Article GOBOARD7, MML version 4.99.1005

:: GOBOARD7:th 1
theorem
for b1, b2, b3 being Element of REAL
      st b3 < abs (b1 - b2) & b2 <= b1 + b3
   holds b2 + b3 < b1;

:: GOBOARD7:th 2
theorem
for b1, b2 being Element of REAL holds
   abs (b1 - b2) = 0
iff
   b1 = b2;

:: GOBOARD7:th 3
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of the carrier of TOP-REAL b1
      st b2 + b3 = b4 + b3
   holds b2 = b4;

:: GOBOARD7:th 4
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of the carrier of TOP-REAL b1
      st b3 + b2 = b3 + b4
   holds b2 = b4;

:: GOBOARD7:th 5
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 in LSeg(b2,b3) & b2 `1 = b3 `1
   holds b1 `1 = b3 `1;

:: GOBOARD7:th 6
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 in LSeg(b2,b3) & b2 `2 = b3 `2
   holds b1 `2 = b3 `2;

:: GOBOARD7:th 7
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
(1 / 2) * (b2 + b3) in LSeg(b2,b3);

:: GOBOARD7:th 8
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 `1 = b2 `1 & b2 `1 = b3 `1 & b1 `2 <= b2 `2 & b2 `2 <= b3 `2
   holds b2 in LSeg(b1,b3);

:: GOBOARD7:th 9
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 `1 <= b2 `1 & b2 `1 <= b3 `1 & b1 `2 = b2 `2 & b2 `2 = b3 `2
   holds b2 in LSeg(b1,b3);

:: GOBOARD7:th 11
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 1 <= b1 & b1 + 1 <= len b3 & 1 <= b2 & b2 + 1 <= width b3
   holds (1 / 2) * ((b3 *(b1,b2)) + (b3 *(b1 + 1,b2 + 1))) = (1 / 2) * ((b3 *(b1,b2 + 1)) + (b3 *(b1 + 1,b2)));

:: GOBOARD7:th 12
theorem
for b1 being non empty FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
      st LSeg(b1,b2) is horizontal
   holds ex b3 being Element of NAT st
      1 <= b3 &
       b3 <= width GoB b1 &
       (for b4 being Element of the carrier of TOP-REAL 2
             st b4 in LSeg(b1,b2)
          holds b4 `2 = ((GoB b1) *(1,b3)) `2);

:: GOBOARD7:th 13
theorem
for b1 being non empty FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
      st LSeg(b1,b2) is vertical
   holds ex b3 being Element of NAT st
      1 <= b3 &
       b3 <= len GoB b1 &
       (for b4 being Element of the carrier of TOP-REAL 2
             st b4 in LSeg(b1,b2)
          holds b4 `1 = ((GoB b1) *(b3,1)) `1);

:: GOBOARD7:th 14
theorem
for b1 being non empty FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
      st b1 is special & b2 <= len GoB b1 & b3 <= width GoB b1
   holds Int cell(GoB b1,b2,b3) misses L~ b1;

:: GOBOARD7:th 15
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 1 <= b1 & b1 <= len b3 & 1 <= b2 & b2 + 2 <= width b3
   holds (LSeg(b3 *(b1,b2),b3 *(b1,b2 + 1))) /\ LSeg(b3 *(b1,b2 + 1),b3 *(b1,b2 + 2)) = {b3 *(b1,b2 + 1)};

:: GOBOARD7:th 16
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 1 <= b1 & b1 + 2 <= len b3 & 1 <= b2 & b2 <= width b3
   holds (LSeg(b3 *(b1,b2),b3 *(b1 + 1,b2))) /\ LSeg(b3 *(b1 + 1,b2),b3 *(b1 + 2,b2)) = {b3 *(b1 + 1,b2)};

:: GOBOARD7: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 1 <= b1 & b1 + 1 <= len b3 & 1 <= b2 & b2 + 1 <= width b3
   holds (LSeg(b3 *(b1,b2),b3 *(b1,b2 + 1))) /\ LSeg(b3 *(b1,b2 + 1),b3 *(b1 + 1,b2 + 1)) = {b3 *(b1,b2 + 1)};

:: GOBOARD7:th 18
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 1 <= b1 & b1 + 1 <= len b3 & 1 <= b2 & b2 + 1 <= width b3
   holds (LSeg(b3 *(b1,b2 + 1),b3 *(b1 + 1,b2 + 1))) /\ LSeg(b3 *(b1 + 1,b2),b3 *(b1 + 1,b2 + 1)) = {b3 *(b1 + 1,b2 + 1)};

:: GOBOARD7:th 19
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 1 <= b1 & b1 + 1 <= len b3 & 1 <= b2 & b2 + 1 <= width b3
   holds (LSeg(b3 *(b1,b2),b3 *(b1 + 1,b2))) /\ LSeg(b3 *(b1,b2),b3 *(b1,b2 + 1)) = {b3 *(b1,b2)};

:: GOBOARD7: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 1 <= b1 & b1 + 1 <= len b3 & 1 <= b2 & b2 + 1 <= width b3
   holds (LSeg(b3 *(b1,b2),b3 *(b1 + 1,b2))) /\ LSeg(b3 *(b1 + 1,b2),b3 *(b1 + 1,b2 + 1)) = {b3 *(b1 + 1,b2)};

:: GOBOARD7:th 21
theorem
for b1 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) *
for b2, b3, b4, b5 being Element of NAT
      st 1 <= b2 &
         b2 <= len b1 &
         1 <= b3 &
         b3 + 1 <= width b1 &
         1 <= b4 &
         b4 <= len b1 &
         1 <= b5 &
         b5 + 1 <= width b1 &
         LSeg(b1 *(b2,b3),b1 *(b2,b3 + 1)) meets LSeg(b1 *(b4,b5),b1 *(b4,b5 + 1))
   holds b2 = b4 & abs (b3 - b5) <= 1;

:: GOBOARD7:th 22
theorem
for b1 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) *
for b2, b3, b4, b5 being Element of NAT
      st 1 <= b2 &
         b2 + 1 <= len b1 &
         1 <= b3 &
         b3 <= width b1 &
         1 <= b4 &
         b4 + 1 <= len b1 &
         1 <= b5 &
         b5 <= width b1 &
         LSeg(b1 *(b2,b3),b1 *(b2 + 1,b3)) meets LSeg(b1 *(b4,b5),b1 *(b4 + 1,b5))
   holds b3 = b5 & abs (b2 - b4) <= 1;

:: GOBOARD7:th 23
theorem
for b1 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) *
for b2, b3, b4, b5 being Element of NAT
      st 1 <= b2 &
         b2 <= len b1 &
         1 <= b3 &
         b3 + 1 <= width b1 &
         1 <= b4 &
         b4 + 1 <= len b1 &
         1 <= b5 &
         b5 <= width b1 &
         LSeg(b1 *(b2,b3),b1 *(b2,b3 + 1)) meets LSeg(b1 *(b4,b5),b1 *(b4 + 1,b5))
   holds (b2 = b4 or b2 = b4 + 1) & (b3 = b5 or b3 + 1 = b5);

:: GOBOARD7:th 24
theorem
for b1 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) *
for b2, b3, b4, b5 being Element of NAT
      st 1 <= b2 &
         b2 <= len b1 &
         1 <= b3 &
         b3 + 1 <= width b1 &
         1 <= b4 &
         b4 <= len b1 &
         1 <= b5 &
         b5 + 1 <= width b1 &
         LSeg(b1 *(b2,b3),b1 *(b2,b3 + 1)) meets LSeg(b1 *(b4,b5),b1 *(b4,b5 + 1)) &
         (b3 = b5 implies LSeg(b1 *(b2,b3),b1 *(b2,b3 + 1)) <> LSeg(b1 *(b4,b5),b1 *(b4,b5 + 1))) &
         (b3 = b5 + 1 implies (LSeg(b1 *(b2,b3),b1 *(b2,b3 + 1))) /\ LSeg(b1 *(b4,b5),b1 *(b4,b5 + 1)) <> {b1 *(b2,b3)})
   holds b3 + 1 = b5 &
    (LSeg(b1 *(b2,b3),b1 *(b2,b3 + 1))) /\ LSeg(b1 *(b4,b5),b1 *(b4,b5 + 1)) = {b1 *(b4,b5)};

:: GOBOARD7:th 25
theorem
for b1 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) *
for b2, b3, b4, b5 being Element of NAT
      st 1 <= b2 &
         b2 + 1 <= len b1 &
         1 <= b3 &
         b3 <= width b1 &
         1 <= b4 &
         b4 + 1 <= len b1 &
         1 <= b5 &
         b5 <= width b1 &
         LSeg(b1 *(b2,b3),b1 *(b2 + 1,b3)) meets LSeg(b1 *(b4,b5),b1 *(b4 + 1,b5)) &
         (b2 = b4 implies LSeg(b1 *(b2,b3),b1 *(b2 + 1,b3)) <> LSeg(b1 *(b4,b5),b1 *(b4 + 1,b5))) &
         (b2 = b4 + 1 implies (LSeg(b1 *(b2,b3),b1 *(b2 + 1,b3))) /\ LSeg(b1 *(b4,b5),b1 *(b4 + 1,b5)) <> {b1 *(b2,b3)})
   holds b2 + 1 = b4 &
    (LSeg(b1 *(b2,b3),b1 *(b2 + 1,b3))) /\ LSeg(b1 *(b4,b5),b1 *(b4 + 1,b5)) = {b1 *(b4,b5)};

:: GOBOARD7:th 26
theorem
for b1 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) *
for b2, b3, b4, b5 being Element of NAT
      st 1 <= b2 &
         b2 <= len b1 &
         1 <= b3 &
         b3 + 1 <= width b1 &
         1 <= b4 &
         b4 + 1 <= len b1 &
         1 <= b5 &
         b5 <= width b1 &
         LSeg(b1 *(b2,b3),b1 *(b2,b3 + 1)) meets LSeg(b1 *(b4,b5),b1 *(b4 + 1,b5)) &
         (b3 = b5 implies (LSeg(b1 *(b2,b3),b1 *(b2,b3 + 1))) /\ LSeg(b1 *(b4,b5),b1 *(b4 + 1,b5)) <> {b1 *(b2,b3)})
   holds b3 + 1 = b5 &
    (LSeg(b1 *(b2,b3),b1 *(b2,b3 + 1))) /\ LSeg(b1 *(b4,b5),b1 *(b4 + 1,b5)) = {b1 *(b2,b3 + 1)};

:: GOBOARD7:th 27
theorem
for b1, b2, b3, b4 being Element of NAT
for b5 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 1 <= b1 &
         b1 <= len b5 &
         1 <= b2 &
         b2 + 1 <= width b5 &
         1 <= b3 &
         b3 <= len b5 &
         1 <= b4 &
         b4 + 1 <= width b5 &
         (1 / 2) * ((b5 *(b1,b2)) + (b5 *(b1,b2 + 1))) in LSeg(b5 *(b3,b4),b5 *(b3,b4 + 1))
   holds b1 = b3 & b2 = b4;

:: GOBOARD7:th 28
theorem
for b1, b2, b3, b4 being Element of NAT
for b5 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 1 <= b1 &
         b1 + 1 <= len b5 &
         1 <= b2 &
         b2 <= width b5 &
         1 <= b3 &
         b3 + 1 <= len b5 &
         1 <= b4 &
         b4 <= width b5 &
         (1 / 2) * ((b5 *(b1,b2)) + (b5 *(b1 + 1,b2))) in LSeg(b5 *(b3,b4),b5 *(b3 + 1,b4))
   holds b1 = b3 & b2 = b4;

:: GOBOARD7:th 29
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 1 <= b1 & b1 + 1 <= len b3 & 1 <= b2 & b2 <= width b3
for b4, b5 being Element of NAT
      st 1 <= b4 & b4 <= len b3 & 1 <= b5 & b5 + 1 <= width b3
   holds not (1 / 2) * ((b3 *(b1,b2)) + (b3 *(b1 + 1,b2))) in LSeg(b3 *(b4,b5),b3 *(b4,b5 + 1));

:: GOBOARD7:th 30
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 1 <= b1 & b1 <= len b3 & 1 <= b2 & b2 + 1 <= width b3
for b4, b5 being Element of NAT
      st 1 <= b4 & b4 + 1 <= len b3 & 1 <= b5 & b5 <= width b3
   holds not (1 / 2) * ((b3 *(b1,b2)) + (b3 *(b1,b2 + 1))) in LSeg(b3 *(b4,b5),b3 *(b4 + 1,b5));

:: GOBOARD7:th 31
theorem
for b1 being Element of NAT
for b2 being non empty standard FinSequence of the carrier of TOP-REAL 2
      st b1 in dom b2 & b1 + 1 in dom b2
   holds b2 /. b1 <> b2 /. (b1 + 1);

:: GOBOARD7:th 32
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
   ex b2 being Element of NAT st
      b2 in dom b1 &
       (b1 /. b2) `1 <> (b1 /. 1) `1;

:: GOBOARD7:th 33
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
   ex b2 being Element of NAT st
      b2 in dom b1 &
       (b1 /. b2) `2 <> (b1 /. 1) `2;

:: GOBOARD7:th 34
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
   1 < len GoB b1;

:: GOBOARD7:th 35
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
   1 < width GoB b1;

:: GOBOARD7:th 36
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
   4 < len b1;

:: GOBOARD7:th 37
theorem
for b1 being circular s.c.c. FinSequence of the carrier of TOP-REAL 2
   st 4 < len b1
for b2, b3 being Element of NAT
      st 1 <= b2 & b2 < b3 & b3 < len b1
   holds b1 /. b2 <> b1 /. b3;

:: GOBOARD7:th 38
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 & b2 < b3 & b3 < len b1
   holds b1 /. b2 <> b1 /. b3;

:: GOBOARD7:th 39
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 & b2 < b3 & b3 <= len b1
   holds b1 /. b2 <> b1 /. b3;

:: GOBOARD7:th 40
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 <= len b1 & b1 /. b2 = b1 /. 1
   holds b2 = len b1;

:: GOBOARD7:th 41
theorem
for b1, b2 being Element of NAT
for b3 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 <= len GoB b3 &
         1 <= b2 &
         b2 + 1 <= width GoB b3 &
         (1 / 2) * (((GoB b3) *(b1,b2)) + ((GoB b3) *(b1,b2 + 1))) in L~ b3
   holds ex b4 being Element of NAT st
      1 <= b4 &
       b4 + 1 <= len b3 &
       LSeg((GoB b3) *(b1,b2),(GoB b3) *(b1,b2 + 1)) = LSeg(b3,b4);

:: GOBOARD7:th 42
theorem
for b1, b2 being Element of NAT
for b3 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 + 1 <= len GoB b3 &
         1 <= b2 &
         b2 <= width GoB b3 &
         (1 / 2) * (((GoB b3) *(b1,b2)) + ((GoB b3) *(b1 + 1,b2))) in L~ b3
   holds ex b4 being Element of NAT st
      1 <= b4 &
       b4 + 1 <= len b3 &
       LSeg((GoB b3) *(b1,b2),(GoB b3) *(b1 + 1,b2)) = LSeg(b3,b4);

:: GOBOARD7:th 43
theorem
for b1, b2, b3 being Element of NAT
for b4 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 + 1 <= len GoB b4 &
         1 <= b2 &
         b2 + 1 <= width GoB b4 &
         1 <= b3 &
         b3 + 1 < len b4 &
         LSeg((GoB b4) *(b1,b2 + 1),(GoB b4) *(b1 + 1,b2 + 1)) = LSeg(b4,b3) &
         LSeg((GoB b4) *(b1 + 1,b2),(GoB b4) *(b1 + 1,b2 + 1)) = LSeg(b4,b3 + 1)
   holds b4 /. b3 = (GoB b4) *(b1,b2 + 1) &
    b4 /. (b3 + 1) = (GoB b4) *(b1 + 1,b2 + 1) &
    b4 /. (b3 + 2) = (GoB b4) *(b1 + 1,b2);

:: GOBOARD7:th 44
theorem
for b1, b2, b3 being Element of NAT
for b4 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 <= len GoB b4 &
         1 <= b2 &
         b2 + 1 < width GoB b4 &
         1 <= b3 &
         b3 + 1 < len b4 &
         LSeg((GoB b4) *(b1,b2 + 1),(GoB b4) *(b1,b2 + 2)) = LSeg(b4,b3) &
         LSeg((GoB b4) *(b1,b2),(GoB b4) *(b1,b2 + 1)) = LSeg(b4,b3 + 1)
   holds b4 /. b3 = (GoB b4) *(b1,b2 + 2) &
    b4 /. (b3 + 1) = (GoB b4) *(b1,b2 + 1) &
    b4 /. (b3 + 2) = (GoB b4) *(b1,b2);

:: GOBOARD7:th 45
theorem
for b1, b2, b3 being Element of NAT
for b4 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 + 1 <= len GoB b4 &
         1 <= b2 &
         b2 + 1 <= width GoB b4 &
         1 <= b3 &
         b3 + 1 < len b4 &
         LSeg((GoB b4) *(b1,b2 + 1),(GoB b4) *(b1 + 1,b2 + 1)) = LSeg(b4,b3) &
         LSeg((GoB b4) *(b1,b2),(GoB b4) *(b1,b2 + 1)) = LSeg(b4,b3 + 1)
   holds b4 /. b3 = (GoB b4) *(b1 + 1,b2 + 1) &
    b4 /. (b3 + 1) = (GoB b4) *(b1,b2 + 1) &
    b4 /. (b3 + 2) = (GoB b4) *(b1,b2);

:: GOBOARD7:th 46
theorem
for b1, b2, b3 being Element of NAT
for b4 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 + 1 <= len GoB b4 &
         1 <= b2 &
         b2 + 1 <= width GoB b4 &
         1 <= b3 &
         b3 + 1 < len b4 &
         LSeg((GoB b4) *(b1 + 1,b2),(GoB b4) *(b1 + 1,b2 + 1)) = LSeg(b4,b3) &
         LSeg((GoB b4) *(b1,b2 + 1),(GoB b4) *(b1 + 1,b2 + 1)) = LSeg(b4,b3 + 1)
   holds b4 /. b3 = (GoB b4) *(b1 + 1,b2) &
    b4 /. (b3 + 1) = (GoB b4) *(b1 + 1,b2 + 1) &
    b4 /. (b3 + 2) = (GoB b4) *(b1,b2 + 1);

:: GOBOARD7:th 47
theorem
for b1, b2, b3 being Element of NAT
for b4 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 + 1 < len GoB b4 &
         1 <= b2 &
         b2 <= width GoB b4 &
         1 <= b3 &
         b3 + 1 < len b4 &
         LSeg((GoB b4) *(b1 + 1,b2),(GoB b4) *(b1 + 2,b2)) = LSeg(b4,b3) &
         LSeg((GoB b4) *(b1,b2),(GoB b4) *(b1 + 1,b2)) = LSeg(b4,b3 + 1)
   holds b4 /. b3 = (GoB b4) *(b1 + 2,b2) &
    b4 /. (b3 + 1) = (GoB b4) *(b1 + 1,b2) &
    b4 /. (b3 + 2) = (GoB b4) *(b1,b2);

:: GOBOARD7:th 48
theorem
for b1, b2, b3 being Element of NAT
for b4 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 + 1 <= len GoB b4 &
         1 <= b2 &
         b2 + 1 <= width GoB b4 &
         1 <= b3 &
         b3 + 1 < len b4 &
         LSeg((GoB b4) *(b1 + 1,b2),(GoB b4) *(b1 + 1,b2 + 1)) = LSeg(b4,b3) &
         LSeg((GoB b4) *(b1,b2),(GoB b4) *(b1 + 1,b2)) = LSeg(b4,b3 + 1)
   holds b4 /. b3 = (GoB b4) *(b1 + 1,b2 + 1) &
    b4 /. (b3 + 1) = (GoB b4) *(b1 + 1,b2) &
    b4 /. (b3 + 2) = (GoB b4) *(b1,b2);

:: GOBOARD7:th 49
theorem
for b1, b2, b3 being Element of NAT
for b4 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 + 1 <= len GoB b4 &
         1 <= b2 &
         b2 + 1 <= width GoB b4 &
         1 <= b3 &
         b3 + 1 < len b4 &
         LSeg((GoB b4) *(b1 + 1,b2),(GoB b4) *(b1 + 1,b2 + 1)) = LSeg(b4,b3) &
         LSeg((GoB b4) *(b1,b2 + 1),(GoB b4) *(b1 + 1,b2 + 1)) = LSeg(b4,b3 + 1)
   holds b4 /. b3 = (GoB b4) *(b1 + 1,b2) &
    b4 /. (b3 + 1) = (GoB b4) *(b1 + 1,b2 + 1) &
    b4 /. (b3 + 2) = (GoB b4) *(b1,b2 + 1);

:: GOBOARD7:th 50
theorem
for b1, b2, b3 being Element of NAT
for b4 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 <= len GoB b4 &
         1 <= b2 &
         b2 + 1 < width GoB b4 &
         1 <= b3 &
         b3 + 1 < len b4 &
         LSeg((GoB b4) *(b1,b2),(GoB b4) *(b1,b2 + 1)) = LSeg(b4,b3) &
         LSeg((GoB b4) *(b1,b2 + 1),(GoB b4) *(b1,b2 + 2)) = LSeg(b4,b3 + 1)
   holds b4 /. b3 = (GoB b4) *(b1,b2) &
    b4 /. (b3 + 1) = (GoB b4) *(b1,b2 + 1) &
    b4 /. (b3 + 2) = (GoB b4) *(b1,b2 + 2);

:: GOBOARD7:th 51
theorem
for b1, b2, b3 being Element of NAT
for b4 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 + 1 <= len GoB b4 &
         1 <= b2 &
         b2 + 1 <= width GoB b4 &
         1 <= b3 &
         b3 + 1 < len b4 &
         LSeg((GoB b4) *(b1,b2),(GoB b4) *(b1,b2 + 1)) = LSeg(b4,b3) &
         LSeg((GoB b4) *(b1,b2 + 1),(GoB b4) *(b1 + 1,b2 + 1)) = LSeg(b4,b3 + 1)
   holds b4 /. b3 = (GoB b4) *(b1,b2) &
    b4 /. (b3 + 1) = (GoB b4) *(b1,b2 + 1) &
    b4 /. (b3 + 2) = (GoB b4) *(b1 + 1,b2 + 1);

:: GOBOARD7:th 52
theorem
for b1, b2, b3 being Element of NAT
for b4 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 + 1 <= len GoB b4 &
         1 <= b2 &
         b2 + 1 <= width GoB b4 &
         1 <= b3 &
         b3 + 1 < len b4 &
         LSeg((GoB b4) *(b1,b2 + 1),(GoB b4) *(b1 + 1,b2 + 1)) = LSeg(b4,b3) &
         LSeg((GoB b4) *(b1 + 1,b2),(GoB b4) *(b1 + 1,b2 + 1)) = LSeg(b4,b3 + 1)
   holds b4 /. b3 = (GoB b4) *(b1,b2 + 1) &
    b4 /. (b3 + 1) = (GoB b4) *(b1 + 1,b2 + 1) &
    b4 /. (b3 + 2) = (GoB b4) *(b1 + 1,b2);

:: GOBOARD7:th 53
theorem
for b1, b2, b3 being Element of NAT
for b4 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 + 1 < len GoB b4 &
         1 <= b2 &
         b2 <= width GoB b4 &
         1 <= b3 &
         b3 + 1 < len b4 &
         LSeg((GoB b4) *(b1,b2),(GoB b4) *(b1 + 1,b2)) = LSeg(b4,b3) &
         LSeg((GoB b4) *(b1 + 1,b2),(GoB b4) *(b1 + 2,b2)) = LSeg(b4,b3 + 1)
   holds b4 /. b3 = (GoB b4) *(b1,b2) &
    b4 /. (b3 + 1) = (GoB b4) *(b1 + 1,b2) &
    b4 /. (b3 + 2) = (GoB b4) *(b1 + 2,b2);

:: GOBOARD7:th 54
theorem
for b1, b2, b3 being Element of NAT
for b4 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 + 1 <= len GoB b4 &
         1 <= b2 &
         b2 + 1 <= width GoB b4 &
         1 <= b3 &
         b3 + 1 < len b4 &
         LSeg((GoB b4) *(b1,b2),(GoB b4) *(b1 + 1,b2)) = LSeg(b4,b3) &
         LSeg((GoB b4) *(b1 + 1,b2),(GoB b4) *(b1 + 1,b2 + 1)) = LSeg(b4,b3 + 1)
   holds b4 /. b3 = (GoB b4) *(b1,b2) &
    b4 /. (b3 + 1) = (GoB b4) *(b1 + 1,b2) &
    b4 /. (b3 + 2) = (GoB b4) *(b1 + 1,b2 + 1);

:: GOBOARD7:th 55
theorem
for b1, b2 being Element of NAT
for b3 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 <= len GoB b3 &
         1 <= b2 &
         b2 + 1 < width GoB b3 &
         LSeg((GoB b3) *(b1,b2),(GoB b3) *(b1,b2 + 1)) c= L~ b3 &
         LSeg((GoB b3) *(b1,b2 + 1),(GoB b3) *(b1,b2 + 2)) c= L~ b3 &
         (b3 /. 1 = (GoB b3) *(b1,b2 + 1) implies (b3 /. 2 = (GoB b3) *(b1,b2) implies b3 /. ((len b3) -' 1) <> (GoB b3) *(b1,b2 + 2)) &
          (b3 /. 2 = (GoB b3) *(b1,b2 + 2) implies b3 /. ((len b3) -' 1) <> (GoB b3) *(b1,b2)))
   holds ex b4 being Element of NAT st
      1 <= b4 &
       b4 + 1 < len b3 &
       b3 /. (b4 + 1) = (GoB b3) *(b1,b2 + 1) &
       (b3 /. b4 = (GoB b3) *(b1,b2) &
        b3 /. (b4 + 2) = (GoB b3) *(b1,b2 + 2) or b3 /. b4 = (GoB b3) *(b1,b2 + 2) &
        b3 /. (b4 + 2) = (GoB b3) *(b1,b2));

:: GOBOARD7:th 56
theorem
for b1, b2 being Element of NAT
for b3 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 + 1 <= len GoB b3 &
         1 <= b2 &
         b2 + 1 <= width GoB b3 &
         LSeg((GoB b3) *(b1,b2),(GoB b3) *(b1,b2 + 1)) c= L~ b3 &
         LSeg((GoB b3) *(b1,b2 + 1),(GoB b3) *(b1 + 1,b2 + 1)) c= L~ b3 &
         (b3 /. 1 = (GoB b3) *(b1,b2 + 1) implies (b3 /. 2 = (GoB b3) *(b1,b2) implies b3 /. ((len b3) -' 1) <> (GoB b3) *(b1 + 1,b2 + 1)) &
          (b3 /. 2 = (GoB b3) *(b1 + 1,b2 + 1) implies b3 /. ((len b3) -' 1) <> (GoB b3) *(b1,b2)))
   holds ex b4 being Element of NAT st
      1 <= b4 &
       b4 + 1 < len b3 &
       b3 /. (b4 + 1) = (GoB b3) *(b1,b2 + 1) &
       (b3 /. b4 = (GoB b3) *(b1,b2) &
        b3 /. (b4 + 2) = (GoB b3) *(b1 + 1,b2 + 1) or b3 /. b4 = (GoB b3) *(b1 + 1,b2 + 1) &
        b3 /. (b4 + 2) = (GoB b3) *(b1,b2));

:: GOBOARD7:th 57
theorem
for b1, b2 being Element of NAT
for b3 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 + 1 <= len GoB b3 &
         1 <= b2 &
         b2 + 1 <= width GoB b3 &
         LSeg((GoB b3) *(b1,b2 + 1),(GoB b3) *(b1 + 1,b2 + 1)) c= L~ b3 &
         LSeg((GoB b3) *(b1 + 1,b2 + 1),(GoB b3) *(b1 + 1,b2)) c= L~ b3 &
         (b3 /. 1 = (GoB b3) *(b1 + 1,b2 + 1) implies (b3 /. 2 = (GoB b3) *(b1,b2 + 1) implies b3 /. ((len b3) -' 1) <> (GoB b3) *(b1 + 1,b2)) &
          (b3 /. 2 = (GoB b3) *(b1 + 1,b2) implies b3 /. ((len b3) -' 1) <> (GoB b3) *(b1,b2 + 1)))
   holds ex b4 being Element of NAT st
      1 <= b4 &
       b4 + 1 < len b3 &
       b3 /. (b4 + 1) = (GoB b3) *(b1 + 1,b2 + 1) &
       (b3 /. b4 = (GoB b3) *(b1,b2 + 1) &
        b3 /. (b4 + 2) = (GoB b3) *(b1 + 1,b2) or b3 /. b4 = (GoB b3) *(b1 + 1,b2) &
        b3 /. (b4 + 2) = (GoB b3) *(b1,b2 + 1));

:: GOBOARD7:th 58
theorem
for b1, b2 being Element of NAT
for b3 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 + 1 < len GoB b3 &
         1 <= b2 &
         b2 <= width GoB b3 &
         LSeg((GoB b3) *(b1,b2),(GoB b3) *(b1 + 1,b2)) c= L~ b3 &
         LSeg((GoB b3) *(b1 + 1,b2),(GoB b3) *(b1 + 2,b2)) c= L~ b3 &
         (b3 /. 1 = (GoB b3) *(b1 + 1,b2) implies (b3 /. 2 = (GoB b3) *(b1,b2) implies b3 /. ((len b3) -' 1) <> (GoB b3) *(b1 + 2,b2)) &
          (b3 /. 2 = (GoB b3) *(b1 + 2,b2) implies b3 /. ((len b3) -' 1) <> (GoB b3) *(b1,b2)))
   holds ex b4 being Element of NAT st
      1 <= b4 &
       b4 + 1 < len b3 &
       b3 /. (b4 + 1) = (GoB b3) *(b1 + 1,b2) &
       (b3 /. b4 = (GoB b3) *(b1,b2) &
        b3 /. (b4 + 2) = (GoB b3) *(b1 + 2,b2) or b3 /. b4 = (GoB b3) *(b1 + 2,b2) &
        b3 /. (b4 + 2) = (GoB b3) *(b1,b2));

:: GOBOARD7:th 59
theorem
for b1, b2 being Element of NAT
for b3 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 + 1 <= len GoB b3 &
         1 <= b2 &
         b2 + 1 <= width GoB b3 &
         LSeg((GoB b3) *(b1,b2),(GoB b3) *(b1 + 1,b2)) c= L~ b3 &
         LSeg((GoB b3) *(b1 + 1,b2),(GoB b3) *(b1 + 1,b2 + 1)) c= L~ b3 &
         (b3 /. 1 = (GoB b3) *(b1 + 1,b2) implies (b3 /. 2 = (GoB b3) *(b1,b2) implies b3 /. ((len b3) -' 1) <> (GoB b3) *(b1 + 1,b2 + 1)) &
          (b3 /. 2 = (GoB b3) *(b1 + 1,b2 + 1) implies b3 /. ((len b3) -' 1) <> (GoB b3) *(b1,b2)))
   holds ex b4 being Element of NAT st
      1 <= b4 &
       b4 + 1 < len b3 &
       b3 /. (b4 + 1) = (GoB b3) *(b1 + 1,b2) &
       (b3 /. b4 = (GoB b3) *(b1,b2) &
        b3 /. (b4 + 2) = (GoB b3) *(b1 + 1,b2 + 1) or b3 /. b4 = (GoB b3) *(b1 + 1,b2 + 1) &
        b3 /. (b4 + 2) = (GoB b3) *(b1,b2));

:: GOBOARD7:th 60
theorem
for b1, b2 being Element of NAT
for b3 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 + 1 <= len GoB b3 &
         1 <= b2 &
         b2 + 1 <= width GoB b3 &
         LSeg((GoB b3) *(b1 + 1,b2),(GoB b3) *(b1 + 1,b2 + 1)) c= L~ b3 &
         LSeg((GoB b3) *(b1 + 1,b2 + 1),(GoB b3) *(b1,b2 + 1)) c= L~ b3 &
         (b3 /. 1 = (GoB b3) *(b1 + 1,b2 + 1) implies (b3 /. 2 = (GoB b3) *(b1 + 1,b2) implies b3 /. ((len b3) -' 1) <> (GoB b3) *(b1,b2 + 1)) &
          (b3 /. 2 = (GoB b3) *(b1,b2 + 1) implies b3 /. ((len b3) -' 1) <> (GoB b3) *(b1 + 1,b2)))
   holds ex b4 being Element of NAT st
      1 <= b4 &
       b4 + 1 < len b3 &
       b3 /. (b4 + 1) = (GoB b3) *(b1 + 1,b2 + 1) &
       (b3 /. b4 = (GoB b3) *(b1 + 1,b2) &
        b3 /. (b4 + 2) = (GoB b3) *(b1,b2 + 1) or b3 /. b4 = (GoB b3) *(b1,b2 + 1) &
        b3 /. (b4 + 2) = (GoB b3) *(b1 + 1,b2));

:: GOBOARD7:th 61
theorem
for b1, b2 being Element of NAT
for b3 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 < len GoB b3 &
         1 <= b2 &
         b2 + 1 < width GoB b3 &
         LSeg((GoB b3) *(b1,b2),(GoB b3) *(b1,b2 + 1)) c= L~ b3 &
         LSeg((GoB b3) *(b1,b2 + 1),(GoB b3) *(b1,b2 + 2)) c= L~ b3
   holds not LSeg((GoB b3) *(b1,b2 + 1),(GoB b3) *(b1 + 1,b2 + 1)) c= L~ b3;

:: GOBOARD7:th 62
theorem
for b1, b2 being Element of NAT
for b3 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 < len GoB b3 &
         1 <= b2 &
         b2 + 1 < width GoB b3 &
         LSeg((GoB b3) *(b1 + 1,b2),(GoB b3) *(b1 + 1,b2 + 1)) c= L~ b3 &
         LSeg((GoB b3) *(b1 + 1,b2 + 1),(GoB b3) *(b1 + 1,b2 + 2)) c= L~ b3
   holds not LSeg((GoB b3) *(b1,b2 + 1),(GoB b3) *(b1 + 1,b2 + 1)) c= L~ b3;

:: GOBOARD7:th 63
theorem
for b1, b2 being Element of NAT
for b3 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 < width GoB b3 &
         1 <= b2 &
         b2 + 1 < len GoB b3 &
         LSeg((GoB b3) *(b2,b1),(GoB b3) *(b2 + 1,b1)) c= L~ b3 &
         LSeg((GoB b3) *(b2 + 1,b1),(GoB b3) *(b2 + 2,b1)) c= L~ b3
   holds not LSeg((GoB b3) *(b2 + 1,b1),(GoB b3) *(b2 + 1,b1 + 1)) c= L~ b3;

:: GOBOARD7:th 64
theorem
for b1, b2 being Element of NAT
for b3 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
      st 1 <= b1 &
         b1 < width GoB b3 &
         1 <= b2 &
         b2 + 1 < len GoB b3 &
         LSeg((GoB b3) *(b2,b1 + 1),(GoB b3) *(b2 + 1,b1 + 1)) c= L~ b3 &
         LSeg((GoB b3) *(b2 + 1,b1 + 1),(GoB b3) *(b2 + 2,b1 + 1)) c= L~ b3
   holds not LSeg((GoB b3) *(b2 + 1,b1),(GoB b3) *(b2 + 1,b1 + 1)) c= L~ b3;

:: GOBOARD7:th 65
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
      st LSeg(b1,b2) is vertical & LSeg(b3,b4) is vertical & b1 `1 = b3 `1 & b1 `2 <= b3 `2 & b3 `2 <= b4 `2 & b4 `2 <= b2 `2
   holds LSeg(b3,b4) c= LSeg(b1,b2);

:: GOBOARD7:th 66
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
      st LSeg(b1,b2) is horizontal & LSeg(b3,b4) is horizontal & b1 `2 = b3 `2 & b1 `1 <= b3 `1 & b3 `1 <= b4 `1 & b4 `1 <= b2 `1
   holds LSeg(b3,b4) c= LSeg(b1,b2);