Article GOBOARD4, MML version 4.99.1005

:: GOBOARD4:th 1
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 being FinSequence of the carrier of TOP-REAL 2
      st 1 <= len b2 &
         1 <= len b3 &
         b2 is_sequence_on b1 &
         b3 is_sequence_on b1 &
         b2 /. 1 in proj2 Line(b1,1) &
         b2 /. len b2 in proj2 Line(b1,len b1) &
         b3 /. 1 in proj2 Col(b1,1) &
         b3 /. len b3 in proj2 Col(b1,width b1)
   holds proj2 b2 meets proj2 b3;

:: GOBOARD4:th 2
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 being FinSequence of the carrier of TOP-REAL 2
      st 2 <= len b2 &
         2 <= len b3 &
         b2 is_sequence_on b1 &
         b3 is_sequence_on b1 &
         b2 /. 1 in proj2 Line(b1,1) &
         b2 /. len b2 in proj2 Line(b1,len b1) &
         b3 /. 1 in proj2 Col(b1,1) &
         b3 /. len b3 in proj2 Col(b1,width b1)
   holds L~ b2 meets L~ b3;

:: GOBOARD4:th 3
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 being FinSequence of the carrier of TOP-REAL 2
      st 2 <= len b2 &
         2 <= len b3 &
         b2 is_sequence_on b1 &
         b3 is_sequence_on b1 &
         b2 /. 1 in proj2 Line(b1,1) &
         b2 /. len b2 in proj2 Line(b1,len b1) &
         b3 /. 1 in proj2 Col(b1,1) &
         b3 /. len b3 in proj2 Col(b1,width b1)
   holds L~ b2 meets L~ b3;

:: GOBOARD4:prednot 1 => GOBOARD4:pred 1
definition
  let a1 be Relation-like set;
  let a2, a3 be Element of REAL;
  pred A1 lies_between A2,A3 means
    proj2 a1 c= [.a2,a3.];
end;

:: GOBOARD4:dfs 1
definiens
  let a1 be Relation-like set;
  let a2, a3 be Element of REAL;
To prove
     a1 lies_between a2,a3
it is sufficient to prove
  thus proj2 a1 c= [.a2,a3.];

:: GOBOARD4:def 1
theorem
for b1 being Relation-like set
for b2, b3 being Element of REAL holds
   b1 lies_between b2,b3
iff
   proj2 b1 c= [.b2,b3.];

:: GOBOARD4:prednot 2 => GOBOARD4:pred 1
definition
  let a1 be Relation-like set;
  let a2, a3 be Element of REAL;
  pred A1 lies_between A2,A3 means
    for b1 being Element of NAT
          st b1 in dom a1
       holds a2 <= a1 . b1 & a1 . b1 <= a3;
end;

:: GOBOARD4:dfs 2
definiens
  let a1 be FinSequence of REAL;
  let a2, a3 be Element of REAL;
To prove
     a1 lies_between a2,a3
it is sufficient to prove
  thus for b1 being Element of NAT
          st b1 in dom a1
       holds a2 <= a1 . b1 & a1 . b1 <= a3;

:: GOBOARD4:def 2
theorem
for b1 being FinSequence of REAL
for b2, b3 being Element of REAL holds
   b1 lies_between b2,b3
iff
   for b4 being Element of NAT
         st b4 in dom b1
      holds b2 <= b1 . b4 & b1 . b4 <= b3;

:: GOBOARD4:th 4
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
      st 2 <= len b1 &
         2 <= len b2 &
         b1 is special &
         b2 is special &
         (for b3 being Element of NAT
               st b3 in dom b1 & b3 + 1 in dom b1
            holds b1 /. b3 <> b1 /. (b3 + 1)) &
         (for b3 being Element of NAT
               st b3 in dom b2 & b3 + 1 in dom b2
            holds b2 /. b3 <> b2 /. (b3 + 1)) &
         X_axis b1 lies_between (X_axis b1) . 1,(X_axis b1) . len b1 &
         X_axis b2 lies_between (X_axis b1) . 1,(X_axis b1) . len b1 &
         Y_axis b1 lies_between (Y_axis b2) . 1,(Y_axis b2) . len b2 &
         Y_axis b2 lies_between (Y_axis b2) . 1,(Y_axis b2) . len b2
   holds L~ b1 meets L~ b2;

:: GOBOARD4:th 5
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
      st b1 is one-to-one &
         b1 is special &
         2 <= len b1 &
         b2 is one-to-one &
         b2 is special &
         2 <= len b2 &
         X_axis b1 lies_between (X_axis b1) . 1,(X_axis b1) . len b1 &
         X_axis b2 lies_between (X_axis b1) . 1,(X_axis b1) . len b1 &
         Y_axis b1 lies_between (Y_axis b2) . 1,(Y_axis b2) . len b2 &
         Y_axis b2 lies_between (Y_axis b2) . 1,(Y_axis b2) . len b2
   holds L~ b1 meets L~ b2;

:: GOBOARD4:th 8
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4, b5, b6 being Element of the carrier of TOP-REAL 2
      st b1 is_S-P_arc_joining b3,b5 &
         b2 is_S-P_arc_joining b4,b6 &
         (for b7 being Element of the carrier of TOP-REAL 2
               st b7 in b1 \/ b2
            holds b3 `1 <= b7 `1 & b7 `1 <= b5 `1) &
         (for b7 being Element of the carrier of TOP-REAL 2
               st b7 in b1 \/ b2
            holds b4 `2 <= b7 `2 & b7 `2 <= b6 `2)
   holds b1 meets b2;