:: by Yatsuka Nakamura

::

:: Received August 21, 1998

:: Copyright (c) 1998-2021 Association of Mizar Users

theorem Th1: :: JGRAPH_1:1

for G being Graph

for IT being oriented Chain of G

for vs being FinSequence of the carrier of G st IT is Simple & vs is_oriented_vertex_seq_of IT holds

for n, m being Nat st 1 <= n & n < m & m <= len vs & vs . n = vs . m holds

( n = 1 & m = len vs )

for IT being oriented Chain of G

for vs being FinSequence of the carrier of G st IT is Simple & vs is_oriented_vertex_seq_of IT holds

for n, m being Nat st 1 <= n & n < m & m <= len vs & vs . n = vs . m holds

( n = 1 & m = len vs )

proof end;

definition

let X be set ;

MultiGraphStruct(# X,[:X,X:],(pr1 (X,X)),(pr2 (X,X)) #) is MultiGraphStruct ;

end;
func PGraph X -> MultiGraphStruct equals :: JGRAPH_1:def 1

MultiGraphStruct(# X,[:X,X:],(pr1 (X,X)),(pr2 (X,X)) #);

coherence MultiGraphStruct(# X,[:X,X:],(pr1 (X,X)),(pr2 (X,X)) #);

MultiGraphStruct(# X,[:X,X:],(pr1 (X,X)),(pr2 (X,X)) #) is MultiGraphStruct ;

:: deftheorem defines PGraph JGRAPH_1:def 1 :

for X being set holds PGraph X = MultiGraphStruct(# X,[:X,X:],(pr1 (X,X)),(pr2 (X,X)) #);

for X being set holds PGraph X = MultiGraphStruct(# X,[:X,X:],(pr1 (X,X)),(pr2 (X,X)) #);

definition

let f be FinSequence;

ex b_{1} being FinSequence st

( len b_{1} = (len f) -' 1 & ( for i being Nat st 1 <= i & i < len f holds

b_{1} . i = [(f . i),(f . (i + 1))] ) )

for b_{1}, b_{2} being FinSequence st len b_{1} = (len f) -' 1 & ( for i being Nat st 1 <= i & i < len f holds

b_{1} . i = [(f . i),(f . (i + 1))] ) & len b_{2} = (len f) -' 1 & ( for i being Nat st 1 <= i & i < len f holds

b_{2} . i = [(f . i),(f . (i + 1))] ) holds

b_{1} = b_{2}

end;
func PairF f -> FinSequence means :Def2: :: JGRAPH_1:def 2

( len it = (len f) -' 1 & ( for i being Nat st 1 <= i & i < len f holds

it . i = [(f . i),(f . (i + 1))] ) );

existence ( len it = (len f) -' 1 & ( for i being Nat st 1 <= i & i < len f holds

it . i = [(f . i),(f . (i + 1))] ) );

ex b

( len b

b

proof end;

uniqueness for b

b

b

b

proof end;

:: deftheorem Def2 defines PairF JGRAPH_1:def 2 :

for f, b_{2} being FinSequence holds

( b_{2} = PairF f iff ( len b_{2} = (len f) -' 1 & ( for i being Nat st 1 <= i & i < len f holds

b_{2} . i = [(f . i),(f . (i + 1))] ) ) );

for f, b

( b

b

registration
end;

theorem :: JGRAPH_1:3

for X being non empty set

for f being FinSequence of X holds f is FinSequence of the carrier of (PGraph X) ;

for f being FinSequence of X holds f is FinSequence of the carrier of (PGraph X) ;

theorem Th4: :: JGRAPH_1:4

for X being non empty set

for f being FinSequence of X holds PairF f is FinSequence of the carrier' of (PGraph X)

for f being FinSequence of X holds PairF f is FinSequence of the carrier' of (PGraph X)

proof end;

definition

let X be non empty set ;

let f be FinSequence of X;

:: original: PairF

redefine func PairF f -> FinSequence of the carrier' of (PGraph X);

coherence

PairF f is FinSequence of the carrier' of (PGraph X) by Th4;

end;
let f be FinSequence of X;

:: original: PairF

redefine func PairF f -> FinSequence of the carrier' of (PGraph X);

coherence

PairF f is FinSequence of the carrier' of (PGraph X) by Th4;

theorem Th5: :: JGRAPH_1:5

for X being non empty set

for n being Nat

for f being FinSequence of X st 1 <= n & n <= len (PairF f) holds

(PairF f) . n in the carrier' of (PGraph X)

for n being Nat

for f being FinSequence of X st 1 <= n & n <= len (PairF f) holds

(PairF f) . n in the carrier' of (PGraph X)

proof end;

definition

let X be non empty set ;

let f be FinSequence of X;

:: original: PairF

redefine func PairF f -> oriented Chain of PGraph X;

coherence

PairF f is oriented Chain of PGraph X by Th6;

end;
let f be FinSequence of X;

:: original: PairF

redefine func PairF f -> oriented Chain of PGraph X;

coherence

PairF f is oriented Chain of PGraph X by Th6;

theorem Th7: :: JGRAPH_1:7

for X being non empty set

for f being FinSequence of X

for f1 being FinSequence of the carrier of (PGraph X) st len f >= 1 & f = f1 holds

f1 is_oriented_vertex_seq_of PairF f

for f being FinSequence of X

for f1 being FinSequence of the carrier of (PGraph X) st len f >= 1 & f = f1 holds

f1 is_oriented_vertex_seq_of PairF f

proof end;

definition

let X be non empty set ;

let f, g be FinSequence of X;

end;
let f, g be FinSequence of X;

pred g is_Shortcut_of f means :: JGRAPH_1:def 3

( f . 1 = g . 1 & f . (len f) = g . (len g) & ex fc being Subset of (PairF f) ex fvs being Subset of f ex sc being oriented simple Chain of PGraph X ex g1 being FinSequence of the carrier of (PGraph X) st

( Seq fc = sc & Seq fvs = g & g1 = g & g1 is_oriented_vertex_seq_of sc ) );

( f . 1 = g . 1 & f . (len f) = g . (len g) & ex fc being Subset of (PairF f) ex fvs being Subset of f ex sc being oriented simple Chain of PGraph X ex g1 being FinSequence of the carrier of (PGraph X) st

( Seq fc = sc & Seq fvs = g & g1 = g & g1 is_oriented_vertex_seq_of sc ) );

:: deftheorem defines is_Shortcut_of JGRAPH_1:def 3 :

for X being non empty set

for f, g being FinSequence of X holds

( g is_Shortcut_of f iff ( f . 1 = g . 1 & f . (len f) = g . (len g) & ex fc being Subset of (PairF f) ex fvs being Subset of f ex sc being oriented simple Chain of PGraph X ex g1 being FinSequence of the carrier of (PGraph X) st

( Seq fc = sc & Seq fvs = g & g1 = g & g1 is_oriented_vertex_seq_of sc ) ) );

for X being non empty set

for f, g being FinSequence of X holds

( g is_Shortcut_of f iff ( f . 1 = g . 1 & f . (len f) = g . (len g) & ex fc being Subset of (PairF f) ex fvs being Subset of f ex sc being oriented simple Chain of PGraph X ex g1 being FinSequence of the carrier of (PGraph X) st

( Seq fc = sc & Seq fvs = g & g1 = g & g1 is_oriented_vertex_seq_of sc ) ) );

theorem Th8: :: JGRAPH_1:8

for X being non empty set

for f, g being FinSequence of X st g is_Shortcut_of f holds

( 1 <= len g & len g <= len f )

for f, g being FinSequence of X st g is_Shortcut_of f holds

( 1 <= len g & len g <= len f )

proof end;

theorem Th9: :: JGRAPH_1:9

for X being non empty set

for f being FinSequence of X st len f >= 1 holds

ex g being FinSequence of X st g is_Shortcut_of f

for f being FinSequence of X st len f >= 1 holds

ex g being FinSequence of X st g is_Shortcut_of f

proof end;

theorem Th10: :: JGRAPH_1:10

for X being non empty set

for f, g being FinSequence of X st g is_Shortcut_of f holds

rng (PairF g) c= rng (PairF f)

for f, g being FinSequence of X st g is_Shortcut_of f holds

rng (PairF g) c= rng (PairF f)

proof end;

theorem Th11: :: JGRAPH_1:11

for X being non empty set

for f, g being FinSequence of X st f . 1 <> f . (len f) & g is_Shortcut_of f holds

g is one-to-one

for f, g being FinSequence of X st f . 1 <> f . (len f) & g is_Shortcut_of f holds

g is one-to-one

proof end;

definition

let n be Nat;

let IT be FinSequence of (TOP-REAL n);

end;
let IT be FinSequence of (TOP-REAL n);

attr IT is nodic means :: JGRAPH_1:def 4

for i, j being Nat holds

( not LSeg (IT,i) meets LSeg (IT,j) or ( (LSeg (IT,i)) /\ (LSeg (IT,j)) = {(IT . i)} & ( IT . i = IT . j or IT . i = IT . (j + 1) ) ) or ( (LSeg (IT,i)) /\ (LSeg (IT,j)) = {(IT . (i + 1))} & ( IT . (i + 1) = IT . j or IT . (i + 1) = IT . (j + 1) ) ) or LSeg (IT,i) = LSeg (IT,j) );

for i, j being Nat holds

( not LSeg (IT,i) meets LSeg (IT,j) or ( (LSeg (IT,i)) /\ (LSeg (IT,j)) = {(IT . i)} & ( IT . i = IT . j or IT . i = IT . (j + 1) ) ) or ( (LSeg (IT,i)) /\ (LSeg (IT,j)) = {(IT . (i + 1))} & ( IT . (i + 1) = IT . j or IT . (i + 1) = IT . (j + 1) ) ) or LSeg (IT,i) = LSeg (IT,j) );

:: deftheorem defines nodic JGRAPH_1:def 4 :

for n being Nat

for IT being FinSequence of (TOP-REAL n) holds

( IT is nodic iff for i, j being Nat holds

( not LSeg (IT,i) meets LSeg (IT,j) or ( (LSeg (IT,i)) /\ (LSeg (IT,j)) = {(IT . i)} & ( IT . i = IT . j or IT . i = IT . (j + 1) ) ) or ( (LSeg (IT,i)) /\ (LSeg (IT,j)) = {(IT . (i + 1))} & ( IT . (i + 1) = IT . j or IT . (i + 1) = IT . (j + 1) ) ) or LSeg (IT,i) = LSeg (IT,j) ) );

for n being Nat

for IT being FinSequence of (TOP-REAL n) holds

( IT is nodic iff for i, j being Nat holds

( not LSeg (IT,i) meets LSeg (IT,j) or ( (LSeg (IT,i)) /\ (LSeg (IT,j)) = {(IT . i)} & ( IT . i = IT . j or IT . i = IT . (j + 1) ) ) or ( (LSeg (IT,i)) /\ (LSeg (IT,j)) = {(IT . (i + 1))} & ( IT . (i + 1) = IT . j or IT . (i + 1) = IT . (j + 1) ) ) or LSeg (IT,i) = LSeg (IT,j) ) );

theorem Th13: :: JGRAPH_1:13

for f being FinSequence of (TOP-REAL 2) st f is s.c.c. & LSeg (f,1) misses LSeg (f,((len f) -' 1)) holds

f is s.n.c.

f is s.n.c.

proof end;

theorem Th15: :: JGRAPH_1:15

for f being FinSequence of (TOP-REAL 2) st f is nodic & PairF f is Simple & f . 1 <> f . (len f) holds

f is s.n.c.

f is s.n.c.

proof end;

theorem Th16: :: JGRAPH_1:16

for n being Nat

for p1, p2, p3 being Point of (TOP-REAL n) holds

( for x being set holds

( not x <> p2 or not x in (LSeg (p1,p2)) /\ (LSeg (p2,p3)) ) or p1 in LSeg (p2,p3) or p3 in LSeg (p1,p2) )

for p1, p2, p3 being Point of (TOP-REAL n) holds

( for x being set holds

( not x <> p2 or not x in (LSeg (p1,p2)) /\ (LSeg (p2,p3)) ) or p1 in LSeg (p2,p3) or p3 in LSeg (p1,p2) )

proof end;

theorem Th17: :: JGRAPH_1:17

for f being FinSequence of (TOP-REAL 2) st f is s.n.c. & (LSeg (f,1)) /\ (LSeg (f,(1 + 1))) c= {(f /. (1 + 1))} & (LSeg (f,((len f) -' 2))) /\ (LSeg (f,((len f) -' 1))) c= {(f /. ((len f) -' 1))} holds

f is unfolded

f is unfolded

proof end;

theorem Th18: :: JGRAPH_1:18

for X being non empty set

for f being FinSequence of X st PairF f is Simple & f . 1 <> f . (len f) holds

( f is one-to-one & len f <> 1 )

for f being FinSequence of X st PairF f is Simple & f . 1 <> f . (len f) holds

( f is one-to-one & len f <> 1 )

proof end;

theorem :: JGRAPH_1:19

for X being non empty set

for f being FinSequence of X st f is one-to-one & len f > 1 holds

( PairF f is Simple & f . 1 <> f . (len f) )

for f being FinSequence of X st f is one-to-one & len f > 1 holds

( PairF f is Simple & f . 1 <> f . (len f) )

proof end;

theorem :: JGRAPH_1:20

for f being FinSequence of (TOP-REAL 2) st f is nodic & PairF f is Simple & f . 1 <> f . (len f) holds

f is unfolded

f is unfolded

proof end;

theorem Th21: :: JGRAPH_1:21

for f, g being FinSequence of (TOP-REAL 2)

for i being Nat st g is_Shortcut_of f & 1 <= i & i + 1 <= len g holds

ex k1 being Element of NAT st

( 1 <= k1 & k1 + 1 <= len f & f /. k1 = g /. i & f /. (k1 + 1) = g /. (i + 1) & f . k1 = g . i & f . (k1 + 1) = g . (i + 1) )

for i being Nat st g is_Shortcut_of f & 1 <= i & i + 1 <= len g holds

ex k1 being Element of NAT st

( 1 <= k1 & k1 + 1 <= len f & f /. k1 = g /. i & f /. (k1 + 1) = g /. (i + 1) & f . k1 = g . i & f . (k1 + 1) = g . (i + 1) )

proof end;

theorem Th25: :: JGRAPH_1:25

for f being FinSequence of (TOP-REAL 2) st f is special & 2 <= len f & f . 1 <> f . (len f) holds

ex g being FinSequence of (TOP-REAL 2) st

( 2 <= len g & g is special & g is one-to-one & L~ g c= L~ f & f . 1 = g . 1 & f . (len f) = g . (len g) & rng g c= rng f )

ex g being FinSequence of (TOP-REAL 2) st

( 2 <= len g & g is special & g is one-to-one & L~ g c= L~ f & f . 1 = g . 1 & f . (len f) = g . (len g) & rng g c= rng f )

proof end;

:: Goboard Theorem for general special sequences

theorem Th26: :: JGRAPH_1:26

for f1, f2 being FinSequence of (TOP-REAL 2) st f1 is special & f2 is special & 2 <= len f1 & 2 <= len f2 & f1 . 1 <> f1 . (len f1) & f2 . 1 <> f2 . (len f2) & X_axis f1 lies_between (X_axis f1) . 1,(X_axis f1) . (len f1) & X_axis f2 lies_between (X_axis f1) . 1,(X_axis f1) . (len f1) & Y_axis f1 lies_between (Y_axis f2) . 1,(Y_axis f2) . (len f2) & Y_axis f2 lies_between (Y_axis f2) . 1,(Y_axis f2) . (len f2) holds

L~ f1 meets L~ f2

L~ f1 meets L~ f2

proof end;

theorem Th28: :: JGRAPH_1:28

for N being Nat

for p1, p2 being Point of (TOP-REAL N)

for x1, x2 being Point of (Euclid N) st x1 = p1 & x2 = p2 holds

|.(p1 - p2).| = dist (x1,x2)

for p1, p2 being Point of (TOP-REAL N)

for x1, x2 being Point of (Euclid N) st x1 = p1 & x2 = p2 holds

|.(p1 - p2).| = dist (x1,x2)

proof end;

theorem Th32: :: JGRAPH_1:32

for p1, p2 being Point of (TOP-REAL 2) holds |.(p1 - p2).| <= |.((p1 `1) - (p2 `1)).| + |.((p1 `2) - (p2 `2)).|

proof end;

theorem Th34: :: JGRAPH_1:34

for p1, p2 being Point of (TOP-REAL 2) holds

( |.((p1 `1) - (p2 `1)).| <= |.(p1 - p2).| & |.((p1 `2) - (p2 `2)).| <= |.(p1 - p2).| )

( |.((p1 `1) - (p2 `1)).| <= |.(p1 - p2).| & |.((p1 `2) - (p2 `2)).| <= |.(p1 - p2).| )

proof end;

::$CT

theorem Th35: :: JGRAPH_1:36

for N being Nat

for p, p1, p2 being Point of (TOP-REAL N) st p in LSeg (p1,p2) holds

( |.(p - p1).| <= |.(p1 - p2).| & |.(p - p2).| <= |.(p1 - p2).| )

for p, p1, p2 being Point of (TOP-REAL N) st p in LSeg (p1,p2) holds

( |.(p - p1).| <= |.(p1 - p2).| & |.(p - p2).| <= |.(p1 - p2).| )

proof end;

theorem Th36: :: JGRAPH_1:37

for M being non empty MetrSpace

for P, Q being Subset of (TopSpaceMetr M) st P <> {} & P is compact & Q <> {} & Q is compact holds

min_dist_min (P,Q) >= 0

for P, Q being Subset of (TopSpaceMetr M) st P <> {} & P is compact & Q <> {} & Q is compact holds

min_dist_min (P,Q) >= 0

proof end;

theorem Th37: :: JGRAPH_1:38

for M being non empty MetrSpace

for P, Q being Subset of (TopSpaceMetr M) st P <> {} & P is compact & Q <> {} & Q is compact holds

( P misses Q iff min_dist_min (P,Q) > 0 )

for P, Q being Subset of (TopSpaceMetr M) st P <> {} & P is compact & Q <> {} & Q is compact holds

( P misses Q iff min_dist_min (P,Q) > 0 )

proof end;

:: Approximation of finite sequence by special finite sequence

theorem Th38: :: JGRAPH_1:39

for f being FinSequence of (TOP-REAL 2)

for a, c, d being Real st 1 <= len f & X_axis f lies_between (X_axis f) . 1,(X_axis f) . (len f) & Y_axis f lies_between c,d & a > 0 & ( for i being Nat st 1 <= i & i + 1 <= len f holds

|.((f /. i) - (f /. (i + 1))).| < a ) holds

ex g being FinSequence of (TOP-REAL 2) st

( g is special & g . 1 = f . 1 & g . (len g) = f . (len f) & len g >= len f & X_axis g lies_between (X_axis f) . 1,(X_axis f) . (len f) & Y_axis g lies_between c,d & ( for j being Nat st j in dom g holds

ex k being Nat st

( k in dom f & |.((g /. j) - (f /. k)).| < a ) ) & ( for j being Nat st 1 <= j & j + 1 <= len g holds

|.((g /. j) - (g /. (j + 1))).| < a ) )

for a, c, d being Real st 1 <= len f & X_axis f lies_between (X_axis f) . 1,(X_axis f) . (len f) & Y_axis f lies_between c,d & a > 0 & ( for i being Nat st 1 <= i & i + 1 <= len f holds

|.((f /. i) - (f /. (i + 1))).| < a ) holds

ex g being FinSequence of (TOP-REAL 2) st

( g is special & g . 1 = f . 1 & g . (len g) = f . (len f) & len g >= len f & X_axis g lies_between (X_axis f) . 1,(X_axis f) . (len f) & Y_axis g lies_between c,d & ( for j being Nat st j in dom g holds

ex k being Nat st

( k in dom f & |.((g /. j) - (f /. k)).| < a ) ) & ( for j being Nat st 1 <= j & j + 1 <= len g holds

|.((g /. j) - (g /. (j + 1))).| < a ) )

proof end;

theorem Th39: :: JGRAPH_1:40

for f being FinSequence of (TOP-REAL 2)

for a, c, d being Real st 1 <= len f & Y_axis f lies_between (Y_axis f) . 1,(Y_axis f) . (len f) & X_axis f lies_between c,d & a > 0 & ( for i being Nat st 1 <= i & i + 1 <= len f holds

|.((f /. i) - (f /. (i + 1))).| < a ) holds

ex g being FinSequence of (TOP-REAL 2) st

( g is special & g . 1 = f . 1 & g . (len g) = f . (len f) & len g >= len f & Y_axis g lies_between (Y_axis f) . 1,(Y_axis f) . (len f) & X_axis g lies_between c,d & ( for j being Nat st j in dom g holds

ex k being Nat st

( k in dom f & |.((g /. j) - (f /. k)).| < a ) ) & ( for j being Nat st 1 <= j & j + 1 <= len g holds

|.((g /. j) - (g /. (j + 1))).| < a ) )

for a, c, d being Real st 1 <= len f & Y_axis f lies_between (Y_axis f) . 1,(Y_axis f) . (len f) & X_axis f lies_between c,d & a > 0 & ( for i being Nat st 1 <= i & i + 1 <= len f holds

|.((f /. i) - (f /. (i + 1))).| < a ) holds

ex g being FinSequence of (TOP-REAL 2) st

( g is special & g . 1 = f . 1 & g . (len g) = f . (len f) & len g >= len f & Y_axis g lies_between (Y_axis f) . 1,(Y_axis f) . (len f) & X_axis g lies_between c,d & ( for j being Nat st j in dom g holds

ex k being Nat st

( k in dom f & |.((g /. j) - (f /. k)).| < a ) ) & ( for j being Nat st 1 <= j & j + 1 <= len g holds

|.((g /. j) - (g /. (j + 1))).| < a ) )

proof end;

theorem Th40: :: JGRAPH_1:41

for f being FinSequence of (TOP-REAL 2) st 1 <= len f holds

( len (X_axis f) = len f & (X_axis f) . 1 = (f /. 1) `1 & (X_axis f) . (len f) = (f /. (len f)) `1 )

( len (X_axis f) = len f & (X_axis f) . 1 = (f /. 1) `1 & (X_axis f) . (len f) = (f /. (len f)) `1 )

proof end;

theorem Th41: :: JGRAPH_1:42

for f being FinSequence of (TOP-REAL 2) st 1 <= len f holds

( len (Y_axis f) = len f & (Y_axis f) . 1 = (f /. 1) `2 & (Y_axis f) . (len f) = (f /. (len f)) `2 )

( len (Y_axis f) = len f & (Y_axis f) . 1 = (f /. 1) `2 & (Y_axis f) . (len f) = (f /. (len f)) `2 )

proof end;

theorem Th42: :: JGRAPH_1:43

for i being Nat

for f being FinSequence of (TOP-REAL 2) st i in dom f holds

( (X_axis f) . i = (f /. i) `1 & (Y_axis f) . i = (f /. i) `2 )

for f being FinSequence of (TOP-REAL 2) st i in dom f holds

( (X_axis f) . i = (f /. i) `1 & (Y_axis f) . i = (f /. i) `2 )

proof end;

:: Goboard Theorem in continuous case

theorem Th43: :: JGRAPH_1:44

for P, Q 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 & Q is_an_arc_of q1,q2 & ( for p being Point of (TOP-REAL 2) st p in P holds

( p1 `1 <= p `1 & p `1 <= p2 `1 ) ) & ( for p being Point of (TOP-REAL 2) st p in Q holds

( p1 `1 <= p `1 & p `1 <= p2 `1 ) ) & ( for p being Point of (TOP-REAL 2) st p in P holds

( q1 `2 <= p `2 & p `2 <= q2 `2 ) ) & ( for p being Point of (TOP-REAL 2) st p in Q holds

( q1 `2 <= p `2 & p `2 <= q2 `2 ) ) holds

P meets Q

for p1, p2, q1, q2 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & Q is_an_arc_of q1,q2 & ( for p being Point of (TOP-REAL 2) st p in P holds

( p1 `1 <= p `1 & p `1 <= p2 `1 ) ) & ( for p being Point of (TOP-REAL 2) st p in Q holds

( p1 `1 <= p `1 & p `1 <= p2 `1 ) ) & ( for p being Point of (TOP-REAL 2) st p in P holds

( q1 `2 <= p `2 & p `2 <= q2 `2 ) ) & ( for p being Point of (TOP-REAL 2) st p in Q holds

( q1 `2 <= p `2 & p `2 <= q2 `2 ) ) holds

P meets Q

proof end;

theorem Th44: :: JGRAPH_1:45

for X, Y being non empty TopSpace

for f being Function of X,Y

for P being non empty Subset of Y

for f1 being Function of X,(Y | P) st f = f1 & f is continuous holds

f1 is continuous

for f being Function of X,Y

for P being non empty Subset of Y

for f1 being Function of X,(Y | P) st f = f1 & f is continuous holds

f1 is continuous

proof end;

theorem Th45: :: JGRAPH_1:46

for X, Y being non empty TopSpace

for f being Function of X,Y

for P being non empty Subset of Y st X is compact & Y is T_2 & f is continuous & f is one-to-one & P = rng f holds

ex f1 being Function of X,(Y | P) st

( f = f1 & f1 is being_homeomorphism )

for f being Function of X,Y

for P being non empty Subset of Y st X is compact & Y is T_2 & f is continuous & f is one-to-one & P = rng f holds

ex f1 being Function of X,(Y | P) st

( f = f1 & f1 is being_homeomorphism )

proof end;

theorem :: JGRAPH_1:47

for f, g being Function of I[01],(TOP-REAL 2)

for a, b, c, d being Real

for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & (f . O) `1 = a & (f . I) `1 = b & (g . O) `2 = c & (g . I) `2 = d & ( for r being Point of I[01] holds

( a <= (f . r) `1 & (f . r) `1 <= b & a <= (g . r) `1 & (g . r) `1 <= b & c <= (f . r) `2 & (f . r) `2 <= d & c <= (g . r) `2 & (g . r) `2 <= d ) ) holds

rng f meets rng g

for a, b, c, d being Real

for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & (f . O) `1 = a & (f . I) `1 = b & (g . O) `2 = c & (g . I) `2 = d & ( for r being Point of I[01] holds

( a <= (f . r) `1 & (f . r) `1 <= b & a <= (g . r) `1 & (g . r) `1 <= b & c <= (f . r) `2 & (f . r) `2 <= d & c <= (g . r) `2 & (g . r) `2 <= d ) ) holds

rng f meets rng g

proof end;