2 st
        a1 ` = b1 \/ b2 & b1 misses b2 & (Cl b1) \ b1 = (Cl b2) \ b2 & b1 is_a_component_of a1 ` & b2 is_a_component_of a1 `);

:: SPRECT_1:def 3
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
      b1 is Jordan
   iff
      b1 ` <> {} &
       (ex b2, b3 being Element of bool the carrier of TOP-REAL 2 st
          b1 ` = b2 \/ b3 & b2 misses b3 & (Cl b2) \ b2 = (Cl b3) \ b3 & b2 is_a_component_of b1 ` & b3 is_a_component_of b1 `);

:: SPRECT_1:th 96
theorem
for b1 being rectangular FinSequence of the carrier of TOP-REAL 2 holds
   LeftComp b1 misses RightComp b1;

:: SPRECT_1:funcreg 12
registration
  let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
  cluster LeftComp a1 -> non empty;
end;

:: SPRECT_1:funcreg 13
registration
  let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
  cluster RightComp a1 -> non empty;
end;

:: SPRECT_1:th 97
theorem
for b1 being rectangular FinSequence of the carrier of TOP-REAL 2 holds
   LeftComp b1 <> RightComp b1;