:: JORDAN1F semantic presentation

theorem Th1: :: JORDAN1F:1
for b1, b2, b3 being Nat
for b4 being FinSequence of the carrier of (TOP-REAL 2)
for b5 being Go-board holds
not ( b4 is_sequence_on b5 & LSeg (b5 * b1,b2),(b5 * b1,b3) meets L~ b4 & [b1,b2] in Indices b5 & [b1,b3] in Indices b5 & b2 <= b3 & ( for b6 being Nat holds
not ( b2 <= b6 & b6 <= b3 & (b5 * b1,b6) `2 = inf (proj2 .: ((LSeg (b5 * b1,b2),(b5 * b1,b3)) /\ (L~ b4))) ) ) )
proof end;

theorem Th2: :: JORDAN1F:2
for b1, b2, b3 being Nat
for b4 being FinSequence of the carrier of (TOP-REAL 2)
for b5 being Go-board holds
not ( b4 is_sequence_on b5 & LSeg (b5 * b1,b2),(b5 * b1,b3) meets L~ b4 & [b1,b2] in Indices b5 & [b1,b3] in Indices b5 & b2 <= b3 & ( for b6 being Nat holds
not ( b2 <= b6 & b6 <= b3 & (b5 * b1,b6) `2 = sup (proj2 .: ((LSeg (b5 * b1,b2),(b5 * b1,b3)) /\ (L~ b4))) ) ) )
proof end;

theorem Th3: :: JORDAN1F:3
for b1, b2, b3 being Nat
for b4 being FinSequence of the carrier of (TOP-REAL 2)
for b5 being Go-board holds
not ( b4 is_sequence_on b5 & LSeg (b5 * b1,b2),(b5 * b3,b2) meets L~ b4 & [b1,b2] in Indices b5 & [b3,b2] in Indices b5 & b1 <= b3 & ( for b6 being Nat holds
not ( b1 <= b6 & b6 <= b3 & (b5 * b6,b2) `1 = inf (proj1 .: ((LSeg (b5 * b1,b2),(b5 * b3,b2)) /\ (L~ b4))) ) ) )
proof end;

theorem Th4: :: JORDAN1F:4
for b1, b2, b3 being Nat
for b4 being FinSequence of the carrier of (TOP-REAL 2)
for b5 being Go-board holds
not ( b4 is_sequence_on b5 & LSeg (b5 * b1,b2),(b5 * b3,b2) meets L~ b4 & [b1,b2] in Indices b5 & [b3,b2] in Indices b5 & b1 <= b3 & ( for b6 being Nat holds
not ( b1 <= b6 & b6 <= b3 & (b5 * b6,b2) `1 = sup (proj1 .: ((LSeg (b5 * b1,b2),(b5 * b3,b2)) /\ (L~ b4))) ) ) )
proof end;

theorem Th5: :: JORDAN1F:5
for b1 being compact non horizontal non vertical Subset of (TOP-REAL 2)
for b2 being Nat holds (Upper_Seq b1,b2) /. 1 = W-min (L~ (Cage b1,b2))
proof end;

theorem Th6: :: JORDAN1F:6
for b1 being compact non horizontal non vertical Subset of (TOP-REAL 2)
for b2 being Nat holds (Lower_Seq b1,b2) /. 1 = E-max (L~ (Cage b1,b2))
proof end;

theorem Th7: :: JORDAN1F:7
for b1 being compact non horizontal non vertical Subset of (TOP-REAL 2)
for b2 being Nat holds (Upper_Seq b1,b2) /. (len (Upper_Seq b1,b2)) = E-max (L~ (Cage b1,b2))
proof end;

theorem Th8: :: JORDAN1F:8
for b1 being compact non horizontal non vertical Subset of (TOP-REAL 2)
for b2 being Nat holds (Lower_Seq b1,b2) /. (len (Lower_Seq b1,b2)) = W-min (L~ (Cage b1,b2))
proof end;

theorem Th9: :: JORDAN1F:9
for b1 being compact non horizontal non vertical Subset of (TOP-REAL 2)
for b2 being Nat holds
( ( L~ (Upper_Seq b1,b2) = Upper_Arc (L~ (Cage b1,b2)) & L~ (Lower_Seq b1,b2) = Lower_Arc (L~ (Cage b1,b2)) ) or ( L~ (Upper_Seq b1,b2) = Lower_Arc (L~ (Cage b1,b2)) & L~ (Lower_Seq b1,b2) = Upper_Arc (L~ (Cage b1,b2)) ) )
proof end;

theorem Th10: :: JORDAN1F:10
for b1 being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for b2 being Nat holds Upper_Seq b1,b2 is_sequence_on Gauge b1,b2
proof end;

theorem Th11: :: JORDAN1F:11
for b1 being Go-board
for b2 being Point of (TOP-REAL 2)
for b3 being FinSequence of (TOP-REAL 2) holds
( b3 is_sequence_on b1 & ex b4, b5 being Nat st
( [b4,b5] in Indices b1 & b2 = b1 * b4,b5 ) & ( for b4, b5, b6, b7 being Nat holds
( [b4,b5] in Indices b1 & [b6,b7] in Indices b1 & b2 = b1 * b4,b5 & b3 /. 1 = b1 * b6,b7 implies (abs (b6 - b4)) + (abs (b7 - b5)) = 1 ) ) implies <*b2*> ^ b3 is_sequence_on b1 )
proof end;

theorem Th12: :: JORDAN1F:12
for b1 being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for b2 being Nat holds Lower_Seq b1,b2 is_sequence_on Gauge b1,b2
proof end;

theorem Th13: :: JORDAN1F:13
for b1 being Nat
for b2 being non empty being_simple_closed_curve compact non horizontal non vertical Subset of (TOP-REAL 2)
for b3 being Point of (TOP-REAL 2) holds
not ( b3 `1 = ((W-bound b2) + (E-bound b2)) / 2 & b3 `2 = inf (proj2 .: ((LSeg ((Gauge b2,1) * (Center (Gauge b2,1)),1),((Gauge b2,1) * (Center (Gauge b2,1)),(width (Gauge b2,1)))) /\ (Upper_Arc (L~ (Cage b2,(b1 + 1)))))) & ( for b4 being Nat holds
not ( 1 <= b4 & b4 <= width (Gauge b2,(b1 + 1)) & b3 = (Gauge b2,(b1 + 1)) * (Center (Gauge b2,(b1 + 1))),b4 ) ) )
proof end;