Article SPPOL_1, MML version 4.99.1005

:: SPPOL_1:th 5
theorem
for b1, b2 being real set
      st b1 <= b2
   holds b1 - 1 <= b2 & b1 - 1 < b2 & b1 <= b2 + 1 & b1 < b2 + 1;

:: SPPOL_1:th 6
theorem
for b1, b2 being natural set
      st b1 < b2
   holds b1 <= b2 - 1;

:: SPPOL_1:th 7
theorem
for b1, b2, b3 being Element of NAT
      st 1 <= b1 - b2 & b1 - b2 <= b3
   holds b1 - b2 in Seg b3 & b1 - b2 is Element of NAT;

:: SPPOL_1:th 12
theorem
for b1, b2, b3 being real set
      st 0 <= b1 &
         b1 <= 1 &
         0 <= b2 &
         0 <= b3 &
         (b1 * b2) + ((1 - b1) * b3) = 0 &
         (b1 = 0 implies b3 <> 0) &
         (b1 = 1 implies b2 <> 0)
   holds b2 = 0 & b3 = 0;

:: SPPOL_1:sch 1
scheme SPPOL_1:sch 1
{F1 -> non empty set,
  F2 -> FinSequence of F1(),
  F3 -> set}:
{F3(F2(), b1) where b1 is Element of NAT: b1 in dom F2() & P1[b1]} is finite


:: SPPOL_1:sch 2
scheme SPPOL_1:sch 2
{F1 -> non empty set,
  F2 -> FinSequence of F1(),
  F3 -> set}:
{F3(F2(), b1) where b1 is Element of NAT: 1 <= b1 & b1 <= len F2() & P1[b1]} is finite


:: SPPOL_1:th 15
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of REAL b1 holds
|.b2 - b3.| - |.b3 - b4.| <= |.b2 - b4.|;

:: SPPOL_1:th 16
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of REAL b1 holds
|.b3 - b2.| - |.b3 - b4.| <= |.b4 - b2.|;

:: SPPOL_1:th 20
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of Euclid b1
for b4, b5 being Element of REAL b1
      st b4 = b2 & b5 = b3
   holds dist(b2,b3) = |.b4 - b5.|;

:: SPPOL_1:th 21
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of the carrier of TOP-REAL b1
      st b2 in LSeg(b3,b4)
   holds ex b5 being Element of REAL st
      0 <= b5 &
       b5 <= 1 &
       b2 = ((1 - b5) * b3) + (b5 * b4);

:: SPPOL_1:th 22
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4 being Element of REAL
      st 0 <= b4 & b4 <= 1
   holds ((1 - b4) * b2) + (b4 * b3) in LSeg(b2,b3);

:: SPPOL_1:th 23
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4 being non empty Element of bool the carrier of TOP-REAL b1
      st b4 is closed(TOP-REAL b1) & b4 c= LSeg(b2,b3)
   holds ex b5 being Element of REAL st
      ((1 - b5) * b2) + (b5 * b3) in b4 &
       (for b6 being Element of REAL
             st 0 <= b6 &
                b6 <= 1 &
                ((1 - b6) * b2) + (b6 * b3) in b4
          holds b5 <= b6);

:: SPPOL_1:th 24
theorem
for b1 being Element of NAT
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL b1
      st LSeg(b4,b5) c= LSeg(b2,b3) & b2 in LSeg(b4,b5) & b2 <> b4
   holds b2 = b5;

:: SPPOL_1:th 25
theorem
for b1 being Element of NAT
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL b1
      st LSeg(b2,b3) = LSeg(b4,b5) & (b2 = b4 implies b3 <> b5)
   holds b2 = b5 & b3 = b4;

:: SPPOL_1:th 26
theorem
for b1 being Element of NAT holds
   TOP-REAL b1 is being_T2;

:: SPPOL_1:th 28
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
LSeg(b2,b3) is compact(TOP-REAL b1);

:: SPPOL_1:th 29
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
LSeg(b2,b3) is closed(TOP-REAL b1);

:: SPPOL_1:prednot 1 => SPPOL_1:pred 1
definition
  let a1 be Element of NAT;
  let a2 be Element of the carrier of TOP-REAL a1;
  let a3 be Element of bool the carrier of TOP-REAL a1;
  pred A2 is_extremal_in A3 means
    a2 in a3 &
     (for b1, b2 being Element of the carrier of TOP-REAL a1
           st a2 in LSeg(b1,b2) & LSeg(b1,b2) c= a3 & a2 <> b1
        holds a2 = b2);
end;

:: SPPOL_1:dfs 1
definiens
  let a1 be Element of NAT;
  let a2 be Element of the carrier of TOP-REAL a1;
  let a3 be Element of bool the carrier of TOP-REAL a1;
To prove
     a2 is_extremal_in a3
it is sufficient to prove
  thus a2 in a3 &
     (for b1, b2 being Element of the carrier of TOP-REAL a1
           st a2 in LSeg(b1,b2) & LSeg(b1,b2) c= a3 & a2 <> b1
        holds a2 = b2);

:: SPPOL_1:def 1
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1
for b3 being Element of bool the carrier of TOP-REAL b1 holds
      b2 is_extremal_in b3
   iff
      b2 in b3 &
       (for b4, b5 being Element of the carrier of TOP-REAL b1
             st b2 in LSeg(b4,b5) & LSeg(b4,b5) c= b3 & b2 <> b4
          holds b2 = b5);

:: SPPOL_1:th 30
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1
for b3, b4 being Element of bool the carrier of TOP-REAL b1
      st b2 is_extremal_in b3 & b4 c= b3 & b2 in b4
   holds b2 is_extremal_in b4;

:: SPPOL_1:th 31
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1 holds
   b2 is_extremal_in {b2};

:: SPPOL_1:th 32
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
b2 is_extremal_in LSeg(b2,b3);

:: SPPOL_1:th 33
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
b2 is_extremal_in LSeg(b3,b2);

:: SPPOL_1:th 34
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of the carrier of TOP-REAL b1
      st b2 is_extremal_in LSeg(b3,b4) & b2 <> b3
   holds b2 = b4;

:: SPPOL_1:th 35
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
      st b1 `1 <> b2 `1 & b1 `2 <> b2 `2
   holds ex b3 being Element of the carrier of TOP-REAL 2 st
      b3 in LSeg(b1,b2) & b3 `1 <> b1 `1 & b3 `1 <> b2 `1 & b3 `2 <> b1 `2 & b3 `2 <> b2 `2;

:: SPPOL_1:attrnot 1 => SPPOL_1:attr 1
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  attr a1 is horizontal means
    for b1, b2 being Element of the carrier of TOP-REAL 2
          st b1 in a1 & b2 in a1
       holds b1 `2 = b2 `2;
end;

:: SPPOL_1:dfs 2
definiens
  let a1 be Element of bool the carrier of TOP-REAL 2;
To prove
     a1 is horizontal
it is sufficient to prove
  thus for b1, b2 being Element of the carrier of TOP-REAL 2
          st b1 in a1 & b2 in a1
       holds b1 `2 = b2 `2;

:: SPPOL_1:def 2
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
      b1 is horizontal
   iff
      for b2, b3 being Element of the carrier of TOP-REAL 2
            st b2 in b1 & b3 in b1
         holds b2 `2 = b3 `2;

:: SPPOL_1:attrnot 2 => SPPOL_1:attr 2
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  attr a1 is vertical means
    for b1, b2 being Element of the carrier of TOP-REAL 2
          st b1 in a1 & b2 in a1
       holds b1 `1 = b2 `1;
end;

:: SPPOL_1:dfs 3
definiens
  let a1 be Element of bool the carrier of TOP-REAL 2;
To prove
     a1 is vertical
it is sufficient to prove
  thus for b1, b2 being Element of the carrier of TOP-REAL 2
          st b1 in a1 & b2 in a1
       holds b1 `1 = b2 `1;

:: SPPOL_1:def 3
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
      b1 is vertical
   iff
      for b2, b3 being Element of the carrier of TOP-REAL 2
            st b2 in b1 & b3 in b1
         holds b2 `1 = b3 `1;

:: SPPOL_1:condreg 1
registration
  cluster non trivial horizontal -> non vertical (Element of bool the carrier of TOP-REAL 2);
end;

:: SPPOL_1:condreg 2
registration
  cluster non trivial vertical -> non horizontal (Element of bool the carrier of TOP-REAL 2);
end;

:: SPPOL_1:th 36
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2 holds
   b1 `2 = b2 `2
iff
   LSeg(b1,b2) is horizontal;

:: SPPOL_1:th 37
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2 holds
   b1 `1 = b2 `1
iff
   LSeg(b1,b2) is vertical;

:: SPPOL_1:th 38
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
      st b1 in LSeg(b2,b3) & b4 in LSeg(b2,b3) & b1 `1 <> b4 `1 & b1 `2 = b4 `2
   holds LSeg(b2,b3) is horizontal;

:: SPPOL_1:th 39
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
      st b1 in LSeg(b2,b3) & b4 in LSeg(b2,b3) & b1 `2 <> b4 `2 & b1 `1 = b4 `1
   holds LSeg(b2,b3) is vertical;

:: SPPOL_1:th 40
theorem
for b1 being Element of NAT
for b2 being FinSequence of the carrier of TOP-REAL 2 holds
   LSeg(b2,b1) is closed(TOP-REAL 2);

:: SPPOL_1:th 41
theorem
for b1 being Element of NAT
for b2 being FinSequence of the carrier of TOP-REAL 2
      st b2 is special & LSeg(b2,b1) is not vertical
   holds LSeg(b2,b1) is horizontal;

:: SPPOL_1:th 42
theorem
for b1 being Element of NAT
for b2 being FinSequence of the carrier of TOP-REAL 2
      st b2 is one-to-one & 1 <= b1 & b1 + 1 <= len b2
   holds LSeg(b2,b1) is not trivial;

:: SPPOL_1:th 43
theorem
for b1 being Element of NAT
for b2 being FinSequence of the carrier of TOP-REAL 2
      st b2 is one-to-one & 1 <= b1 & b1 + 1 <= len b2 & LSeg(b2,b1) is vertical
   holds LSeg(b2,b1) is not horizontal;

:: SPPOL_1:th 44
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2 holds
   {LSeg(b1,b2) where b2 is Element of NAT: 1 <= b2 & b2 <= len b1} is finite;

:: SPPOL_1:th 45
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2 holds
   {LSeg(b1,b2) where b2 is Element of NAT: 1 <= b2 & b2 + 1 <= len b1} is finite;

:: SPPOL_1:th 46
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2 holds
   {LSeg(b1,b2) where b2 is Element of NAT: 1 <= b2 & b2 <= len b1} is Element of bool bool the carrier of TOP-REAL 2;

:: SPPOL_1:th 47
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2 holds
   {LSeg(b1,b2) where b2 is Element of NAT: 1 <= b2 & b2 + 1 <= len b1} is Element of bool bool the carrier of TOP-REAL 2;

:: SPPOL_1:th 48
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2 being FinSequence of the carrier of TOP-REAL 2
      st b1 = union {LSeg(b2,b3) where b3 is Element of NAT: 1 <= b3 & b3 + 1 <= len b2}
   holds b1 is closed(TOP-REAL 2);

:: SPPOL_1:th 49
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2 holds
   L~ b1 is closed(TOP-REAL 2);

:: SPPOL_1:attrnot 3 => SPPOL_1:attr 3
definition
  let a1 be FinSequence of the carrier of TOP-REAL 2;
  attr a1 is alternating means
    for b1 being Element of NAT
          st 1 <= b1 & b1 + 2 <= len a1
       holds (a1 /. b1) `1 <> (a1 /. (b1 + 2)) `1 &
        (a1 /. b1) `2 <> (a1 /. (b1 + 2)) `2;
end;

:: SPPOL_1:dfs 4
definiens
  let a1 be FinSequence of the carrier of TOP-REAL 2;
To prove
     a1 is alternating
it is sufficient to prove
  thus for b1 being Element of NAT
          st 1 <= b1 & b1 + 2 <= len a1
       holds (a1 /. b1) `1 <> (a1 /. (b1 + 2)) `1 &
        (a1 /. b1) `2 <> (a1 /. (b1 + 2)) `2;

:: SPPOL_1:def 4
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2 holds
      b1 is alternating
   iff
      for b2 being Element of NAT
            st 1 <= b2 & b2 + 2 <= len b1
         holds (b1 /. b2) `1 <> (b1 /. (b2 + 2)) `1 &
          (b1 /. b2) `2 <> (b1 /. (b2 + 2)) `2;

:: SPPOL_1:th 50
theorem
for b1 being Element of NAT
for b2 being FinSequence of the carrier of TOP-REAL 2
      st b2 is special &
         b2 is alternating &
         1 <= b1 &
         b1 + 2 <= len b2 &
         (b2 /. b1) `1 = (b2 /. (b1 + 1)) `1
   holds (b2 /. (b1 + 1)) `2 = (b2 /. (b1 + 2)) `2;

:: SPPOL_1:th 51
theorem
for b1 being Element of NAT
for b2 being FinSequence of the carrier of TOP-REAL 2
      st b2 is special &
         b2 is alternating &
         1 <= b1 &
         b1 + 2 <= len b2 &
         (b2 /. b1) `2 = (b2 /. (b1 + 1)) `2
   holds (b2 /. (b1 + 1)) `1 = (b2 /. (b1 + 2)) `1;

:: SPPOL_1:th 52
theorem
for b1 being Element of NAT
for b2 being FinSequence of the carrier of TOP-REAL 2
for b3, b4, b5 being Element of the carrier of TOP-REAL 2
      st b2 is special &
         b2 is alternating &
         1 <= b1 &
         b1 + 2 <= len b2 &
         b3 = b2 /. b1 &
         b4 = b2 /. (b1 + 1) &
         b5 = b2 /. (b1 + 2) &
         (b3 `1 = b4 `1 implies b5 `1 = b4 `1)
   holds b3 `2 = b4 `2 & b5 `2 <> b4 `2;

:: SPPOL_1:th 53
theorem
for b1 being Element of NAT
for b2 being FinSequence of the carrier of TOP-REAL 2
for b3, b4, b5 being Element of the carrier of TOP-REAL 2
      st b2 is special &
         b2 is alternating &
         1 <= b1 &
         b1 + 2 <= len b2 &
         b3 = b2 /. b1 &
         b4 = b2 /. (b1 + 1) &
         b5 = b2 /. (b1 + 2) &
         (b4 `1 = b5 `1 implies b3 `1 = b4 `1)
   holds b4 `2 = b5 `2 & b3 `2 <> b4 `2;

:: SPPOL_1:th 54
theorem
for b1 being Element of NAT
for b2 being FinSequence of the carrier of TOP-REAL 2
      st b2 is special & b2 is alternating & 1 <= b1 & b1 + 2 <= len b2
   holds not LSeg(b2 /. b1,b2 /. (b1 + 2)) c= (LSeg(b2,b1)) \/ LSeg(b2,b1 + 1);

:: SPPOL_1:th 55
theorem
for b1 being Element of NAT
for b2 being FinSequence of the carrier of TOP-REAL 2
      st b2 is special & b2 is alternating & 1 <= b1 & b1 + 2 <= len b2 & LSeg(b2,b1) is vertical
   holds LSeg(b2,b1 + 1) is horizontal;

:: SPPOL_1:th 56
theorem
for b1 being Element of NAT
for b2 being FinSequence of the carrier of TOP-REAL 2
      st b2 is special & b2 is alternating & 1 <= b1 & b1 + 2 <= len b2 & LSeg(b2,b1) is horizontal
   holds LSeg(b2,b1 + 1) is vertical;

:: SPPOL_1:th 57
theorem
for b1 being Element of NAT
for b2 being FinSequence of the carrier of TOP-REAL 2
      st b2 is special &
         b2 is alternating &
         1 <= b1 &
         b1 + 2 <= len b2 &
         (LSeg(b2,b1) is vertical implies LSeg(b2,b1 + 1) is not horizontal)
   holds LSeg(b2,b1) is horizontal & LSeg(b2,b1 + 1) is vertical;

:: SPPOL_1:th 58
theorem
for b1 being Element of NAT
for b2 being FinSequence of the carrier of TOP-REAL 2
for b3, b4 being Element of the carrier of TOP-REAL 2
      st b2 is special &
         b2 is alternating &
         1 <= b1 &
         b1 + 2 <= len b2 &
         b2 /. (b1 + 1) in LSeg(b3,b4) &
         LSeg(b3,b4) c= (LSeg(b2,b1)) \/ LSeg(b2,b1 + 1) &
         b2 /. (b1 + 1) <> b3
   holds b2 /. (b1 + 1) = b4;

:: SPPOL_1:th 59
theorem
for b1 being Element of NAT
for b2 being FinSequence of the carrier of TOP-REAL 2
      st b2 is special & b2 is alternating & 1 <= b1 & b1 + 2 <= len b2
   holds b2 /. (b1 + 1) is_extremal_in (LSeg(b2,b1)) \/ LSeg(b2,b1 + 1);

:: SPPOL_1:th 60
theorem
for b1 being Element of NAT
for b2 being FinSequence of the carrier of TOP-REAL 2
for b3, b4 being Element of the carrier of TOP-REAL 2
for b5 being Element of the carrier of Euclid 2
   st b2 is special &
      b2 is alternating &
      1 <= b1 &
      b1 + 2 <= len b2 &
      b5 = b2 /. (b1 + 1) &
      b2 /. (b1 + 1) in LSeg(b3,b4) &
      b2 /. (b1 + 1) <> b4 &
      not b3 in (LSeg(b2,b1)) \/ LSeg(b2,b1 + 1)
for b6 being Element of REAL
      st 0 < b6
   holds ex b7 being Element of the carrier of TOP-REAL 2 st
      not b7 in (LSeg(b2,b1)) \/ LSeg(b2,b1 + 1) &
       b7 in LSeg(b3,b4) &
       b7 in Ball(b5,b6);

:: SPPOL_1:prednot 2 => SPPOL_1:pred 2
definition
  let a1, a2 be FinSequence of the carrier of TOP-REAL 2;
  let a3 be Element of bool the carrier of TOP-REAL 2;
  pred A1,A2 are_generators_of A3 means
    a1 is alternating &
     a1 is being_S-Seq &
     a2 is alternating &
     a2 is being_S-Seq &
     a1 /. 1 = a2 /. 1 &
     a1 /. len a1 = a2 /. len a2 &
     <*a1 /. 2,a1 /. 1,a2 /. 2*> is alternating &
     <*a1 /. ((len a1) - 1),a1 /. len a1,a2 /. ((len a2) - 1)*> is alternating &
     a1 /. 1 <> a1 /. len a1 &
     (L~ a1) /\ L~ a2 = {a1 /. 1,a1 /. len a1} &
     a3 = (L~ a1) \/ L~ a2;
end;

:: SPPOL_1:dfs 5
definiens
  let a1, a2 be FinSequence of the carrier of TOP-REAL 2;
  let a3 be Element of bool the carrier of TOP-REAL 2;
To prove
     a1,a2 are_generators_of a3
it is sufficient to prove
  thus a1 is alternating &
     a1 is being_S-Seq &
     a2 is alternating &
     a2 is being_S-Seq &
     a1 /. 1 = a2 /. 1 &
     a1 /. len a1 = a2 /. len a2 &
     <*a1 /. 2,a1 /. 1,a2 /. 2*> is alternating &
     <*a1 /. ((len a1) - 1),a1 /. len a1,a2 /. ((len a2) - 1)*> is alternating &
     a1 /. 1 <> a1 /. len a1 &
     (L~ a1) /\ L~ a2 = {a1 /. 1,a1 /. len a1} &
     a3 = (L~ a1) \/ L~ a2;

:: SPPOL_1:def 5
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
for b3 being Element of bool the carrier of TOP-REAL 2 holds
      b1,b2 are_generators_of b3
   iff
      b1 is alternating &
       b1 is being_S-Seq &
       b2 is alternating &
       b2 is being_S-Seq &
       b1 /. 1 = b2 /. 1 &
       b1 /. len b1 = b2 /. len b2 &
       <*b1 /. 2,b1 /. 1,b2 /. 2*> is alternating &
       <*b1 /. ((len b1) - 1),b1 /. len b1,b2 /. ((len b2) - 1)*> is alternating &
       b1 /. 1 <> b1 /. len b1 &
       (L~ b1) /\ L~ b2 = {b1 /. 1,b1 /. len b1} &
       b3 = (L~ b1) \/ L~ b2;

:: SPPOL_1:th 61
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4 being FinSequence of the carrier of TOP-REAL 2
      st b3,b4 are_generators_of b2 & 1 < b1 & b1 < len b3
   holds b3 /. b1 is_extremal_in b2;

:: SPPOL_1:th 62
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4, b5 being Element of the carrier of Euclid b1
      st b2 = b4 & b3 = b5
   holds dist(b4,b5) = |.b2 - b3.|;

:: SPPOL_1:th 63
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
      st LSeg(b1,b2) is horizontal & b3 in LSeg(b1,b2)
   holds b1 `2 = b3 `2;

:: SPPOL_1:th 64
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
      st LSeg(b1,b2) is vertical & b3 in LSeg(b1,b2)
   holds b1 `1 = b3 `1;

:: SPPOL_1:exreg 1
registration
  cluster non empty compact horizontal Element of bool the carrier of TOP-REAL 2;
end;

:: SPPOL_1:exreg 2
registration
  cluster non empty compact vertical Element of bool the carrier of TOP-REAL 2;
end;