:: JORDAN20 semantic presentation  Show TPTP formulae Show IDV graph for whole article:: Showing IDV graph ... (Click the Palm Trees again to close it)

theorem Th1: :: JORDAN20:1  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for P being Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & p in P holds
Segment P,p1,p2,p,p = {p}
proof end;

theorem Th2: :: JORDAN20:2  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for p1, p2, p being Point of (TOP-REAL 2)
for a being Real st p in LSeg p1,p2 & p1 `1 <= a & p2 `1 <= a holds
p `1 <= a
proof end;

theorem Th3: :: JORDAN20:3  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for p1, p2, p being Point of (TOP-REAL 2)
for a being Real st p in LSeg p1,p2 & p1 `1 >= a & p2 `1 >= a holds
p `1 >= a
proof end;

theorem :: JORDAN20:4  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for p1, p2, p being Point of (TOP-REAL 2)
for a being Real st p in LSeg p1,p2 & p1 `1 < a & p2 `1 < a holds
p `1 < a
proof end;

theorem :: JORDAN20:5  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for p1, p2, p being Point of (TOP-REAL 2)
for a being Real st p in LSeg p1,p2 & p1 `1 > a & p2 `1 > a holds
p `1 > a
proof end;

theorem Th6: :: JORDAN20:6  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for j being Nat
for f being S-Sequence_in_R2
for p, q being Point of (TOP-REAL 2) st 1 <= j & j < len f & p in LSeg f,j & q in LSeg f,j & (f /. j) `2 = (f /. (j + 1)) `2 & (f /. j) `1 > (f /. (j + 1)) `1 & LE p,q, L~ f,f /. 1,f /. (len f) holds
p `1 >= q `1
proof end;

theorem Th7: :: JORDAN20:7  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for j being Nat
for f being S-Sequence_in_R2
for p, q being Point of (TOP-REAL 2) st 1 <= j & j < len f & p in LSeg f,j & q in LSeg f,j & (f /. j) `2 = (f /. (j + 1)) `2 & (f /. j) `1 < (f /. (j + 1)) `1 & LE p,q, L~ f,f /. 1,f /. (len f) holds
p `1 <= q `1
proof end;

definition
let P be Subset of (TOP-REAL 2);
let p1, p2, p be Point of (TOP-REAL 2);
let e be Real;
pred p is_Lin P,p1,p2,e means :Def1: :: JORDAN20:def 1
( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p4 being Point of (TOP-REAL 2) st
( p4 `1 < e & LE p4,p,P,p1,p2 & ( for p5 being Point of (TOP-REAL 2) st LE p4,p5,P,p1,p2 & LE p5,p,P,p1,p2 holds
p5 `1 <= e ) ) );
pred p is_Rin P,p1,p2,e means :Def2: :: JORDAN20:def 2
( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p4 being Point of (TOP-REAL 2) st
( p4 `1 > e & LE p4,p,P,p1,p2 & ( for p5 being Point of (TOP-REAL 2) st LE p4,p5,P,p1,p2 & LE p5,p,P,p1,p2 holds
p5 `1 >= e ) ) );
pred p is_Lout P,p1,p2,e means :Def3: :: JORDAN20:def 3
( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p4 being Point of (TOP-REAL 2) st
( p4 `1 < e & LE p,p4,P,p1,p2 & ( for p5 being Point of (TOP-REAL 2) st LE p5,p4,P,p1,p2 & LE p,p5,P,p1,p2 holds
p5 `1 <= e ) ) );
pred p is_Rout P,p1,p2,e means :Def4: :: JORDAN20:def 4
( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p4 being Point of (TOP-REAL 2) st
( p4 `1 > e & LE p,p4,P,p1,p2 & ( for p5 being Point of (TOP-REAL 2) st LE p5,p4,P,p1,p2 & LE p,p5,P,p1,p2 holds
p5 `1 >= e ) ) );
pred p is_OSin P,p1,p2,e means :Def5: :: JORDAN20:def 5
( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p7 being Point of (TOP-REAL 2) st
( LE p7,p,P,p1,p2 & ( for p8 being Point of (TOP-REAL 2) st LE p7,p8,P,p1,p2 & LE p8,p,P,p1,p2 holds
p8 `1 = e ) & ( for p4 being Point of (TOP-REAL 2) st LE p4,p7,P,p1,p2 & p4 <> p7 holds
( ex p5 being Point of (TOP-REAL 2) st
( LE p4,p5,P,p1,p2 & LE p5,p7,P,p1,p2 & p5 `1 > e ) & ex p6 being Point of (TOP-REAL 2) st
( LE p4,p6,P,p1,p2 & LE p6,p7,P,p1,p2 & p6 `1 < e ) ) ) ) );
pred p is_OSout P,p1,p2,e means :Def6: :: JORDAN20:def 6
( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p7 being Point of (TOP-REAL 2) st
( LE p,p7,P,p1,p2 & ( for p8 being Point of (TOP-REAL 2) st LE p8,p7,P,p1,p2 & LE p,p8,P,p1,p2 holds
p8 `1 = e ) & ( for p4 being Point of (TOP-REAL 2) st LE p7,p4,P,p1,p2 & p4 <> p7 holds
( ex p5 being Point of (TOP-REAL 2) st
( LE p5,p4,P,p1,p2 & LE p7,p5,P,p1,p2 & p5 `1 > e ) & ex p6 being Point of (TOP-REAL 2) st
( LE p6,p4,P,p1,p2 & LE p7,p6,P,p1,p2 & p6 `1 < e ) ) ) ) );
correctness
;
end;

:: deftheorem Def1 defines is_Lin JORDAN20:def 1 :
for P being Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2)
for e being Real holds
( p is_Lin P,p1,p2,e iff ( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p4 being Point of (TOP-REAL 2) st
( p4 `1 < e & LE p4,p,P,p1,p2 & ( for p5 being Point of (TOP-REAL 2) st LE p4,p5,P,p1,p2 & LE p5,p,P,p1,p2 holds
p5 `1 <= e ) ) ) );

:: deftheorem Def2 defines is_Rin JORDAN20:def 2 :
for P being Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2)
for e being Real holds
( p is_Rin P,p1,p2,e iff ( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p4 being Point of (TOP-REAL 2) st
( p4 `1 > e & LE p4,p,P,p1,p2 & ( for p5 being Point of (TOP-REAL 2) st LE p4,p5,P,p1,p2 & LE p5,p,P,p1,p2 holds
p5 `1 >= e ) ) ) );

:: deftheorem Def3 defines is_Lout JORDAN20:def 3 :
for P being Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2)
for e being Real holds
( p is_Lout P,p1,p2,e iff ( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p4 being Point of (TOP-REAL 2) st
( p4 `1 < e & LE p,p4,P,p1,p2 & ( for p5 being Point of (TOP-REAL 2) st LE p5,p4,P,p1,p2 & LE p,p5,P,p1,p2 holds
p5 `1 <= e ) ) ) );

:: deftheorem Def4 defines is_Rout JORDAN20:def 4 :
for P being Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2)
for e being Real holds
( p is_Rout P,p1,p2,e iff ( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p4 being Point of (TOP-REAL 2) st
( p4 `1 > e & LE p,p4,P,p1,p2 & ( for p5 being Point of (TOP-REAL 2) st LE p5,p4,P,p1,p2 & LE p,p5,P,p1,p2 holds
p5 `1 >= e ) ) ) );

:: deftheorem Def5 defines is_OSin JORDAN20:def 5 :
for P being Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2)
for e being Real holds
( p is_OSin P,p1,p2,e iff ( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p7 being Point of (TOP-REAL 2) st
( LE p7,p,P,p1,p2 & ( for p8 being Point of (TOP-REAL 2) st LE p7,p8,P,p1,p2 & LE p8,p,P,p1,p2 holds
p8 `1 = e ) & ( for p4 being Point of (TOP-REAL 2) st LE p4,p7,P,p1,p2 & p4 <> p7 holds
( ex p5 being Point of (TOP-REAL 2) st
( LE p4,p5,P,p1,p2 & LE p5,p7,P,p1,p2 & p5 `1 > e ) & ex p6 being Point of (TOP-REAL 2) st
( LE p4,p6,P,p1,p2 & LE p6,p7,P,p1,p2 & p6 `1 < e ) ) ) ) ) );

:: deftheorem Def6 defines is_OSout JORDAN20:def 6 :
for P being Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2)
for e being Real holds
( p is_OSout P,p1,p2,e iff ( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p7 being Point of (TOP-REAL 2) st
( LE p,p7,P,p1,p2 & ( for p8 being Point of (TOP-REAL 2) st LE p8,p7,P,p1,p2 & LE p,p8,P,p1,p2 holds
p8 `1 = e ) & ( for p4 being Point of (TOP-REAL 2) st LE p7,p4,P,p1,p2 & p4 <> p7 holds
( ex p5 being Point of (TOP-REAL 2) st
( LE p5,p4,P,p1,p2 & LE p7,p5,P,p1,p2 & p5 `1 > e ) & ex p6 being Point of (TOP-REAL 2) st
( LE p6,p4,P,p1,p2 & LE p7,p6,P,p1,p2 & p6 `1 < e ) ) ) ) ) );

theorem :: JORDAN20:8  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for P being Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2)
for e being Real st P is_an_arc_of p1,p2 & p1 `1 <= e & p2 `1 >= e holds
ex p3 being Point of (TOP-REAL 2) st
( p3 in P & p3 `1 = e )
proof end;

theorem :: JORDAN20:9  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for P being non empty Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2)
for e being Real st P is_an_arc_of p1,p2 & p1 `1 < e & p2 `1 > e & p in P & p `1 = e & not p is_Lin P,p1,p2,e & not p is_Rin P,p1,p2,e holds
p is_OSin P,p1,p2,e
proof end;

theorem :: JORDAN20:10  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for P being non empty Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2)
for e being Real st P is_an_arc_of p1,p2 & p1 `1 < e & p2 `1 > e & p in P & p `1 = e & not p is_Lout P,p1,p2,e & not p is_Rout P,p1,p2,e holds
p is_OSout P,p1,p2,e
proof end;

theorem Th11: :: JORDAN20:11  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for P being Subset of I[01]
for s being Real st P = [.0,s.[ holds
P is open
proof end;

theorem Th12: :: JORDAN20:12  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for P being Subset of I[01]
for s being Real st P = ].s,1.] holds
P is open
proof end;

theorem Th13: :: JORDAN20:13  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for P being non empty Subset of (TOP-REAL 2)
for P1 being Subset of ((TOP-REAL 2) | P)
for Q being Subset of I[01]
for f being Function of I[01] ,((TOP-REAL 2) | P)
for s being Real st s <= 1 & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( 0 <= ss & ss < s & q0 = f . ss ) } & Q = [.0,s.[ holds
f .: Q = P1
proof end;

theorem Th14: :: JORDAN20:14  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for P being non empty Subset of (TOP-REAL 2)
for P1 being Subset of ((TOP-REAL 2) | P)
for Q being Subset of I[01]
for f being Function of I[01] ,((TOP-REAL 2) | P)
for s being Real st s >= 0 & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( s < ss & ss <= 1 & q0 = f . ss ) } & Q = ].s,1.] holds
f .: Q = P1
proof end;

theorem Th15: :: JORDAN20:15  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for P being non empty Subset of (TOP-REAL 2)
for P1 being Subset of ((TOP-REAL 2) | P)
for f being Function of I[01] ,((TOP-REAL 2) | P)
for s being Real st s <= 1 & f is_homeomorphism & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( 0 <= ss & ss < s & q0 = f . ss ) } holds
P1 is open
proof end;

theorem Th16: :: JORDAN20:16  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for P being non empty Subset of (TOP-REAL 2)
for P1 being Subset of ((TOP-REAL 2) | P)
for f being Function of I[01] ,((TOP-REAL 2) | P)
for s being Real st s >= 0 & f is_homeomorphism & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( s < ss & ss <= 1 & q0 = f . ss ) } holds
P1 is open
proof end;

theorem Th17: :: JORDAN20:17  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopStruct
for Q1, Q2 being Subset of T
for p1, p2 being Point of T st Q1 /\ Q2 = {} & Q1 \/ Q2 = the carrier of T & p1 in Q1 & p2 in Q2 & Q1 is open & Q2 is open holds
for P being Function of I[01] ,T holds
( not P is continuous or not P . 0 = p1 or not P . 1 = p2 )
proof end;

theorem Th18: :: JORDAN20:18  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for P being non empty Subset of (TOP-REAL 2)
for Q being Subset of ((TOP-REAL 2) | P)
for p1, p2, q being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & q in P & q <> p1 & q <> p2 & Q = P \ {q} holds
( not Q is connected & ( for R being Function of I[01] ,(((TOP-REAL 2) | P) | Q) holds
( not R is continuous or not R . 0 = p1 or not R . 1 = p2 ) ) )
proof end;

theorem Th19: :: JORDAN20:19  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for P being non empty Subset of (TOP-REAL 2)
for p1, p2, q1, q2 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & q1 in P & q2 in P & not LE q1,q2,P,p1,p2 holds
LE q2,q1,P,p1,p2
proof end;

theorem Th20: :: JORDAN20:20  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for P being non empty Subset of (TOP-REAL 2)
for p1, p2, q1 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & q1 in P & p1 <> q1 holds
Segment P,p1,p2,p1,q1 is_an_arc_of p1,q1
proof end;

theorem Th21: :: JORDAN20:21  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for n being Nat
for p1, p2 being Point of (TOP-REAL n)
for P, P1 being non empty Subset of (TOP-REAL n) st P is_an_arc_of p1,p2 & P1 is_an_arc_of p1,p2 & P1 c= P holds
P1 = P
proof end;

theorem Th22: :: JORDAN20:22  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for P being non empty Subset of (TOP-REAL 2)
for p1, p2, q1 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & q1 in P & p2 <> q1 holds
Segment P,p1,p2,q1,p2 is_an_arc_of q1,p2
proof end;

theorem Th23: :: JORDAN20:23  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for P being non empty Subset of (TOP-REAL 2)
for p1, p2, q1, q2, q3 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & LE q1,q2,P,p1,p2 & LE q2,q3,P,p1,p2 holds
(Segment P,p1,p2,q1,q2) \/ (Segment P,p1,p2,q2,q3) = Segment P,p1,p2,q1,q3
proof end;

theorem :: JORDAN20:24  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for P being non empty Subset of (TOP-REAL 2)
for p1, p2, q1, q2, q3 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & LE q1,q2,P,p1,p2 & LE q2,q3,P,p1,p2 holds
(Segment P,p1,p2,q1,q2) /\ (Segment P,p1,p2,q2,q3) = {q2}
proof end;

theorem Th25: :: JORDAN20:25  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for P being non empty Subset of (TOP-REAL 2)
for p1, p2 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 holds
Segment P,p1,p2,p1,p2 = P
proof end;

theorem :: JORDAN20:26  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem Th27: :: JORDAN20:27  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for T being non empty TopSpace
for a, b, c being Point of T
for G1 being Path of a,b
for G2 being Path of b,c st G1 is continuous & G2 is continuous & G1 . 0 = a & G1 . 1 = b & G2 . 0 = b & G2 . 1 = c holds
( G1 + G2 is continuous & (G1 + G2) . 0 = a & (G1 + G2) . 1 = c )
proof end;

theorem Th28: :: JORDAN20:28  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for P, Q1 being non empty Subset of (TOP-REAL 2)
for p1, p2, q1, q2 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & Q1 is_an_arc_of q1,q2 & LE q1,q2,P,p1,p2 & Q1 c= P holds
Q1 = Segment P,p1,p2,q1,q2
proof end;

theorem :: JORDAN20:29  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for P being non empty Subset of (TOP-REAL 2)
for p1, p2, q1, q2, p being Point of (TOP-REAL 2)
for e being Real st p1 `1 < e & p2 `1 > e & q1 is_Lin P,p1,p2,e & q2 `1 = e & LSeg q1,q2 c= P & p in LSeg q1,q2 holds
p is_Lin P,p1,p2,e
proof end;

theorem :: JORDAN20:30  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for P being non empty Subset of (TOP-REAL 2)
for p1, p2, q1, q2, p being Point of (TOP-REAL 2)
for e being Real st p1 `1 < e & p2 `1 > e & q1 is_Rin P,p1,p2,e & q2 `1 = e & LSeg q1,q2 c= P & p in LSeg q1,q2 holds
p is_Rin P,p1,p2,e
proof end;

theorem :: JORDAN20:31  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for P being non empty Subset of (TOP-REAL 2)
for p1, p2, q1, q2, p being Point of (TOP-REAL 2)
for e being Real st p1 `1 < e & p2 `1 > e & q1 is_Lout P,p1,p2,e & q2 `1 = e & LSeg q1,q2 c= P & p in LSeg q1,q2 holds
p is_Lout P,p1,p2,e
proof end;

theorem :: JORDAN20:32  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for P being non empty Subset of (TOP-REAL 2)
for p1, p2, q1, q2, p being Point of (TOP-REAL 2)
for e being Real st p1 `1 < e & p2 `1 > e & q1 is_Rout P,p1,p2,e & q2 `1 = e & LSeg q1,q2 c= P & p in LSeg q1,q2 holds
p is_Rout P,p1,p2,e
proof end;

theorem :: JORDAN20:33  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for P being non empty Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2)
for e being Real st P is_S-P_arc_joining p1,p2 & p1 `1 < e & p2 `1 > e & p in P & p `1 = e & not p is_Lin P,p1,p2,e holds
p is_Rin P,p1,p2,e
proof end;

theorem :: JORDAN20:34  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for P being non empty Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2)
for e being Real st P is_S-P_arc_joining p1,p2 & p1 `1 < e & p2 `1 > e & p in P & p `1 = e & not p is_Lout P,p1,p2,e holds
p is_Rout P,p1,p2,e
proof end;