Article GOBOARD9, MML version 4.99.1005

:: GOBOARD9:th 3
theorem
for b1 being TopSpace-like TopStruct
for b2, b3, b4 being Element of bool the carrier of b1
      st b2 is_a_component_of b4 & b3 is_a_component_of b4 & b2 <> b3
   holds b2 misses b3;

:: GOBOARD9:th 4
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b1 | b3
      st b2 = b4
   holds b1 | b2 = (b1 | b3) | b4;

:: GOBOARD9:th 5
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty Element of bool the carrier of b1
      st b2 c= b3 & b2 is connected(b1)
   holds ex b4 being Element of bool the carrier of b1 st
      b4 is_a_component_of b3 & b2 c= b4;

:: GOBOARD9:th 6
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being Element of bool the carrier of b1
      st b3 is connected(b1) & b4 is_a_component_of b5 & b2 c= b4 & b2 meets b3 & b3 c= b5
   holds b3 c= b4;

:: GOBOARD9:th 7
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2 holds
LSeg(b1,b2) is convex(2);

:: GOBOARD9:th 8
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2 holds
LSeg(b1,b2) is connected(TOP-REAL 2);

:: GOBOARD9:exreg 1
registration
  cluster non empty convex Element of bool the carrier of TOP-REAL 2;
end;

:: GOBOARD9:th 9
theorem
for b1, b2 being convex Element of bool the carrier of TOP-REAL 2 holds
b1 /\ b2 is convex(2);

:: GOBOARD9:th 10
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2 holds
   Rev X_axis b1 = X_axis Rev b1;

:: GOBOARD9:th 11
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2 holds
   Rev Y_axis b1 = Y_axis Rev b1;

:: GOBOARD9:exreg 2
registration
  cluster Relation-like Function-like non constant finite FinSequence-like set;
end;

:: GOBOARD9:funcreg 1
registration
  let a1 be Relation-like Function-like non constant FinSequence-like set;
  cluster Rev a1 -> Relation-like Function-like non constant FinSequence-like;
end;

:: GOBOARD9:funcnot 1 => GOBOARD9:func 1
definition
  let a1 be non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
  redefine func Rev a1 -> non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
end;

:: GOBOARD9:th 12
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
      st 1 <= b2 & 1 <= b3 & b2 + b3 = len b1
   holds left_cell(b1,b2) = right_cell(Rev b1,b3);

:: GOBOARD9:th 13
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
      st 1 <= b2 & 1 <= b3 & b2 + b3 = len b1
   holds left_cell(Rev b1,b2) = right_cell(b1,b3);

:: GOBOARD9:th 14
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
      st 1 <= b2 & b2 + 1 <= len b1
   holds ex b3, b4 being Element of NAT st
      b3 <= len GoB b1 & b4 <= width GoB b1 & cell(GoB b1,b3,b4) = left_cell(b1,b2);

:: GOBOARD9:th 15
theorem
for b1 being Element of NAT
for b2 being non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of (the carrier of TOP-REAL 2) *
      st b1 <= width b2
   holds Int h_strip(b2,b1) is convex(2);

:: GOBOARD9:th 16
theorem
for b1 being Element of NAT
for b2 being non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of (the carrier of TOP-REAL 2) *
      st b1 <= len b2
   holds Int v_strip(b2,b1) is convex(2);

:: GOBOARD9:th 17
theorem
for b1, b2 being Element of NAT
for b3 being non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of (the carrier of TOP-REAL 2) *
      st b1 <= len b3 & b2 <= width b3
   holds Int cell(b3,b1,b2) <> {};

:: GOBOARD9:th 18
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
      st 1 <= b2 & b2 + 1 <= len b1
   holds Int left_cell(b1,b2) <> {};

:: GOBOARD9:th 19
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
      st 1 <= b2 & b2 + 1 <= len b1
   holds Int right_cell(b1,b2) <> {};

:: GOBOARD9:th 20
theorem
for b1, b2 being Element of NAT
for b3 being non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of (the carrier of TOP-REAL 2) *
      st b1 <= len b3 & b2 <= width b3
   holds Int cell(b3,b1,b2) is convex(2);

:: GOBOARD9:th 21
theorem
for b1, b2 being Element of NAT
for b3 being non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of (the carrier of TOP-REAL 2) *
      st b1 <= len b3 & b2 <= width b3
   holds Int cell(b3,b1,b2) is connected(TOP-REAL 2);

:: GOBOARD9:th 22
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
      st 1 <= b2 & b2 + 1 <= len b1
   holds Int left_cell(b1,b2) is connected(TOP-REAL 2);

:: GOBOARD9:th 23
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
      st 1 <= b2 & b2 + 1 <= len b1
   holds Int right_cell(b1,b2) is connected(TOP-REAL 2);

:: GOBOARD9:funcnot 2 => GOBOARD9:func 2
definition
  let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
  func LeftComp A1 -> Element of bool the carrier of TOP-REAL 2 means
    it is_a_component_of (L~ a1) ` & Int left_cell(a1,1) c= it;
end;

:: GOBOARD9:def 1
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2 holds
      b2 = LeftComp b1
   iff
      b2 is_a_component_of (L~ b1) ` & Int left_cell(b1,1) c= b2;

:: GOBOARD9:funcnot 3 => GOBOARD9:func 3
definition
  let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
  func RightComp A1 -> Element of bool the carrier of TOP-REAL 2 means
    it is_a_component_of (L~ a1) ` & Int right_cell(a1,1) c= it;
end;

:: GOBOARD9:def 2
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2 holds
      b2 = RightComp b1
   iff
      b2 is_a_component_of (L~ b1) ` & Int right_cell(b1,1) c= b2;

:: GOBOARD9:th 24
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
      st 1 <= b2 & b2 + 1 <= len b1
   holds Int left_cell(b1,b2) c= LeftComp b1;

:: GOBOARD9:th 25
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
   GoB Rev b1 = GoB b1;

:: GOBOARD9:th 26
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
   RightComp b1 = LeftComp Rev b1;

:: GOBOARD9:th 27
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
   RightComp Rev b1 = LeftComp b1;

:: GOBOARD9:th 28
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
      st 1 <= b2 & b2 + 1 <= len b1
   holds Int right_cell(b1,b2) c= RightComp b1;