Article GOBRD11, MML version 4.99.1005

:: GOBRD11:th 1
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
      st b3 in b2 & b2 is connected(b1)
   holds b2 c= Component_of b3;

:: GOBRD11:th 2
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Element of bool the carrier of b1
      st b4 is_a_component_of b1 & b2 c= b4 & b3 is connected(b1) & Cl b2 meets Cl b3
   holds b3 c= b4;

:: GOBRD11:th 3
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is_a_component_of b1 & b3 is_a_component_of b1
   holds b2 \/ b3 is a_union_of_components of b1;

:: GOBRD11:th 4
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Element of bool the carrier of b1 holds
Down(b2 \/ b3,b4) = (Down(b2,b4)) \/ Down(b3,b4);

:: GOBRD11:th 5
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Element of bool the carrier of b1 holds
Down(b2 /\ b3,b4) = (Down(b2,b4)) /\ Down(b3,b4);

:: GOBRD11:th 6
theorem
for b1 being non empty non constant circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
   (L~ b1) ` <> {};

:: GOBRD11:funcreg 1
registration
  let a1 be non empty non constant circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
  cluster (L~ a1) ` -> non empty;
end;

:: GOBRD11:th 7
theorem
for b1 being non empty non constant circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
   the carrier of TOP-REAL 2 = union {cell(GoB b1,b2,b3) where b2 is Element of NAT, b3 is Element of NAT: b2 <= len GoB b1 & b3 <= width GoB b1};

:: GOBRD11:th 8
theorem
for b1 being Element of REAL
for b2, b3 being Element of bool the carrier of TOP-REAL 2
      st b2 = {|[b4,b5]| where b4 is Element of REAL, b5 is Element of REAL: b5 <= b1} &
         b3 = {|[b4,b5]| where b4 is Element of REAL, b5 is Element of REAL: b1 < b5}
   holds b2 = b3 `;

:: GOBRD11:th 9
theorem
for b1 being Element of REAL
for b2, b3 being Element of bool the carrier of TOP-REAL 2
      st b2 = {|[b4,b5]| where b4 is Element of REAL, b5 is Element of REAL: b1 <= b5} &
         b3 = {|[b4,b5]| where b4 is Element of REAL, b5 is Element of REAL: b5 < b1}
   holds b2 = b3 `;

:: GOBRD11:th 10
theorem
for b1 being Element of REAL
for b2, b3 being Element of bool the carrier of TOP-REAL 2
      st b2 = {|[b4,b5]| where b4 is Element of REAL, b5 is Element of REAL: b1 <= b4} &
         b3 = {|[b4,b5]| where b4 is Element of REAL, b5 is Element of REAL: b4 < b1}
   holds b2 = b3 `;

:: GOBRD11:th 11
theorem
for b1 being Element of REAL
for b2, b3 being Element of bool the carrier of TOP-REAL 2
      st b2 = {|[b4,b5]| where b4 is Element of REAL, b5 is Element of REAL: b4 <= b1} &
         b3 = {|[b4,b5]| where b4 is Element of REAL, b5 is Element of REAL: b1 < b4}
   holds b2 = b3 `;

:: GOBRD11:th 12
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2 being Element of REAL
      st b1 = {|[b3,b4]| where b3 is Element of REAL, b4 is Element of REAL: b4 <= b2}
   holds b1 is closed(TOP-REAL 2);

:: GOBRD11:th 13
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2 being Element of REAL
      st b1 = {|[b3,b4]| where b3 is Element of REAL, b4 is Element of REAL: b2 <= b4}
   holds b1 is closed(TOP-REAL 2);

:: GOBRD11:th 14
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2 being Element of REAL
      st b1 = {|[b3,b4]| where b3 is Element of REAL, b4 is Element of REAL: b3 <= b2}
   holds b1 is closed(TOP-REAL 2);

:: GOBRD11:th 15
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2 being Element of REAL
      st b1 = {|[b3,b4]| where b3 is Element of REAL, b4 is Element of REAL: b2 <= b3}
   holds b1 is closed(TOP-REAL 2);

:: GOBRD11:th 16
theorem
for b1 being Element of NAT
for b2 being tabular FinSequence of (the carrier of TOP-REAL 2) * holds
   h_strip(b2,b1) is closed(TOP-REAL 2);

:: GOBRD11:th 17
theorem
for b1 being Element of NAT
for b2 being tabular FinSequence of (the carrier of TOP-REAL 2) * holds
   v_strip(b2,b1) is closed(TOP-REAL 2);

:: GOBRD11:th 18
theorem
for b1 being non empty-yielding tabular FinSequence of (the carrier of TOP-REAL 2) *
      st b1 is X_equal-in-line
   holds v_strip(b1,0) = {|[b2,b3]| where b2 is Element of REAL, b3 is Element of REAL: b2 <= (b1 *(1,1)) `1};

:: GOBRD11:th 19
theorem
for b1 being non empty-yielding tabular FinSequence of (the carrier of TOP-REAL 2) *
      st b1 is X_equal-in-line
   holds v_strip(b1,len b1) = {|[b2,b3]| where b2 is Element of REAL, b3 is Element of REAL: (b1 *(len b1,1)) `1 <= b2};

:: GOBRD11:th 20
theorem
for b1 being Element of NAT
for b2 being non empty-yielding tabular FinSequence of (the carrier of TOP-REAL 2) *
      st b2 is X_equal-in-line & 1 <= b1 & b1 < len b2
   holds v_strip(b2,b1) = {|[b3,b4]| where b3 is Element of REAL, b4 is Element of REAL: (b2 *(b1,1)) `1 <= b3 & b3 <= (b2 *(b1 + 1,1)) `1};

:: GOBRD11:th 21
theorem
for b1 being non empty-yielding tabular FinSequence of (the carrier of TOP-REAL 2) *
      st b1 is Y_equal-in-column
   holds h_strip(b1,0) = {|[b2,b3]| where b2 is Element of REAL, b3 is Element of REAL: b3 <= (b1 *(1,1)) `2};

:: GOBRD11:th 22
theorem
for b1 being non empty-yielding tabular FinSequence of (the carrier of TOP-REAL 2) *
      st b1 is Y_equal-in-column
   holds h_strip(b1,width b1) = {|[b2,b3]| where b2 is Element of REAL, b3 is Element of REAL: (b1 *(1,width b1)) `2 <= b3};

:: GOBRD11:th 23
theorem
for b1 being Element of NAT
for b2 being non empty-yielding tabular FinSequence of (the carrier of TOP-REAL 2) *
      st b2 is Y_equal-in-column & 1 <= b1 & b1 < width b2
   holds h_strip(b2,b1) = {|[b3,b4]| where b3 is Element of REAL, b4 is Element of REAL: (b2 *(1,b1)) `2 <= b4 & b4 <= (b2 *(1,b1 + 1)) `2};

:: GOBRD11:th 24
theorem
for b1 being non empty-yielding tabular X_equal-in-line Y_equal-in-column FinSequence of (the carrier of TOP-REAL 2) * holds
   cell(b1,0,0) = {|[b2,b3]| where b2 is Element of REAL, b3 is Element of REAL: b2 <= (b1 *(1,1)) `1 & b3 <= (b1 *(1,1)) `2};

:: GOBRD11:th 25
theorem
for b1 being non empty-yielding tabular X_equal-in-line Y_equal-in-column FinSequence of (the carrier of TOP-REAL 2) * holds
   cell(b1,0,width b1) = {|[b2,b3]| where b2 is Element of REAL, b3 is Element of REAL: b2 <= (b1 *(1,1)) `1 & (b1 *(1,width b1)) `2 <= b3};

:: GOBRD11:th 26
theorem
for b1 being Element of NAT
for b2 being non empty-yielding tabular X_equal-in-line Y_equal-in-column FinSequence of (the carrier of TOP-REAL 2) *
      st 1 <= b1 & b1 < width b2
   holds cell(b2,0,b1) = {|[b3,b4]| where b3 is Element of REAL, b4 is Element of REAL: b3 <= (b2 *(1,1)) `1 & (b2 *(1,b1)) `2 <= b4 & b4 <= (b2 *(1,b1 + 1)) `2};

:: GOBRD11:th 27
theorem
for b1 being non empty-yielding tabular X_equal-in-line Y_equal-in-column FinSequence of (the carrier of TOP-REAL 2) * holds
   cell(b1,len b1,0) = {|[b2,b3]| where b2 is Element of REAL, b3 is Element of REAL: (b1 *(len b1,1)) `1 <= b2 & b3 <= (b1 *(1,1)) `2};

:: GOBRD11:th 28
theorem
for b1 being non empty-yielding tabular X_equal-in-line Y_equal-in-column FinSequence of (the carrier of TOP-REAL 2) * holds
   cell(b1,len b1,width b1) = {|[b2,b3]| where b2 is Element of REAL, b3 is Element of REAL: (b1 *(len b1,1)) `1 <= b2 & (b1 *(1,width b1)) `2 <= b3};

:: GOBRD11:th 29
theorem
for b1 being Element of NAT
for b2 being non empty-yielding tabular X_equal-in-line Y_equal-in-column FinSequence of (the carrier of TOP-REAL 2) *
      st 1 <= b1 & b1 < width b2
   holds cell(b2,len b2,b1) = {|[b3,b4]| where b3 is Element of REAL, b4 is Element of REAL: (b2 *(len b2,1)) `1 <= b3 & (b2 *(1,b1)) `2 <= b4 & b4 <= (b2 *(1,b1 + 1)) `2};

:: GOBRD11:th 30
theorem
for b1 being Element of NAT
for b2 being non empty-yielding tabular X_equal-in-line Y_equal-in-column FinSequence of (the carrier of TOP-REAL 2) *
      st 1 <= b1 & b1 < len b2
   holds cell(b2,b1,0) = {|[b3,b4]| where b3 is Element of REAL, b4 is Element of REAL: (b2 *(b1,1)) `1 <= b3 & b3 <= (b2 *(b1 + 1,1)) `1 & b4 <= (b2 *(1,1)) `2};

:: GOBRD11:th 31
theorem
for b1 being Element of NAT
for b2 being non empty-yielding tabular X_equal-in-line Y_equal-in-column FinSequence of (the carrier of TOP-REAL 2) *
      st 1 <= b1 & b1 < len b2
   holds cell(b2,b1,width b2) = {|[b3,b4]| where b3 is Element of REAL, b4 is Element of REAL: (b2 *(b1,1)) `1 <= b3 & b3 <= (b2 *(b1 + 1,1)) `1 & (b2 *(1,width b2)) `2 <= b4};

:: GOBRD11:th 32
theorem
for b1, b2 being Element of NAT
for b3 being non empty-yielding tabular X_equal-in-line Y_equal-in-column FinSequence of (the carrier of TOP-REAL 2) *
      st 1 <= b1 & b1 < len b3 & 1 <= b2 & b2 < width b3
   holds cell(b3,b1,b2) = {|[b4,b5]| where b4 is Element of REAL, b5 is Element of REAL: (b3 *(b1,1)) `1 <= b4 & b4 <= (b3 *(b1 + 1,1)) `1 & (b3 *(1,b2)) `2 <= b5 & b5 <= (b3 *(1,b2 + 1)) `2};

:: GOBRD11:th 33
theorem
for b1, b2 being Element of NAT
for b3 being tabular FinSequence of (the carrier of TOP-REAL 2) * holds
   cell(b3,b1,b2) is closed(TOP-REAL 2);

:: GOBRD11:th 34
theorem
for b1 being non empty-yielding tabular FinSequence of (the carrier of TOP-REAL 2) * holds
   1 <= len b1 & 1 <= width b1;

:: GOBRD11:th 35
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 cell(b3,b1,b2) = Cl Int cell(b3,b1,b2);