Article JGRAPH_8, MML version 4.99.1005
:: JGRAPH_8:funcreg 1
registration
let a1, a2, a3, a4 be real set;
cluster closed_inside_of_rectangle(a1,a2,a3,a4) -> convex;
end;
:: JGRAPH_8:funcreg 2
registration
let a1, a2, a3, a4 be real set;
cluster Trectangle(a1,a2,a3,a4) -> convex;
end;
:: JGRAPH_8:th 1
theorem
for b1 being Element of NAT
for b2 being real positive set
for b3 being Function-like quasi_total continuous Relation of the carrier of I[01],the carrier of TOP-REAL b1 holds
ex b4 being FinSequence of REAL st
b4 . 1 = 0 &
b4 . len b4 = 1 &
5 <= len b4 &
rng b4 c= the carrier of I[01] &
b4 is increasing &
(for b5 being Element of NAT
for b6 being Element of bool the carrier of I[01]
for b7 being Element of bool the carrier of Euclid b1
st 1 <= b5 &
b5 < len b4 &
b6 = [.b4 /. b5,b4 /. (b5 + 1).] &
b7 = b3 .: b6
holds diameter b7 < b2);
:: JGRAPH_8:th 2
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4 being Element of bool the carrier of TOP-REAL b1
st b4 c= LSeg(b2,b3) & b2 in b4 & b3 in b4 & b4 is connected(TOP-REAL b1)
holds b4 = LSeg(b2,b3);
:: JGRAPH_8:th 3
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4 being Path of b2,b3
st rng b4 c= LSeg(b2,b3)
holds rng b4 = LSeg(b2,b3);
:: JGRAPH_8:th 4
theorem
for b1, b2 being non empty Element of bool the carrier of TOP-REAL 2
for b3, b4, b5, b6 being Element of the carrier of TOP-REAL 2
for b7 being Path of b3,b4
for b8 being Path of b5,b6
st rng b7 = b1 &
rng b8 = b2 &
(for b9 being Element of the carrier of TOP-REAL 2
st b9 in b1
holds b3 `1 <= b9 `1 & b9 `1 <= b4 `1) &
(for b9 being Element of the carrier of TOP-REAL 2
st b9 in b2
holds b3 `1 <= b9 `1 & b9 `1 <= b4 `1) &
(for b9 being Element of the carrier of TOP-REAL 2
st b9 in b1
holds b5 `2 <= b9 `2 & b9 `2 <= b6 `2) &
(for b9 being Element of the carrier of TOP-REAL 2
st b9 in b2
holds b5 `2 <= b9 `2 & b9 `2 <= b6 `2)
holds b1 meets b2;
:: JGRAPH_8:th 5
theorem
for b1, b2, b3, b4 being real set
for b5, b6 being Function-like quasi_total continuous Relation of the carrier of I[01],the carrier of TOP-REAL 2
for b7, b8 being Element of the carrier of I[01]
st b7 = 0 &
b8 = 1 &
(b5 . b7) `1 = b1 &
(b5 . b8) `1 = b2 &
(b6 . b7) `2 = b3 &
(b6 . b8) `2 = b4 &
(for b9 being Element of the carrier of I[01] holds
b1 <= (b5 . b9) `1 & (b5 . b9) `1 <= b2 & b1 <= (b6 . b9) `1 & (b6 . b9) `1 <= b2 & b3 <= (b5 . b9) `2 & (b5 . b9) `2 <= b4 & b3 <= (b6 . b9) `2 & (b6 . b9) `2 <= b4)
holds rng b5 meets rng b6;
:: JGRAPH_8:th 6
theorem
for b1, b2, b3, b4 being real set
for b5, b6, b7, b8 being Element of the carrier of Trectangle(b1,b2,b3,b4)
for b9 being Path of b5,b6
for b10 being Path of b8,b7
for b11, b12, b13, b14 being Element of the carrier of TOP-REAL 2
st b11 `1 = b1 & b12 `1 = b2 & b13 `2 = b3 & b14 `2 = b4 & b5 = b11 & b6 = b12 & b7 = b13 & b8 = b14
holds ex b15, b16 being Element of the carrier of I[01] st
b9 . b15 = b10 . b16;