:: JGRAPH_1 semantic presentation
begin
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 Element of NAT st 1 <= n & n < m & m <= len vs & vs . n = vs . m holds
( n = 1 & m = len vs )
proof
let G be Graph; ::_thesis: 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 Element of NAT st 1 <= n & n < m & m <= len vs & vs . n = vs . m holds
( n = 1 & m = len vs )
let IT be oriented Chain of G; ::_thesis: 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 Element of NAT st 1 <= n & n < m & m <= len vs & vs . n = vs . m holds
( n = 1 & m = len vs )
let vs be FinSequence of the carrier of G; ::_thesis: ( IT is Simple & vs is_oriented_vertex_seq_of IT implies for n, m being Element of NAT st 1 <= n & n < m & m <= len vs & vs . n = vs . m holds
( n = 1 & m = len vs ) )
assume that
A1: IT is Simple and
A2: vs is_oriented_vertex_seq_of IT ; ::_thesis: for n, m being Element of NAT st 1 <= n & n < m & m <= len vs & vs . n = vs . m holds
( n = 1 & m = len vs )
A3: len vs = (len IT) + 1 by A2, GRAPH_4:def_5;
consider vs9 being FinSequence of the carrier of G such that
A4: vs9 is_oriented_vertex_seq_of IT and
A5: for n, m being Element of NAT st 1 <= n & n < m & m <= len vs9 & vs9 . n = vs9 . m holds
( n = 1 & m = len vs9 ) by A1, GRAPH_4:def_7;
percases ( IT <> {} or IT = {} ) ;
suppose IT <> {} ; ::_thesis: for n, m being Element of NAT st 1 <= n & n < m & m <= len vs & vs . n = vs . m holds
( n = 1 & m = len vs )
then vs = vs9 by A2, A4, GRAPH_4:10;
hence for n, m being Element of NAT st 1 <= n & n < m & m <= len vs & vs . n = vs . m holds
( n = 1 & m = len vs ) by A5; ::_thesis: verum
end;
suppose IT = {} ; ::_thesis: for n, m being Element of NAT st 1 <= n & n < m & m <= len vs & vs . n = vs . m holds
( n = 1 & m = len vs )
then len IT = 0 ;
hence for n, m being Element of NAT st 1 <= n & n < m & m <= len vs & vs . n = vs . m holds
( n = 1 & m = len vs ) by A3, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
definition
let X be set ;
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)) #) is MultiGraphStruct ;
end;
:: deftheorem defines PGraph JGRAPH_1:def_1_:_
for X being set holds PGraph X = MultiGraphStruct(# X,[:X,X:],(pr1 (X,X)),(pr2 (X,X)) #);
theorem :: JGRAPH_1:2
for X being set holds the carrier of (PGraph X) = X ;
definition
let f be FinSequence;
func PairF f -> FinSequence means :Def2: :: JGRAPH_1:def 2
( len it = (len f) -' 1 & ( for i being Element of NAT st 1 <= i & i < len f holds
it . i = [(f . i),(f . (i + 1))] ) );
existence
ex b1 being FinSequence st
( len b1 = (len f) -' 1 & ( for i being Element of NAT st 1 <= i & i < len f holds
b1 . i = [(f . i),(f . (i + 1))] ) )
proof
deffunc H1( Nat) -> set = [(f . $1),(f . ($1 + 1))];
ex p being FinSequence st
( len p = (len f) -' 1 & ( for k being Nat st k in dom p holds
p . k = H1(k) ) ) from FINSEQ_1:sch_2();
then consider p being FinSequence such that
A1: len p = (len f) -' 1 and
A2: for k being Nat st k in dom p holds
p . k = [(f . k),(f . (k + 1))] ;
for i being Element of NAT st 1 <= i & i < len f holds
p . i = [(f . i),(f . (i + 1))]
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i < len f implies p . i = [(f . i),(f . (i + 1))] )
assume that
A3: 1 <= i and
A4: i < len f ; ::_thesis: p . i = [(f . i),(f . (i + 1))]
i + 1 <= len f by A4, NAT_1:13;
then A5: (i + 1) - 1 <= (len f) - 1 by XREAL_1:9;
(len f) - 1 = (len f) -' 1 by A3, A4, XREAL_1:233, XXREAL_0:2;
then i in dom p by A1, A3, A5, FINSEQ_3:25;
hence p . i = [(f . i),(f . (i + 1))] by A2; ::_thesis: verum
end;
hence ex b1 being FinSequence st
( len b1 = (len f) -' 1 & ( for i being Element of NAT st 1 <= i & i < len f holds
b1 . i = [(f . i),(f . (i + 1))] ) ) by A1; ::_thesis: verum
end;
uniqueness
for b1, b2 being FinSequence st len b1 = (len f) -' 1 & ( for i being Element of NAT st 1 <= i & i < len f holds
b1 . i = [(f . i),(f . (i + 1))] ) & len b2 = (len f) -' 1 & ( for i being Element of NAT st 1 <= i & i < len f holds
b2 . i = [(f . i),(f . (i + 1))] ) holds
b1 = b2
proof
let g1, g2 be FinSequence; ::_thesis: ( len g1 = (len f) -' 1 & ( for i being Element of NAT st 1 <= i & i < len f holds
g1 . i = [(f . i),(f . (i + 1))] ) & len g2 = (len f) -' 1 & ( for i being Element of NAT st 1 <= i & i < len f holds
g2 . i = [(f . i),(f . (i + 1))] ) implies g1 = g2 )
assume that
A6: len g1 = (len f) -' 1 and
A7: for i being Element of NAT st 1 <= i & i < len f holds
g1 . i = [(f . i),(f . (i + 1))] and
A8: len g2 = (len f) -' 1 and
A9: for i being Element of NAT st 1 <= i & i < len f holds
g2 . i = [(f . i),(f . (i + 1))] ; ::_thesis: g1 = g2
percases ( len f >= 1 or len f < 1 ) ;
supposeA10: len f >= 1 ; ::_thesis: g1 = g2
for j being Nat st 1 <= j & j <= len g1 holds
g1 . j = g2 . j
proof
let j be Nat; ::_thesis: ( 1 <= j & j <= len g1 implies g1 . j = g2 . j )
assume that
A11: 1 <= j and
A12: j <= len g1 ; ::_thesis: g1 . j = g2 . j
len f < (len f) + 1 by NAT_1:13;
then (len f) - 1 < len f by XREAL_1:19;
then len g1 < len f by A6, A10, XREAL_1:233;
then A13: ( j in NAT & j < len f ) by A12, ORDINAL1:def_12, XXREAL_0:2;
then g1 . j = [(f . j),(f . (j + 1))] by A7, A11;
hence g1 . j = g2 . j by A9, A11, A13; ::_thesis: verum
end;
hence g1 = g2 by A6, A8, FINSEQ_1:14; ::_thesis: verum
end;
suppose len f < 1 ; ::_thesis: g1 = g2
then (len f) + 1 <= 1 by NAT_1:13;
then ((len f) + 1) - 1 <= 1 - 1 by XREAL_1:9;
then A14: len f = 0 ;
0 - 1 < 0 ;
then g1 = {} by A6, A14, XREAL_0:def_2;
hence g1 = g2 by A6, A8; ::_thesis: verum
end;
end;
end;
end;
:: deftheorem Def2 defines PairF JGRAPH_1:def_2_:_
for f, b2 being FinSequence holds
( b2 = PairF f iff ( len b2 = (len f) -' 1 & ( for i being Element of NAT st 1 <= i & i < len f holds
b2 . i = [(f . i),(f . (i + 1))] ) ) );
registration
let X be non empty set ;
cluster PGraph X -> non empty ;
coherence
not PGraph X is empty ;
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) ;
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)
proof
let X be non empty set ; ::_thesis: for f being FinSequence of X holds PairF f is FinSequence of the carrier' of (PGraph X)
let f be FinSequence of X; ::_thesis: PairF f is FinSequence of the carrier' of (PGraph X)
rng (PairF f) c= [:X,X:]
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (PairF f) or y in [:X,X:] )
A1: (len f) -' 1 < ((len f) -' 1) + 1 by NAT_1:13;
assume y in rng (PairF f) ; ::_thesis: y in [:X,X:]
then consider x being set such that
A2: x in dom (PairF f) and
A3: y = (PairF f) . x by FUNCT_1:def_3;
reconsider n = x as Element of NAT by A2;
A4: x in Seg (len (PairF f)) by A2, FINSEQ_1:def_3;
then A5: 1 <= n by FINSEQ_1:1;
A6: len (PairF f) = (len f) -' 1 by Def2;
A7: n <= len (PairF f) by A4, FINSEQ_1:1;
then 1 <= (len f) -' 1 by A5, A6, XXREAL_0:2;
then (len f) -' 1 = (len f) - 1 by NAT_D:39;
then A8: n < len f by A7, A6, A1, XXREAL_0:2;
then A9: n + 1 <= len f by NAT_1:13;
1 < n + 1 by A5, NAT_1:13;
then n + 1 in dom f by A9, FINSEQ_3:25;
then A10: f . (n + 1) in rng f by FUNCT_1:def_3;
n in dom f by A5, A8, FINSEQ_3:25;
then A11: f . n in rng f by FUNCT_1:def_3;
(PairF f) . n = [(f . n),(f . (n + 1))] by A5, A8, Def2;
hence y in [:X,X:] by A3, A11, A10, ZFMISC_1:def_2; ::_thesis: verum
end;
hence PairF f is FinSequence of the carrier' of (PGraph X) by FINSEQ_1:def_4; ::_thesis: verum
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;
theorem Th5: :: JGRAPH_1:5
for X being non empty set
for n being Element of 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
let X be non empty set ; ::_thesis: for n being Element of NAT
for f being FinSequence of X st 1 <= n & n <= len (PairF f) holds
(PairF f) . n in the carrier' of (PGraph X)
let n be Element of NAT ; ::_thesis: for f being FinSequence of X st 1 <= n & n <= len (PairF f) holds
(PairF f) . n in the carrier' of (PGraph X)
let f be FinSequence of X; ::_thesis: ( 1 <= n & n <= len (PairF f) implies (PairF f) . n in the carrier' of (PGraph X) )
assume that
A1: 1 <= n and
A2: n <= len (PairF f) ; ::_thesis: (PairF f) . n in the carrier' of (PGraph X)
A3: (len f) -' 1 < ((len f) -' 1) + 1 by NAT_1:13;
A4: len (PairF f) = (len f) -' 1 by Def2;
then 1 <= (len f) -' 1 by A1, A2, XXREAL_0:2;
then (len f) -' 1 = (len f) - 1 by NAT_D:39;
then A5: n < len f by A2, A4, A3, XXREAL_0:2;
then A6: n + 1 <= len f by NAT_1:13;
1 < n + 1 by A1, NAT_1:13;
then n + 1 in dom f by A6, FINSEQ_3:25;
then A7: f . (n + 1) in rng f by FUNCT_1:def_3;
n in dom f by A1, A5, FINSEQ_3:25;
then A8: f . n in rng f by FUNCT_1:def_3;
(PairF f) . n = [(f . n),(f . (n + 1))] by A1, A5, Def2;
hence (PairF f) . n in the carrier' of (PGraph X) by A8, A7, ZFMISC_1:def_2; ::_thesis: verum
end;
theorem Th6: :: JGRAPH_1:6
for X being non empty set
for f being FinSequence of X holds PairF f is oriented Chain of PGraph X
proof
let X be non empty set ; ::_thesis: for f being FinSequence of X holds PairF f is oriented Chain of PGraph X
let f be FinSequence of X; ::_thesis: PairF f is oriented Chain of PGraph X
A1: now__::_thesis:_(_(_len_f_>=_1_&_PairF_f_is_Chain_of_PGraph_X_)_or_(_len_f_<_1_&_PairF_f_is_oriented_Chain_of_PGraph_X_)_)
percases ( len f >= 1 or len f < 1 ) ;
case len f >= 1 ; ::_thesis: PairF f is Chain of PGraph X
then ((len f) - 1) + 1 = ((len f) -' 1) + 1 by XREAL_1:233;
then A2: len f = (len (PairF f)) + 1 by Def2;
A3: for n being Element of NAT st 1 <= n & n <= len f holds
f . n in the carrier of (PGraph X)
proof
let n be Element of NAT ; ::_thesis: ( 1 <= n & n <= len f implies f . n in the carrier of (PGraph X) )
assume ( 1 <= n & n <= len f ) ; ::_thesis: f . n in the carrier of (PGraph X)
then n in dom f by FINSEQ_3:25;
then f . n in rng f by FUNCT_1:def_3;
hence f . n in the carrier of (PGraph X) ; ::_thesis: verum
end;
A4: for n being Element of NAT st 1 <= n & n <= len (PairF f) holds
ex x9, y9 being Element of (PGraph X) st
( x9 = f . n & y9 = f . (n + 1) & (PairF f) . n joins x9,y9 )
proof
A5: (len f) -' 1 < ((len f) -' 1) + 1 by NAT_1:13;
let n be Element of NAT ; ::_thesis: ( 1 <= n & n <= len (PairF f) implies ex x9, y9 being Element of (PGraph X) st
( x9 = f . n & y9 = f . (n + 1) & (PairF f) . n joins x9,y9 ) )
assume that
A6: 1 <= n and
A7: n <= len (PairF f) ; ::_thesis: ex x9, y9 being Element of (PGraph X) st
( x9 = f . n & y9 = f . (n + 1) & (PairF f) . n joins x9,y9 )
A8: 1 < n + 1 by A6, NAT_1:13;
A9: len (PairF f) = (len f) -' 1 by Def2;
then 1 <= (len f) -' 1 by A6, A7, XXREAL_0:2;
then (len f) -' 1 = (len f) - 1 by NAT_D:39;
then A10: n < len f by A7, A9, A5, XXREAL_0:2;
then n + 1 <= len f by NAT_1:13;
then n + 1 in dom f by A8, FINSEQ_3:25;
then A11: f . (n + 1) in rng f by FUNCT_1:def_3;
n in dom f by A6, A10, FINSEQ_3:25;
then A12: f . n in rng f by FUNCT_1:def_3;
then reconsider a = f . n, b = f . (n + 1) as Element of (PGraph X) by A11;
(pr2 (X,X)) . ((f . n),(f . (n + 1))) = f . (n + 1) by A12, A11, FUNCT_3:def_5;
then A13: the Target of (PGraph X) . ((PairF f) . n) = b by A6, A10, Def2;
(pr1 (X,X)) . ((f . n),(f . (n + 1))) = f . n by A12, A11, FUNCT_3:def_4;
then the Source of (PGraph X) . ((PairF f) . n) = a by A6, A10, Def2;
then (PairF f) . n joins a,b by A13, GRAPH_1:def_12;
hence ex x9, y9 being Element of (PGraph X) st
( x9 = f . n & y9 = f . (n + 1) & (PairF f) . n joins x9,y9 ) ; ::_thesis: verum
end;
for n being Element of NAT st 1 <= n & n <= len (PairF f) holds
(PairF f) . n in the carrier' of (PGraph X) by Th5;
hence PairF f is Chain of PGraph X by A2, A3, A4, GRAPH_1:def_14; ::_thesis: verum
end;
case len f < 1 ; ::_thesis: PairF f is oriented Chain of PGraph X
then (len f) + 1 <= 1 by NAT_1:13;
then ((len f) + 1) - 1 <= 1 - 1 by XREAL_1:9;
then A14: len f = 0 ;
0 - 1 < 0 ;
then (len f) -' 1 = 0 by A14, XREAL_0:def_2;
then len (PairF f) = 0 by Def2;
then PairF f = {} ;
hence PairF f is oriented Chain of PGraph X by GRAPH_1:14; ::_thesis: verum
end;
end;
end;
for n being Element of NAT st 1 <= n & n < len (PairF f) holds
the Source of (PGraph X) . ((PairF f) . (n + 1)) = the Target of (PGraph X) . ((PairF f) . n)
proof
A15: (len f) -' 1 < ((len f) -' 1) + 1 by NAT_1:13;
let n be Element of NAT ; ::_thesis: ( 1 <= n & n < len (PairF f) implies the Source of (PGraph X) . ((PairF f) . (n + 1)) = the Target of (PGraph X) . ((PairF f) . n) )
assume that
A16: 1 <= n and
A17: n < len (PairF f) ; ::_thesis: the Source of (PGraph X) . ((PairF f) . (n + 1)) = the Target of (PGraph X) . ((PairF f) . n)
A18: len (PairF f) = (len f) -' 1 by Def2;
then 1 <= (len f) -' 1 by A16, A17, XXREAL_0:2;
then A19: (len f) -' 1 = (len f) - 1 by NAT_D:39;
then A20: n < len f by A17, A18, A15, XXREAL_0:2;
then n in dom f by A16, FINSEQ_3:25;
then A21: f . n in rng f by FUNCT_1:def_3;
n + 1 <= len (PairF f) by A17, NAT_1:13;
then A22: n + 1 < len f by A18, A19, A15, XXREAL_0:2;
then A23: (n + 1) + 1 <= len f by NAT_1:13;
A24: 1 < n + 1 by A16, NAT_1:13;
then 1 < (n + 1) + 1 by NAT_1:13;
then (n + 1) + 1 in dom f by A23, FINSEQ_3:25;
then A25: f . ((n + 1) + 1) in rng f by FUNCT_1:def_3;
n + 1 <= len f by A20, NAT_1:13;
then n + 1 in dom f by A24, FINSEQ_3:25;
then A26: f . (n + 1) in rng f by FUNCT_1:def_3;
then reconsider b = f . (n + 1) as Element of (PGraph X) ;
(pr2 (X,X)) . ((f . n),(f . (n + 1))) = f . (n + 1) by A21, A26, FUNCT_3:def_5;
then A27: the Target of (PGraph X) . ((PairF f) . n) = b by A16, A20, Def2;
n + 1 in dom f by A22, A24, FINSEQ_3:25;
then f . (n + 1) in rng f by FUNCT_1:def_3;
then (pr1 (X,X)) . ((f . (n + 1)),(f . ((n + 1) + 1))) = f . (n + 1) by A25, FUNCT_3:def_4;
hence the Source of (PGraph X) . ((PairF f) . (n + 1)) = the Target of (PGraph X) . ((PairF f) . n) by A22, A24, A27, Def2; ::_thesis: verum
end;
hence PairF f is oriented Chain of PGraph X by A1, GRAPH_1:def_15; ::_thesis: verum
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;
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
proof
let X be non empty set ; ::_thesis: 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
let f be FinSequence of X; ::_thesis: 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
let f1 be FinSequence of the carrier of (PGraph X); ::_thesis: ( len f >= 1 & f = f1 implies f1 is_oriented_vertex_seq_of PairF f )
assume that
A1: len f >= 1 and
A2: f = f1 ; ::_thesis: f1 is_oriented_vertex_seq_of PairF f
A3: for n being Element of NAT st 1 <= n & n <= len (PairF f) holds
(PairF f) . n orientedly_joins f1 /. n,f1 /. (n + 1)
proof
A4: (len f) -' 1 < ((len f) -' 1) + 1 by NAT_1:13;
let n be Element of NAT ; ::_thesis: ( 1 <= n & n <= len (PairF f) implies (PairF f) . n orientedly_joins f1 /. n,f1 /. (n + 1) )
assume that
A5: 1 <= n and
A6: n <= len (PairF f) ; ::_thesis: (PairF f) . n orientedly_joins f1 /. n,f1 /. (n + 1)
A7: 1 < n + 1 by A5, NAT_1:13;
A8: len (PairF f) = (len f) -' 1 by Def2;
then 1 <= (len f) -' 1 by A5, A6, XXREAL_0:2;
then (len f) -' 1 = (len f) - 1 by NAT_D:39;
then A9: n < len f by A6, A8, A4, XXREAL_0:2;
then n + 1 <= len f by NAT_1:13;
then A10: n + 1 in dom f by A7, FINSEQ_3:25;
then A11: f . (n + 1) in rng f by FUNCT_1:def_3;
A12: n in dom f by A5, A9, FINSEQ_3:25;
then A13: f . n in rng f by FUNCT_1:def_3;
then A14: (pr1 (X,X)) . ((f . n),(f . (n + 1))) = f . n by A11, FUNCT_3:def_4;
A15: (pr2 (X,X)) . ((f . n),(f . (n + 1))) = f . (n + 1) by A13, A11, FUNCT_3:def_5;
f1 /. (n + 1) = f1 . (n + 1) by A2, A10, PARTFUN1:def_6;
then A16: the Target of (PGraph X) . ((PairF f) . n) = f1 /. (n + 1) by A2, A5, A9, A15, Def2;
f1 /. n = f1 . n by A2, A12, PARTFUN1:def_6;
then the Source of (PGraph X) . ((PairF f) . n) = f1 /. n by A2, A5, A9, A14, Def2;
hence (PairF f) . n orientedly_joins f1 /. n,f1 /. (n + 1) by A16, GRAPH_4:def_1; ::_thesis: verum
end;
(len f) -' 1 = (len f) - 1 by A1, XREAL_1:233;
then ((len f) - 1) + 1 = (len (PairF f)) + 1 by Def2;
hence f1 is_oriented_vertex_seq_of PairF f by A2, A3, GRAPH_4:def_5; ::_thesis: verum
end;
begin
definition
let X be non empty set ;
let f, g be FinSequence of X;
predg is_Shortcut_of f means :Def3: :: 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 ) );
end;
:: deftheorem Def3 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 ) ) );
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 )
proof
let X be non empty set ; ::_thesis: for f, g being FinSequence of X st g is_Shortcut_of f holds
( 1 <= len g & len g <= len f )
let f, g be FinSequence of X; ::_thesis: ( g is_Shortcut_of f implies ( 1 <= len g & len g <= len f ) )
reconsider df = dom f as finite set ;
A1: card df = card (Seg (len f)) by FINSEQ_1:def_3
.= len f by FINSEQ_1:57 ;
assume g is_Shortcut_of f ; ::_thesis: ( 1 <= len g & len g <= len f )
then consider fc being Subset of (PairF f), fvs being Subset of f, sc being oriented simple Chain of PGraph X, g1 being FinSequence of the carrier of (PGraph X) such that
Seq fc = sc and
A2: Seq fvs = g and
A3: g1 = g and
A4: g1 is_oriented_vertex_seq_of sc by Def3;
A5: len g1 = (len sc) + 1 by A4, GRAPH_4:def_5;
reconsider dfvs = dom fvs as finite set ;
A6: rng (Sgm (dom fvs)) c= dom fvs by FINSEQ_1:50;
A7: dom fvs c= dom f by RELAT_1:11;
then A8: dom fvs c= Seg (len f) by FINSEQ_1:def_3;
g = fvs * (Sgm (dom fvs)) by A2, FINSEQ_1:def_14;
then dom g = dom (Sgm (dom fvs)) by A6, RELAT_1:27;
then len g = len (Sgm (dom fvs)) by FINSEQ_3:29
.= card dfvs by A8, FINSEQ_3:39 ;
hence ( 1 <= len g & len g <= len f ) by A3, A5, A7, A1, NAT_1:12, NAT_1:43; ::_thesis: verum
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
proof
let X be non empty set ; ::_thesis: for f being FinSequence of X st len f >= 1 holds
ex g being FinSequence of X st g is_Shortcut_of f
let f be FinSequence of X; ::_thesis: ( len f >= 1 implies ex g being FinSequence of X st g is_Shortcut_of f )
reconsider f1 = f as FinSequence of the carrier of (PGraph X) ;
assume len f >= 1 ; ::_thesis: ex g being FinSequence of X st g is_Shortcut_of f
then consider fc being Subset of (PairF f), fvs being Subset of f1, sc being oriented simple Chain of PGraph X, vs1 being FinSequence of the carrier of (PGraph X) such that
A1: ( Seq fc = sc & Seq fvs = vs1 & vs1 is_oriented_vertex_seq_of sc & f1 . 1 = vs1 . 1 & f1 . (len f1) = vs1 . (len vs1) ) by Th7, GRAPH_4:21;
reconsider g1 = vs1 as FinSequence of X ;
g1 is_Shortcut_of f by A1, Def3;
hence ex g being FinSequence of X st g is_Shortcut_of f ; ::_thesis: verum
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)
proof
let X be non empty set ; ::_thesis: for f, g being FinSequence of X st g is_Shortcut_of f holds
rng (PairF g) c= rng (PairF f)
let f, g be FinSequence of X; ::_thesis: ( g is_Shortcut_of f implies rng (PairF g) c= rng (PairF f) )
A1: len (PairF g) = (len g) -' 1 by Def2;
len g < (len g) + 1 by NAT_1:13;
then A2: (len g) - 1 < len g by XREAL_1:19;
assume g is_Shortcut_of f ; ::_thesis: rng (PairF g) c= rng (PairF f)
then consider fc being Subset of (PairF f), fvs being Subset of f, sc being oriented simple Chain of PGraph X, g1 being FinSequence of the carrier of (PGraph X) such that
A3: Seq fc = sc and
Seq fvs = g and
A4: g1 = g and
A5: g1 is_oriented_vertex_seq_of sc by Def3;
A6: rng (Sgm (dom fc)) = dom fc by FINSEQ_1:50;
fc * (Sgm (dom fc)) = sc by A3, FINSEQ_1:def_14;
then rng fc = rng sc by A6, RELAT_1:28;
then A7: rng sc c= rng (PairF f) by RELAT_1:11;
len f < (len f) + 1 by NAT_1:13;
then A8: (len f) - 1 < len f by XREAL_1:19;
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (PairF g) or y in rng (PairF f) )
assume y in rng (PairF g) ; ::_thesis: y in rng (PairF f)
then consider x being set such that
A9: x in dom (PairF g) and
A10: y = (PairF g) . x by FUNCT_1:def_3;
reconsider n = x as Element of NAT by A9;
A11: x in Seg (len (PairF g)) by A9, FINSEQ_1:def_3;
then A12: 1 <= n by FINSEQ_1:1;
then A13: 1 <= n + 1 by NAT_1:13;
A14: n <= len (PairF g) by A11, FINSEQ_1:1;
then 1 <= len (PairF g) by A12, XXREAL_0:2;
then (len g) -' 1 = (len g) - 1 by A1, NAT_D:39;
then A15: n < len g by A14, A1, A2, XXREAL_0:2;
then n + 1 <= len g by NAT_1:13;
then n + 1 <= (len sc) + 1 by A4, A5, GRAPH_4:def_5;
then A16: n <= len sc by XREAL_1:6;
then A17: sc . n orientedly_joins g1 /. n,g1 /. (n + 1) by A5, A12, GRAPH_4:def_5;
n in dom sc by A12, A16, FINSEQ_3:25;
then sc . n in rng sc by FUNCT_1:def_3;
then consider z being set such that
A18: z in dom (PairF f) and
A19: (PairF f) . z = sc . n by A7, FUNCT_1:def_3;
reconsider m = z as Element of NAT by A18;
A20: z in Seg (len (PairF f)) by A18, FINSEQ_1:def_3;
then A21: 1 <= m by FINSEQ_1:1;
m <= len (PairF f) by A20, FINSEQ_1:1;
then A22: m <= (len f) -' 1 by Def2;
then 1 <= (len f) -' 1 by A21, XXREAL_0:2;
then (len f) -' 1 = (len f) - 1 by NAT_D:39;
then A23: m < len f by A22, A8, XXREAL_0:2;
then A24: m + 1 <= len f by NAT_1:13;
A25: 1 <= m by A20, FINSEQ_1:1;
then A26: [(f . m),(f . (m + 1))] = sc . n by A19, A23, Def2;
1 < m + 1 by A21, NAT_1:13;
then m + 1 in dom f by A24, FINSEQ_3:25;
then A27: f . (m + 1) in rng f by FUNCT_1:def_3;
m in dom f by A25, A23, FINSEQ_3:25;
then A28: f . m in rng f by FUNCT_1:def_3;
then the Source of (PGraph X) . ((f . m),(f . (m + 1))) = f . m by A27, FUNCT_3:def_4;
then g1 /. n = f . m by A17, A26, GRAPH_4:def_1;
then A29: g . n = f . m by A4, A12, A15, FINSEQ_4:15;
A30: (PairF g) . x = [(g . n),(g . (n + 1))] by A12, A15, Def2;
the Target of (PGraph X) . ((f . m),(f . (m + 1))) = f . (m + 1) by A28, A27, FUNCT_3:def_5;
then A31: g1 /. (n + 1) = f . (m + 1) by A17, A26, GRAPH_4:def_1;
n + 1 <= len g1 by A4, A15, NAT_1:13;
then g . (n + 1) = f . (m + 1) by A4, A31, A13, FINSEQ_4:15;
hence y in rng (PairF f) by A10, A30, A18, A19, A26, A29, FUNCT_1:def_3; ::_thesis: verum
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
proof
let X be non empty set ; ::_thesis: for f, g being FinSequence of X st f . 1 <> f . (len f) & g is_Shortcut_of f holds
g is one-to-one
let f, g be FinSequence of X; ::_thesis: ( f . 1 <> f . (len f) & g is_Shortcut_of f implies g is one-to-one )
assume that
A1: f . 1 <> f . (len f) and
A2: g is_Shortcut_of f ; ::_thesis: g is one-to-one
A3: ( f . 1 = g . 1 & f . (len f) = g . (len g) ) by A2, Def3;
consider fc being Subset of (PairF f), fvs being Subset of f, sc being oriented simple Chain of PGraph X, g1 being FinSequence of the carrier of (PGraph X) such that
Seq fc = sc and
Seq fvs = g and
A4: g1 = g and
A5: g1 is_oriented_vertex_seq_of sc by A2, Def3;
sc is Simple by GRAPH_4:18;
then consider vs being FinSequence of the carrier of (PGraph X) such that
A6: vs is_oriented_vertex_seq_of sc and
A7: for n, m being Element of NAT st 1 <= n & n < m & m <= len vs & vs . n = vs . m holds
( n = 1 & m = len vs ) by GRAPH_4:def_7;
A8: len g1 = (len sc) + 1 by A5, GRAPH_4:def_5;
then 1 <= len g1 by NAT_1:11;
then 1 < len g1 by A1, A3, A4, XXREAL_0:1;
then sc <> {} by A8, NAT_1:13;
then A9: g1 = vs by A5, A6, GRAPH_4:10;
let x, y be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not x in K48(g) or not y in K48(g) or not g . x = g . y or x = y )
assume that
A10: x in dom g and
A11: y in dom g and
A12: g . x = g . y ; ::_thesis: x = y
assume A13: x <> y ; ::_thesis: contradiction
reconsider i1 = x as Element of NAT by A10;
A14: x in Seg (len g) by A10, FINSEQ_1:def_3;
then A15: 1 <= i1 by FINSEQ_1:1;
A16: i1 <= len g by A14, FINSEQ_1:1;
reconsider i2 = y as Element of NAT by A11;
A17: y in Seg (len g) by A11, FINSEQ_1:def_3;
then A18: 1 <= i2 by FINSEQ_1:1;
A19: i2 <= len g by A17, FINSEQ_1:1;
percases ( i1 <= i2 or i1 > i2 ) ;
suppose i1 <= i2 ; ::_thesis: contradiction
then A20: i1 < i2 by A13, XXREAL_0:1;
then i1 = 1 by A4, A7, A9, A12, A15, A19;
hence contradiction by A1, A3, A4, A7, A9, A12, A19, A20; ::_thesis: verum
end;
supposeA21: i1 > i2 ; ::_thesis: contradiction
then i2 = 1 by A4, A7, A9, A12, A16, A18;
hence contradiction by A1, A3, A4, A7, A9, A12, A16, A21; ::_thesis: verum
end;
end;
end;
definition
let n be Element of NAT ;
let IT be FinSequence of (TOP-REAL n);
attrIT is nodic means :Def4: :: JGRAPH_1:def 4
for i, j being Element of 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) );
end;
:: deftheorem Def4 defines nodic JGRAPH_1:def_4_:_
for n being Element of NAT
for IT being FinSequence of (TOP-REAL n) holds
( IT is nodic iff for i, j being Element of 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 :: JGRAPH_1:12
for f being FinSequence of (TOP-REAL 2) st f is s.n.c. holds
f is s.c.c.
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: ( f is s.n.c. implies f is s.c.c. )
assume f is s.n.c. ; ::_thesis: f is s.c.c.
then for i, j being Element of NAT st i + 1 < j & ( ( i > 1 & j < len f ) or j + 1 < len f ) holds
LSeg (f,i) misses LSeg (f,j) by TOPREAL1:def_7;
hence f is s.c.c. by GOBOARD5:def_4; ::_thesis: verum
end;
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.
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: ( f is s.c.c. & LSeg (f,1) misses LSeg (f,((len f) -' 1)) implies f is s.n.c. )
assume that
A1: f is s.c.c. and
A2: LSeg (f,1) misses LSeg (f,((len f) -' 1)) ; ::_thesis: f is s.n.c.
for i, j being Nat st i + 1 < j holds
LSeg (f,i) misses LSeg (f,j)
proof
let i, j be Nat; ::_thesis: ( i + 1 < j implies LSeg (f,i) misses LSeg (f,j) )
assume A3: i + 1 < j ; ::_thesis: LSeg (f,i) misses LSeg (f,j)
A4: ( i in NAT & j in NAT ) by ORDINAL1:def_12;
percases ( len f <> 0 or len f = 0 ) ;
suppose len f <> 0 ; ::_thesis: LSeg (f,i) misses LSeg (f,j)
then A5: len f >= 0 + 1 by NAT_1:13;
now__::_thesis:_(_(_1_<=_i_&_j_+_1_<=_len_f_&_(LSeg_(f,i))_/\_(LSeg_(f,j))_=_{}_)_or_(_(_not_1_<=_i_or_not_j_+_1_<=_len_f_)_&_(LSeg_(f,i))_/\_(LSeg_(f,j))_=_{}_)_)
percases ( ( 1 <= i & j + 1 <= len f ) or not 1 <= i or not j + 1 <= len f ) ;
caseA6: ( 1 <= i & j + 1 <= len f ) ; ::_thesis: (LSeg (f,i)) /\ (LSeg (f,j)) = {}
then A7: j < len f by NAT_1:13;
now__::_thesis:_(_(_i_=_1_&_j_+_1_=_len_f_&_(LSeg_(f,i))_/\_(LSeg_(f,j))_=_{}_)_or_(_(_not_i_=_1_or_not_j_+_1_=_len_f_)_&_(LSeg_(f,i))_/\_(LSeg_(f,j))_=_{}_)_)
percases ( ( i = 1 & j + 1 = len f ) or not i = 1 or not j + 1 = len f ) ;
caseA8: ( i = 1 & j + 1 = len f ) ; ::_thesis: (LSeg (f,i)) /\ (LSeg (f,j)) = {}
then j = (len f) - 1 ;
then LSeg (f,i) misses LSeg (f,j) by A2, A5, A8, XREAL_1:233;
hence (LSeg (f,i)) /\ (LSeg (f,j)) = {} by XBOOLE_0:def_7; ::_thesis: verum
end;
case ( not i = 1 or not j + 1 = len f ) ; ::_thesis: (LSeg (f,i)) /\ (LSeg (f,j)) = {}
then ( i > 1 or j + 1 < len f ) by A6, XXREAL_0:1;
then LSeg (f,i) misses LSeg (f,j) by A1, A3, A4, A7, GOBOARD5:def_4;
hence (LSeg (f,i)) /\ (LSeg (f,j)) = {} by XBOOLE_0:def_7; ::_thesis: verum
end;
end;
end;
hence (LSeg (f,i)) /\ (LSeg (f,j)) = {} ; ::_thesis: verum
end;
caseA9: ( not 1 <= i or not j + 1 <= len f ) ; ::_thesis: (LSeg (f,i)) /\ (LSeg (f,j)) = {}
now__::_thesis:_(_(_1_>_i_&_(LSeg_(f,i))_/\_(LSeg_(f,j))_=_{}_)_or_(_j_+_1_>_len_f_&_(LSeg_(f,i))_/\_(LSeg_(f,j))_=_{}_)_)
percases ( 1 > i or j + 1 > len f ) by A9;
case 1 > i ; ::_thesis: (LSeg (f,i)) /\ (LSeg (f,j)) = {}
then LSeg (f,i) = {} by TOPREAL1:def_3;
hence (LSeg (f,i)) /\ (LSeg (f,j)) = {} ; ::_thesis: verum
end;
case j + 1 > len f ; ::_thesis: (LSeg (f,i)) /\ (LSeg (f,j)) = {}
then LSeg (f,j) = {} by TOPREAL1:def_3;
hence (LSeg (f,i)) /\ (LSeg (f,j)) = {} ; ::_thesis: verum
end;
end;
end;
hence (LSeg (f,i)) /\ (LSeg (f,j)) = {} ; ::_thesis: verum
end;
end;
end;
hence LSeg (f,i) misses LSeg (f,j) by XBOOLE_0:def_7; ::_thesis: verum
end;
supposeA10: len f = 0 ; ::_thesis: LSeg (f,i) misses LSeg (f,j)
now__::_thesis:_(_(_i_>=_1_&_(LSeg_(f,i))_/\_(LSeg_(f,j))_=_{}_)_or_(_i_<_1_&_(LSeg_(f,i))_/\_(LSeg_(f,j))_=_{}_)_)
percases ( i >= 1 or i < 1 ) ;
case i >= 1 ; ::_thesis: (LSeg (f,i)) /\ (LSeg (f,j)) = {}
i + 1 > len f by A10;
then LSeg (f,i) = {} by TOPREAL1:def_3;
hence (LSeg (f,i)) /\ (LSeg (f,j)) = {} ; ::_thesis: verum
end;
case i < 1 ; ::_thesis: (LSeg (f,i)) /\ (LSeg (f,j)) = {}
then LSeg (f,i) = {} by TOPREAL1:def_3;
hence (LSeg (f,i)) /\ (LSeg (f,j)) = {} ; ::_thesis: verum
end;
end;
end;
hence LSeg (f,i) misses LSeg (f,j) by XBOOLE_0:def_7; ::_thesis: verum
end;
end;
end;
hence f is s.n.c. by TOPREAL1:def_7; ::_thesis: verum
end;
theorem Th14: :: JGRAPH_1:14
for f being FinSequence of (TOP-REAL 2) st f is nodic & PairF f is Simple holds
f is s.c.c.
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: ( f is nodic & PairF f is Simple implies f is s.c.c. )
assume that
A1: f is nodic and
A2: PairF f is Simple ; ::_thesis: f is s.c.c.
reconsider f1 = f as FinSequence of the carrier of (PGraph the carrier of (TOP-REAL 2)) ;
percases ( len f >= 1 or len f < 1 ) ;
suppose len f >= 1 ; ::_thesis: f is s.c.c.
then A3: f1 is_oriented_vertex_seq_of PairF f by Th7;
for i, j being Element of NAT st i + 1 < j & ( ( i > 1 & j < len f ) or j + 1 < len f ) holds
LSeg (f,i) misses LSeg (f,j)
proof
let i, j be Element of NAT ; ::_thesis: ( i + 1 < j & ( ( i > 1 & j < len f ) or j + 1 < len f ) implies LSeg (f,i) misses LSeg (f,j) )
assume that
A4: i + 1 < j and
A5: ( ( i > 1 & j < len f ) or j + 1 < len f ) ; ::_thesis: LSeg (f,i) misses LSeg (f,j)
percases ( i >= 1 or i < 1 ) ;
supposeA6: i >= 1 ; ::_thesis: LSeg (f,i) misses LSeg (f,j)
A7: i < j by A4, NAT_1:13;
then A8: 1 <= j by A6, XXREAL_0:2;
then A9: 1 < j + 1 by NAT_1:13;
A10: i + 1 < j + 1 by A4, NAT_1:13;
A11: 1 < i + 1 by A6, NAT_1:13;
A12: j < len f by A5, NAT_1:13;
then A13: i + 1 < len f by A4, XXREAL_0:2;
A14: j + 1 <= len f by A5, NAT_1:13;
A15: i < j + 1 by A7, NAT_1:13;
then A16: i < len f by A14, XXREAL_0:2;
now__::_thesis:_not_LSeg_(f,i)_meets_LSeg_(f,j)
assume A17: LSeg (f,i) meets LSeg (f,j) ; ::_thesis: contradiction
now__::_thesis:_(_(_(LSeg_(f,i))_/\_(LSeg_(f,j))_=_{(f_._i)}_&_(_f_._i_=_f_._j_or_f_._i_=_f_._(j_+_1)_)_&_LSeg_(f,i)_<>_LSeg_(f,j)_&_contradiction_)_or_(_(LSeg_(f,i))_/\_(LSeg_(f,j))_=_{(f_._(i_+_1))}_&_(_f_._(i_+_1)_=_f_._j_or_f_._(i_+_1)_=_f_._(j_+_1)_)_&_LSeg_(f,i)_<>_LSeg_(f,j)_&_contradiction_)_or_(_LSeg_(f,i)_=_LSeg_(f,j)_&_contradiction_)_)
percases ( ( (LSeg (f,i)) /\ (LSeg (f,j)) = {(f . i)} & ( f . i = f . j or f . i = f . (j + 1) ) & LSeg (f,i) <> LSeg (f,j) ) or ( (LSeg (f,i)) /\ (LSeg (f,j)) = {(f . (i + 1))} & ( f . (i + 1) = f . j or f . (i + 1) = f . (j + 1) ) & LSeg (f,i) <> LSeg (f,j) ) or LSeg (f,i) = LSeg (f,j) ) by A1, A17, Def4;
caseA18: ( (LSeg (f,i)) /\ (LSeg (f,j)) = {(f . i)} & ( f . i = f . j or f . i = f . (j + 1) ) & LSeg (f,i) <> LSeg (f,j) ) ; ::_thesis: contradiction
now__::_thesis:_(_(_f_._i_=_f_._j_&_contradiction_)_or_(_f_._i_=_f_._(j_+_1)_&_contradiction_)_)
percases ( f . i = f . j or f . i = f . (j + 1) ) by A18;
case f . i = f . j ; ::_thesis: contradiction
hence contradiction by A2, A3, A6, A7, A12, Th1; ::_thesis: verum
end;
case f . i = f . (j + 1) ; ::_thesis: contradiction
hence contradiction by A2, A3, A5, A6, A15, A14, Th1; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
caseA19: ( (LSeg (f,i)) /\ (LSeg (f,j)) = {(f . (i + 1))} & ( f . (i + 1) = f . j or f . (i + 1) = f . (j + 1) ) & LSeg (f,i) <> LSeg (f,j) ) ; ::_thesis: contradiction
now__::_thesis:_(_(_f_._(i_+_1)_=_f_._j_&_contradiction_)_or_(_f_._(i_+_1)_=_f_._(j_+_1)_&_contradiction_)_)
percases ( f . (i + 1) = f . j or f . (i + 1) = f . (j + 1) ) by A19;
case f . (i + 1) = f . j ; ::_thesis: contradiction
hence contradiction by A2, A3, A4, A12, A11, Th1; ::_thesis: verum
end;
case f . (i + 1) = f . (j + 1) ; ::_thesis: contradiction
hence contradiction by A2, A3, A10, A14, A11, Th1; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
case LSeg (f,i) = LSeg (f,j) ; ::_thesis: contradiction
then LSeg ((f /. i),(f /. (i + 1))) = LSeg (f,j) by A6, A13, TOPREAL1:def_3;
then A20: LSeg ((f /. i),(f /. (i + 1))) = LSeg ((f /. j),(f /. (j + 1))) by A8, A14, TOPREAL1:def_3;
A21: ( f /. j = f . j & f /. (j + 1) = f . (j + 1) ) by A8, A12, A14, A9, FINSEQ_4:15;
A22: ( f /. i = f . i & f /. (i + 1) = f . (i + 1) ) by A6, A13, A16, A11, FINSEQ_4:15;
now__::_thesis:_(_(_f_._i_=_f_._j_&_f_._(i_+_1)_=_f_._(j_+_1)_&_contradiction_)_or_(_f_._i_=_f_._(j_+_1)_&_f_._(i_+_1)_=_f_._j_&_contradiction_)_)
percases ( ( f . i = f . j & f . (i + 1) = f . (j + 1) ) or ( f . i = f . (j + 1) & f . (i + 1) = f . j ) ) by A20, A22, A21, SPPOL_1:8;
case ( f . i = f . j & f . (i + 1) = f . (j + 1) ) ; ::_thesis: contradiction
hence contradiction by A2, A3, A10, A14, A11, Th1; ::_thesis: verum
end;
case ( f . i = f . (j + 1) & f . (i + 1) = f . j ) ; ::_thesis: contradiction
hence contradiction by A2, A3, A4, A12, A11, Th1; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
hence LSeg (f,i) misses LSeg (f,j) ; ::_thesis: verum
end;
suppose i < 1 ; ::_thesis: LSeg (f,i) misses LSeg (f,j)
then LSeg (f,i) = {} by TOPREAL1:def_3;
then (LSeg (f,i)) /\ (LSeg (f,j)) = {} ;
hence LSeg (f,i) misses LSeg (f,j) by XBOOLE_0:def_7; ::_thesis: verum
end;
end;
end;
hence f is s.c.c. by GOBOARD5:def_4; ::_thesis: verum
end;
supposeA23: len f < 1 ; ::_thesis: f is s.c.c.
for i, j being Element of NAT st i + 1 < j & ( ( i > 1 & j < len f ) or j + 1 < len f ) holds
LSeg (f,i) misses LSeg (f,j)
proof
let i, j be Element of NAT ; ::_thesis: ( i + 1 < j & ( ( i > 1 & j < len f ) or j + 1 < len f ) implies LSeg (f,i) misses LSeg (f,j) )
assume that
i + 1 < j and
( ( i > 1 & j < len f ) or j + 1 < len f ) ; ::_thesis: LSeg (f,i) misses LSeg (f,j)
percases ( i >= 1 or i < 1 ) ;
suppose i >= 1 ; ::_thesis: LSeg (f,i) misses LSeg (f,j)
then i > len f by A23, XXREAL_0:2;
then i + 1 > len f by NAT_1:13;
then LSeg (f,i) = {} by TOPREAL1:def_3;
then (LSeg (f,i)) /\ (LSeg (f,j)) = {} ;
hence LSeg (f,i) misses LSeg (f,j) by XBOOLE_0:def_7; ::_thesis: verum
end;
suppose i < 1 ; ::_thesis: LSeg (f,i) misses LSeg (f,j)
then LSeg (f,i) = {} by TOPREAL1:def_3;
then (LSeg (f,i)) /\ (LSeg (f,j)) = {} ;
hence LSeg (f,i) misses LSeg (f,j) by XBOOLE_0:def_7; ::_thesis: verum
end;
end;
end;
hence f is s.c.c. by GOBOARD5:def_4; ::_thesis: verum
end;
end;
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.
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: ( f is nodic & PairF f is Simple & f . 1 <> f . (len f) implies f is s.n.c. )
assume that
A1: f is nodic and
A2: PairF f is Simple and
A3: f . 1 <> f . (len f) ; ::_thesis: f is s.n.c.
reconsider f1 = f as FinSequence of the carrier of (PGraph the carrier of (TOP-REAL 2)) ;
A4: (len f) -' 1 <= len f by NAT_D:44;
percases ( (len f) -' 1 > 2 or (len f) -' 1 <= 2 ) ;
supposeA5: (len f) -' 1 > 2 ; ::_thesis: f is s.n.c.
then A6: (len f) -' 1 > 1 by XXREAL_0:2;
then A7: 1 < len f by NAT_D:44;
len f >= 1 by A6, NAT_D:44;
then A8: f1 is_oriented_vertex_seq_of PairF f by Th7;
A9: 1 + 1 < len f by A5, NAT_D:44;
A10: (len f) -' 1 = (len f) - 1 by A6, NAT_D:39;
then A11: ((len f) -' 1) + 1 = len f ;
now__::_thesis:_not_LSeg_(f,1)_meets_LSeg_(f,((len_f)_-'_1))
assume A12: LSeg (f,1) meets LSeg (f,((len f) -' 1)) ; ::_thesis: contradiction
now__::_thesis:_(_(_(LSeg_(f,1))_/\_(LSeg_(f,((len_f)_-'_1)))_=_{(f_._1)}_&_(_f_._1_=_f_._((len_f)_-'_1)_or_f_._1_=_f_._(((len_f)_-'_1)_+_1)_)_&_contradiction_)_or_(_(LSeg_(f,1))_/\_(LSeg_(f,((len_f)_-'_1)))_=_{(f_._(1_+_1))}_&_(_f_._(1_+_1)_=_f_._((len_f)_-'_1)_or_f_._(1_+_1)_=_f_._(((len_f)_-'_1)_+_1)_)_&_contradiction_)_or_(_LSeg_(f,1)_=_LSeg_(f,((len_f)_-'_1))_&_contradiction_)_)
percases ( ( (LSeg (f,1)) /\ (LSeg (f,((len f) -' 1))) = {(f . 1)} & ( f . 1 = f . ((len f) -' 1) or f . 1 = f . (((len f) -' 1) + 1) ) ) or ( (LSeg (f,1)) /\ (LSeg (f,((len f) -' 1))) = {(f . (1 + 1))} & ( f . (1 + 1) = f . ((len f) -' 1) or f . (1 + 1) = f . (((len f) -' 1) + 1) ) ) or LSeg (f,1) = LSeg (f,((len f) -' 1)) ) by A1, A12, Def4;
case ( (LSeg (f,1)) /\ (LSeg (f,((len f) -' 1))) = {(f . 1)} & ( f . 1 = f . ((len f) -' 1) or f . 1 = f . (((len f) -' 1) + 1) ) ) ; ::_thesis: contradiction
hence contradiction by A2, A3, A4, A6, A8, A10, Th1; ::_thesis: verum
end;
caseA13: ( (LSeg (f,1)) /\ (LSeg (f,((len f) -' 1))) = {(f . (1 + 1))} & ( f . (1 + 1) = f . ((len f) -' 1) or f . (1 + 1) = f . (((len f) -' 1) + 1) ) ) ; ::_thesis: contradiction
now__::_thesis:_(_(_f_._(1_+_1)_=_f_._((len_f)_-'_1)_&_contradiction_)_or_(_f_._(1_+_1)_=_f_._(((len_f)_-'_1)_+_1)_&_contradiction_)_)
percases ( f . (1 + 1) = f . ((len f) -' 1) or f . (1 + 1) = f . (((len f) -' 1) + 1) ) by A13;
case f . (1 + 1) = f . ((len f) -' 1) ; ::_thesis: contradiction
hence contradiction by A2, A4, A5, A8, Th1; ::_thesis: verum
end;
case f . (1 + 1) = f . (((len f) -' 1) + 1) ; ::_thesis: contradiction
hence contradiction by A2, A8, A9, A10, Th1; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
case LSeg (f,1) = LSeg (f,((len f) -' 1)) ; ::_thesis: contradiction
then LSeg ((f /. 1),(f /. (1 + 1))) = LSeg (f,((len f) -' 1)) by A9, TOPREAL1:def_3;
then A14: LSeg ((f /. 1),(f /. (1 + 1))) = LSeg ((f /. ((len f) -' 1)),(f /. (((len f) -' 1) + 1))) by A6, A10, TOPREAL1:def_3;
A15: 1 + 1 < ((len f) -' 1) + 1 by A6, XREAL_1:6;
(len f) -' 1 < len f by A11, NAT_1:13;
then A16: f /. ((len f) -' 1) = f . ((len f) -' 1) by A6, FINSEQ_4:15;
1 < len f by A6, NAT_D:44;
then A17: f /. 1 = f . 1 by FINSEQ_4:15;
A18: ( f /. (1 + 1) = f . (1 + 1) & f /. (((len f) -' 1) + 1) = f . (((len f) -' 1) + 1) ) by A7, A9, A10, FINSEQ_4:15;
now__::_thesis:_(_(_f_._1_=_f_._((len_f)_-'_1)_&_f_._(1_+_1)_=_f_._(((len_f)_-'_1)_+_1)_&_contradiction_)_or_(_f_._1_=_f_._(((len_f)_-'_1)_+_1)_&_f_._(1_+_1)_=_f_._((len_f)_-'_1)_&_contradiction_)_)
percases ( ( f . 1 = f . ((len f) -' 1) & f . (1 + 1) = f . (((len f) -' 1) + 1) ) or ( f . 1 = f . (((len f) -' 1) + 1) & f . (1 + 1) = f . ((len f) -' 1) ) ) by A14, A17, A16, A18, SPPOL_1:8;
case ( f . 1 = f . ((len f) -' 1) & f . (1 + 1) = f . (((len f) -' 1) + 1) ) ; ::_thesis: contradiction
hence contradiction by A2, A8, A10, A15, Th1; ::_thesis: verum
end;
case ( f . 1 = f . (((len f) -' 1) + 1) & f . (1 + 1) = f . ((len f) -' 1) ) ; ::_thesis: contradiction
hence contradiction by A3, A10; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
hence f is s.n.c. by A1, A2, Th13, Th14; ::_thesis: verum
end;
supposeA19: (len f) -' 1 <= 2 ; ::_thesis: f is s.n.c.
for i, j being Nat st i + 1 < j holds
LSeg (f,i) misses LSeg (f,j)
proof
let i, j be Nat; ::_thesis: ( i + 1 < j implies LSeg (f,i) misses LSeg (f,j) )
assume A20: i + 1 < j ; ::_thesis: LSeg (f,i) misses LSeg (f,j)
percases ( ( 1 <= i & j + 1 <= len f ) or 1 > i or j + 1 > len f ) ;
supposeA21: ( 1 <= i & j + 1 <= len f ) ; ::_thesis: LSeg (f,i) misses LSeg (f,j)
then 1 < i + 1 by NAT_1:13;
then 1 + 1 < (i + 1) + 1 by XREAL_1:8;
then A22: (1 + 1) + 1 <= (i + 1) + 1 by NAT_1:13;
(i + 1) + 1 < j + 1 by A20, XREAL_1:6;
then (i + 1) + 1 < len f by A21, XXREAL_0:2;
then (1 + 1) + 1 < len f by A22, XXREAL_0:2;
then A23: ((1 + 1) + 1) - 1 < (len f) - 1 by XREAL_1:9;
then 1 < (len f) - 1 by XXREAL_0:2;
hence LSeg (f,i) misses LSeg (f,j) by A19, A23, NAT_D:39; ::_thesis: verum
end;
supposeA24: ( 1 > i or j + 1 > len f ) ; ::_thesis: LSeg (f,i) misses LSeg (f,j)
now__::_thesis:_(_(_1_>_i_&_(LSeg_(f,i))_/\_(LSeg_(f,j))_=_{}_)_or_(_j_+_1_>_len_f_&_(LSeg_(f,i))_/\_(LSeg_(f,j))_=_{}_)_)
percases ( 1 > i or j + 1 > len f ) by A24;
case 1 > i ; ::_thesis: (LSeg (f,i)) /\ (LSeg (f,j)) = {}
then LSeg (f,i) = {} by TOPREAL1:def_3;
hence (LSeg (f,i)) /\ (LSeg (f,j)) = {} ; ::_thesis: verum
end;
case j + 1 > len f ; ::_thesis: (LSeg (f,i)) /\ (LSeg (f,j)) = {}
then LSeg (f,j) = {} by TOPREAL1:def_3;
hence (LSeg (f,i)) /\ (LSeg (f,j)) = {} ; ::_thesis: verum
end;
end;
end;
hence LSeg (f,i) misses LSeg (f,j) by XBOOLE_0:def_7; ::_thesis: verum
end;
end;
end;
hence f is s.n.c. by TOPREAL1:def_7; ::_thesis: verum
end;
end;
end;
theorem Th16: :: JGRAPH_1:16
for n being Element of 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) )
proof
let n be Element of NAT ; ::_thesis: 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) )
let p1, p2, p3 be Point of (TOP-REAL n); ::_thesis: ( 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) )
given x being set such that A1: x <> p2 and
A2: x in (LSeg (p1,p2)) /\ (LSeg (p2,p3)) ; ::_thesis: ( p1 in LSeg (p2,p3) or p3 in LSeg (p1,p2) )
reconsider p = x as Point of (TOP-REAL n) by A2;
A3: p in { (((1 - r1) * p1) + (r1 * p2)) where r1 is Real : ( 0 <= r1 & r1 <= 1 ) } by A2, XBOOLE_0:def_4;
A4: p in { (((1 - r2) * p2) + (r2 * p3)) where r2 is Real : ( 0 <= r2 & r2 <= 1 ) } by A2, XBOOLE_0:def_4;
consider r1 being Real such that
A5: p = ((1 - r1) * p1) + (r1 * p2) and
0 <= r1 and
A6: r1 <= 1 by A3;
consider r2 being Real such that
A7: p = ((1 - r2) * p2) + (r2 * p3) and
A8: 0 <= r2 and
r2 <= 1 by A4;
percases ( r1 >= 1 - r2 or r1 < 1 - r2 ) ;
supposeA9: r1 >= 1 - r2 ; ::_thesis: ( p1 in LSeg (p2,p3) or p3 in LSeg (p1,p2) )
now__::_thesis:_(_(_r2_<>_0_&_(_p1_in_LSeg_(p2,p3)_or_p3_in_LSeg_(p1,p2)_)_)_or_(_r2_=_0_&_(_p1_in_LSeg_(p2,p3)_or_p3_in_LSeg_(p1,p2)_)_)_)
percases ( r2 <> 0 or r2 = 0 ) ;
caseA10: r2 <> 0 ; ::_thesis: ( p1 in LSeg (p2,p3) or p3 in LSeg (p1,p2) )
r2 * p3 = (((1 - r1) * p1) + (r1 * p2)) - ((1 - r2) * p2) by A5, A7, EUCLID:48;
then r2 * p3 = ((1 - r1) * p1) + ((r1 * p2) - ((1 - r2) * p2)) by EUCLID:45;
then (r2 ") * (r2 * p3) = (r2 ") * (((1 - r1) * p1) + ((r1 - (1 - r2)) * p2)) by EUCLID:50;
then ((r2 ") * r2) * p3 = (r2 ") * (((1 - r1) * p1) + ((r1 - (1 - r2)) * p2)) by EUCLID:30;
then 1 * p3 = (r2 ") * (((1 - r1) * p1) + ((r1 - (1 - r2)) * p2)) by A10, XCMPLX_0:def_7;
then A11: p3 = (r2 ") * (((1 - r1) * p1) + ((r1 - (1 - r2)) * p2)) by EUCLID:29
.= ((r2 ") * ((1 - r1) * p1)) + ((r2 ") * ((r1 - (1 - r2)) * p2)) by EUCLID:32
.= (((r2 ") * (1 - r1)) * p1) + ((r2 ") * ((r1 - (1 - r2)) * p2)) by EUCLID:30
.= (((r2 ") * (1 - r1)) * p1) + (((r2 ") * (r1 - (1 - r2))) * p2) by EUCLID:30 ;
r1 <= 1 + 0 by A6;
then r1 - 1 <= 0 by XREAL_1:20;
then (r1 - 1) + r2 <= 0 + r2 by XREAL_1:6;
then A12: (r2 ") * (r1 - (1 - r2)) <= (r2 ") * r2 by A8, XREAL_1:64;
((r2 ") * (1 - r1)) + ((r2 ") * (r1 - (1 - r2))) = (r2 ") * ((1 - 1) + r2)
.= 1 by A10, XCMPLX_0:def_7 ;
then A13: (r2 ") * (1 - r1) = 1 - ((r2 ") * (r1 - (1 - r2))) ;
r1 - (1 - r2) >= 0 by A9, XREAL_1:48;
hence ( p1 in LSeg (p2,p3) or p3 in LSeg (p1,p2) ) by A8, A11, A13, A12; ::_thesis: verum
end;
case r2 = 0 ; ::_thesis: ( p1 in LSeg (p2,p3) or p3 in LSeg (p1,p2) )
then p = (1 * p2) + (0. (TOP-REAL n)) by A7, EUCLID:29;
then p = p2 + (0. (TOP-REAL n)) by EUCLID:29;
hence ( p1 in LSeg (p2,p3) or p3 in LSeg (p1,p2) ) by A1, EUCLID:27; ::_thesis: verum
end;
end;
end;
hence ( p1 in LSeg (p2,p3) or p3 in LSeg (p1,p2) ) ; ::_thesis: verum
end;
supposeA14: r1 < 1 - r2 ; ::_thesis: ( p1 in LSeg (p2,p3) or p3 in LSeg (p1,p2) )
set s2 = 1 - r1;
set s1 = 1 - r2;
(1 - (1 - r1)) + (1 - r1) <= 1 + (1 - r1) by A6, XREAL_1:6;
then A15: 1 - 1 <= 1 - r1 by XREAL_1:20;
A16: 0 + (1 - r2) <= (1 - (1 - r2)) + (1 - r2) by A8, XREAL_1:6;
now__::_thesis:_(_(_1_-_r1_<>_0_&_(_p1_in_LSeg_(p2,p3)_or_p3_in_LSeg_(p1,p2)_)_)_or_(_1_-_r1_=_0_&_(_p1_in_LSeg_(p2,p3)_or_p3_in_LSeg_(p1,p2)_)_)_)
percases ( 1 - r1 <> 0 or 1 - r1 = 0 ) ;
caseA17: 1 - r1 <> 0 ; ::_thesis: ( p1 in LSeg (p2,p3) or p3 in LSeg (p1,p2) )
(1 - r1) * p1 = (((1 - (1 - r2)) * p3) + ((1 - r2) * p2)) - ((1 - (1 - r1)) * p2) by A5, A7, EUCLID:48
.= ((1 - (1 - r2)) * p3) + (((1 - r2) * p2) - ((1 - (1 - r1)) * p2)) by EUCLID:45
.= ((1 - (1 - r2)) * p3) + (((1 - r2) - (1 - (1 - r1))) * p2) by EUCLID:50 ;
then (((1 - r1) ") * (1 - r1)) * p1 = ((1 - r1) ") * (((1 - (1 - r2)) * p3) + (((1 - r2) - (1 - (1 - r1))) * p2)) by EUCLID:30;
then 1 * p1 = ((1 - r1) ") * (((1 - (1 - r2)) * p3) + (((1 - r2) - (1 - (1 - r1))) * p2)) by A17, XCMPLX_0:def_7;
then p1 = ((1 - r1) ") * (((1 - (1 - r2)) * p3) + (((1 - r2) - (1 - (1 - r1))) * p2)) by EUCLID:29
.= (((1 - r1) ") * ((1 - (1 - r2)) * p3)) + (((1 - r1) ") * (((1 - r2) - (1 - (1 - r1))) * p2)) by EUCLID:32
.= ((((1 - r1) ") * (1 - (1 - r2))) * p3) + (((1 - r1) ") * (((1 - r2) - (1 - (1 - r1))) * p2)) by EUCLID:30 ;
then A18: p1 = ((((1 - r1) ") * (1 - (1 - r2))) * p3) + ((((1 - r1) ") * ((1 - r2) - (1 - (1 - r1)))) * p2) by EUCLID:30;
1 - r2 <= 1 + 0 by A16;
then (1 - r2) - 1 <= 0 by XREAL_1:20;
then ((1 - r2) - 1) + (1 - r1) <= 0 + (1 - r1) by XREAL_1:6;
then A19: ((1 - r1) ") * ((1 - r2) - (1 - (1 - r1))) <= ((1 - r1) ") * (1 - r1) by A15, XREAL_1:64;
(((1 - r1) ") * (1 - (1 - r2))) + (((1 - r1) ") * ((1 - r2) - (1 - (1 - r1)))) = ((1 - r1) ") * ((1 - 1) + (1 - r1))
.= 1 by A17, XCMPLX_0:def_7 ;
then A20: ((1 - r1) ") * (1 - (1 - r2)) = 1 - (((1 - r1) ") * ((1 - r2) - (1 - (1 - r1)))) ;
(1 - r2) - (1 - (1 - r1)) >= 0 by A14, XREAL_1:48;
then p1 in { (((1 - r) * p3) + (r * p2)) where r is Real : ( 0 <= r & r <= 1 ) } by A15, A18, A20, A19;
hence ( p1 in LSeg (p2,p3) or p3 in LSeg (p1,p2) ) by RLTOPSP1:def_2; ::_thesis: verum
end;
case 1 - r1 = 0 ; ::_thesis: ( p1 in LSeg (p2,p3) or p3 in LSeg (p1,p2) )
then p = (1 * p2) + (0. (TOP-REAL n)) by A5, EUCLID:29;
then p = p2 + (0. (TOP-REAL n)) by EUCLID:29;
hence ( p1 in LSeg (p2,p3) or p3 in LSeg (p1,p2) ) by A1, EUCLID:27; ::_thesis: verum
end;
end;
end;
hence ( p1 in LSeg (p2,p3) or p3 in LSeg (p1,p2) ) ; ::_thesis: verum
end;
end;
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
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: ( 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))} implies f is unfolded )
assume that
A1: f is s.n.c. and
A2: (LSeg (f,1)) /\ (LSeg (f,(1 + 1))) c= {(f /. (1 + 1))} and
A3: (LSeg (f,((len f) -' 2))) /\ (LSeg (f,((len f) -' 1))) c= {(f /. ((len f) -' 1))} ; ::_thesis: f is unfolded
for i being Nat st 1 <= i & i + 2 <= len f holds
(LSeg (f,i)) /\ (LSeg (f,(i + 1))) = {(f /. (i + 1))}
proof
let i be Nat; ::_thesis: ( 1 <= i & i + 2 <= len f implies (LSeg (f,i)) /\ (LSeg (f,(i + 1))) = {(f /. (i + 1))} )
assume that
A4: 1 <= i and
A5: i + 2 <= len f ; ::_thesis: (LSeg (f,i)) /\ (LSeg (f,(i + 1))) = {(f /. (i + 1))}
A6: 1 < i + 1 by A4, NAT_1:13;
then A7: LSeg (f,(i + 1)) = LSeg ((f /. (i + 1)),(f /. ((i + 1) + 1))) by A5, TOPREAL1:def_3;
A8: 1 < (i + 1) + 1 by A6, NAT_1:13;
(i + 1) + 1 = i + 2 ;
then A9: i + 1 < len f by A5, NAT_1:13;
then A10: LSeg (f,i) = LSeg ((f /. i),(f /. (i + 1))) by A4, TOPREAL1:def_3;
( f /. (i + 1) in LSeg ((f /. i),(f /. (i + 1))) & f /. (i + 1) in LSeg ((f /. (i + 1)),(f /. ((i + 1) + 1))) ) by RLTOPSP1:68;
then f /. (i + 1) in (LSeg (f,i)) /\ (LSeg (f,(i + 1))) by A10, A7, XBOOLE_0:def_4;
then A11: {(f /. (i + 1))} c= (LSeg (f,i)) /\ (LSeg (f,(i + 1))) by ZFMISC_1:31;
A12: i < len f by A9, NAT_1:13;
percases ( i = 1 or i <> 1 ) ;
suppose i = 1 ; ::_thesis: (LSeg (f,i)) /\ (LSeg (f,(i + 1))) = {(f /. (i + 1))}
hence (LSeg (f,i)) /\ (LSeg (f,(i + 1))) = {(f /. (i + 1))} by A2, A11, XBOOLE_0:def_10; ::_thesis: verum
end;
supposeA13: i <> 1 ; ::_thesis: (LSeg (f,i)) /\ (LSeg (f,(i + 1))) = {(f /. (i + 1))}
now__::_thesis:_(_(_i_+_2_=_len_f_&_(LSeg_(f,i))_/\_(LSeg_(f,(i_+_1)))_=_{(f_/._(i_+_1))}_)_or_(_i_+_2_<>_len_f_&_(LSeg_(f,i))_/\_(LSeg_(f,(i_+_1)))_=_{(f_/._(i_+_1))}_)_)
percases ( i + 2 = len f or i + 2 <> len f ) ;
caseA14: i + 2 = len f ; ::_thesis: (LSeg (f,i)) /\ (LSeg (f,(i + 1))) = {(f /. (i + 1))}
then ( (len f) - 2 = (len f) -' 2 & (len f) - 1 = (len f) -' 1 ) by A4, A6, NAT_D:39;
hence (LSeg (f,i)) /\ (LSeg (f,(i + 1))) = {(f /. (i + 1))} by A3, A11, A14, XBOOLE_0:def_10; ::_thesis: verum
end;
caseA15: i + 2 <> len f ; ::_thesis: (LSeg (f,i)) /\ (LSeg (f,(i + 1))) = {(f /. (i + 1))}
1 < i by A4, A13, XXREAL_0:1;
then 1 + 1 <= i by NAT_1:13;
then A16: (1 + 1) - 1 <= i - 1 by XREAL_1:9;
i + 2 < len f by A5, A15, XXREAL_0:1;
then A17: ((i + 1) + 1) + 1 <= len f by NAT_1:13;
now__::_thesis:_not_(LSeg_(f,i))_/\_(LSeg_(f,(i_+_1)))_<>_{(f_/._(i_+_1))}
( f /. (i + 1) in LSeg (f,(i + 1)) & f /. (i + 1) in LSeg (f,i) ) by A10, A7, RLTOPSP1:68;
then f /. (i + 1) in (LSeg (f,i)) /\ (LSeg (f,(i + 1))) by XBOOLE_0:def_4;
then A18: {(f /. (i + 1))} c= (LSeg (f,i)) /\ (LSeg (f,(i + 1))) by ZFMISC_1:31;
assume (LSeg (f,i)) /\ (LSeg (f,(i + 1))) <> {(f /. (i + 1))} ; ::_thesis: contradiction
then not (LSeg (f,i)) /\ (LSeg (f,(i + 1))) c= {(f /. (i + 1))} by A18, XBOOLE_0:def_10;
then consider x being set such that
A19: x in (LSeg (f,i)) /\ (LSeg (f,(i + 1))) and
A20: not x in {(f /. (i + 1))} by TARSKI:def_3;
A21: LSeg (f,((i + 1) + 1)) = LSeg ((f /. ((i + 1) + 1)),(f /. (((i + 1) + 1) + 1))) by A8, A17, TOPREAL1:def_3;
A22: x <> f /. (i + 1) by A20, TARSKI:def_1;
now__::_thesis:_(_(_f_/._i_in_LSeg_((f_/._(i_+_1)),(f_/._((i_+_1)_+_1)))_&_contradiction_)_or_(_f_/._((i_+_1)_+_1)_in_LSeg_((f_/._i),(f_/._(i_+_1)))_&_contradiction_)_)
percases ( f /. i in LSeg ((f /. (i + 1)),(f /. ((i + 1) + 1))) or f /. ((i + 1) + 1) in LSeg ((f /. i),(f /. (i + 1))) ) by A10, A7, A19, A22, Th16;
caseA23: f /. i in LSeg ((f /. (i + 1)),(f /. ((i + 1) + 1))) ; ::_thesis: contradiction
A24: i -' 1 = i - 1 by A4, XREAL_1:233;
then (i -' 1) + 1 < i + 1 by NAT_1:13;
then LSeg (f,(i -' 1)) misses LSeg (f,(i + 1)) by A1, TOPREAL1:def_7;
then A25: (LSeg (f,(i -' 1))) /\ (LSeg (f,(i + 1))) = {} by XBOOLE_0:def_7;
LSeg (f,(i -' 1)) = LSeg ((f /. (i -' 1)),(f /. ((i -' 1) + 1))) by A12, A16, A24, TOPREAL1:def_3;
then f /. i in LSeg (f,(i -' 1)) by A24, RLTOPSP1:68;
hence contradiction by A7, A23, A25, XBOOLE_0:def_4; ::_thesis: verum
end;
caseA26: f /. ((i + 1) + 1) in LSeg ((f /. i),(f /. (i + 1))) ; ::_thesis: contradiction
i + 1 < (i + 1) + 1 by NAT_1:13;
then LSeg (f,i) misses LSeg (f,((i + 1) + 1)) by A1, TOPREAL1:def_7;
then A27: (LSeg (f,i)) /\ (LSeg (f,((i + 1) + 1))) = {} by XBOOLE_0:def_7;
f /. ((i + 1) + 1) in LSeg (f,((i + 1) + 1)) by A21, RLTOPSP1:68;
hence contradiction by A10, A26, A27, XBOOLE_0:def_4; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
hence (LSeg (f,i)) /\ (LSeg (f,(i + 1))) = {(f /. (i + 1))} ; ::_thesis: verum
end;
end;
end;
hence (LSeg (f,i)) /\ (LSeg (f,(i + 1))) = {(f /. (i + 1))} ; ::_thesis: verum
end;
end;
end;
hence f is unfolded by TOPREAL1:def_6; ::_thesis: verum
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 )
proof
let X be non empty set ; ::_thesis: 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 )
let f be FinSequence of X; ::_thesis: ( PairF f is Simple & f . 1 <> f . (len f) implies ( f is one-to-one & len f <> 1 ) )
thus ( PairF f is Simple & f . 1 <> f . (len f) implies ( f is one-to-one & len f <> 1 ) ) ::_thesis: verum
proof
reconsider f1 = f as FinSequence of the carrier of (PGraph X) ;
assume that
A1: PairF f is Simple and
A2: f . 1 <> f . (len f) ; ::_thesis: ( f is one-to-one & len f <> 1 )
consider vs being FinSequence of the carrier of (PGraph X) such that
A3: vs is_oriented_vertex_seq_of PairF f and
A4: for n, m being Element of NAT st 1 <= n & n < m & m <= len vs & vs . n = vs . m holds
( n = 1 & m = len vs ) by A1, GRAPH_4:def_7;
now__::_thesis:_(_(_len_f_>=_1_&_f_is_one-to-one_)_or_(_len_f_<_1_&_f_is_one-to-one_)_)
percases ( len f >= 1 or len f < 1 ) ;
caseA5: len f >= 1 ; ::_thesis: f is one-to-one
now__::_thesis:_(_(_len_f_>_1_&_f_is_one-to-one_)_or_(_len_f_=_1_&_f_is_one-to-one_)_)
percases ( len f > 1 or len f = 1 ) by A5, XXREAL_0:1;
caseA6: len f > 1 ; ::_thesis: f is one-to-one
A7: f1 is_oriented_vertex_seq_of PairF f by A5, Th7;
then len f1 = (len (PairF f)) + 1 by GRAPH_4:def_5;
then 1 - 1 < ((len (PairF f)) + 1) - 1 by A6, XREAL_1:9;
then PairF f <> {} ;
then A8: vs = f1 by A3, A7, GRAPH_4:10;
for x, y being set st x in dom f & y in dom f & f . x = f . y holds
x = y
proof
let x, y be set ; ::_thesis: ( x in dom f & y in dom f & f . x = f . y implies x = y )
assume that
A9: x in dom f and
A10: y in dom f and
A11: f . x = f . y ; ::_thesis: x = y
reconsider i = x, j = y as Element of NAT by A9, A10;
A12: dom f = Seg (len f) by FINSEQ_1:def_3;
then A13: i <= len f by A9, FINSEQ_1:1;
A14: j <= len f by A10, A12, FINSEQ_1:1;
A15: 1 <= j by A10, A12, FINSEQ_1:1;
A16: 1 <= i by A9, A12, FINSEQ_1:1;
now__::_thesis:_not_i_<>_j
assume A17: i <> j ; ::_thesis: contradiction
now__::_thesis:_(_(_i_<_j_&_contradiction_)_or_(_j_<_i_&_contradiction_)_)
percases ( i < j or j < i ) by A17, XXREAL_0:1;
caseA18: i < j ; ::_thesis: contradiction
then i = 1 by A4, A8, A11, A16, A14;
hence contradiction by A2, A4, A8, A11, A14, A18; ::_thesis: verum
end;
caseA19: j < i ; ::_thesis: contradiction
then j = 1 by A4, A8, A11, A13, A15;
hence contradiction by A2, A4, A8, A11, A13, A19; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
hence x = y ; ::_thesis: verum
end;
hence f is one-to-one by FUNCT_1:def_4; ::_thesis: verum
end;
case len f = 1 ; ::_thesis: f is one-to-one
hence f is one-to-one by A2; ::_thesis: verum
end;
end;
end;
hence f is one-to-one ; ::_thesis: verum
end;
case len f < 1 ; ::_thesis: f is one-to-one
then (len f) + 1 <= 1 by NAT_1:13;
then ((len f) + 1) - 1 <= 1 - 1 by XREAL_1:9;
then len f = 0 ;
then Seg (len f) = {} ;
then for x, y being set st x in dom f & y in dom f & f . x = f . y holds
x = y by FINSEQ_1:def_3;
hence f is one-to-one by FUNCT_1:def_4; ::_thesis: verum
end;
end;
end;
hence ( f is one-to-one & len f <> 1 ) by A2; ::_thesis: verum
end;
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) )
proof
let X be non empty set ; ::_thesis: 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) )
let f be FinSequence of X; ::_thesis: ( f is one-to-one & len f > 1 implies ( PairF f is Simple & f . 1 <> f . (len f) ) )
assume that
A1: f is one-to-one and
A2: len f > 1 ; ::_thesis: ( PairF f is Simple & f . 1 <> f . (len f) )
A3: ( 1 in dom f & len f in dom f ) by A2, FINSEQ_3:25;
reconsider f1 = f as FinSequence of the carrier of (PGraph X) ;
A4: for i, j being Element of NAT st 1 <= i & i < j & j <= len f1 & f1 . i = f1 . j holds
( i = 1 & j = len f1 )
proof
let i, j be Element of NAT ; ::_thesis: ( 1 <= i & i < j & j <= len f1 & f1 . i = f1 . j implies ( i = 1 & j = len f1 ) )
assume that
A5: 1 <= i and
A6: i < j and
A7: j <= len f1 and
A8: f1 . i = f1 . j ; ::_thesis: ( i = 1 & j = len f1 )
1 < j by A5, A6, XXREAL_0:2;
then j in Seg (len f) by A7, FINSEQ_1:1;
then A9: j in dom f by FINSEQ_1:def_3;
i < len f by A6, A7, XXREAL_0:2;
then i in Seg (len f) by A5, FINSEQ_1:1;
then i in dom f by FINSEQ_1:def_3;
hence ( i = 1 & j = len f1 ) by A1, A6, A8, A9, FUNCT_1:def_4; ::_thesis: verum
end;
f1 is_oriented_vertex_seq_of PairF f by A2, Th7;
hence ( PairF f is Simple & f . 1 <> f . (len f) ) by A1, A2, A3, A4, FUNCT_1:def_4, GRAPH_4:def_7; ::_thesis: verum
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
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: ( f is nodic & PairF f is Simple & f . 1 <> f . (len f) implies f is unfolded )
assume that
A1: f is nodic and
A2: ( PairF f is Simple & f . 1 <> f . (len f) ) ; ::_thesis: f is unfolded
percases ( 2 < len f or len f <= 2 ) ;
supposeA3: 2 < len f ; ::_thesis: f is unfolded
then A4: 2 + 1 <= len f by NAT_1:13;
then A5: 2 + 1 in dom f by FINSEQ_3:25;
A6: f . 2 = f /. 2 by A3, FINSEQ_4:15;
then A7: f . 2 in LSeg (f,2) by A4, TOPREAL1:21;
1 + 1 <= len f by A3;
then f . 2 in LSeg (f,1) by A6, TOPREAL1:21;
then (LSeg (f,1)) /\ (LSeg (f,2)) <> {} by A7, XBOOLE_0:def_4;
then A8: LSeg (f,1) meets LSeg (f,2) by XBOOLE_0:def_7;
A9: len f < (len f) + 1 by NAT_1:13;
then A10: (len f) - 1 < len f by XREAL_1:19;
A11: 2 in dom f by A3, FINSEQ_3:25;
A12: LSeg (f,1) = LSeg ((f /. 1),(f /. (1 + 1))) by A3, TOPREAL1:def_3;
A13: f is one-to-one by A2, Th18;
A14: 1 < len f by A3, XXREAL_0:2;
then A15: 1 in dom f by FINSEQ_3:25;
A16: f . 1 = f /. 1 by A14, FINSEQ_4:15;
A17: (len f) -' 2 = (len f) - 2 by A3, XREAL_1:233;
A18: LSeg (f,2) = LSeg ((f /. 2),(f /. (2 + 1))) by A4, TOPREAL1:def_3;
now__::_thesis:_not_LSeg_(f,1)_=_LSeg_(f,2)
assume A19: LSeg (f,1) = LSeg (f,2) ; ::_thesis: contradiction
now__::_thesis:_(_(_f_/._1_=_f_/._2_&_f_/._(1_+_1)_=_f_/._(2_+_1)_&_contradiction_)_or_(_f_/._1_=_f_/._(2_+_1)_&_f_/._(1_+_1)_=_f_/._2_&_contradiction_)_)
percases ( ( f /. 1 = f /. 2 & f /. (1 + 1) = f /. (2 + 1) ) or ( f /. 1 = f /. (2 + 1) & f /. (1 + 1) = f /. 2 ) ) by A12, A18, A19, SPPOL_1:8;
caseA20: ( f /. 1 = f /. 2 & f /. (1 + 1) = f /. (2 + 1) ) ; ::_thesis: contradiction
( f . 1 = f /. 1 & f . 2 = f /. 2 ) by A3, A14, FINSEQ_4:15;
hence contradiction by A13, A15, A11, A20, FUNCT_1:def_4; ::_thesis: verum
end;
caseA21: ( f /. 1 = f /. (2 + 1) & f /. (1 + 1) = f /. 2 ) ; ::_thesis: contradiction
f . (2 + 1) = f /. (2 + 1) by A4, FINSEQ_4:15;
hence contradiction by A13, A15, A16, A5, A21, FUNCT_1:def_4; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
then ( ( (LSeg (f,1)) /\ (LSeg (f,2)) = {(f . 1)} & ( f . 1 = f . 2 or f . 1 = f . (2 + 1) ) ) or ( (LSeg (f,1)) /\ (LSeg (f,2)) = {(f . (1 + 1))} & ( f . (1 + 1) = f . 2 or f . (1 + 1) = f . (2 + 1) ) ) ) by A1, A8, Def4;
then A22: (LSeg (f,1)) /\ (LSeg (f,(1 + 1))) c= {(f /. (1 + 1))} by A3, A13, A15, A5, FINSEQ_4:15, FUNCT_1:def_4;
A23: (len f) - 1 = (len f) -' 1 by A3, XREAL_1:233, XXREAL_0:2;
then A24: ((len f) -' 1) + 1 in dom f by A14, FINSEQ_3:25;
A25: (1 + 1) - 1 <= (len f) - 1 by A3, XREAL_1:9;
then A26: f . ((len f) -' 1) = f /. ((len f) -' 1) by A23, A10, FINSEQ_4:15;
A27: (2 + 1) - 2 <= (len f) - 2 by A4, XREAL_1:9;
then A28: LSeg (f,((len f) -' 2)) = LSeg ((f /. ((len f) -' 2)),(f /. (((len f) -' 2) + 1))) by A17, A10, TOPREAL1:def_3;
A29: ((len f) - 1) - 1 < (len f) - 1 by A10, XREAL_1:9;
then (len f) - 2 < len f by A10, XXREAL_0:2;
then A30: f . ((len f) -' 2) = f /. ((len f) -' 2) by A27, A17, FINSEQ_4:15;
A31: (len f) -' 2 < len f by A17, A10, A29, XXREAL_0:2;
then A32: (len f) -' 2 in dom f by A27, A17, FINSEQ_3:25;
A33: LSeg (f,((len f) -' 1)) = LSeg ((f /. ((len f) -' 1)),(f /. (((len f) -' 1) + 1))) by A25, A23, TOPREAL1:def_3;
A34: f . ((len f) -' 2) = f /. ((len f) -' 2) by A27, A17, A31, FINSEQ_4:15;
A35: now__::_thesis:_not_LSeg_(f,((len_f)_-'_2))_=_LSeg_(f,((len_f)_-'_1))
assume A36: LSeg (f,((len f) -' 2)) = LSeg (f,((len f) -' 1)) ; ::_thesis: contradiction
A37: (len f) -' 2 in dom f by A27, A17, A31, FINSEQ_3:25;
A38: (len f) -' 1 in dom f by A25, A23, A10, FINSEQ_3:25;
now__::_thesis:_(_(_f_/._((len_f)_-'_2)_=_f_/._((len_f)_-'_1)_&_f_/._(((len_f)_-'_2)_+_1)_=_f_/._(((len_f)_-'_1)_+_1)_&_contradiction_)_or_(_f_/._((len_f)_-'_2)_=_f_/._(((len_f)_-'_1)_+_1)_&_f_/._(((len_f)_-'_2)_+_1)_=_f_/._((len_f)_-'_1)_&_contradiction_)_)
percases ( ( f /. ((len f) -' 2) = f /. ((len f) -' 1) & f /. (((len f) -' 2) + 1) = f /. (((len f) -' 1) + 1) ) or ( f /. ((len f) -' 2) = f /. (((len f) -' 1) + 1) & f /. (((len f) -' 2) + 1) = f /. ((len f) -' 1) ) ) by A28, A33, A36, SPPOL_1:8;
caseA39: ( f /. ((len f) -' 2) = f /. ((len f) -' 1) & f /. (((len f) -' 2) + 1) = f /. (((len f) -' 1) + 1) ) ; ::_thesis: contradiction
f . ((len f) -' 1) = f /. ((len f) -' 1) by A25, A23, A10, FINSEQ_4:15;
hence contradiction by A13, A23, A17, A9, A34, A37, A38, A39, FUNCT_1:def_4; ::_thesis: verum
end;
caseA40: ( f /. ((len f) -' 2) = f /. (((len f) -' 1) + 1) & f /. (((len f) -' 2) + 1) = f /. ((len f) -' 1) ) ; ::_thesis: contradiction
((len f) -' 1) + 1 in Seg (len f) by A14, A23, FINSEQ_1:1;
then A41: ((len f) -' 1) + 1 in dom f by FINSEQ_1:def_3;
f . (((len f) -' 1) + 1) = f /. (((len f) -' 1) + 1) by A14, A23, FINSEQ_4:15;
hence contradiction by A13, A23, A17, A10, A29, A30, A37, A40, A41, FUNCT_1:def_4; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
((len f) -' 1) + 1 = len f by A23;
then A42: f . ((len f) -' 1) in LSeg (f,((len f) -' 1)) by A25, A26, TOPREAL1:21;
((len f) -' 2) + 1 = (len f) - ((1 + 1) - 1) by A17;
then f . ((len f) -' 1) in LSeg (f,((len f) -' 2)) by A27, A23, A10, A26, TOPREAL1:21;
then (LSeg (f,((len f) -' 2))) /\ (LSeg (f,((len f) -' 1))) <> {} by A42, XBOOLE_0:def_4;
then LSeg (f,((len f) -' 2)) meets LSeg (f,((len f) -' 1)) by XBOOLE_0:def_7;
then ( ( (LSeg (f,((len f) -' 2))) /\ (LSeg (f,((len f) -' 1))) = {(f . ((len f) -' 2))} & ( f . ((len f) -' 2) = f . ((len f) -' 1) or f . ((len f) -' 2) = f . (((len f) -' 1) + 1) ) ) or ( (LSeg (f,((len f) -' 2))) /\ (LSeg (f,((len f) -' 1))) = {(f . (((len f) -' 2) + 1))} & ( f . (((len f) -' 2) + 1) = f . ((len f) -' 1) or f . (((len f) -' 2) + 1) = f . (((len f) -' 1) + 1) ) ) ) by A1, A35, Def4;
then (LSeg (f,((len f) -' 2))) /\ (LSeg (f,((len f) -' 1))) c= {(f /. ((len f) -' 1))} by A13, A25, A23, A17, A10, A29, A32, A24, FINSEQ_4:15, FUNCT_1:def_4;
hence f is unfolded by A1, A2, A22, Th15, Th17; ::_thesis: verum
end;
supposeA43: len f <= 2 ; ::_thesis: f is unfolded
for i being Nat st 1 <= i & i + 2 <= len f holds
(LSeg (f,i)) /\ (LSeg (f,(i + 1))) = {(f /. (i + 1))}
proof
let i be Nat; ::_thesis: ( 1 <= i & i + 2 <= len f implies (LSeg (f,i)) /\ (LSeg (f,(i + 1))) = {(f /. (i + 1))} )
assume that
A44: 1 <= i and
A45: i + 2 <= len f ; ::_thesis: (LSeg (f,i)) /\ (LSeg (f,(i + 1))) = {(f /. (i + 1))}
i + 2 <= 2 by A43, A45, XXREAL_0:2;
then i <= 2 - 2 by XREAL_1:19;
hence (LSeg (f,i)) /\ (LSeg (f,(i + 1))) = {(f /. (i + 1))} by A44, XXREAL_0:2; ::_thesis: verum
end;
hence f is unfolded by TOPREAL1:def_6; ::_thesis: verum
end;
end;
end;
theorem Th21: :: JGRAPH_1:21
for f, g being FinSequence of (TOP-REAL 2)
for i being Element of 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
let f, g be FinSequence of (TOP-REAL 2); ::_thesis: for i being Element of 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) )
let i be Element of NAT ; ::_thesis: ( g is_Shortcut_of f & 1 <= i & i + 1 <= len g implies 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) ) )
assume that
A1: g is_Shortcut_of f and
A2: 1 <= i and
A3: i + 1 <= len g ; ::_thesis: 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) )
A4: len (PairF g) = (len g) -' 1 by Def2;
A5: i < len g by A3, NAT_1:13;
then A6: (PairF g) . i = [(g . i),(g . (i + 1))] by A2, Def2;
1 <= i + 1 by A2, NAT_1:13;
then A7: g /. (i + 1) = g . (i + 1) by A3, FINSEQ_4:15;
i <= len g by A3, NAT_1:13;
then A8: g /. i = g . i by A2, FINSEQ_4:15;
A9: len g <= len f by A1, Th8;
1 < len g by A2, A5, XXREAL_0:2;
then A10: (len f) -' 1 = (len f) - 1 by A9, XREAL_1:233, XXREAL_0:2;
A11: len (PairF f) = (len f) -' 1 by Def2;
A12: i <= (len g) - 1 by A3, XREAL_1:19;
then 1 <= (len g) - 1 by A2, XXREAL_0:2;
then (len g) -' 1 = (len g) - 1 by NAT_D:39;
then i in dom (PairF g) by A2, A4, A12, FINSEQ_3:25;
then A13: [(g . i),(g . (i + 1))] in rng (PairF g) by A6, FUNCT_1:def_3;
rng (PairF g) c= rng (PairF f) by A1, Th10;
then consider x being set such that
A14: x in dom (PairF f) and
A15: (PairF f) . x = [(g . i),(g . (i + 1))] by A13, FUNCT_1:def_3;
reconsider k = x as Element of NAT by A14;
A16: x in Seg (len (PairF f)) by A14, FINSEQ_1:def_3;
then A17: 1 <= k by FINSEQ_1:1;
k <= len (PairF f) by A16, FINSEQ_1:1;
then A18: k + 1 <= ((len f) - 1) + 1 by A11, A10, XREAL_1:6;
then A19: k < len f by NAT_1:13;
then [(g . i),(g . (i + 1))] = [(f . k),(f . (k + 1))] by A15, A17, Def2;
then A20: ( g . i = f . k & g . (i + 1) = f . (k + 1) ) by XTUPLE_0:1;
1 < k + 1 by A17, NAT_1:13;
then A21: f /. (k + 1) = f . (k + 1) by A18, FINSEQ_4:15;
f /. k = f . k by A17, A19, FINSEQ_4:15;
hence 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) ) by A17, A18, A20, A8, A7, A21; ::_thesis: verum
end;
theorem Th22: :: JGRAPH_1:22
for f, g being FinSequence of (TOP-REAL 2) st g is_Shortcut_of f holds
rng g c= rng f
proof
let f, g be FinSequence of (TOP-REAL 2); ::_thesis: ( g is_Shortcut_of f implies rng g c= rng f )
assume A1: g is_Shortcut_of f ; ::_thesis: rng g c= rng f
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng g or x in rng f )
assume x in rng g ; ::_thesis: x in rng f
then consider z being set such that
A2: z in dom g and
A3: x = g . z by FUNCT_1:def_3;
reconsider i = z as Element of NAT by A2;
A4: z in Seg (len g) by A2, FINSEQ_1:def_3;
then A5: 1 <= i by FINSEQ_1:1;
A6: i <= len g by A4, FINSEQ_1:1;
percases ( i < len g or i >= len g ) ;
suppose i < len g ; ::_thesis: x in rng f
then i + 1 <= len g by NAT_1:13;
then consider k1 being Element of NAT such that
A7: 1 <= k1 and
A8: k1 + 1 <= len f and
f /. k1 = g /. i and
f /. (k1 + 1) = g /. (i + 1) and
A9: f . k1 = g . i and
f . (k1 + 1) = g . (i + 1) by A1, A5, Th21;
k1 < len f by A8, NAT_1:13;
then k1 in dom f by A7, FINSEQ_3:25;
hence x in rng f by A3, A9, FUNCT_1:def_3; ::_thesis: verum
end;
suppose i >= len g ; ::_thesis: x in rng f
then A10: i = len g by A6, XXREAL_0:1;
now__::_thesis:_(_(_1_<_i_&_x_in_rng_f_)_or_(_1_>=_i_&_x_in_rng_f_)_)
percases ( 1 < i or 1 >= i ) ;
caseA11: 1 < i ; ::_thesis: x in rng f
then A12: i -' 1 = i - 1 by XREAL_1:233;
1 - 1 < i - 1 by A11, XREAL_1:9;
then 0 < i -' 1 by A11, XREAL_1:233;
then 0 + 1 <= i -' 1 by NAT_1:13;
then consider k1 being Element of NAT such that
A13: 1 <= k1 and
A14: k1 + 1 <= len f and
f /. k1 = g /. (i -' 1) and
f /. (k1 + 1) = g /. ((i -' 1) + 1) and
f . k1 = g . (i -' 1) and
A15: f . (k1 + 1) = g . ((i -' 1) + 1) by A1, A6, A12, Th21;
1 < k1 + 1 by A13, NAT_1:13;
then k1 + 1 in dom f by A14, FINSEQ_3:25;
hence x in rng f by A3, A12, A15, FUNCT_1:def_3; ::_thesis: verum
end;
case 1 >= i ; ::_thesis: x in rng f
then A16: i = 1 by A5, XXREAL_0:1;
A17: f . 1 = g . 1 by A1, Def3;
len g <= len f by A1, Th8;
then 1 in dom f by A10, A16, FINSEQ_3:25;
hence x in rng f by A3, A16, A17, FUNCT_1:def_3; ::_thesis: verum
end;
end;
end;
hence x in rng f ; ::_thesis: verum
end;
end;
end;
theorem Th23: :: JGRAPH_1:23
for f, g being FinSequence of (TOP-REAL 2) st g is_Shortcut_of f holds
L~ g c= L~ f
proof
let f, g be FinSequence of (TOP-REAL 2); ::_thesis: ( g is_Shortcut_of f implies L~ g c= L~ f )
assume A1: g is_Shortcut_of f ; ::_thesis: L~ g c= L~ f
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in L~ g or x in L~ f )
assume x in L~ g ; ::_thesis: x in L~ f
then x in union { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } by TOPREAL1:def_4;
then consider y being set such that
A2: ( x in y & y in { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } ) by TARSKI:def_4;
consider i being Element of NAT such that
A3: y = LSeg (g,i) and
A4: ( 1 <= i & i + 1 <= len g ) by A2;
consider k1 being Element of NAT such that
A5: ( 1 <= k1 & k1 + 1 <= len f ) and
A6: ( f /. k1 = g /. i & f /. (k1 + 1) = g /. (i + 1) ) and
f . k1 = g . i and
f . (k1 + 1) = g . (i + 1) by A1, A4, Th21;
A7: LSeg (f,k1) in { (LSeg (f,k)) where k is Element of NAT : ( 1 <= k & k + 1 <= len f ) } by A5;
x in LSeg ((g /. i),(g /. (i + 1))) by A2, A3, A4, TOPREAL1:def_3;
then x in LSeg (f,k1) by A5, A6, TOPREAL1:def_3;
then x in union { (LSeg (f,k)) where k is Element of NAT : ( 1 <= k & k + 1 <= len f ) } by A7, TARSKI:def_4;
hence x in L~ f by TOPREAL1:def_4; ::_thesis: verum
end;
theorem Th24: :: JGRAPH_1:24
for f, g being FinSequence of (TOP-REAL 2) st f is special & g is_Shortcut_of f holds
g is special
proof
let f, g be FinSequence of (TOP-REAL 2); ::_thesis: ( f is special & g is_Shortcut_of f implies g is special )
assume that
A1: f is special and
A2: g is_Shortcut_of f ; ::_thesis: g is special
for i being Nat st 1 <= i & i + 1 <= len g & not (g /. i) `1 = (g /. (i + 1)) `1 holds
(g /. i) `2 = (g /. (i + 1)) `2
proof
A3: len g <= len f by A2, Th8;
A4: len (PairF g) = (len g) -' 1 by Def2;
let i be Nat; ::_thesis: ( 1 <= i & i + 1 <= len g & not (g /. i) `1 = (g /. (i + 1)) `1 implies (g /. i) `2 = (g /. (i + 1)) `2 )
assume that
A5: 1 <= i and
A6: i + 1 <= len g ; ::_thesis: ( (g /. i) `1 = (g /. (i + 1)) `1 or (g /. i) `2 = (g /. (i + 1)) `2 )
i <= len g by A6, NAT_1:13;
then A7: g /. i = g . i by A5, FINSEQ_4:15;
A8: i < len g by A6, NAT_1:13;
then 1 < len g by A5, XXREAL_0:2;
then A9: (len f) -' 1 = (len f) - 1 by A3, XREAL_1:233, XXREAL_0:2;
A10: i <= (len g) - 1 by A6, XREAL_1:19;
then 1 <= (len g) - 1 by A5, XXREAL_0:2;
then (len g) -' 1 = (len g) - 1 by NAT_D:39;
then A11: i in dom (PairF g) by A5, A4, A10, FINSEQ_3:25;
i in NAT by ORDINAL1:def_12;
then (PairF g) . i = [(g . i),(g . (i + 1))] by A5, A8, Def2;
then A12: [(g . i),(g . (i + 1))] in rng (PairF g) by A11, FUNCT_1:def_3;
rng (PairF g) c= rng (PairF f) by A2, Th10;
then consider x being set such that
A13: x in dom (PairF f) and
A14: (PairF f) . x = [(g . i),(g . (i + 1))] by A12, FUNCT_1:def_3;
reconsider k = x as Element of NAT by A13;
A15: x in Seg (len (PairF f)) by A13, FINSEQ_1:def_3;
then A16: 1 <= k by FINSEQ_1:1;
1 <= i + 1 by A5, NAT_1:13;
then A17: g /. (i + 1) = g . (i + 1) by A6, FINSEQ_4:15;
A18: len (PairF f) = (len f) -' 1 by Def2;
k <= len (PairF f) by A15, FINSEQ_1:1;
then A19: k + 1 <= ((len f) - 1) + 1 by A18, A9, XREAL_1:6;
then A20: k < len f by NAT_1:13;
then (PairF f) . k = [(f . k),(f . (k + 1))] by A16, Def2;
then A21: ( g . i = f . k & g . (i + 1) = f . (k + 1) ) by A14, XTUPLE_0:1;
1 < k + 1 by A16, NAT_1:13;
then A22: f /. (k + 1) = f . (k + 1) by A19, FINSEQ_4:15;
f /. k = f . k by A16, A20, FINSEQ_4:15;
hence ( (g /. i) `1 = (g /. (i + 1)) `1 or (g /. i) `2 = (g /. (i + 1)) `2 ) by A1, A16, A19, A21, A7, A17, A22, TOPREAL1:def_5; ::_thesis: verum
end;
hence g is special by TOPREAL1:def_5; ::_thesis: verum
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 )
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: ( f is special & 2 <= len f & f . 1 <> f . (len f) implies 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 ) )
assume that
A1: f is special and
A2: 2 <= len f and
A3: f . 1 <> f . (len f) ; ::_thesis: 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 )
consider g being FinSequence of (TOP-REAL 2) such that
A4: g is_Shortcut_of f by A2, Th9, XXREAL_0:2;
A5: ( g . 1 = f . 1 & g . (len g) = f . (len f) ) by A4, Def3;
1 <= len g by A4, Th8;
then 1 < len g by A3, A5, XXREAL_0:1;
then A6: 1 + 1 <= len g by NAT_1:13;
A7: ( L~ g c= L~ f & rng g c= rng f ) by A4, Th22, Th23;
g is one-to-one by A3, A4, Th11;
hence 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 ) by A1, A4, A5, A7, A6, Th24; ::_thesis: verum
end;
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
proof
let f1, f2 be FinSequence of (TOP-REAL 2); ::_thesis: ( 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) implies L~ f1 meets L~ f2 )
assume that
A1: f1 is special and
A2: f2 is special and
A3: 2 <= len f1 and
A4: 2 <= len f2 and
A5: f1 . 1 <> f1 . (len f1) and
A6: f2 . 1 <> f2 . (len f2) and
A7: X_axis f1 lies_between (X_axis f1) . 1,(X_axis f1) . (len f1) and
A8: X_axis f2 lies_between (X_axis f1) . 1,(X_axis f1) . (len f1) and
A9: Y_axis f1 lies_between (Y_axis f2) . 1,(Y_axis f2) . (len f2) and
A10: Y_axis f2 lies_between (Y_axis f2) . 1,(Y_axis f2) . (len f2) ; ::_thesis: L~ f1 meets L~ f2
consider g1 being FinSequence of (TOP-REAL 2) such that
A11: 2 <= len g1 and
A12: g1 is special and
A13: g1 is one-to-one and
A14: L~ g1 c= L~ f1 and
A15: f1 . 1 = g1 . 1 and
A16: f1 . (len f1) = g1 . (len g1) and
A17: rng g1 c= rng f1 by A1, A3, A5, Th25;
consider g2 being FinSequence of (TOP-REAL 2) such that
A18: 2 <= len g2 and
A19: g2 is special and
A20: g2 is one-to-one and
A21: L~ g2 c= L~ f2 and
A22: f2 . 1 = g2 . 1 and
A23: f2 . (len f2) = g2 . (len g2) and
A24: rng g2 c= rng f2 by A2, A4, A6, Th25;
A25: for k being Element of NAT st k in dom g2 & k + 1 in dom g2 holds
g2 /. k <> g2 /. (k + 1)
proof
let k be Element of NAT ; ::_thesis: ( k in dom g2 & k + 1 in dom g2 implies g2 /. k <> g2 /. (k + 1) )
assume that
A26: k in dom g2 and
A27: k + 1 in dom g2 ; ::_thesis: g2 /. k <> g2 /. (k + 1)
A28: g2 . k = g2 /. k by A26, PARTFUN1:def_6;
k < k + 1 by NAT_1:13;
then g2 . k <> g2 . (k + 1) by A20, A26, A27, FUNCT_1:def_4;
hence g2 /. k <> g2 /. (k + 1) by A27, A28, PARTFUN1:def_6; ::_thesis: verum
end;
for i being Element of NAT st i in dom (Y_axis g1) holds
( (Y_axis g2) . 1 <= (Y_axis g1) . i & (Y_axis g1) . i <= (Y_axis g2) . (len g2) )
proof
let i be Element of NAT ; ::_thesis: ( i in dom (Y_axis g1) implies ( (Y_axis g2) . 1 <= (Y_axis g1) . i & (Y_axis g1) . i <= (Y_axis g2) . (len g2) ) )
A29: len (Y_axis f2) = len f2 by GOBOARD1:def_2;
A30: 1 <= len f2 by A4, XXREAL_0:2;
then 1 in Seg (len f2) by FINSEQ_1:1;
then A31: 1 in dom (Y_axis f2) by A29, FINSEQ_1:def_3;
len f2 in Seg (len f2) by A30, FINSEQ_1:1;
then A32: len f2 in dom (Y_axis f2) by A29, FINSEQ_1:def_3;
A33: len (Y_axis g2) = len g2 by GOBOARD1:def_2;
A34: 1 <= len g2 by A18, XXREAL_0:2;
then 1 in Seg (len g2) by FINSEQ_1:1;
then A35: 1 in dom (Y_axis g2) by A33, FINSEQ_1:def_3;
g2 /. 1 = g2 . 1 by A34, FINSEQ_4:15;
then A36: g2 /. 1 = f2 /. 1 by A22, A30, FINSEQ_4:15;
len g2 in Seg (len g2) by A34, FINSEQ_1:1;
then A37: len g2 in dom (Y_axis g2) by A33, FINSEQ_1:def_3;
g2 /. (len g2) = g2 . (len g2) by A34, FINSEQ_4:15;
then A38: g2 /. (len g2) = f2 /. (len f2) by A23, A30, FINSEQ_4:15;
assume A39: i in dom (Y_axis g1) ; ::_thesis: ( (Y_axis g2) . 1 <= (Y_axis g1) . i & (Y_axis g1) . i <= (Y_axis g2) . (len g2) )
then A40: (Y_axis g1) . i = (g1 /. i) `2 by GOBOARD1:def_2;
len (Y_axis g1) = len g1 by GOBOARD1:def_2;
then i in Seg (len g1) by A39, FINSEQ_1:def_3;
then A41: i in dom g1 by FINSEQ_1:def_3;
then g1 . i in rng g1 by FUNCT_1:def_3;
then consider y being set such that
A42: y in dom f1 and
A43: g1 . i = f1 . y by A17, FUNCT_1:def_3;
reconsider j = y as Element of NAT by A42;
f1 . j = f1 /. j by A42, PARTFUN1:def_6;
then A44: g1 /. i = f1 /. j by A41, A43, PARTFUN1:def_6;
( len (Y_axis f1) = len f1 & j in Seg (len f1) ) by A42, FINSEQ_1:def_3, GOBOARD1:def_2;
then A45: j in dom (Y_axis f1) by FINSEQ_1:def_3;
then A46: (Y_axis f1) . j = (f1 /. j) `2 by GOBOARD1:def_2;
(Y_axis f1) . j <= (Y_axis f2) . (len f2) by A9, A45, GOBOARD4:def_2;
then A47: (g1 /. i) `2 <= (g2 /. (len g2)) `2 by A38, A44, A46, A32, GOBOARD1:def_2;
(Y_axis f2) . 1 <= (Y_axis f1) . j by A9, A45, GOBOARD4:def_2;
then (g2 /. 1) `2 <= (g1 /. i) `2 by A36, A31, A44, A46, GOBOARD1:def_2;
hence ( (Y_axis g2) . 1 <= (Y_axis g1) . i & (Y_axis g1) . i <= (Y_axis g2) . (len g2) ) by A35, A47, A40, A37, GOBOARD1:def_2; ::_thesis: verum
end;
then A48: Y_axis g1 lies_between (Y_axis g2) . 1,(Y_axis g2) . (len g2) by GOBOARD4:def_2;
for i being Element of NAT st i in dom (X_axis g2) holds
( (X_axis g1) . 1 <= (X_axis g2) . i & (X_axis g2) . i <= (X_axis g1) . (len g1) )
proof
let i be Element of NAT ; ::_thesis: ( i in dom (X_axis g2) implies ( (X_axis g1) . 1 <= (X_axis g2) . i & (X_axis g2) . i <= (X_axis g1) . (len g1) ) )
A49: len (X_axis f1) = len f1 by GOBOARD1:def_1;
A50: 1 <= len f1 by A3, XXREAL_0:2;
then 1 in Seg (len f1) by FINSEQ_1:1;
then A51: 1 in dom (X_axis f1) by A49, FINSEQ_1:def_3;
len f1 in Seg (len f1) by A50, FINSEQ_1:1;
then A52: len f1 in dom (X_axis f1) by A49, FINSEQ_1:def_3;
A53: len (X_axis g1) = len g1 by GOBOARD1:def_1;
A54: 1 <= len g1 by A11, XXREAL_0:2;
then 1 in Seg (len g1) by FINSEQ_1:1;
then A55: 1 in dom (X_axis g1) by A53, FINSEQ_1:def_3;
g1 /. 1 = g1 . 1 by A54, FINSEQ_4:15;
then A56: g1 /. 1 = f1 /. 1 by A15, A50, FINSEQ_4:15;
len g1 in Seg (len g1) by A54, FINSEQ_1:1;
then A57: len g1 in dom (X_axis g1) by A53, FINSEQ_1:def_3;
g1 /. (len g1) = g1 . (len g1) by A54, FINSEQ_4:15;
then A58: g1 /. (len g1) = f1 /. (len f1) by A16, A50, FINSEQ_4:15;
assume A59: i in dom (X_axis g2) ; ::_thesis: ( (X_axis g1) . 1 <= (X_axis g2) . i & (X_axis g2) . i <= (X_axis g1) . (len g1) )
then A60: (X_axis g2) . i = (g2 /. i) `1 by GOBOARD1:def_1;
len (X_axis g2) = len g2 by GOBOARD1:def_1;
then i in Seg (len g2) by A59, FINSEQ_1:def_3;
then A61: i in dom g2 by FINSEQ_1:def_3;
then g2 . i in rng g2 by FUNCT_1:def_3;
then consider y being set such that
A62: y in dom f2 and
A63: g2 . i = f2 . y by A24, FUNCT_1:def_3;
reconsider j = y as Element of NAT by A62;
f2 . j = f2 /. j by A62, PARTFUN1:def_6;
then A64: g2 /. i = f2 /. j by A61, A63, PARTFUN1:def_6;
( len (X_axis f2) = len f2 & j in Seg (len f2) ) by A62, FINSEQ_1:def_3, GOBOARD1:def_1;
then A65: j in dom (X_axis f2) by FINSEQ_1:def_3;
then A66: (X_axis f2) . j = (f2 /. j) `1 by GOBOARD1:def_1;
(X_axis f2) . j <= (X_axis f1) . (len f1) by A8, A65, GOBOARD4:def_2;
then A67: (g2 /. i) `1 <= (g1 /. (len g1)) `1 by A58, A64, A66, A52, GOBOARD1:def_1;
(X_axis f1) . 1 <= (X_axis f2) . j by A8, A65, GOBOARD4:def_2;
then (g1 /. 1) `1 <= (g2 /. i) `1 by A56, A51, A64, A66, GOBOARD1:def_1;
hence ( (X_axis g1) . 1 <= (X_axis g2) . i & (X_axis g2) . i <= (X_axis g1) . (len g1) ) by A55, A67, A60, A57, GOBOARD1:def_1; ::_thesis: verum
end;
then A68: X_axis g2 lies_between (X_axis g1) . 1,(X_axis g1) . (len g1) by GOBOARD4:def_2;
for i being Element of NAT st i in dom (Y_axis g2) holds
( (Y_axis g2) . 1 <= (Y_axis g2) . i & (Y_axis g2) . i <= (Y_axis g2) . (len g2) )
proof
let i be Element of NAT ; ::_thesis: ( i in dom (Y_axis g2) implies ( (Y_axis g2) . 1 <= (Y_axis g2) . i & (Y_axis g2) . i <= (Y_axis g2) . (len g2) ) )
A69: len (Y_axis f2) = len f2 by GOBOARD1:def_2;
A70: 1 <= len f2 by A4, XXREAL_0:2;
then 1 in Seg (len f2) by FINSEQ_1:1;
then A71: 1 in dom (Y_axis f2) by A69, FINSEQ_1:def_3;
A72: 1 <= len g2 by A18, XXREAL_0:2;
then g2 /. 1 = g2 . 1 by FINSEQ_4:15;
then A73: g2 /. 1 = f2 /. 1 by A22, A70, FINSEQ_4:15;
A74: len (Y_axis g2) = len g2 by GOBOARD1:def_2;
then len g2 in Seg (len (Y_axis g2)) by A72, FINSEQ_1:1;
then A75: len g2 in dom (Y_axis g2) by FINSEQ_1:def_3;
g2 /. (len g2) = g2 . (len g2) by A72, FINSEQ_4:15;
then A76: g2 /. (len g2) = f2 /. (len f2) by A23, A70, FINSEQ_4:15;
1 in Seg (len g2) by A72, FINSEQ_1:1;
then A77: 1 in dom (Y_axis g2) by A74, FINSEQ_1:def_3;
len f2 in Seg (len f2) by A70, FINSEQ_1:1;
then A78: len f2 in dom (Y_axis f2) by A69, FINSEQ_1:def_3;
assume A79: i in dom (Y_axis g2) ; ::_thesis: ( (Y_axis g2) . 1 <= (Y_axis g2) . i & (Y_axis g2) . i <= (Y_axis g2) . (len g2) )
then A80: (Y_axis g2) . i = (g2 /. i) `2 by GOBOARD1:def_2;
i in Seg (len g2) by A79, A74, FINSEQ_1:def_3;
then A81: i in dom g2 by FINSEQ_1:def_3;
then g2 . i in rng g2 by FUNCT_1:def_3;
then consider y being set such that
A82: y in dom f2 and
A83: g2 . i = f2 . y by A24, FUNCT_1:def_3;
reconsider j = y as Element of NAT by A82;
f2 . j = f2 /. j by A82, PARTFUN1:def_6;
then A84: g2 /. i = f2 /. j by A81, A83, PARTFUN1:def_6;
j in Seg (len f2) by A82, FINSEQ_1:def_3;
then A85: j in dom (Y_axis f2) by A69, FINSEQ_1:def_3;
then A86: (Y_axis f2) . j = (f2 /. j) `2 by GOBOARD1:def_2;
(Y_axis f2) . j <= (Y_axis f2) . (len f2) by A10, A85, GOBOARD4:def_2;
then A87: (g2 /. i) `2 <= (g2 /. (len g2)) `2 by A76, A84, A86, A78, GOBOARD1:def_2;
(Y_axis f2) . 1 <= (Y_axis f2) . j by A10, A85, GOBOARD4:def_2;
then (g2 /. 1) `2 <= (g2 /. i) `2 by A73, A71, A84, A86, GOBOARD1:def_2;
hence ( (Y_axis g2) . 1 <= (Y_axis g2) . i & (Y_axis g2) . i <= (Y_axis g2) . (len g2) ) by A77, A87, A80, A75, GOBOARD1:def_2; ::_thesis: verum
end;
then A88: Y_axis g2 lies_between (Y_axis g2) . 1,(Y_axis g2) . (len g2) by GOBOARD4:def_2;
for i being Element of NAT st i in dom (X_axis g1) holds
( (X_axis g1) . 1 <= (X_axis g1) . i & (X_axis g1) . i <= (X_axis g1) . (len g1) )
proof
let i be Element of NAT ; ::_thesis: ( i in dom (X_axis g1) implies ( (X_axis g1) . 1 <= (X_axis g1) . i & (X_axis g1) . i <= (X_axis g1) . (len g1) ) )
A89: len (X_axis f1) = len f1 by GOBOARD1:def_1;
assume A90: i in dom (X_axis g1) ; ::_thesis: ( (X_axis g1) . 1 <= (X_axis g1) . i & (X_axis g1) . i <= (X_axis g1) . (len g1) )
then A91: (X_axis g1) . i = (g1 /. i) `1 by GOBOARD1:def_1;
A92: 1 <= len f1 by A3, XXREAL_0:2;
then len f1 in Seg (len f1) by FINSEQ_1:1;
then A93: len f1 in dom (X_axis f1) by A89, FINSEQ_1:def_3;
A94: 1 <= len g1 by A11, XXREAL_0:2;
then g1 /. 1 = g1 . 1 by FINSEQ_4:15;
then A95: g1 /. 1 = f1 /. 1 by A15, A92, FINSEQ_4:15;
g1 /. (len g1) = g1 . (len g1) by A94, FINSEQ_4:15;
then A96: g1 /. (len g1) = f1 /. (len f1) by A16, A92, FINSEQ_4:15;
A97: len (X_axis g1) = len g1 by GOBOARD1:def_1;
then A98: 1 in dom (X_axis g1) by A94, FINSEQ_3:25;
i in Seg (len g1) by A90, A97, FINSEQ_1:def_3;
then A99: i in dom g1 by FINSEQ_1:def_3;
then g1 . i in rng g1 by FUNCT_1:def_3;
then consider y being set such that
A100: y in dom f1 and
A101: g1 . i = f1 . y by A17, FUNCT_1:def_3;
reconsider j = y as Element of NAT by A100;
f1 . j = f1 /. j by A100, PARTFUN1:def_6;
then A102: g1 /. i = f1 /. j by A99, A101, PARTFUN1:def_6;
len g1 in Seg (len g1) by A94, FINSEQ_1:1;
then A103: len g1 in dom (X_axis g1) by A97, FINSEQ_1:def_3;
j in Seg (len f1) by A100, FINSEQ_1:def_3;
then A104: j in dom (X_axis f1) by A89, FINSEQ_1:def_3;
then A105: (X_axis f1) . j = (f1 /. j) `1 by GOBOARD1:def_1;
(X_axis f1) . j <= (X_axis f1) . (len f1) by A7, A104, GOBOARD4:def_2;
then A106: (g1 /. i) `1 <= (g1 /. (len g1)) `1 by A96, A102, A105, A93, GOBOARD1:def_1;
A107: (X_axis f1) . 1 <= (X_axis f1) . j by A7, A104, GOBOARD4:def_2;
1 in dom (X_axis f1) by A89, A92, FINSEQ_3:25;
then (g1 /. 1) `1 <= (g1 /. i) `1 by A95, A102, A107, A105, GOBOARD1:def_1;
hence ( (X_axis g1) . 1 <= (X_axis g1) . i & (X_axis g1) . i <= (X_axis g1) . (len g1) ) by A98, A106, A91, A103, GOBOARD1:def_1; ::_thesis: verum
end;
then A108: X_axis g1 lies_between (X_axis g1) . 1,(X_axis g1) . (len g1) by GOBOARD4:def_2;
for k being Element of NAT st k in dom g1 & k + 1 in dom g1 holds
g1 /. k <> g1 /. (k + 1)
proof
let k be Element of NAT ; ::_thesis: ( k in dom g1 & k + 1 in dom g1 implies g1 /. k <> g1 /. (k + 1) )
assume that
A109: k in dom g1 and
A110: k + 1 in dom g1 ; ::_thesis: g1 /. k <> g1 /. (k + 1)
A111: g1 . k = g1 /. k by A109, PARTFUN1:def_6;
k < k + 1 by NAT_1:13;
then g1 . k <> g1 . (k + 1) by A13, A109, A110, FUNCT_1:def_4;
hence g1 /. k <> g1 /. (k + 1) by A110, A111, PARTFUN1:def_6; ::_thesis: verum
end;
then L~ g1 meets L~ g2 by A11, A12, A18, A19, A108, A68, A48, A88, A25, GOBOARD4:4;
then (L~ g1) /\ (L~ g2) <> {} by XBOOLE_0:def_7;
then (L~ f1) /\ (L~ f2) <> {} by A14, A21, XBOOLE_1:3, XBOOLE_1:27;
hence L~ f1 meets L~ f2 by XBOOLE_0:def_7; ::_thesis: verum
end;
begin
theorem Th27: :: JGRAPH_1:27
for a, b, r1, r2 being Real st a <= r1 & r1 <= b & a <= r2 & r2 <= b holds
abs (r1 - r2) <= b - a
proof
let a, b, r1, r2 be Real; ::_thesis: ( a <= r1 & r1 <= b & a <= r2 & r2 <= b implies abs (r1 - r2) <= b - a )
assume that
A1: a <= r1 and
A2: ( r1 <= b & a <= r2 ) and
A3: r2 <= b ; ::_thesis: abs (r1 - r2) <= b - a
percases ( r1 - r2 >= 0 or r1 - r2 < 0 ) ;
supposeA4: r1 - r2 >= 0 ; ::_thesis: abs (r1 - r2) <= b - a
A5: ( r1 - r2 <= b - r2 & b - r2 <= b - a ) by A2, XREAL_1:9, XREAL_1:10;
abs (r1 - r2) = r1 - r2 by A4, ABSVALUE:def_1;
hence abs (r1 - r2) <= b - a by A5, XXREAL_0:2; ::_thesis: verum
end;
suppose r1 - r2 < 0 ; ::_thesis: abs (r1 - r2) <= b - a
then A6: abs (r1 - r2) = - (r1 - r2) by ABSVALUE:def_1
.= r2 - r1 ;
( r2 - r1 <= b - r1 & b - r1 <= b - a ) by A1, A3, XREAL_1:9, XREAL_1:10;
hence abs (r1 - r2) <= b - a by A6, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
theorem Th28: :: JGRAPH_1:28
for n being Element of 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)
proof
let n be Element of NAT ; ::_thesis: 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)
let p1, p2 be Point of (TOP-REAL n); ::_thesis: for x1, x2 being Point of (Euclid n) st x1 = p1 & x2 = p2 holds
|.(p1 - p2).| = dist (x1,x2)
let x1, x2 be Point of (Euclid n); ::_thesis: ( x1 = p1 & x2 = p2 implies |.(p1 - p2).| = dist (x1,x2) )
assume A1: ( x1 = p1 & x2 = p2 ) ; ::_thesis: |.(p1 - p2).| = dist (x1,x2)
reconsider x19 = x1, x29 = x2 as Element of REAL n ;
(Pitag_dist n) . (x19,x29) = |.(x19 - x29).| by EUCLID:def_6
.= |.(p1 - p2).| by A1 ;
hence |.(p1 - p2).| = dist (x1,x2) by METRIC_1:def_1; ::_thesis: verum
end;
theorem Th29: :: JGRAPH_1:29
for p being Point of (TOP-REAL 2) holds |.p.| ^2 = ((p `1) ^2) + ((p `2) ^2)
proof
let p be Point of (TOP-REAL 2); ::_thesis: |.p.| ^2 = ((p `1) ^2) + ((p `2) ^2)
reconsider w = p as Element of REAL 2 by EUCLID:22;
A1: (sqr w) . 2 = (p `2) ^2 by VALUED_1:11;
0 <= Sum (sqr w) by RVSUM_1:86;
then A2: |.p.| ^2 = Sum (sqr w) by SQUARE_1:def_2;
( len (sqr w) = 2 & (sqr w) . 1 = (p `1) ^2 ) by CARD_1:def_7, VALUED_1:11;
then sqr w = <*((p `1) ^2),((p `2) ^2)*> by A1, FINSEQ_1:44;
hence |.p.| ^2 = ((p `1) ^2) + ((p `2) ^2) by A2, RVSUM_1:77; ::_thesis: verum
end;
theorem Th30: :: JGRAPH_1:30
for p being Point of (TOP-REAL 2) holds |.p.| = sqrt (((p `1) ^2) + ((p `2) ^2))
proof
let p be Point of (TOP-REAL 2); ::_thesis: |.p.| = sqrt (((p `1) ^2) + ((p `2) ^2))
|.p.| ^2 = ((p `1) ^2) + ((p `2) ^2) by Th29;
hence |.p.| = sqrt (((p `1) ^2) + ((p `2) ^2)) by SQUARE_1:22; ::_thesis: verum
end;
theorem Th31: :: JGRAPH_1:31
for p being Point of (TOP-REAL 2) holds |.p.| <= (abs (p `1)) + (abs (p `2))
proof
let p be Point of (TOP-REAL 2); ::_thesis: |.p.| <= (abs (p `1)) + (abs (p `2))
|.p.| ^2 = ((p `1) ^2) + ((p `2) ^2) by Th29;
then sqrt (((p `1) ^2) + ((p `2) ^2)) = |.p.| by SQUARE_1:22;
hence |.p.| <= (abs (p `1)) + (abs (p `2)) by COMPLEX1:78; ::_thesis: verum
end;
theorem Th32: :: JGRAPH_1:32
for p1, p2 being Point of (TOP-REAL 2) holds |.(p1 - p2).| <= (abs ((p1 `1) - (p2 `1))) + (abs ((p1 `2) - (p2 `2)))
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: |.(p1 - p2).| <= (abs ((p1 `1) - (p2 `1))) + (abs ((p1 `2) - (p2 `2)))
( (p1 `1) - (p2 `1) = (p1 - p2) `1 & (p1 `2) - (p2 `2) = (p1 - p2) `2 ) by TOPREAL3:3;
hence |.(p1 - p2).| <= (abs ((p1 `1) - (p2 `1))) + (abs ((p1 `2) - (p2 `2))) by Th31; ::_thesis: verum
end;
theorem Th33: :: JGRAPH_1:33
for p being Point of (TOP-REAL 2) holds
( abs (p `1) <= |.p.| & abs (p `2) <= |.p.| )
proof
let p be Point of (TOP-REAL 2); ::_thesis: ( abs (p `1) <= |.p.| & abs (p `2) <= |.p.| )
|.p.| = sqrt (((p `1) ^2) + ((p `2) ^2)) by Th30;
hence ( abs (p `1) <= |.p.| & abs (p `2) <= |.p.| ) by COMPLEX1:79; ::_thesis: verum
end;
theorem Th34: :: JGRAPH_1:34
for p1, p2 being Point of (TOP-REAL 2) holds
( abs ((p1 `1) - (p2 `1)) <= |.(p1 - p2).| & abs ((p1 `2) - (p2 `2)) <= |.(p1 - p2).| )
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( abs ((p1 `1) - (p2 `1)) <= |.(p1 - p2).| & abs ((p1 `2) - (p2 `2)) <= |.(p1 - p2).| )
( (p1 `1) - (p2 `1) = (p1 - p2) `1 & (p1 `2) - (p2 `2) = (p1 - p2) `2 ) by TOPREAL3:3;
hence ( abs ((p1 `1) - (p2 `1)) <= |.(p1 - p2).| & abs ((p1 `2) - (p2 `2)) <= |.(p1 - p2).| ) by Th33; ::_thesis: verum
end;
theorem Th35: :: JGRAPH_1:35
for n being Element of NAT
for p, p1, p2 being Point of (TOP-REAL n) st p in LSeg (p1,p2) holds
ex r being Real st
( 0 <= r & r <= 1 & p = ((1 - r) * p1) + (r * p2) )
proof
let n be Element of NAT ; ::_thesis: for p, p1, p2 being Point of (TOP-REAL n) st p in LSeg (p1,p2) holds
ex r being Real st
( 0 <= r & r <= 1 & p = ((1 - r) * p1) + (r * p2) )
let p, p1, p2 be Point of (TOP-REAL n); ::_thesis: ( p in LSeg (p1,p2) implies ex r being Real st
( 0 <= r & r <= 1 & p = ((1 - r) * p1) + (r * p2) ) )
assume p in LSeg (p1,p2) ; ::_thesis: ex r being Real st
( 0 <= r & r <= 1 & p = ((1 - r) * p1) + (r * p2) )
then ex r1 being Real st
( p = ((1 - r1) * p1) + (r1 * p2) & 0 <= r1 & r1 <= 1 ) ;
hence ex r being Real st
( 0 <= r & r <= 1 & p = ((1 - r) * p1) + (r * p2) ) ; ::_thesis: verum
end;
theorem Th36: :: JGRAPH_1:36
for n being Element of 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).| )
proof
let n be Element of NAT ; ::_thesis: 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).| )
let p, p1, p2 be Point of (TOP-REAL n); ::_thesis: ( p in LSeg (p1,p2) implies ( |.(p - p1).| <= |.(p1 - p2).| & |.(p - p2).| <= |.(p1 - p2).| ) )
assume A1: p in LSeg (p1,p2) ; ::_thesis: ( |.(p - p1).| <= |.(p1 - p2).| & |.(p - p2).| <= |.(p1 - p2).| )
then consider r being Real such that
A2: 0 <= r and
A3: r <= 1 and
A4: p = ((1 - r) * p1) + (r * p2) by Th35;
A5: 0 <= 1 - r by A3, XREAL_1:48;
p - p1 = (((1 - r) * p1) - p1) + (r * p2) by A4, EUCLID:26
.= (((1 - r) * p1) - (1 * p1)) + (r * p2) by EUCLID:29
.= (((1 + (- r)) - 1) * p1) + (r * p2) by EUCLID:50
.= (r * p2) + (- (r * p1)) by EUCLID:40
.= (r * p2) + (r * (- p1)) by EUCLID:40
.= r * (p2 - p1) by EUCLID:32 ;
then |.(p - p1).| = (abs r) * |.(p2 - p1).| by TOPRNS_1:7
.= (abs r) * |.(p1 - p2).| by TOPRNS_1:27
.= r * |.(p1 - p2).| by A2, ABSVALUE:def_1 ;
then A6: |.(p1 - p2).| - |.(p - p1).| = (1 - r) * |.(p1 - p2).| ;
consider r being Real such that
A7: 0 <= r and
A8: r <= 1 and
A9: p = ((1 - r) * p2) + (r * p1) by A1, Th35;
p - p2 = (((1 - r) * p2) + (- p2)) + (r * p1) by A9, EUCLID:26
.= (((1 - r) * p2) - (1 * p2)) + (r * p1) by EUCLID:29
.= (((1 + (- r)) - 1) * p2) + (r * p1) by EUCLID:50
.= (r * p1) + (- (r * p2)) by EUCLID:40
.= (r * p1) + (r * (- p2)) by EUCLID:40
.= r * (p1 - p2) by EUCLID:32 ;
then |.(p - p2).| = (abs r) * |.(p1 - p2).| by TOPRNS_1:7
.= r * |.(p1 - p2).| by A7, ABSVALUE:def_1 ;
then A10: |.(p1 - p2).| - |.(p - p2).| = (1 - r) * |.(p1 - p2).| ;
0 <= 1 - r by A8, XREAL_1:48;
hence ( |.(p - p1).| <= |.(p1 - p2).| & |.(p - p2).| <= |.(p1 - p2).| ) by A6, A5, A10, XREAL_1:49; ::_thesis: verum
end;
begin
theorem Th37: :: 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
proof
let M be non empty MetrSpace; ::_thesis: for P, Q being Subset of (TopSpaceMetr M) st P <> {} & P is compact & Q <> {} & Q is compact holds
min_dist_min (P,Q) >= 0
let P, Q be Subset of (TopSpaceMetr M); ::_thesis: ( P <> {} & P is compact & Q <> {} & Q is compact implies min_dist_min (P,Q) >= 0 )
assume ( P <> {} & P is compact & Q <> {} & Q is compact ) ; ::_thesis: min_dist_min (P,Q) >= 0
then ex x1, x2 being Point of M st
( x1 in P & x2 in Q & dist (x1,x2) = min_dist_min (P,Q) ) by WEIERSTR:30;
hence min_dist_min (P,Q) >= 0 by METRIC_1:5; ::_thesis: verum
end;
theorem Th38: :: 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 )
proof
let M be non empty MetrSpace; ::_thesis: 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 )
let P, Q be Subset of (TopSpaceMetr M); ::_thesis: ( P <> {} & P is compact & Q <> {} & Q is compact implies ( P misses Q iff min_dist_min (P,Q) > 0 ) )
assume that
A1: P <> {} and
A2: P is compact and
A3: Q <> {} and
A4: Q is compact ; ::_thesis: ( P misses Q iff min_dist_min (P,Q) > 0 )
A5: now__::_thesis:_(_P_/\_Q_<>_{}_implies_not_min_dist_min_(P,Q)_>_0_)
set p = the Element of P /\ Q;
assume A6: P /\ Q <> {} ; ::_thesis: not min_dist_min (P,Q) > 0
then A7: the Element of P /\ Q in P by XBOOLE_0:def_4;
then reconsider p9 = the Element of P /\ Q as Element of (TopSpaceMetr M) ;
reconsider q = p9 as Point of M by TOPMETR:12;
the distance of M is Reflexive by METRIC_1:def_6;
then the distance of M . (q,q) = 0 by METRIC_1:def_2;
then A8: dist (q,q) = 0 by METRIC_1:def_1;
the Element of P /\ Q in Q by A6, XBOOLE_0:def_4;
hence not min_dist_min (P,Q) > 0 by A2, A4, A7, A8, WEIERSTR:34; ::_thesis: verum
end;
consider x1, x2 being Point of M such that
A9: ( x1 in P & x2 in Q ) and
A10: dist (x1,x2) = min_dist_min (P,Q) by A1, A2, A3, A4, WEIERSTR:30;
A11: the distance of M is discerning by METRIC_1:def_7;
now__::_thesis:_(_not_min_dist_min_(P,Q)_>_0_implies_P_/\_Q_<>_{}_)
assume not min_dist_min (P,Q) > 0 ; ::_thesis: P /\ Q <> {}
then dist (x1,x2) = 0 by A1, A2, A3, A4, A10, Th37;
then the distance of M . (x1,x2) = 0 by METRIC_1:def_1;
then x1 = x2 by A11, METRIC_1:def_3;
hence P /\ Q <> {} by A9, XBOOLE_0:def_4; ::_thesis: verum
end;
hence ( P misses Q iff min_dist_min (P,Q) > 0 ) by A5, XBOOLE_0:def_7; ::_thesis: verum
end;
theorem Th39: :: 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 Element of 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 Element of NAT st j in dom g holds
ex k being Element of NAT st
( k in dom f & |.((g /. j) - (f /. k)).| < a ) ) & ( for j being Element of NAT st 1 <= j & j + 1 <= len g holds
|.((g /. j) - (g /. (j + 1))).| < a ) )
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: 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 Element of 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 Element of NAT st j in dom g holds
ex k being Element of NAT st
( k in dom f & |.((g /. j) - (f /. k)).| < a ) ) & ( for j being Element of NAT st 1 <= j & j + 1 <= len g holds
|.((g /. j) - (g /. (j + 1))).| < a ) )
let a, c, d be Real; ::_thesis: ( 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 Element of NAT st 1 <= i & i + 1 <= len f holds
|.((f /. i) - (f /. (i + 1))).| < a ) implies 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 Element of NAT st j in dom g holds
ex k being Element of NAT st
( k in dom f & |.((g /. j) - (f /. k)).| < a ) ) & ( for j being Element of NAT st 1 <= j & j + 1 <= len g holds
|.((g /. j) - (g /. (j + 1))).| < a ) ) )
assume that
A1: 1 <= len f and
A2: X_axis f lies_between (X_axis f) . 1,(X_axis f) . (len f) and
A3: Y_axis f lies_between c,d and
A4: a > 0 and
A5: for i being Element of NAT st 1 <= i & i + 1 <= len f holds
|.((f /. i) - (f /. (i + 1))).| < a ; ::_thesis: 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 Element of NAT st j in dom g holds
ex k being Element of NAT st
( k in dom f & |.((g /. j) - (f /. k)).| < a ) ) & ( for j being Element of NAT st 1 <= j & j + 1 <= len g holds
|.((g /. j) - (g /. (j + 1))).| < a ) )
A6: f . (len f) = f /. (len f) by A1, FINSEQ_4:15;
defpred S1[ set , set ] means for j being Element of NAT st ( $1 = 2 * j or $1 = (2 * j) -' 1 ) holds
( ( $1 = 2 * j implies $2 = |[((f /. j) `1),((f /. (j + 1)) `2)]| ) & ( $1 = (2 * j) -' 1 implies $2 = f /. j ) );
A7: for k being Nat st k in Seg ((2 * (len f)) -' 1) holds
ex x being set st S1[k,x]
proof
let k be Nat; ::_thesis: ( k in Seg ((2 * (len f)) -' 1) implies ex x being set st S1[k,x] )
assume A8: k in Seg ((2 * (len f)) -' 1) ; ::_thesis: ex x being set st S1[k,x]
then A9: 1 <= k by FINSEQ_1:1;
percases ( k mod 2 = 0 or k mod 2 = 1 ) by NAT_D:12;
supposeA10: k mod 2 = 0 ; ::_thesis: ex x being set st S1[k,x]
consider i being Nat such that
A11: k = (2 * i) + (k mod 2) and
k mod 2 < 2 by NAT_D:def_2;
for j being Element of NAT st ( k = 2 * j or k = (2 * j) -' 1 ) holds
( ( k = 2 * j implies |[((f /. i) `1),((f /. (i + 1)) `2)]| = |[((f /. j) `1),((f /. (j + 1)) `2)]| ) & ( k = (2 * j) -' 1 implies |[((f /. i) `1),((f /. (i + 1)) `2)]| = f /. j ) )
proof
let j be Element of NAT ; ::_thesis: ( ( k = 2 * j or k = (2 * j) -' 1 ) implies ( ( k = 2 * j implies |[((f /. i) `1),((f /. (i + 1)) `2)]| = |[((f /. j) `1),((f /. (j + 1)) `2)]| ) & ( k = (2 * j) -' 1 implies |[((f /. i) `1),((f /. (i + 1)) `2)]| = f /. j ) ) )
assume ( k = 2 * j or k = (2 * j) -' 1 ) ; ::_thesis: ( ( k = 2 * j implies |[((f /. i) `1),((f /. (i + 1)) `2)]| = |[((f /. j) `1),((f /. (j + 1)) `2)]| ) & ( k = (2 * j) -' 1 implies |[((f /. i) `1),((f /. (i + 1)) `2)]| = f /. j ) )
now__::_thesis:_not_k_=_(2_*_j)_-'_1
assume A12: k = (2 * j) -' 1 ; ::_thesis: contradiction
now__::_thesis:_not_j_=_0
0 - 1 < 0 ;
then A13: 0 -' 1 = 0 by XREAL_0:def_2;
assume j = 0 ; ::_thesis: contradiction
hence contradiction by A8, A12, A13, FINSEQ_1:1; ::_thesis: verum
end;
then A14: j >= 0 + 1 by NAT_1:13;
k = ((2 * (j - 1)) + (1 + 1)) - 1 by A9, A12, NAT_D:39
.= (2 * (j - 1)) + 1 ;
then k = (2 * (j -' 1)) + 1 by A14, XREAL_1:233;
hence contradiction by A10, NAT_D:def_2; ::_thesis: verum
end;
hence ( ( k = 2 * j implies |[((f /. i) `1),((f /. (i + 1)) `2)]| = |[((f /. j) `1),((f /. (j + 1)) `2)]| ) & ( k = (2 * j) -' 1 implies |[((f /. i) `1),((f /. (i + 1)) `2)]| = f /. j ) ) by A10, A11; ::_thesis: verum
end;
hence ex x being set st S1[k,x] ; ::_thesis: verum
end;
supposeA15: k mod 2 = 1 ; ::_thesis: ex x being set st S1[k,x]
consider i being Nat such that
A16: k = (2 * i) + (k mod 2) and
A17: k mod 2 < 2 by NAT_D:def_2;
for j being Element of NAT st ( k = 2 * j or k = (2 * j) -' 1 ) holds
( ( k = 2 * j implies f /. (i + 1) = |[((f /. j) `1),((f /. (j + 1)) `2)]| ) & ( k = (2 * j) -' 1 implies f /. (i + 1) = f /. j ) )
proof
let j be Element of NAT ; ::_thesis: ( ( k = 2 * j or k = (2 * j) -' 1 ) implies ( ( k = 2 * j implies f /. (i + 1) = |[((f /. j) `1),((f /. (j + 1)) `2)]| ) & ( k = (2 * j) -' 1 implies f /. (i + 1) = f /. j ) ) )
assume A18: ( k = 2 * j or k = (2 * j) -' 1 ) ; ::_thesis: ( ( k = 2 * j implies f /. (i + 1) = |[((f /. j) `1),((f /. (j + 1)) `2)]| ) & ( k = (2 * j) -' 1 implies f /. (i + 1) = f /. j ) )
percases ( k = (2 * j) -' 1 or k = 2 * j ) by A18;
supposeA19: k = (2 * j) -' 1 ; ::_thesis: ( ( k = 2 * j implies f /. (i + 1) = |[((f /. j) `1),((f /. (j + 1)) `2)]| ) & ( k = (2 * j) -' 1 implies f /. (i + 1) = f /. j ) )
A20: now__::_thesis:_(_k_=_(2_*_j)_-'_1_implies_f_/._(i_+_1)_=_f_/._j_)
assume k = (2 * j) -' 1 ; ::_thesis: f /. (i + 1) = f /. j
then k = ((2 * (j - 1)) + (1 + 1)) - 1 by A9, NAT_D:39
.= (2 * (j - 1)) + 1 ;
hence f /. (i + 1) = f /. j by A15, A16; ::_thesis: verum
end;
k = (2 * j) - 1 by A9, A19, NAT_D:39;
hence ( ( k = 2 * j implies f /. (i + 1) = |[((f /. j) `1),((f /. (j + 1)) `2)]| ) & ( k = (2 * j) -' 1 implies f /. (i + 1) = f /. j ) ) by A20; ::_thesis: verum
end;
supposeA21: k = 2 * j ; ::_thesis: ( ( k = 2 * j implies f /. (i + 1) = |[((f /. j) `1),((f /. (j + 1)) `2)]| ) & ( k = (2 * j) -' 1 implies f /. (i + 1) = f /. j ) )
then A22: 2 * (j - i) = 1 by A15, A16;
then j - i >= 0 ;
then A23: j - i = j -' i by XREAL_0:def_2;
( j - i = 0 or j - i > 0 ) by A22;
then j - i >= 0 + 1 by A15, A16, A21, A23, NAT_1:13;
then 2 * (j - i) >= 2 * 1 by XREAL_1:64;
hence ( ( k = 2 * j implies f /. (i + 1) = |[((f /. j) `1),((f /. (j + 1)) `2)]| ) & ( k = (2 * j) -' 1 implies f /. (i + 1) = f /. j ) ) by A16, A17, A21; ::_thesis: verum
end;
end;
end;
hence ex x being set st S1[k,x] ; ::_thesis: verum
end;
end;
end;
ex p being FinSequence st
( dom p = Seg ((2 * (len f)) -' 1) & ( for k being Nat st k in Seg ((2 * (len f)) -' 1) holds
S1[k,p . k] ) ) from FINSEQ_1:sch_1(A7);
then consider p being FinSequence such that
A24: dom p = Seg ((2 * (len f)) -' 1) and
A25: for k being Nat st k in Seg ((2 * (len f)) -' 1) holds
for j being Element of NAT st ( k = 2 * j or k = (2 * j) -' 1 ) holds
( ( k = 2 * j implies p . k = |[((f /. j) `1),((f /. (j + 1)) `2)]| ) & ( k = (2 * j) -' 1 implies p . k = f /. j ) ) ;
rng p c= the carrier of (TOP-REAL 2)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng p or y in the carrier of (TOP-REAL 2) )
assume y in rng p ; ::_thesis: y in the carrier of (TOP-REAL 2)
then consider x being set such that
A26: x in dom p and
A27: y = p . x by FUNCT_1:def_3;
reconsider i = x as Element of NAT by A26;
x in Seg (len p) by A26, FINSEQ_1:def_3;
then A28: 1 <= i by FINSEQ_1:1;
percases ( i mod 2 = 0 or i mod 2 = 1 ) by NAT_D:12;
supposeA29: i mod 2 = 0 ; ::_thesis: y in the carrier of (TOP-REAL 2)
consider j being Nat such that
A30: i = (2 * j) + (i mod 2) and
i mod 2 < 2 by NAT_D:def_2;
reconsider j = j as Element of NAT by ORDINAL1:def_12;
p . i = |[((f /. j) `1),((f /. (j + 1)) `2)]| by A24, A25, A26, A29, A30;
hence y in the carrier of (TOP-REAL 2) by A27; ::_thesis: verum
end;
supposeA31: i mod 2 = 1 ; ::_thesis: y in the carrier of (TOP-REAL 2)
consider j being Nat such that
A32: i = (2 * j) + (i mod 2) and
i mod 2 < 2 by NAT_D:def_2;
reconsider j = j as Element of NAT by ORDINAL1:def_12;
(2 * (j + 1)) -' 1 = (2 * (j + 1)) - 1 by A28, A31, A32, NAT_D:39;
then p . i = f /. (j + 1) by A24, A25, A26, A31, A32;
hence y in the carrier of (TOP-REAL 2) by A27; ::_thesis: verum
end;
end;
end;
then reconsider g1 = p as FinSequence of (TOP-REAL 2) by FINSEQ_1:def_4;
A33: len p = (2 * (len f)) -' 1 by A24, FINSEQ_1:def_3;
for i being Nat st 1 <= i & i + 1 <= len g1 & not (g1 /. i) `1 = (g1 /. (i + 1)) `1 holds
(g1 /. i) `2 = (g1 /. (i + 1)) `2
proof
let i be Nat; ::_thesis: ( 1 <= i & i + 1 <= len g1 & not (g1 /. i) `1 = (g1 /. (i + 1)) `1 implies (g1 /. i) `2 = (g1 /. (i + 1)) `2 )
assume that
A34: 1 <= i and
A35: i + 1 <= len g1 ; ::_thesis: ( (g1 /. i) `1 = (g1 /. (i + 1)) `1 or (g1 /. i) `2 = (g1 /. (i + 1)) `2 )
A36: i < len g1 by A35, NAT_1:13;
then A37: g1 . i = g1 /. i by A34, FINSEQ_4:15;
A38: 1 < i + 1 by A34, NAT_1:13;
then A39: i + 1 in Seg (len g1) by A35, FINSEQ_1:1;
A40: i in Seg ((2 * (len f)) -' 1) by A33, A34, A36, FINSEQ_1:1;
A41: g1 . (i + 1) = g1 /. (i + 1) by A35, A38, FINSEQ_4:15;
percases ( i mod 2 = 0 or i mod 2 = 1 ) by NAT_D:12;
supposeA42: i mod 2 = 0 ; ::_thesis: ( (g1 /. i) `1 = (g1 /. (i + 1)) `1 or (g1 /. i) `2 = (g1 /. (i + 1)) `2 )
consider j being Nat such that
A43: i = (2 * j) + (i mod 2) and
i mod 2 < 2 by NAT_D:def_2;
reconsider j = j as Element of NAT by ORDINAL1:def_12;
1 <= 1 + i by NAT_1:11;
then (2 * (j + 1)) -' 1 = (2 * (j + 1)) - 1 by A42, A43, NAT_D:39;
then A44: g1 . (i + 1) = f /. (j + 1) by A25, A33, A39, A42, A43;
g1 . i = |[((f /. j) `1),((f /. (j + 1)) `2)]| by A25, A40, A42, A43;
hence ( (g1 /. i) `1 = (g1 /. (i + 1)) `1 or (g1 /. i) `2 = (g1 /. (i + 1)) `2 ) by A37, A41, A44, EUCLID:52; ::_thesis: verum
end;
supposeA45: i mod 2 = 1 ; ::_thesis: ( (g1 /. i) `1 = (g1 /. (i + 1)) `1 or (g1 /. i) `2 = (g1 /. (i + 1)) `2 )
consider j being Nat such that
A46: i = (2 * j) + (i mod 2) and
i mod 2 < 2 by NAT_D:def_2;
reconsider j = j as Element of NAT by ORDINAL1:def_12;
i + 1 = 2 * (j + 1) by A45, A46;
then A47: g1 . (i + 1) = |[((f /. (j + 1)) `1),((f /. ((j + 1) + 1)) `2)]| by A25, A33, A39;
(2 * (j + 1)) -' 1 = (2 * (j + 1)) - 1 by A34, A45, A46, NAT_D:39;
then g1 . i = f /. (j + 1) by A25, A40, A45, A46;
hence ( (g1 /. i) `1 = (g1 /. (i + 1)) `1 or (g1 /. i) `2 = (g1 /. (i + 1)) `2 ) by A37, A41, A47, EUCLID:52; ::_thesis: verum
end;
end;
end;
then A48: g1 is special by TOPREAL1:def_5;
A49: 2 * (len f) >= 2 * 1 by A1, XREAL_1:64;
then A50: (2 * (len f)) -' 1 = (2 * (len f)) - 1 by XREAL_1:233, XXREAL_0:2;
for i being Element of NAT st i in dom (Y_axis g1) holds
( c <= (Y_axis g1) . i & (Y_axis g1) . i <= d )
proof
let i be Element of NAT ; ::_thesis: ( i in dom (Y_axis g1) implies ( c <= (Y_axis g1) . i & (Y_axis g1) . i <= d ) )
A51: len (Y_axis f) = len f by GOBOARD1:def_2;
assume A52: i in dom (Y_axis g1) ; ::_thesis: ( c <= (Y_axis g1) . i & (Y_axis g1) . i <= d )
then A53: i in Seg (len (Y_axis g1)) by FINSEQ_1:def_3;
then A54: i in Seg (len g1) by GOBOARD1:def_2;
i in Seg (len g1) by A53, GOBOARD1:def_2;
then A55: i <= len g1 by FINSEQ_1:1;
A56: 1 <= i by A53, FINSEQ_1:1;
then A57: g1 /. i = g1 . i by A55, FINSEQ_4:15;
A58: (Y_axis g1) . i = (g1 /. i) `2 by A52, GOBOARD1:def_2;
percases ( i mod 2 = 0 or i mod 2 = 1 ) by NAT_D:12;
supposeA59: i mod 2 = 0 ; ::_thesis: ( c <= (Y_axis g1) . i & (Y_axis g1) . i <= d )
consider j being Nat such that
A60: i = (2 * j) + (i mod 2) and
i mod 2 < 2 by NAT_D:def_2;
reconsider j = j as Element of NAT by ORDINAL1:def_12;
g1 . i = |[((f /. j) `1),((f /. (j + 1)) `2)]| by A25, A33, A54, A59, A60;
then A61: (g1 /. i) `2 = (f /. (j + 1)) `2 by A57, EUCLID:52;
(2 * j) + 1 <= ((2 * (len f)) - 1) + 1 by A33, A50, A55, A59, A60, XREAL_1:6;
then 2 * j < 2 * (len f) by NAT_1:13;
then (2 * j) / 2 < (2 * (len f)) / 2 by XREAL_1:74;
then ( 1 <= j + 1 & j + 1 <= len f ) by NAT_1:11, NAT_1:13;
then j + 1 in Seg (len f) by FINSEQ_1:1;
then A62: j + 1 in dom (Y_axis f) by A51, FINSEQ_1:def_3;
then (Y_axis f) . (j + 1) = (f /. (j + 1)) `2 by GOBOARD1:def_2;
hence ( c <= (Y_axis g1) . i & (Y_axis g1) . i <= d ) by A3, A58, A61, A62, GOBOARD4:def_2; ::_thesis: verum
end;
supposeA63: i mod 2 = 1 ; ::_thesis: ( c <= (Y_axis g1) . i & (Y_axis g1) . i <= d )
consider j being Nat such that
A64: i = (2 * j) + (i mod 2) and
i mod 2 < 2 by NAT_D:def_2;
reconsider j = j as Element of NAT by ORDINAL1:def_12;
(2 * (j + 1)) -' 1 = (2 * (j + 1)) - 1 by A56, A63, A64, NAT_D:39;
then A65: (g1 /. i) `2 = (f /. (j + 1)) `2 by A25, A33, A54, A57, A63, A64;
(2 * j) + 1 <= ((2 * (len f)) - 1) + 1 by A33, A50, A55, A63, A64, NAT_1:13;
then 2 * j < 2 * (len f) by NAT_1:13;
then (2 * j) / 2 < (2 * (len f)) / 2 by XREAL_1:74;
then ( 1 <= j + 1 & j + 1 <= len f ) by NAT_1:11, NAT_1:13;
then j + 1 in Seg (len f) by FINSEQ_1:1;
then A66: j + 1 in dom (Y_axis f) by A51, FINSEQ_1:def_3;
then (Y_axis f) . (j + 1) = (f /. (j + 1)) `2 by GOBOARD1:def_2;
hence ( c <= (Y_axis g1) . i & (Y_axis g1) . i <= d ) by A3, A58, A65, A66, GOBOARD4:def_2; ::_thesis: verum
end;
end;
end;
then A67: Y_axis g1 lies_between c,d by GOBOARD4:def_2;
for i being Element of NAT st i in dom (X_axis g1) holds
( (X_axis f) . 1 <= (X_axis g1) . i & (X_axis g1) . i <= (X_axis f) . (len f) )
proof
let i be Element of NAT ; ::_thesis: ( i in dom (X_axis g1) implies ( (X_axis f) . 1 <= (X_axis g1) . i & (X_axis g1) . i <= (X_axis f) . (len f) ) )
A68: len (X_axis f) = len f by GOBOARD1:def_1;
assume A69: i in dom (X_axis g1) ; ::_thesis: ( (X_axis f) . 1 <= (X_axis g1) . i & (X_axis g1) . i <= (X_axis f) . (len f) )
then A70: i in Seg (len (X_axis g1)) by FINSEQ_1:def_3;
then A71: i in Seg (len g1) by GOBOARD1:def_1;
i in Seg (len g1) by A70, GOBOARD1:def_1;
then A72: i <= len g1 by FINSEQ_1:1;
A73: 1 <= i by A70, FINSEQ_1:1;
then A74: g1 /. i = g1 . i by A72, FINSEQ_4:15;
A75: (X_axis g1) . i = (g1 /. i) `1 by A69, GOBOARD1:def_1;
percases ( i mod 2 = 0 or i mod 2 = 1 ) by NAT_D:12;
supposeA76: i mod 2 = 0 ; ::_thesis: ( (X_axis f) . 1 <= (X_axis g1) . i & (X_axis g1) . i <= (X_axis f) . (len f) )
consider j being Nat such that
A77: i = (2 * j) + (i mod 2) and
i mod 2 < 2 by NAT_D:def_2;
reconsider j = j as Element of NAT by ORDINAL1:def_12;
g1 . i = |[((f /. j) `1),((f /. (j + 1)) `2)]| by A25, A33, A71, A76, A77;
then A78: (g1 /. i) `1 = (f /. j) `1 by A74, EUCLID:52;
(2 * j) + 1 <= ((2 * (len f)) - 1) + 1 by A33, A50, A72, A76, A77, XREAL_1:6;
then 2 * j < 2 * (len f) by NAT_1:13;
then A79: (2 * j) / 2 < (2 * (len f)) / 2 by XREAL_1:74;
j > 0 by A70, A76, A77, FINSEQ_1:1;
then j >= 0 + 1 by NAT_1:13;
then j in Seg (len f) by A79, FINSEQ_1:1;
then A80: j in dom (X_axis f) by A68, FINSEQ_1:def_3;
then (X_axis f) . j = (f /. j) `1 by GOBOARD1:def_1;
hence ( (X_axis f) . 1 <= (X_axis g1) . i & (X_axis g1) . i <= (X_axis f) . (len f) ) by A2, A75, A78, A80, GOBOARD4:def_2; ::_thesis: verum
end;
supposeA81: i mod 2 = 1 ; ::_thesis: ( (X_axis f) . 1 <= (X_axis g1) . i & (X_axis g1) . i <= (X_axis f) . (len f) )
consider j being Nat such that
A82: i = (2 * j) + (i mod 2) and
i mod 2 < 2 by NAT_D:def_2;
reconsider j = j as Element of NAT by ORDINAL1:def_12;
(2 * (j + 1)) -' 1 = (2 * (j + 1)) - 1 by A73, A81, A82, NAT_D:39;
then A83: (g1 /. i) `1 = (f /. (j + 1)) `1 by A25, A33, A71, A74, A81, A82;
(2 * j) + 1 <= ((2 * (len f)) - 1) + 1 by A33, A50, A72, A81, A82, NAT_1:13;
then 2 * j < 2 * (len f) by NAT_1:13;
then (2 * j) / 2 < (2 * (len f)) / 2 by XREAL_1:74;
then ( 1 <= j + 1 & j + 1 <= len f ) by NAT_1:11, NAT_1:13;
then j + 1 in Seg (len f) by FINSEQ_1:1;
then A84: j + 1 in dom (X_axis f) by A68, FINSEQ_1:def_3;
then (X_axis f) . (j + 1) = (f /. (j + 1)) `1 by GOBOARD1:def_1;
hence ( (X_axis f) . 1 <= (X_axis g1) . i & (X_axis g1) . i <= (X_axis f) . (len f) ) by A2, A75, A83, A84, GOBOARD4:def_2; ::_thesis: verum
end;
end;
end;
then A85: X_axis g1 lies_between (X_axis f) . 1,(X_axis f) . (len f) by GOBOARD4:def_2;
(len f) + (len f) >= (len f) + 1 by A1, XREAL_1:6;
then A86: (2 * (len f)) - 1 >= ((len f) + 1) - 1 by XREAL_1:9;
A87: (2 * 1) -' 1 = (1 + 1) -' 1
.= 1 by NAT_D:34 ;
A88: (2 * (len f)) - 1 >= (1 + 1) - 1 by A49, XREAL_1:9;
then 1 in Seg ((2 * (len f)) -' 1) by A50, FINSEQ_1:1;
then A89: p . 1 = f /. 1 by A25, A87;
A90: for i being Element of NAT st 1 <= i & i + 1 <= len g1 holds
|.((g1 /. i) - (g1 /. (i + 1))).| < a
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i + 1 <= len g1 implies |.((g1 /. i) - (g1 /. (i + 1))).| < a )
assume that
A91: 1 <= i and
A92: i + 1 <= len g1 ; ::_thesis: |.((g1 /. i) - (g1 /. (i + 1))).| < a
A93: g1 . (i + 1) = g1 /. (i + 1) by A92, FINSEQ_4:15, NAT_1:11;
i <= len g1 by A92, NAT_1:13;
then A94: i in Seg (len g1) by A91, FINSEQ_1:1;
i <= len g1 by A92, NAT_1:13;
then A95: g1 . i = g1 /. i by A91, FINSEQ_4:15;
1 <= i + 1 by NAT_1:11;
then A96: i + 1 in Seg ((2 * (len f)) -' 1) by A33, A92, FINSEQ_1:1;
percases ( i mod 2 = 0 or i mod 2 = 1 ) by NAT_D:12;
supposeA97: i mod 2 = 0 ; ::_thesis: |.((g1 /. i) - (g1 /. (i + 1))).| < a
consider j being Nat such that
A98: i = (2 * j) + (i mod 2) and
i mod 2 < 2 by NAT_D:def_2;
reconsider j = j as Element of NAT by ORDINAL1:def_12;
A99: g1 . i = |[((f /. j) `1),((f /. (j + 1)) `2)]| by A25, A33, A94, A97, A98;
then A100: (g1 /. i) `2 = (f /. (j + 1)) `2 by A95, EUCLID:52;
1 <= 1 + i by NAT_1:11;
then (2 * (j + 1)) -' 1 = (2 * (j + 1)) - 1 by A97, A98, NAT_D:39;
then g1 . (i + 1) = f /. (j + 1) by A25, A96, A97, A98;
then A101: ( (g1 /. (i + 1)) `1 = (f /. (j + 1)) `1 & (g1 /. (i + 1)) `2 = (f /. (j + 1)) `2 ) by A92, FINSEQ_4:15, NAT_1:11;
A102: (g1 /. i) - (g1 /. (i + 1)) = |[(((g1 /. i) `1) - ((g1 /. (i + 1)) `1)),(((g1 /. i) `2) - ((g1 /. (i + 1)) `2))]| by EUCLID:61
.= |[(((f /. j) `1) - ((f /. (j + 1)) `1)),0]| by A95, A99, A100, A101, EUCLID:52 ;
then A103: ((g1 /. i) - (g1 /. (i + 1))) `1 = ((f /. j) `1) - ((f /. (j + 1)) `1) by EUCLID:52;
(2 * j) + 1 <= ((2 * (len f)) - 1) + 1 by A33, A50, A92, A97, A98, NAT_1:13;
then 2 * j < 2 * (len f) by NAT_1:13;
then (2 * j) / 2 < (2 * (len f)) / 2 by XREAL_1:74;
then A104: j + 1 <= len f by NAT_1:13;
|.((g1 /. i) - (g1 /. (i + 1))).| = sqrt (((((g1 /. i) - (g1 /. (i + 1))) `1) ^2) + ((((g1 /. i) - (g1 /. (i + 1))) `2) ^2)) by Th30
.= sqrt (((((f /. j) `1) - ((f /. (j + 1)) `1)) ^2) + (0 ^2)) by A102, A103, EUCLID:52
.= sqrt ((((f /. j) `1) - ((f /. (j + 1)) `1)) ^2) ;
then |.((g1 /. i) - (g1 /. (i + 1))).| = abs (((f /. j) `1) - ((f /. (j + 1)) `1)) by COMPLEX1:72;
then A105: |.((g1 /. i) - (g1 /. (i + 1))).| <= |.((f /. j) - (f /. (j + 1))).| by Th34;
j > 0 by A91, A97, A98;
then j >= 0 + 1 by NAT_1:13;
then |.((f /. j) - (f /. (j + 1))).| < a by A5, A104;
hence |.((g1 /. i) - (g1 /. (i + 1))).| < a by A105, XXREAL_0:2; ::_thesis: verum
end;
supposeA106: i mod 2 = 1 ; ::_thesis: |.((g1 /. i) - (g1 /. (i + 1))).| < a
consider j being Nat such that
A107: i = (2 * j) + (i mod 2) and
i mod 2 < 2 by NAT_D:def_2;
reconsider j = j as Element of NAT by ORDINAL1:def_12;
(2 * (j + 1)) -' 1 = (2 * (j + 1)) - 1 by A91, A106, A107, NAT_D:39;
then A108: ( (g1 /. i) `1 = (f /. (j + 1)) `1 & (g1 /. i) `2 = (f /. (j + 1)) `2 ) by A25, A33, A94, A95, A106, A107;
i + 1 = 2 * (j + 1) by A106, A107;
then A109: g1 /. (i + 1) = |[((f /. (j + 1)) `1),((f /. ((j + 1) + 1)) `2)]| by A25, A96, A93;
then A110: (g1 /. (i + 1)) `1 = (f /. (j + 1)) `1 by EUCLID:52;
A111: (g1 /. i) - (g1 /. (i + 1)) = |[(((g1 /. i) `1) - ((g1 /. (i + 1)) `1)),(((g1 /. i) `2) - ((g1 /. (i + 1)) `2))]| by EUCLID:61
.= |[0,(((f /. (j + 1)) `2) - ((f /. ((j + 1) + 1)) `2))]| by A109, A110, A108, EUCLID:52 ;
then A112: ((g1 /. i) - (g1 /. (i + 1))) `1 = 0 by EUCLID:52;
(2 * (j + 1)) + 1 <= ((2 * (len f)) - 1) + 1 by A33, A50, A92, A106, A107, XREAL_1:6;
then 2 * (j + 1) < 2 * (len f) by NAT_1:13;
then (2 * (j + 1)) / 2 < (2 * (len f)) / 2 by XREAL_1:74;
then (j + 1) + 1 <= len f by NAT_1:13;
then A113: |.((f /. (j + 1)) - (f /. ((j + 1) + 1))).| < a by A5, NAT_1:11;
|.((g1 /. i) - (g1 /. (i + 1))).| = sqrt (((((g1 /. i) - (g1 /. (i + 1))) `1) ^2) + ((((g1 /. i) - (g1 /. (i + 1))) `2) ^2)) by Th30
.= sqrt ((((f /. (j + 1)) `2) - ((f /. ((j + 1) + 1)) `2)) ^2) by A111, A112, EUCLID:52 ;
then |.((g1 /. i) - (g1 /. (i + 1))).| = abs (((f /. (j + 1)) `2) - ((f /. ((j + 1) + 1)) `2)) by COMPLEX1:72;
then |.((g1 /. i) - (g1 /. (i + 1))).| <= |.((f /. (j + 1)) - (f /. ((j + 1) + 1))).| by Th34;
hence |.((g1 /. i) - (g1 /. (i + 1))).| < a by A113, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
A114: for i being Element of NAT st i in dom g1 holds
ex k being Element of NAT st
( k in dom f & |.((g1 /. i) - (f /. k)).| < a )
proof
let i be Element of NAT ; ::_thesis: ( i in dom g1 implies ex k being Element of NAT st
( k in dom f & |.((g1 /. i) - (f /. k)).| < a ) )
assume A115: i in dom g1 ; ::_thesis: ex k being Element of NAT st
( k in dom f & |.((g1 /. i) - (f /. k)).| < a )
then A116: i in Seg (len g1) by FINSEQ_1:def_3;
then A117: i <= len g1 by FINSEQ_1:1;
A118: 1 <= i by A116, FINSEQ_1:1;
then A119: g1 . i = g1 /. i by A117, FINSEQ_4:15;
percases ( i mod 2 = 0 or i mod 2 = 1 ) by NAT_D:12;
supposeA120: i mod 2 = 0 ; ::_thesis: ex k being Element of NAT st
( k in dom f & |.((g1 /. i) - (f /. k)).| < a )
consider j being Nat such that
A121: i = (2 * j) + (i mod 2) and
i mod 2 < 2 by NAT_D:def_2;
reconsider j = j as Element of NAT by ORDINAL1:def_12;
j > 0 by A116, A120, A121, FINSEQ_1:1;
then A122: j >= 0 + 1 by NAT_1:13;
A123: g1 . i = |[((f /. j) `1),((f /. (j + 1)) `2)]| by A24, A25, A115, A120, A121;
then A124: (g1 /. i) `1 = (f /. j) `1 by A119, EUCLID:52;
A125: (g1 /. i) `2 = (f /. (j + 1)) `2 by A119, A123, EUCLID:52;
A126: (g1 /. i) - (f /. j) = |[(((g1 /. i) `1) - ((f /. j) `1)),(((g1 /. i) `2) - ((f /. j) `2))]| by EUCLID:61
.= |[0,(((g1 /. i) `2) - ((f /. j) `2))]| by A124 ;
then ((g1 /. i) - (f /. j)) `2 = ((g1 /. i) `2) - ((f /. j) `2) by EUCLID:52;
then |.((g1 /. i) - (f /. j)).| = sqrt (((((g1 /. i) - (f /. j)) `1) ^2) + ((((g1 /. i) `2) - ((f /. j) `2)) ^2)) by Th30
.= sqrt ((0 ^2) + ((((f /. (j + 1)) `2) - ((f /. j) `2)) ^2)) by A125, A126, EUCLID:52
.= sqrt ((((f /. (j + 1)) `2) - ((f /. j) `2)) ^2) ;
then |.((g1 /. i) - (f /. j)).| = abs (((f /. (j + 1)) `2) - ((f /. j) `2)) by COMPLEX1:72
.= abs (((f /. j) `2) - ((f /. (j + 1)) `2)) by UNIFORM1:11 ;
then A127: |.((g1 /. i) - (f /. j)).| <= |.((f /. j) - (f /. (j + 1))).| by Th34;
(2 * j) + 1 <= ((2 * (len f)) - 1) + 1 by A33, A50, A117, A120, A121, XREAL_1:6;
then 2 * j < 2 * (len f) by NAT_1:13;
then A128: (2 * j) / 2 < (2 * (len f)) / 2 by XREAL_1:74;
then j + 1 <= len f by NAT_1:13;
then |.((f /. j) - (f /. (j + 1))).| < a by A5, A122;
then A129: |.((g1 /. i) - (f /. j)).| < a by A127, XXREAL_0:2;
j in dom f by A122, A128, FINSEQ_3:25;
hence ex k being Element of NAT st
( k in dom f & |.((g1 /. i) - (f /. k)).| < a ) by A129; ::_thesis: verum
end;
supposeA130: i mod 2 = 1 ; ::_thesis: ex k being Element of NAT st
( k in dom f & |.((g1 /. i) - (f /. k)).| < a )
consider j being Nat such that
A131: i = (2 * j) + (i mod 2) and
i mod 2 < 2 by NAT_D:def_2;
reconsider j = j as Element of NAT by ORDINAL1:def_12;
(2 * (j + 1)) -' 1 = (2 * (j + 1)) - 1 by A118, A130, A131, NAT_D:39;
then g1 . i = f /. (j + 1) by A24, A25, A115, A130, A131;
then A132: |.((g1 /. i) - (f /. (j + 1))).| = |.(0. (TOP-REAL 2)).| by A119, EUCLID:42
.= 0 by TOPRNS_1:23 ;
((2 * j) + 1) + 1 <= ((2 * (len f)) - 1) + 1 by A33, A50, A117, A130, A131, XREAL_1:6;
then (2 * j) + 1 < 2 * (len f) by NAT_1:13;
then 2 * j < 2 * (len f) by NAT_1:13;
then (2 * j) / 2 < (2 * (len f)) / 2 by XREAL_1:74;
then A133: j + 1 <= len f by NAT_1:13;
1 <= j + 1 by NAT_1:11;
then j + 1 in dom f by A133, FINSEQ_3:25;
hence ex k being Element of NAT st
( k in dom f & |.((g1 /. i) - (f /. k)).| < a ) by A4, A132; ::_thesis: verum
end;
end;
end;
(2 * (len f)) -' 1 in Seg ((2 * (len f)) -' 1) by A50, A88, FINSEQ_1:1;
then g1 . (len g1) = f . (len f) by A25, A33, A6;
hence 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 Element of NAT st j in dom g holds
ex k being Element of NAT st
( k in dom f & |.((g /. j) - (f /. k)).| < a ) ) & ( for j being Element of NAT st 1 <= j & j + 1 <= len g holds
|.((g /. j) - (g /. (j + 1))).| < a ) ) by A1, A33, A48, A50, A89, A86, A85, A67, A114, A90, FINSEQ_4:15; ::_thesis: verum
end;
theorem Th40: :: 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 Element of 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 Element of NAT st j in dom g holds
ex k being Element of NAT st
( k in dom f & |.((g /. j) - (f /. k)).| < a ) ) & ( for j being Element of NAT st 1 <= j & j + 1 <= len g holds
|.((g /. j) - (g /. (j + 1))).| < a ) )
proof
A1: (2 * 1) -' 1 = (1 + 1) -' 1
.= 1 by NAT_D:34 ;
let f be FinSequence of (TOP-REAL 2); ::_thesis: 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 Element of 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 Element of NAT st j in dom g holds
ex k being Element of NAT st
( k in dom f & |.((g /. j) - (f /. k)).| < a ) ) & ( for j being Element of NAT st 1 <= j & j + 1 <= len g holds
|.((g /. j) - (g /. (j + 1))).| < a ) )
let a, c, d be Real; ::_thesis: ( 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 Element of NAT st 1 <= i & i + 1 <= len f holds
|.((f /. i) - (f /. (i + 1))).| < a ) implies 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 Element of NAT st j in dom g holds
ex k being Element of NAT st
( k in dom f & |.((g /. j) - (f /. k)).| < a ) ) & ( for j being Element of NAT st 1 <= j & j + 1 <= len g holds
|.((g /. j) - (g /. (j + 1))).| < a ) ) )
assume that
A2: 1 <= len f and
A3: Y_axis f lies_between (Y_axis f) . 1,(Y_axis f) . (len f) and
A4: X_axis f lies_between c,d and
A5: a > 0 and
A6: for i being Element of NAT st 1 <= i & i + 1 <= len f holds
|.((f /. i) - (f /. (i + 1))).| < a ; ::_thesis: 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 Element of NAT st j in dom g holds
ex k being Element of NAT st
( k in dom f & |.((g /. j) - (f /. k)).| < a ) ) & ( for j being Element of NAT st 1 <= j & j + 1 <= len g holds
|.((g /. j) - (g /. (j + 1))).| < a ) )
(len f) + (len f) >= (len f) + 1 by A2, XREAL_1:6;
then A7: (2 * (len f)) - 1 >= ((len f) + 1) - 1 by XREAL_1:9;
defpred S1[ set , set ] means for j being Element of NAT st ( $1 = 2 * j or $1 = (2 * j) -' 1 ) holds
( ( $1 = 2 * j implies $2 = |[((f /. j) `1),((f /. (j + 1)) `2)]| ) & ( $1 = (2 * j) -' 1 implies $2 = f /. j ) );
A8: for k being Nat st k in Seg ((2 * (len f)) -' 1) holds
ex x being set st S1[k,x]
proof
let k be Nat; ::_thesis: ( k in Seg ((2 * (len f)) -' 1) implies ex x being set st S1[k,x] )
assume A9: k in Seg ((2 * (len f)) -' 1) ; ::_thesis: ex x being set st S1[k,x]
then A10: 1 <= k by FINSEQ_1:1;
percases ( k mod 2 = 0 or k mod 2 = 1 ) by NAT_D:12;
supposeA11: k mod 2 = 0 ; ::_thesis: ex x being set st S1[k,x]
consider i being Nat such that
A12: k = (2 * i) + (k mod 2) and
k mod 2 < 2 by NAT_D:def_2;
for j being Element of NAT st ( k = 2 * j or k = (2 * j) -' 1 ) holds
( ( k = 2 * j implies |[((f /. i) `1),((f /. (i + 1)) `2)]| = |[((f /. j) `1),((f /. (j + 1)) `2)]| ) & ( k = (2 * j) -' 1 implies |[((f /. i) `1),((f /. (i + 1)) `2)]| = f /. j ) )
proof
let j be Element of NAT ; ::_thesis: ( ( k = 2 * j or k = (2 * j) -' 1 ) implies ( ( k = 2 * j implies |[((f /. i) `1),((f /. (i + 1)) `2)]| = |[((f /. j) `1),((f /. (j + 1)) `2)]| ) & ( k = (2 * j) -' 1 implies |[((f /. i) `1),((f /. (i + 1)) `2)]| = f /. j ) ) )
assume ( k = 2 * j or k = (2 * j) -' 1 ) ; ::_thesis: ( ( k = 2 * j implies |[((f /. i) `1),((f /. (i + 1)) `2)]| = |[((f /. j) `1),((f /. (j + 1)) `2)]| ) & ( k = (2 * j) -' 1 implies |[((f /. i) `1),((f /. (i + 1)) `2)]| = f /. j ) )
now__::_thesis:_not_k_=_(2_*_j)_-'_1
assume A13: k = (2 * j) -' 1 ; ::_thesis: contradiction
now__::_thesis:_not_j_=_0
0 - 1 < 0 ;
then A14: 0 -' 1 = 0 by XREAL_0:def_2;
assume j = 0 ; ::_thesis: contradiction
hence contradiction by A9, A13, A14, FINSEQ_1:1; ::_thesis: verum
end;
then A15: j >= 0 + 1 by NAT_1:13;
k = ((2 * (j - 1)) + (1 + 1)) - 1 by A10, A13, NAT_D:39
.= (2 * (j - 1)) + 1 ;
then k = (2 * (j -' 1)) + 1 by A15, XREAL_1:233;
hence contradiction by A11, NAT_D:def_2; ::_thesis: verum
end;
hence ( ( k = 2 * j implies |[((f /. i) `1),((f /. (i + 1)) `2)]| = |[((f /. j) `1),((f /. (j + 1)) `2)]| ) & ( k = (2 * j) -' 1 implies |[((f /. i) `1),((f /. (i + 1)) `2)]| = f /. j ) ) by A11, A12; ::_thesis: verum
end;
hence ex x being set st S1[k,x] ; ::_thesis: verum
end;
supposeA16: k mod 2 = 1 ; ::_thesis: ex x being set st S1[k,x]
consider i being Nat such that
A17: k = (2 * i) + (k mod 2) and
A18: k mod 2 < 2 by NAT_D:def_2;
for j being Element of NAT st ( k = 2 * j or k = (2 * j) -' 1 ) holds
( ( k = 2 * j implies f /. (i + 1) = |[((f /. j) `1),((f /. (j + 1)) `2)]| ) & ( k = (2 * j) -' 1 implies f /. (i + 1) = f /. j ) )
proof
let j be Element of NAT ; ::_thesis: ( ( k = 2 * j or k = (2 * j) -' 1 ) implies ( ( k = 2 * j implies f /. (i + 1) = |[((f /. j) `1),((f /. (j + 1)) `2)]| ) & ( k = (2 * j) -' 1 implies f /. (i + 1) = f /. j ) ) )
assume A19: ( k = 2 * j or k = (2 * j) -' 1 ) ; ::_thesis: ( ( k = 2 * j implies f /. (i + 1) = |[((f /. j) `1),((f /. (j + 1)) `2)]| ) & ( k = (2 * j) -' 1 implies f /. (i + 1) = f /. j ) )
percases ( k = (2 * j) -' 1 or k = 2 * j ) by A19;
supposeA20: k = (2 * j) -' 1 ; ::_thesis: ( ( k = 2 * j implies f /. (i + 1) = |[((f /. j) `1),((f /. (j + 1)) `2)]| ) & ( k = (2 * j) -' 1 implies f /. (i + 1) = f /. j ) )
A21: now__::_thesis:_(_k_=_(2_*_j)_-'_1_implies_f_/._(i_+_1)_=_f_/._j_)
assume k = (2 * j) -' 1 ; ::_thesis: f /. (i + 1) = f /. j
then k = ((2 * (j - 1)) + (1 + 1)) - 1 by A10, NAT_D:39
.= (2 * (j - 1)) + 1 ;
hence f /. (i + 1) = f /. j by A16, A17; ::_thesis: verum
end;
k = (2 * j) - 1 by A10, A20, NAT_D:39;
hence ( ( k = 2 * j implies f /. (i + 1) = |[((f /. j) `1),((f /. (j + 1)) `2)]| ) & ( k = (2 * j) -' 1 implies f /. (i + 1) = f /. j ) ) by A21; ::_thesis: verum
end;
supposeA22: k = 2 * j ; ::_thesis: ( ( k = 2 * j implies f /. (i + 1) = |[((f /. j) `1),((f /. (j + 1)) `2)]| ) & ( k = (2 * j) -' 1 implies f /. (i + 1) = f /. j ) )
then A23: 2 * (j - i) = 1 by A16, A17;
then j - i >= 0 ;
then A24: j - i = j -' i by XREAL_0:def_2;
( j - i = 0 or j - i > 0 ) by A23;
then j - i >= 0 + 1 by A16, A17, A22, A24, NAT_1:13;
then 2 * (j - i) >= 2 * 1 by XREAL_1:64;
hence ( ( k = 2 * j implies f /. (i + 1) = |[((f /. j) `1),((f /. (j + 1)) `2)]| ) & ( k = (2 * j) -' 1 implies f /. (i + 1) = f /. j ) ) by A17, A18, A22; ::_thesis: verum
end;
end;
end;
hence ex x being set st S1[k,x] ; ::_thesis: verum
end;
end;
end;
ex p being FinSequence st
( dom p = Seg ((2 * (len f)) -' 1) & ( for k being Nat st k in Seg ((2 * (len f)) -' 1) holds
S1[k,p . k] ) ) from FINSEQ_1:sch_1(A8);
then consider p being FinSequence such that
A25: dom p = Seg ((2 * (len f)) -' 1) and
A26: for k being Nat st k in Seg ((2 * (len f)) -' 1) holds
for j being Element of NAT st ( k = 2 * j or k = (2 * j) -' 1 ) holds
( ( k = 2 * j implies p . k = |[((f /. j) `1),((f /. (j + 1)) `2)]| ) & ( k = (2 * j) -' 1 implies p . k = f /. j ) ) ;
rng p c= the carrier of (TOP-REAL 2)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng p or y in the carrier of (TOP-REAL 2) )
assume y in rng p ; ::_thesis: y in the carrier of (TOP-REAL 2)
then consider x being set such that
A27: x in dom p and
A28: y = p . x by FUNCT_1:def_3;
reconsider i = x as Element of NAT by A27;
x in Seg (len p) by A27, FINSEQ_1:def_3;
then A29: 1 <= i by FINSEQ_1:1;
percases ( i mod 2 = 0 or i mod 2 = 1 ) by NAT_D:12;
supposeA30: i mod 2 = 0 ; ::_thesis: y in the carrier of (TOP-REAL 2)
consider j being Nat such that
A31: i = (2 * j) + (i mod 2) and
i mod 2 < 2 by NAT_D:def_2;
reconsider j = j as Element of NAT by ORDINAL1:def_12;
p . i = |[((f /. j) `1),((f /. (j + 1)) `2)]| by A25, A26, A27, A30, A31;
hence y in the carrier of (TOP-REAL 2) by A28; ::_thesis: verum
end;
supposeA32: i mod 2 = 1 ; ::_thesis: y in the carrier of (TOP-REAL 2)
consider j being Nat such that
A33: i = (2 * j) + (i mod 2) and
i mod 2 < 2 by NAT_D:def_2;
reconsider j = j as Element of NAT by ORDINAL1:def_12;
(2 * (j + 1)) - 1 = (2 * (j + 1)) -' 1 by A29, A32, A33, NAT_D:39;
then p . i = f /. (j + 1) by A25, A26, A27, A32, A33;
hence y in the carrier of (TOP-REAL 2) by A28; ::_thesis: verum
end;
end;
end;
then reconsider g1 = p as FinSequence of (TOP-REAL 2) by FINSEQ_1:def_4;
A34: len p = (2 * (len f)) -' 1 by A25, FINSEQ_1:def_3;
for i being Nat st 1 <= i & i + 1 <= len g1 & not (g1 /. i) `1 = (g1 /. (i + 1)) `1 holds
(g1 /. i) `2 = (g1 /. (i + 1)) `2
proof
let i be Nat; ::_thesis: ( 1 <= i & i + 1 <= len g1 & not (g1 /. i) `1 = (g1 /. (i + 1)) `1 implies (g1 /. i) `2 = (g1 /. (i + 1)) `2 )
assume that
A35: 1 <= i and
A36: i + 1 <= len g1 ; ::_thesis: ( (g1 /. i) `1 = (g1 /. (i + 1)) `1 or (g1 /. i) `2 = (g1 /. (i + 1)) `2 )
1 < i + 1 by A35, NAT_1:13;
then A37: ( i + 1 in Seg (len g1) & g1 /. (i + 1) = g1 . (i + 1) ) by A36, FINSEQ_1:1, FINSEQ_4:15;
i < len g1 by A36, NAT_1:13;
then A38: ( i in Seg ((2 * (len f)) -' 1) & g1 . i = g1 /. i ) by A34, A35, FINSEQ_1:1, FINSEQ_4:15;
percases ( i mod 2 = 0 or i mod 2 = 1 ) by NAT_D:12;
supposeA39: i mod 2 = 0 ; ::_thesis: ( (g1 /. i) `1 = (g1 /. (i + 1)) `1 or (g1 /. i) `2 = (g1 /. (i + 1)) `2 )
consider j being Nat such that
A40: i = (2 * j) + (i mod 2) and
i mod 2 < 2 by NAT_D:def_2;
reconsider j = j as Element of NAT by ORDINAL1:def_12;
1 <= 1 + i by NAT_1:11;
then (2 * (j + 1)) -' 1 = (2 * (j + 1)) - 1 by A39, A40, NAT_D:39;
then A41: g1 /. (i + 1) = f /. (j + 1) by A26, A34, A37, A39, A40;
g1 /. i = |[((f /. j) `1),((f /. (j + 1)) `2)]| by A26, A38, A39, A40;
hence ( (g1 /. i) `1 = (g1 /. (i + 1)) `1 or (g1 /. i) `2 = (g1 /. (i + 1)) `2 ) by A41, EUCLID:52; ::_thesis: verum
end;
supposeA42: i mod 2 = 1 ; ::_thesis: ( (g1 /. i) `1 = (g1 /. (i + 1)) `1 or (g1 /. i) `2 = (g1 /. (i + 1)) `2 )
consider j being Nat such that
A43: i = (2 * j) + (i mod 2) and
i mod 2 < 2 by NAT_D:def_2;
reconsider j = j as Element of NAT by ORDINAL1:def_12;
i + 1 = 2 * (j + 1) by A42, A43;
then A44: g1 /. (i + 1) = |[((f /. (j + 1)) `1),((f /. ((j + 1) + 1)) `2)]| by A26, A34, A37;
(2 * (j + 1)) -' 1 = (2 * (j + 1)) - 1 by A35, A42, A43, NAT_D:39;
then g1 /. i = f /. (j + 1) by A26, A38, A42, A43;
hence ( (g1 /. i) `1 = (g1 /. (i + 1)) `1 or (g1 /. i) `2 = (g1 /. (i + 1)) `2 ) by A44, EUCLID:52; ::_thesis: verum
end;
end;
end;
then A45: g1 is special by TOPREAL1:def_5;
A46: 2 * (len f) >= 2 * 1 by A2, XREAL_1:64;
then A47: (2 * (len f)) -' 1 = (2 * (len f)) - 1 by XREAL_1:233, XXREAL_0:2;
for i being Element of NAT st i in dom (Y_axis g1) holds
( (Y_axis f) . 1 <= (Y_axis g1) . i & (Y_axis g1) . i <= (Y_axis f) . (len f) )
proof
let i be Element of NAT ; ::_thesis: ( i in dom (Y_axis g1) implies ( (Y_axis f) . 1 <= (Y_axis g1) . i & (Y_axis g1) . i <= (Y_axis f) . (len f) ) )
A48: len (Y_axis f) = len f by GOBOARD1:def_2;
assume A49: i in dom (Y_axis g1) ; ::_thesis: ( (Y_axis f) . 1 <= (Y_axis g1) . i & (Y_axis g1) . i <= (Y_axis f) . (len f) )
then A50: i in Seg (len (Y_axis g1)) by FINSEQ_1:def_3;
then A51: i in Seg (len g1) by GOBOARD1:def_2;
i in Seg (len g1) by A50, GOBOARD1:def_2;
then A52: i <= len g1 by FINSEQ_1:1;
A53: 1 <= i by A50, FINSEQ_1:1;
then A54: g1 /. i = g1 . i by A52, FINSEQ_4:15;
A55: (Y_axis g1) . i = (g1 /. i) `2 by A49, GOBOARD1:def_2;
percases ( i mod 2 = 0 or i mod 2 = 1 ) by NAT_D:12;
supposeA56: i mod 2 = 0 ; ::_thesis: ( (Y_axis f) . 1 <= (Y_axis g1) . i & (Y_axis g1) . i <= (Y_axis f) . (len f) )
consider j being Nat such that
A57: i = (2 * j) + (i mod 2) and
i mod 2 < 2 by NAT_D:def_2;
reconsider j = j as Element of NAT by ORDINAL1:def_12;
g1 /. i = |[((f /. j) `1),((f /. (j + 1)) `2)]| by A26, A34, A51, A54, A56, A57;
then A58: (g1 /. i) `2 = (f /. (j + 1)) `2 by EUCLID:52;
(2 * j) + 1 <= ((2 * (len f)) - 1) + 1 by A34, A47, A52, A56, A57, XREAL_1:6;
then 2 * j < 2 * (len f) by NAT_1:13;
then (2 * j) / 2 < (2 * (len f)) / 2 by XREAL_1:74;
then ( 1 <= j + 1 & j + 1 <= len f ) by NAT_1:11, NAT_1:13;
then j + 1 in Seg (len f) by FINSEQ_1:1;
then A59: j + 1 in dom (Y_axis f) by A48, FINSEQ_1:def_3;
then (Y_axis f) . (j + 1) = (f /. (j + 1)) `2 by GOBOARD1:def_2;
hence ( (Y_axis f) . 1 <= (Y_axis g1) . i & (Y_axis g1) . i <= (Y_axis f) . (len f) ) by A3, A55, A58, A59, GOBOARD4:def_2; ::_thesis: verum
end;
supposeA60: i mod 2 = 1 ; ::_thesis: ( (Y_axis f) . 1 <= (Y_axis g1) . i & (Y_axis g1) . i <= (Y_axis f) . (len f) )
consider j being Nat such that
A61: i = (2 * j) + (i mod 2) and
i mod 2 < 2 by NAT_D:def_2;
reconsider j = j as Element of NAT by ORDINAL1:def_12;
(2 * (j + 1)) -' 1 = (2 * (j + 1)) - 1 by A53, A60, A61, NAT_D:39;
then A62: (g1 /. i) `2 = (f /. (j + 1)) `2 by A26, A34, A51, A54, A60, A61;
(2 * j) + 1 <= ((2 * (len f)) - 1) + 1 by A34, A47, A52, A60, A61, NAT_1:13;
then 2 * j < 2 * (len f) by NAT_1:13;
then (2 * j) / 2 < (2 * (len f)) / 2 by XREAL_1:74;
then ( 1 <= j + 1 & j + 1 <= len f ) by NAT_1:11, NAT_1:13;
then j + 1 in Seg (len f) by FINSEQ_1:1;
then A63: j + 1 in dom (Y_axis f) by A48, FINSEQ_1:def_3;
then (Y_axis f) . (j + 1) = (f /. (j + 1)) `2 by GOBOARD1:def_2;
hence ( (Y_axis f) . 1 <= (Y_axis g1) . i & (Y_axis g1) . i <= (Y_axis f) . (len f) ) by A3, A55, A62, A63, GOBOARD4:def_2; ::_thesis: verum
end;
end;
end;
then A64: Y_axis g1 lies_between (Y_axis f) . 1,(Y_axis f) . (len f) by GOBOARD4:def_2;
A65: (2 * (len f)) - 1 >= (1 + 1) - 1 by A46, XREAL_1:9;
then 1 in Seg ((2 * (len f)) -' 1) by A47, FINSEQ_1:1;
then p . 1 = f /. 1 by A26, A1;
then A66: g1 . 1 = f . 1 by A2, FINSEQ_4:15;
A67: for i being Element of NAT st 1 <= i & i + 1 <= len g1 holds
|.((g1 /. i) - (g1 /. (i + 1))).| < a
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i + 1 <= len g1 implies |.((g1 /. i) - (g1 /. (i + 1))).| < a )
assume that
A68: 1 <= i and
A69: i + 1 <= len g1 ; ::_thesis: |.((g1 /. i) - (g1 /. (i + 1))).| < a
A70: g1 . (i + 1) = g1 /. (i + 1) by A69, FINSEQ_4:15, NAT_1:11;
i <= len g1 by A69, NAT_1:13;
then A71: i in Seg (len g1) by A68, FINSEQ_1:1;
1 <= i + 1 by NAT_1:11;
then A72: i + 1 in Seg ((2 * (len f)) -' 1) by A34, A69, FINSEQ_1:1;
i <= len g1 by A69, NAT_1:13;
then A73: g1 . i = g1 /. i by A68, FINSEQ_4:15;
percases ( i mod 2 = 0 or i mod 2 = 1 ) by NAT_D:12;
supposeA74: i mod 2 = 0 ; ::_thesis: |.((g1 /. i) - (g1 /. (i + 1))).| < a
consider j being Nat such that
A75: i = (2 * j) + (i mod 2) and
i mod 2 < 2 by NAT_D:def_2;
reconsider j = j as Element of NAT by ORDINAL1:def_12;
A76: g1 /. i = |[((f /. j) `1),((f /. (j + 1)) `2)]| by A26, A34, A71, A73, A74, A75;
then A77: (g1 /. i) `2 = (f /. (j + 1)) `2 by EUCLID:52;
1 <= 1 + i by NAT_1:11;
then (2 * (j + 1)) -' 1 = (2 * (j + 1)) - 1 by A74, A75, NAT_D:39;
then A78: g1 /. (i + 1) = f /. (j + 1) by A26, A72, A70, A74, A75;
(g1 /. i) - (g1 /. (i + 1)) = |[(((g1 /. i) `1) - ((g1 /. (i + 1)) `1)),(((g1 /. i) `2) - ((g1 /. (i + 1)) `2))]| by EUCLID:61
.= |[(((f /. j) `1) - ((f /. (j + 1)) `1)),0]| by A76, A78, A77, EUCLID:52 ;
then ( ((g1 /. i) - (g1 /. (i + 1))) `1 = ((f /. j) `1) - ((f /. (j + 1)) `1) & ((g1 /. i) - (g1 /. (i + 1))) `2 = 0 ) by EUCLID:52;
then |.((g1 /. i) - (g1 /. (i + 1))).| = sqrt (((((f /. j) `1) - ((f /. (j + 1)) `1)) ^2) + (0 ^2)) by Th30
.= sqrt ((((f /. j) `1) - ((f /. (j + 1)) `1)) ^2) ;
then |.((g1 /. i) - (g1 /. (i + 1))).| = abs (((f /. j) `1) - ((f /. (j + 1)) `1)) by COMPLEX1:72;
then A79: |.((g1 /. i) - (g1 /. (i + 1))).| <= |.((f /. j) - (f /. (j + 1))).| by Th34;
(2 * j) + 1 <= ((2 * (len f)) - 1) + 1 by A34, A47, A69, A74, A75, NAT_1:13;
then 2 * j < 2 * (len f) by NAT_1:13;
then (2 * j) / 2 < (2 * (len f)) / 2 by XREAL_1:74;
then A80: j + 1 <= len f by NAT_1:13;
j > 0 by A68, A74, A75;
then j >= 0 + 1 by NAT_1:13;
then |.((f /. j) - (f /. (j + 1))).| < a by A6, A80;
hence |.((g1 /. i) - (g1 /. (i + 1))).| < a by A79, XXREAL_0:2; ::_thesis: verum
end;
supposeA81: i mod 2 = 1 ; ::_thesis: |.((g1 /. i) - (g1 /. (i + 1))).| < a
consider j being Nat such that
A82: i = (2 * j) + (i mod 2) and
i mod 2 < 2 by NAT_D:def_2;
reconsider j = j as Element of NAT by ORDINAL1:def_12;
(2 * (j + 1)) -' 1 = (2 * (j + 1)) - 1 by A68, A81, A82, NAT_D:39;
then A83: g1 /. i = f /. (j + 1) by A26, A34, A71, A73, A81, A82;
i + 1 = 2 * (j + 1) by A81, A82;
then A84: g1 /. (i + 1) = |[((f /. (j + 1)) `1),((f /. ((j + 1) + 1)) `2)]| by A26, A72, A70;
then A85: (g1 /. (i + 1)) `1 = (f /. (j + 1)) `1 by EUCLID:52;
(g1 /. i) - (g1 /. (i + 1)) = |[(((g1 /. i) `1) - ((g1 /. (i + 1)) `1)),(((g1 /. i) `2) - ((g1 /. (i + 1)) `2))]| by EUCLID:61
.= |[0,(((f /. (j + 1)) `2) - ((f /. ((j + 1) + 1)) `2))]| by A83, A84, A85, EUCLID:52 ;
then ( ((g1 /. i) - (g1 /. (i + 1))) `1 = 0 & ((g1 /. i) - (g1 /. (i + 1))) `2 = ((f /. (j + 1)) `2) - ((f /. ((j + 1) + 1)) `2) ) by EUCLID:52;
then |.((g1 /. i) - (g1 /. (i + 1))).| = sqrt ((0 ^2) + ((((f /. (j + 1)) `2) - ((f /. ((j + 1) + 1)) `2)) ^2)) by Th30
.= sqrt ((((f /. (j + 1)) `2) - ((f /. ((j + 1) + 1)) `2)) ^2) ;
then |.((g1 /. i) - (g1 /. (i + 1))).| = abs (((f /. (j + 1)) `2) - ((f /. ((j + 1) + 1)) `2)) by COMPLEX1:72;
then A86: |.((g1 /. i) - (g1 /. (i + 1))).| <= |.((f /. (j + 1)) - (f /. ((j + 1) + 1))).| by Th34;
(2 * (j + 1)) + 1 <= ((2 * (len f)) - 1) + 1 by A34, A47, A69, A81, A82, XREAL_1:6;
then 2 * (j + 1) < 2 * (len f) by NAT_1:13;
then (2 * (j + 1)) / 2 < (2 * (len f)) / 2 by XREAL_1:74;
then (j + 1) + 1 <= len f by NAT_1:13;
then |.((f /. (j + 1)) - (f /. ((j + 1) + 1))).| < a by A6, NAT_1:11;
hence |.((g1 /. i) - (g1 /. (i + 1))).| < a by A86, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
A87: for i being Element of NAT st i in dom g1 holds
ex k being Element of NAT st
( k in dom f & |.((g1 /. i) - (f /. k)).| < a )
proof
let i be Element of NAT ; ::_thesis: ( i in dom g1 implies ex k being Element of NAT st
( k in dom f & |.((g1 /. i) - (f /. k)).| < a ) )
assume A88: i in dom g1 ; ::_thesis: ex k being Element of NAT st
( k in dom f & |.((g1 /. i) - (f /. k)).| < a )
then A89: i <= len g1 by FINSEQ_3:25;
A90: 1 <= i by A88, FINSEQ_3:25;
then A91: g1 . i = g1 /. i by A89, FINSEQ_4:15;
percases ( i mod 2 = 0 or i mod 2 = 1 ) by NAT_D:12;
supposeA92: i mod 2 = 0 ; ::_thesis: ex k being Element of NAT st
( k in dom f & |.((g1 /. i) - (f /. k)).| < a )
consider j being Nat such that
A93: i = (2 * j) + (i mod 2) and
i mod 2 < 2 by NAT_D:def_2;
reconsider j = j as Element of NAT by ORDINAL1:def_12;
j > 0 by A88, A92, A93, FINSEQ_3:25;
then A94: j >= 0 + 1 by NAT_1:13;
A95: g1 /. i = |[((f /. j) `1),((f /. (j + 1)) `2)]| by A25, A26, A88, A91, A92, A93;
then A96: (g1 /. i) `1 = (f /. j) `1 by EUCLID:52;
A97: (g1 /. i) - (f /. j) = |[(((g1 /. i) `1) - ((f /. j) `1)),(((g1 /. i) `2) - ((f /. j) `2))]| by EUCLID:61
.= |[0,(((g1 /. i) `2) - ((f /. j) `2))]| by A96 ;
then A98: ((g1 /. i) - (f /. j)) `1 = 0 by EUCLID:52;
((g1 /. i) - (f /. j)) `2 = ((g1 /. i) `2) - ((f /. j) `2) by A97, EUCLID:52;
then |.((g1 /. i) - (f /. j)).| = sqrt (((((g1 /. i) - (f /. j)) `1) ^2) + ((((g1 /. i) `2) - ((f /. j) `2)) ^2)) by Th30
.= sqrt ((((f /. (j + 1)) `2) - ((f /. j) `2)) ^2) by A95, A98, EUCLID:52 ;
then |.((g1 /. i) - (f /. j)).| = abs (((f /. (j + 1)) `2) - ((f /. j) `2)) by COMPLEX1:72
.= abs (((f /. j) `2) - ((f /. (j + 1)) `2)) by UNIFORM1:11 ;
then A99: |.((g1 /. i) - (f /. j)).| <= |.((f /. j) - (f /. (j + 1))).| by Th34;
(2 * j) + 1 <= ((2 * (len f)) - 1) + 1 by A34, A47, A89, A92, A93, XREAL_1:6;
then 2 * j < 2 * (len f) by NAT_1:13;
then A100: (2 * j) / 2 < (2 * (len f)) / 2 by XREAL_1:74;
then j + 1 <= len f by NAT_1:13;
then |.((f /. j) - (f /. (j + 1))).| < a by A6, A94;
then A101: |.((g1 /. i) - (f /. j)).| < a by A99, XXREAL_0:2;
j in dom f by A94, A100, FINSEQ_3:25;
hence ex k being Element of NAT st
( k in dom f & |.((g1 /. i) - (f /. k)).| < a ) by A101; ::_thesis: verum
end;
supposeA102: i mod 2 = 1 ; ::_thesis: ex k being Element of NAT st
( k in dom f & |.((g1 /. i) - (f /. k)).| < a )
consider j being Nat such that
A103: i = (2 * j) + (i mod 2) and
i mod 2 < 2 by NAT_D:def_2;
reconsider j = j as Element of NAT by ORDINAL1:def_12;
(2 * (j + 1)) -' 1 = (2 * (j + 1)) - 1 by A90, A102, A103, NAT_D:39;
then g1 /. i = f /. (j + 1) by A25, A26, A88, A91, A102, A103;
then A104: |.((g1 /. i) - (f /. (j + 1))).| = |.(0. (TOP-REAL 2)).| by EUCLID:42
.= 0 by TOPRNS_1:23 ;
((2 * j) + 1) + 1 <= ((2 * (len f)) - 1) + 1 by A34, A47, A89, A102, A103, XREAL_1:6;
then (2 * j) + 1 < 2 * (len f) by NAT_1:13;
then 2 * j < 2 * (len f) by NAT_1:13;
then (2 * j) / 2 < (2 * (len f)) / 2 by XREAL_1:74;
then A105: j + 1 <= len f by NAT_1:13;
1 <= j + 1 by NAT_1:11;
then j + 1 in dom f by A105, FINSEQ_3:25;
hence ex k being Element of NAT st
( k in dom f & |.((g1 /. i) - (f /. k)).| < a ) by A5, A104; ::_thesis: verum
end;
end;
end;
for i being Element of NAT st i in dom (X_axis g1) holds
( c <= (X_axis g1) . i & (X_axis g1) . i <= d )
proof
let i be Element of NAT ; ::_thesis: ( i in dom (X_axis g1) implies ( c <= (X_axis g1) . i & (X_axis g1) . i <= d ) )
A106: len (X_axis f) = len f by GOBOARD1:def_1;
assume A107: i in dom (X_axis g1) ; ::_thesis: ( c <= (X_axis g1) . i & (X_axis g1) . i <= d )
then A108: i in Seg (len (X_axis g1)) by FINSEQ_1:def_3;
then A109: i in Seg (len g1) by GOBOARD1:def_1;
then A110: i <= len g1 by FINSEQ_1:1;
A111: 1 <= i by A108, FINSEQ_1:1;
then A112: g1 /. i = g1 . i by A110, FINSEQ_4:15;
A113: (X_axis g1) . i = (g1 /. i) `1 by A107, GOBOARD1:def_1;
percases ( i mod 2 = 0 or i mod 2 = 1 ) by NAT_D:12;
supposeA114: i mod 2 = 0 ; ::_thesis: ( c <= (X_axis g1) . i & (X_axis g1) . i <= d )
consider j being Nat such that
A115: i = (2 * j) + (i mod 2) and
i mod 2 < 2 by NAT_D:def_2;
reconsider j = j as Element of NAT by ORDINAL1:def_12;
g1 /. i = |[((f /. j) `1),((f /. (j + 1)) `2)]| by A26, A34, A109, A112, A114, A115;
then A116: (g1 /. i) `1 = (f /. j) `1 by EUCLID:52;
(2 * j) + 1 <= ((2 * (len f)) - 1) + 1 by A34, A47, A110, A114, A115, XREAL_1:6;
then 2 * j < 2 * (len f) by NAT_1:13;
then A117: (2 * j) / 2 < (2 * (len f)) / 2 by XREAL_1:74;
j > 0 by A108, A114, A115, FINSEQ_1:1;
then j >= 0 + 1 by NAT_1:13;
then j in Seg (len f) by A117, FINSEQ_1:1;
then A118: j in dom (X_axis f) by A106, FINSEQ_1:def_3;
then (X_axis f) . j = (f /. j) `1 by GOBOARD1:def_1;
hence ( c <= (X_axis g1) . i & (X_axis g1) . i <= d ) by A4, A113, A116, A118, GOBOARD4:def_2; ::_thesis: verum
end;
supposeA119: i mod 2 = 1 ; ::_thesis: ( c <= (X_axis g1) . i & (X_axis g1) . i <= d )
consider j being Nat such that
A120: i = (2 * j) + (i mod 2) and
i mod 2 < 2 by NAT_D:def_2;
reconsider j = j as Element of NAT by ORDINAL1:def_12;
(2 * (j + 1)) -' 1 = (2 * (j + 1)) - 1 by A111, A119, A120, NAT_D:39;
then A121: (g1 /. i) `1 = (f /. (j + 1)) `1 by A26, A34, A109, A112, A119, A120;
(2 * j) + 1 <= ((2 * (len f)) - 1) + 1 by A34, A47, A110, A119, A120, NAT_1:13;
then 2 * j < 2 * (len f) by NAT_1:13;
then (2 * j) / 2 < (2 * (len f)) / 2 by XREAL_1:74;
then ( 1 <= j + 1 & j + 1 <= len f ) by NAT_1:11, NAT_1:13;
then j + 1 in Seg (len f) by FINSEQ_1:1;
then A122: j + 1 in dom (X_axis f) by A106, FINSEQ_1:def_3;
then (X_axis f) . (j + 1) = (f /. (j + 1)) `1 by GOBOARD1:def_1;
hence ( c <= (X_axis g1) . i & (X_axis g1) . i <= d ) by A4, A113, A121, A122, GOBOARD4:def_2; ::_thesis: verum
end;
end;
end;
then A123: X_axis g1 lies_between c,d by GOBOARD4:def_2;
(2 * (len f)) -' 1 in Seg ((2 * (len f)) -' 1) by A47, A65, FINSEQ_1:1;
then p . ((2 * (len f)) -' 1) = f /. (len f) by A26;
hence 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 Element of NAT st j in dom g holds
ex k being Element of NAT st
( k in dom f & |.((g /. j) - (f /. k)).| < a ) ) & ( for j being Element of NAT st 1 <= j & j + 1 <= len g holds
|.((g /. j) - (g /. (j + 1))).| < a ) ) by A2, A34, A45, A47, A66, A7, A64, A123, A87, A67, FINSEQ_4:15; ::_thesis: verum
end;
theorem Th41: :: 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 )
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: ( 1 <= len f implies ( len (X_axis f) = len f & (X_axis f) . 1 = (f /. 1) `1 & (X_axis f) . (len f) = (f /. (len f)) `1 ) )
A1: len (X_axis f) = len f by GOBOARD1:def_1;
assume A2: 1 <= len f ; ::_thesis: ( len (X_axis f) = len f & (X_axis f) . 1 = (f /. 1) `1 & (X_axis f) . (len f) = (f /. (len f)) `1 )
then len f in Seg (len f) by FINSEQ_1:1;
then A3: len f in dom (X_axis f) by A1, FINSEQ_1:def_3;
1 in Seg (len f) by A2, FINSEQ_1:1;
then 1 in dom (X_axis f) by A1, FINSEQ_1:def_3;
hence ( len (X_axis f) = len f & (X_axis f) . 1 = (f /. 1) `1 & (X_axis f) . (len f) = (f /. (len f)) `1 ) by A3, GOBOARD1:def_1; ::_thesis: verum
end;
theorem Th42: :: 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 )
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: ( 1 <= len f implies ( len (Y_axis f) = len f & (Y_axis f) . 1 = (f /. 1) `2 & (Y_axis f) . (len f) = (f /. (len f)) `2 ) )
A1: len (Y_axis f) = len f by GOBOARD1:def_2;
assume A2: 1 <= len f ; ::_thesis: ( len (Y_axis f) = len f & (Y_axis f) . 1 = (f /. 1) `2 & (Y_axis f) . (len f) = (f /. (len f)) `2 )
then len f in Seg (len f) by FINSEQ_1:1;
then A3: len f in dom (Y_axis f) by A1, FINSEQ_1:def_3;
1 in Seg (len f) by A2, FINSEQ_1:1;
then 1 in dom (Y_axis f) by A1, FINSEQ_1:def_3;
hence ( len (Y_axis f) = len f & (Y_axis f) . 1 = (f /. 1) `2 & (Y_axis f) . (len f) = (f /. (len f)) `2 ) by A3, GOBOARD1:def_2; ::_thesis: verum
end;
theorem Th43: :: JGRAPH_1:43
for i being Element of 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 )
proof
let i be Element of NAT ; ::_thesis: 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 )
let f be FinSequence of (TOP-REAL 2); ::_thesis: ( i in dom f implies ( (X_axis f) . i = (f /. i) `1 & (Y_axis f) . i = (f /. i) `2 ) )
assume A1: i in dom f ; ::_thesis: ( (X_axis f) . i = (f /. i) `1 & (Y_axis f) . i = (f /. i) `2 )
len (X_axis f) = len f by GOBOARD1:def_1;
then i in dom (X_axis f) by A1, FINSEQ_3:29;
hence (X_axis f) . i = (f /. i) `1 by GOBOARD1:def_1; ::_thesis: (Y_axis f) . i = (f /. i) `2
len (Y_axis f) = len f by GOBOARD1:def_2;
then i in dom (Y_axis f) by A1, FINSEQ_3:29;
hence (Y_axis f) . i = (f /. i) `2 by GOBOARD1:def_2; ::_thesis: verum
end;
theorem Th44: :: 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
proof
let P, Q be non empty Subset of (TOP-REAL 2); ::_thesis: 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
let p1, p2, q1, q2 be Point of (TOP-REAL 2); ::_thesis: ( 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 ) ) implies P meets Q )
assume that
A1: P is_an_arc_of p1,p2 and
A2: Q is_an_arc_of q1,q2 and
A3: for p being Point of (TOP-REAL 2) st p in P holds
( p1 `1 <= p `1 & p `1 <= p2 `1 ) and
A4: for p being Point of (TOP-REAL 2) st p in Q holds
( p1 `1 <= p `1 & p `1 <= p2 `1 ) and
A5: for p being Point of (TOP-REAL 2) st p in P holds
( q1 `2 <= p `2 & p `2 <= q2 `2 ) and
A6: for p being Point of (TOP-REAL 2) st p in Q holds
( q1 `2 <= p `2 & p `2 <= q2 `2 ) ; ::_thesis: P meets Q
consider g being Function of I[01],((TOP-REAL 2) | Q) such that
A7: g is being_homeomorphism and
A8: g . 0 = q1 and
A9: g . 1 = q2 by A2, TOPREAL1:def_1;
A10: TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopSpaceMetr (Euclid 2) by EUCLID:def_8;
then reconsider P9 = P, Q9 = Q as Subset of (TopSpaceMetr (Euclid 2)) ;
P is compact by A1, JORDAN5A:1;
then A11: P9 is compact by A10, COMPTS_1:23;
Q is compact by A2, JORDAN5A:1;
then A12: Q9 is compact by A10, COMPTS_1:23;
set e = (min_dist_min (P9,Q9)) / 5;
consider f being Function of I[01],((TOP-REAL 2) | P) such that
A13: f is being_homeomorphism and
A14: f . 0 = p1 and
A15: f . 1 = p2 by A1, TOPREAL1:def_1;
consider f1 being Function of I[01],(TOP-REAL 2) such that
A16: f = f1 and
A17: f1 is continuous and
f1 is one-to-one by A13, JORDAN7:15;
consider g1 being Function of I[01],(TOP-REAL 2) such that
A18: g = g1 and
A19: g1 is continuous and
g1 is one-to-one by A7, JORDAN7:15;
assume P /\ Q = {} ; :: according to XBOOLE_0:def_7 ::_thesis: contradiction
then P misses Q by XBOOLE_0:def_7;
then A20: min_dist_min (P9,Q9) > 0 by A11, A12, Th38;
then A21: (min_dist_min (P9,Q9)) / 5 > 0 / 5 by XREAL_1:74;
then consider hb being FinSequence of REAL such that
A22: hb . 1 = 0 and
A23: hb . (len hb) = 1 and
A24: 5 <= len hb and
A25: rng hb c= the carrier of I[01] and
A26: hb is increasing and
A27: for i being Element of NAT
for R being Subset of I[01]
for W being Subset of (Euclid 2) st 1 <= i & i < len hb & R = [.(hb /. i),(hb /. (i + 1)).] & W = g1 .: R holds
diameter W < (min_dist_min (P9,Q9)) / 5 by A19, UNIFORM1:13;
consider h being FinSequence of REAL such that
A28: h . 1 = 0 and
A29: h . (len h) = 1 and
A30: 5 <= len h and
A31: rng h c= the carrier of I[01] and
A32: h is increasing and
A33: for i being Element of NAT
for R being Subset of I[01]
for W being Subset of (Euclid 2) st 1 <= i & i < len h & R = [.(h /. i),(h /. (i + 1)).] & W = f1 .: R holds
diameter W < (min_dist_min (P9,Q9)) / 5 by A17, A21, UNIFORM1:13;
deffunc H1( Nat) -> set = f1 . (h . $1);
ex h19 being FinSequence st
( len h19 = len h & ( for i being Nat st i in dom h19 holds
h19 . i = H1(i) ) ) from FINSEQ_1:sch_2();
then consider h19 being FinSequence such that
A34: len h19 = len h and
A35: for i being Nat st i in dom h19 holds
h19 . i = f1 . (h . i) ;
A36: dom g1 = [#] I[01] by A7, A18, TOPS_2:def_5
.= the carrier of I[01] ;
rng h19 c= the carrier of (TOP-REAL 2)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng h19 or y in the carrier of (TOP-REAL 2) )
assume y in rng h19 ; ::_thesis: y in the carrier of (TOP-REAL 2)
then consider x being set such that
A37: x in dom h19 and
A38: y = h19 . x by FUNCT_1:def_3;
reconsider i = x as Element of NAT by A37;
dom h19 = Seg (len h19) by FINSEQ_1:def_3;
then i in dom h by A34, A37, FINSEQ_1:def_3;
then A39: h . i in rng h by FUNCT_1:def_3;
A40: dom f1 = [#] I[01] by A13, A16, TOPS_2:def_5
.= the carrier of I[01] ;
A41: rng f = [#] ((TOP-REAL 2) | P) by A13, TOPS_2:def_5
.= P by PRE_TOPC:def_5 ;
h19 . i = f1 . (h . i) by A35, A37;
then h19 . i in rng f by A16, A31, A39, A40, FUNCT_1:def_3;
hence y in the carrier of (TOP-REAL 2) by A38, A41; ::_thesis: verum
end;
then reconsider h1 = h19 as FinSequence of (TOP-REAL 2) by FINSEQ_1:def_4;
A42: len h1 >= 1 by A30, A34, XXREAL_0:2;
then A43: h1 . 1 = h1 /. 1 by FINSEQ_4:15;
A44: for i being Element of NAT st 1 <= i & i + 1 <= len h1 holds
|.((h1 /. i) - (h1 /. (i + 1))).| < (min_dist_min (P9,Q9)) / 5
proof
reconsider Pa = P as Subset of (TOP-REAL 2) ;
reconsider W1 = P as Subset of (Euclid 2) by TOPREAL3:8;
let i be Element of NAT ; ::_thesis: ( 1 <= i & i + 1 <= len h1 implies |.((h1 /. i) - (h1 /. (i + 1))).| < (min_dist_min (P9,Q9)) / 5 )
assume that
A45: 1 <= i and
A46: i + 1 <= len h1 ; ::_thesis: |.((h1 /. i) - (h1 /. (i + 1))).| < (min_dist_min (P9,Q9)) / 5
A47: 1 < i + 1 by A45, NAT_1:13;
then A48: h . (i + 1) = h /. (i + 1) by A34, A46, FINSEQ_4:15;
A49: i + 1 in dom h19 by A46, A47, FINSEQ_3:25;
then A50: h19 . (i + 1) = f1 . (h . (i + 1)) by A35;
then A51: h1 /. (i + 1) = f1 . (h . (i + 1)) by A46, A47, FINSEQ_4:15;
A52: i < len h1 by A46, NAT_1:13;
then A53: h . i = h /. i by A34, A45, FINSEQ_4:15;
A54: i in dom h by A34, A45, A52, FINSEQ_3:25;
then A55: h . i in rng h by FUNCT_1:def_3;
A56: i + 1 in dom h by A34, A46, A47, FINSEQ_3:25;
then h . (i + 1) in rng h by FUNCT_1:def_3;
then reconsider R = [.(h /. i),(h /. (i + 1)).] as Subset of I[01] by A31, A55, A53, A48, BORSUK_1:40, XXREAL_2:def_12;
reconsider W = f1 .: R as Subset of (Euclid 2) by TOPREAL3:8;
A57: Pa is compact by A1, JORDAN5A:1;
reconsider Pa = Pa as non empty Subset of (TOP-REAL 2) ;
A58: rng f = [#] ((TOP-REAL 2) | P) by A13, TOPS_2:def_5
.= P by PRE_TOPC:def_5 ;
set r1 = (((E-bound Pa) - (W-bound Pa)) + ((N-bound Pa) - (S-bound Pa))) + 1;
A59: for x, y being Point of (Euclid 2) st x in W1 & y in W1 holds
dist (x,y) <= (((E-bound Pa) - (W-bound Pa)) + ((N-bound Pa) - (S-bound Pa))) + 1
proof
let x, y be Point of (Euclid 2); ::_thesis: ( x in W1 & y in W1 implies dist (x,y) <= (((E-bound Pa) - (W-bound Pa)) + ((N-bound Pa) - (S-bound Pa))) + 1 )
assume that
A60: x in W1 and
A61: y in W1 ; ::_thesis: dist (x,y) <= (((E-bound Pa) - (W-bound Pa)) + ((N-bound Pa) - (S-bound Pa))) + 1
reconsider pw1 = x, pw2 = y as Point of (TOP-REAL 2) by A60, A61;
A62: ( S-bound Pa <= pw2 `2 & pw2 `2 <= N-bound Pa ) by A57, A61, PSCOMP_1:24;
( S-bound Pa <= pw1 `2 & pw1 `2 <= N-bound Pa ) by A57, A60, PSCOMP_1:24;
then A63: abs ((pw1 `2) - (pw2 `2)) <= (N-bound Pa) - (S-bound Pa) by A62, Th27;
A64: ( W-bound Pa <= pw2 `1 & pw2 `1 <= E-bound Pa ) by A57, A61, PSCOMP_1:24;
( W-bound Pa <= pw1 `1 & pw1 `1 <= E-bound Pa ) by A57, A60, PSCOMP_1:24;
then abs ((pw1 `1) - (pw2 `1)) <= (E-bound Pa) - (W-bound Pa) by A64, Th27;
then A65: (abs ((pw1 `1) - (pw2 `1))) + (abs ((pw1 `2) - (pw2 `2))) <= ((E-bound Pa) - (W-bound Pa)) + ((N-bound Pa) - (S-bound Pa)) by A63, XREAL_1:7;
((E-bound Pa) - (W-bound Pa)) + ((N-bound Pa) - (S-bound Pa)) <= (((E-bound Pa) - (W-bound Pa)) + ((N-bound Pa) - (S-bound Pa))) + 1 by XREAL_1:29;
then A66: (abs ((pw1 `1) - (pw2 `1))) + (abs ((pw1 `2) - (pw2 `2))) <= (((E-bound Pa) - (W-bound Pa)) + ((N-bound Pa) - (S-bound Pa))) + 1 by A65, XXREAL_0:2;
( dist (x,y) = |.(pw1 - pw2).| & |.(pw1 - pw2).| <= (abs ((pw1 `1) - (pw2 `1))) + (abs ((pw1 `2) - (pw2 `2))) ) by Th28, Th32;
hence dist (x,y) <= (((E-bound Pa) - (W-bound Pa)) + ((N-bound Pa) - (S-bound Pa))) + 1 by A66, XXREAL_0:2; ::_thesis: verum
end;
A67: p1 in Pa by A1, TOPREAL1:1;
then ( S-bound Pa <= p1 `2 & p1 `2 <= N-bound Pa ) by A57, PSCOMP_1:24;
then S-bound Pa <= N-bound Pa by XXREAL_0:2;
then A68: 0 <= (N-bound Pa) - (S-bound Pa) by XREAL_1:48;
( W-bound Pa <= p1 `1 & p1 `1 <= E-bound Pa ) by A57, A67, PSCOMP_1:24;
then W-bound Pa <= E-bound Pa by XXREAL_0:2;
then 0 <= (E-bound Pa) - (W-bound Pa) by XREAL_1:48;
then A69: W1 is bounded by A68, A59, TBSP_1:def_7;
A70: dom f1 = [#] I[01] by A13, A16, TOPS_2:def_5
.= the carrier of I[01] ;
i + 1 in dom h by A34, A46, A47, FINSEQ_3:25;
then h . (i + 1) in rng h by FUNCT_1:def_3;
then h19 . (i + 1) in rng f by A16, A31, A50, A70, FUNCT_1:def_3;
then A71: f1 . (h . (i + 1)) is Element of (TOP-REAL 2) by A35, A49, A58;
A72: i in dom h19 by A45, A52, FINSEQ_3:25;
then A73: h19 . i = f1 . (h . i) by A35;
then h19 . i in rng f by A16, A31, A55, A70, FUNCT_1:def_3;
then f1 . (h . i) is Element of (TOP-REAL 2) by A35, A72, A58;
then reconsider w1 = f1 . (h . i), w2 = f1 . (h . (i + 1)) as Point of (Euclid 2) by A71, TOPREAL3:8;
i < i + 1 by NAT_1:13;
then A74: h /. i <= h /. (i + 1) by A32, A54, A53, A56, A48, SEQM_3:def_1;
then h . i in R by A53, XXREAL_1:1;
then A75: w1 in f1 .: R by A70, FUNCT_1:def_6;
h . (i + 1) in R by A48, A74, XXREAL_1:1;
then A76: w2 in f1 .: R by A70, FUNCT_1:def_6;
dom f1 = [#] I[01] by A13, A16, TOPS_2:def_5;
then P = f1 .: [.0,1.] by A16, A58, BORSUK_1:40, RELAT_1:113;
then W is bounded by A69, BORSUK_1:40, RELAT_1:123, TBSP_1:14;
then A77: dist (w1,w2) <= diameter W by A75, A76, TBSP_1:def_8;
diameter W < (min_dist_min (P9,Q9)) / 5 by A33, A34, A45, A52;
then A78: dist (w1,w2) < (min_dist_min (P9,Q9)) / 5 by A77, XXREAL_0:2;
h1 /. i = f1 . (h . i) by A45, A52, A73, FINSEQ_4:15;
hence |.((h1 /. i) - (h1 /. (i + 1))).| < (min_dist_min (P9,Q9)) / 5 by A51, A78, Th28; ::_thesis: verum
end;
A79: for i being Element of NAT st i in dom h1 holds
( (h1 /. 1) `1 <= (h1 /. i) `1 & (h1 /. i) `1 <= (h1 /. (len h1)) `1 )
proof
len h in dom h19 by A34, A42, FINSEQ_3:25;
then h1 . (len h1) = f1 . (h . (len h)) by A34, A35;
then A80: h1 /. (len h1) = f1 . (h . (len h)) by A42, FINSEQ_4:15;
let i be Element of NAT ; ::_thesis: ( i in dom h1 implies ( (h1 /. 1) `1 <= (h1 /. i) `1 & (h1 /. i) `1 <= (h1 /. (len h1)) `1 ) )
assume A81: i in dom h1 ; ::_thesis: ( (h1 /. 1) `1 <= (h1 /. i) `1 & (h1 /. i) `1 <= (h1 /. (len h1)) `1 )
then h1 . i = f1 . (h . i) by A35;
then A82: h1 /. i = f1 . (h . i) by A81, PARTFUN1:def_6;
i in Seg (len h) by A34, A81, FINSEQ_1:def_3;
then i in dom h by FINSEQ_1:def_3;
then A83: h . i in rng h by FUNCT_1:def_3;
dom f1 = [#] I[01] by A13, A16, TOPS_2:def_5
.= the carrier of I[01] ;
then A84: h1 /. i in rng f by A16, A31, A82, A83, FUNCT_1:def_3;
1 in dom h19 by A42, FINSEQ_3:25;
then h1 . 1 = f1 . (h . 1) by A35;
then A85: h1 /. 1 = f1 . (h . 1) by A42, FINSEQ_4:15;
rng f = [#] ((TOP-REAL 2) | P) by A13, TOPS_2:def_5
.= P by PRE_TOPC:def_5 ;
hence ( (h1 /. 1) `1 <= (h1 /. i) `1 & (h1 /. i) `1 <= (h1 /. (len h1)) `1 ) by A3, A14, A15, A16, A28, A29, A85, A80, A84; ::_thesis: verum
end;
for i being Element of NAT st i in dom (X_axis h1) holds
( (X_axis h1) . 1 <= (X_axis h1) . i & (X_axis h1) . i <= (X_axis h1) . (len h1) )
proof
let i be Element of NAT ; ::_thesis: ( i in dom (X_axis h1) implies ( (X_axis h1) . 1 <= (X_axis h1) . i & (X_axis h1) . i <= (X_axis h1) . (len h1) ) )
A86: ( (X_axis h1) . 1 = (h1 /. 1) `1 & (X_axis h1) . (len h1) = (h1 /. (len h1)) `1 ) by A42, Th41;
assume i in dom (X_axis h1) ; ::_thesis: ( (X_axis h1) . 1 <= (X_axis h1) . i & (X_axis h1) . i <= (X_axis h1) . (len h1) )
then i in Seg (len (X_axis h1)) by FINSEQ_1:def_3;
then i in Seg (len h1) by A42, Th41;
then A87: i in dom h1 by FINSEQ_1:def_3;
then (X_axis h1) . i = (h1 /. i) `1 by Th43;
hence ( (X_axis h1) . 1 <= (X_axis h1) . i & (X_axis h1) . i <= (X_axis h1) . (len h1) ) by A79, A87, A86; ::_thesis: verum
end;
then A88: X_axis h1 lies_between (X_axis h1) . 1,(X_axis h1) . (len h1) by GOBOARD4:def_2;
A89: for i being Element of NAT st i in dom h1 holds
( q1 `2 <= (h1 /. i) `2 & (h1 /. i) `2 <= q2 `2 )
proof
let i be Element of NAT ; ::_thesis: ( i in dom h1 implies ( q1 `2 <= (h1 /. i) `2 & (h1 /. i) `2 <= q2 `2 ) )
A90: rng f = [#] ((TOP-REAL 2) | P) by A13, TOPS_2:def_5
.= P by PRE_TOPC:def_5 ;
assume A91: i in dom h1 ; ::_thesis: ( q1 `2 <= (h1 /. i) `2 & (h1 /. i) `2 <= q2 `2 )
then h1 . i = f1 . (h . i) by A35;
then A92: h1 /. i = f1 . (h . i) by A91, PARTFUN1:def_6;
i in Seg (len h1) by A91, FINSEQ_1:def_3;
then i in dom h by A34, FINSEQ_1:def_3;
then A93: h . i in rng h by FUNCT_1:def_3;
dom f1 = [#] I[01] by A13, A16, TOPS_2:def_5
.= the carrier of I[01] ;
then h1 /. i in rng f by A16, A31, A92, A93, FUNCT_1:def_3;
hence ( q1 `2 <= (h1 /. i) `2 & (h1 /. i) `2 <= q2 `2 ) by A5, A90; ::_thesis: verum
end;
for i being Element of NAT st i in dom (Y_axis h1) holds
( q1 `2 <= (Y_axis h1) . i & (Y_axis h1) . i <= q2 `2 )
proof
let i be Element of NAT ; ::_thesis: ( i in dom (Y_axis h1) implies ( q1 `2 <= (Y_axis h1) . i & (Y_axis h1) . i <= q2 `2 ) )
assume i in dom (Y_axis h1) ; ::_thesis: ( q1 `2 <= (Y_axis h1) . i & (Y_axis h1) . i <= q2 `2 )
then i in Seg (len (Y_axis h1)) by FINSEQ_1:def_3;
then i in Seg (len h1) by A42, Th42;
then A94: i in dom h1 by FINSEQ_1:def_3;
then (Y_axis h1) . i = (h1 /. i) `2 by Th43;
hence ( q1 `2 <= (Y_axis h1) . i & (Y_axis h1) . i <= q2 `2 ) by A89, A94; ::_thesis: verum
end;
then Y_axis h1 lies_between q1 `2 ,q2 `2 by GOBOARD4:def_2;
then consider f2 being FinSequence of (TOP-REAL 2) such that
A95: f2 is special and
A96: f2 . 1 = h1 . 1 and
A97: f2 . (len f2) = h1 . (len h1) and
A98: len f2 >= len h1 and
A99: ( X_axis f2 lies_between (X_axis h1) . 1,(X_axis h1) . (len h1) & Y_axis f2 lies_between q1 `2 ,q2 `2 ) and
A100: for j being Element of NAT st j in dom f2 holds
ex k being Element of NAT st
( k in dom h1 & |.((f2 /. j) - (h1 /. k)).| < (min_dist_min (P9,Q9)) / 5 ) and
A101: for j being Element of NAT st 1 <= j & j + 1 <= len f2 holds
|.((f2 /. j) - (f2 /. (j + 1))).| < (min_dist_min (P9,Q9)) / 5 by A21, A44, A42, A88, Th39;
A102: len f2 >= 1 by A42, A98, XXREAL_0:2;
then A103: f2 . (len f2) = f2 /. (len f2) by FINSEQ_4:15;
deffunc H2( Nat) -> set = g1 . (hb . $1);
ex h29 being FinSequence st
( len h29 = len hb & ( for i being Nat st i in dom h29 holds
h29 . i = H2(i) ) ) from FINSEQ_1:sch_2();
then consider h29 being FinSequence such that
A104: len h29 = len hb and
A105: for i being Nat st i in dom h29 holds
h29 . i = g1 . (hb . i) ;
rng h29 c= the carrier of (TOP-REAL 2)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng h29 or y in the carrier of (TOP-REAL 2) )
assume y in rng h29 ; ::_thesis: y in the carrier of (TOP-REAL 2)
then consider x being set such that
A106: x in dom h29 and
A107: y = h29 . x by FUNCT_1:def_3;
reconsider i = x as Element of NAT by A106;
dom h29 = Seg (len h29) by FINSEQ_1:def_3;
then i in dom hb by A104, A106, FINSEQ_1:def_3;
then A108: hb . i in rng hb by FUNCT_1:def_3;
A109: dom g1 = [#] I[01] by A7, A18, TOPS_2:def_5
.= the carrier of I[01] ;
A110: rng g = [#] ((TOP-REAL 2) | Q) by A7, TOPS_2:def_5
.= Q by PRE_TOPC:def_5 ;
h29 . i = g1 . (hb . i) by A105, A106;
then h29 . i in rng g by A18, A25, A108, A109, FUNCT_1:def_3;
hence y in the carrier of (TOP-REAL 2) by A107, A110; ::_thesis: verum
end;
then reconsider h2 = h29 as FinSequence of (TOP-REAL 2) by FINSEQ_1:def_4;
A111: rng f = [#] ((TOP-REAL 2) | P) by A13, TOPS_2:def_5
.= P by PRE_TOPC:def_5 ;
A112: for i being Element of NAT st 1 <= i & i + 1 <= len h2 holds
|.((h2 /. i) - (h2 /. (i + 1))).| < (min_dist_min (P9,Q9)) / 5
proof
reconsider Qa = Q as Subset of (TOP-REAL 2) ;
reconsider W1 = Q as Subset of (Euclid 2) by TOPREAL3:8;
let i be Element of NAT ; ::_thesis: ( 1 <= i & i + 1 <= len h2 implies |.((h2 /. i) - (h2 /. (i + 1))).| < (min_dist_min (P9,Q9)) / 5 )
assume that
A113: 1 <= i and
A114: i + 1 <= len h2 ; ::_thesis: |.((h2 /. i) - (h2 /. (i + 1))).| < (min_dist_min (P9,Q9)) / 5
A115: Qa is compact by A2, JORDAN5A:1;
reconsider Qa = Qa as non empty Subset of (TOP-REAL 2) ;
A116: rng g = [#] ((TOP-REAL 2) | Q) by A7, TOPS_2:def_5
.= Q by PRE_TOPC:def_5 ;
set r1 = (((E-bound Qa) - (W-bound Qa)) + ((N-bound Qa) - (S-bound Qa))) + 1;
A117: for x, y being Point of (Euclid 2) st x in W1 & y in W1 holds
dist (x,y) <= (((E-bound Qa) - (W-bound Qa)) + ((N-bound Qa) - (S-bound Qa))) + 1
proof
let x, y be Point of (Euclid 2); ::_thesis: ( x in W1 & y in W1 implies dist (x,y) <= (((E-bound Qa) - (W-bound Qa)) + ((N-bound Qa) - (S-bound Qa))) + 1 )
assume that
A118: x in W1 and
A119: y in W1 ; ::_thesis: dist (x,y) <= (((E-bound Qa) - (W-bound Qa)) + ((N-bound Qa) - (S-bound Qa))) + 1
reconsider pw1 = x, pw2 = y as Point of (TOP-REAL 2) by A118, A119;
A120: ( S-bound Qa <= pw2 `2 & pw2 `2 <= N-bound Qa ) by A115, A119, PSCOMP_1:24;
( S-bound Qa <= pw1 `2 & pw1 `2 <= N-bound Qa ) by A115, A118, PSCOMP_1:24;
then A121: abs ((pw1 `2) - (pw2 `2)) <= (N-bound Qa) - (S-bound Qa) by A120, Th27;
A122: ( W-bound Qa <= pw2 `1 & pw2 `1 <= E-bound Qa ) by A115, A119, PSCOMP_1:24;
( W-bound Qa <= pw1 `1 & pw1 `1 <= E-bound Qa ) by A115, A118, PSCOMP_1:24;
then abs ((pw1 `1) - (pw2 `1)) <= (E-bound Qa) - (W-bound Qa) by A122, Th27;
then A123: (abs ((pw1 `1) - (pw2 `1))) + (abs ((pw1 `2) - (pw2 `2))) <= ((E-bound Qa) - (W-bound Qa)) + ((N-bound Qa) - (S-bound Qa)) by A121, XREAL_1:7;
((E-bound Qa) - (W-bound Qa)) + ((N-bound Qa) - (S-bound Qa)) <= (((E-bound Qa) - (W-bound Qa)) + ((N-bound Qa) - (S-bound Qa))) + 1 by XREAL_1:29;
then A124: (abs ((pw1 `1) - (pw2 `1))) + (abs ((pw1 `2) - (pw2 `2))) <= (((E-bound Qa) - (W-bound Qa)) + ((N-bound Qa) - (S-bound Qa))) + 1 by A123, XXREAL_0:2;
( dist (x,y) = |.(pw1 - pw2).| & |.(pw1 - pw2).| <= (abs ((pw1 `1) - (pw2 `1))) + (abs ((pw1 `2) - (pw2 `2))) ) by Th28, Th32;
hence dist (x,y) <= (((E-bound Qa) - (W-bound Qa)) + ((N-bound Qa) - (S-bound Qa))) + 1 by A124, XXREAL_0:2; ::_thesis: verum
end;
A125: q1 in Qa by A2, TOPREAL1:1;
then ( S-bound Qa <= q1 `2 & q1 `2 <= N-bound Qa ) by A115, PSCOMP_1:24;
then S-bound Qa <= N-bound Qa by XXREAL_0:2;
then A126: 0 <= (N-bound Qa) - (S-bound Qa) by XREAL_1:48;
( W-bound Qa <= q1 `1 & q1 `1 <= E-bound Qa ) by A115, A125, PSCOMP_1:24;
then W-bound Qa <= E-bound Qa by XXREAL_0:2;
then 0 <= (E-bound Qa) - (W-bound Qa) by XREAL_1:48;
then A127: W1 is bounded by A126, A117, TBSP_1:def_7;
A128: dom g1 = [#] I[01] by A7, A18, TOPS_2:def_5
.= the carrier of I[01] ;
A129: 1 < i + 1 by A113, NAT_1:13;
then i + 1 in Seg (len hb) by A104, A114, FINSEQ_1:1;
then i + 1 in dom hb by FINSEQ_1:def_3;
then A130: hb . (i + 1) in rng hb by FUNCT_1:def_3;
A131: i < len h2 by A114, NAT_1:13;
then A132: hb . i = hb /. i by A104, A113, FINSEQ_4:15;
A133: i + 1 in dom h29 by A114, A129, FINSEQ_3:25;
then h29 . (i + 1) = g1 . (hb . (i + 1)) by A105;
then h29 . (i + 1) in rng g by A18, A25, A128, A130, FUNCT_1:def_3;
then A134: g1 . (hb . (i + 1)) is Element of (TOP-REAL 2) by A105, A133, A116;
A135: hb . (i + 1) = hb /. (i + 1) by A104, A114, A129, FINSEQ_4:15;
i in dom h29 by A113, A131, FINSEQ_3:25;
then A136: h29 . i = g1 . (hb . i) by A105;
i in Seg (len hb) by A104, A113, A131, FINSEQ_1:1;
then A137: i in dom hb by FINSEQ_1:def_3;
then A138: hb . i in rng hb by FUNCT_1:def_3;
then h29 . i in rng g by A18, A25, A136, A128, FUNCT_1:def_3;
then reconsider w1 = g1 . (hb . i), w2 = g1 . (hb . (i + 1)) as Point of (Euclid 2) by A136, A116, A134, TOPREAL3:8;
i + 1 in Seg (len hb) by A104, A114, A129, FINSEQ_1:1;
then A139: i + 1 in dom hb by FINSEQ_1:def_3;
then hb . (i + 1) in rng hb by FUNCT_1:def_3;
then reconsider R = [.(hb /. i),(hb /. (i + 1)).] as Subset of I[01] by A25, A138, A132, A135, BORSUK_1:40, XXREAL_2:def_12;
i < i + 1 by NAT_1:13;
then A140: hb /. i <= hb /. (i + 1) by A26, A137, A132, A139, A135, SEQM_3:def_1;
then hb . i in R by A132, XXREAL_1:1;
then A141: w1 in g1 .: R by A128, FUNCT_1:def_6;
hb . (i + 1) in R by A135, A140, XXREAL_1:1;
then A142: w2 in g1 .: R by A128, FUNCT_1:def_6;
reconsider W = g1 .: R as Subset of (Euclid 2) by TOPREAL3:8;
dom g1 = [#] I[01] by A7, A18, TOPS_2:def_5;
then Q = g1 .: [.0,1.] by A18, A116, BORSUK_1:40, RELAT_1:113;
then W is bounded by A127, BORSUK_1:40, RELAT_1:123, TBSP_1:14;
then A143: dist (w1,w2) <= diameter W by A141, A142, TBSP_1:def_8;
diameter W < (min_dist_min (P9,Q9)) / 5 by A27, A104, A113, A131;
then A144: dist (w1,w2) < (min_dist_min (P9,Q9)) / 5 by A143, XXREAL_0:2;
h2 . (i + 1) = h2 /. (i + 1) by A114, A129, FINSEQ_4:15;
then A145: h2 /. (i + 1) = g1 . (hb . (i + 1)) by A105, A133;
h2 /. i = g1 . (hb . i) by A113, A131, A136, FINSEQ_4:15;
hence |.((h2 /. i) - (h2 /. (i + 1))).| < (min_dist_min (P9,Q9)) / 5 by A145, A144, Th28; ::_thesis: verum
end;
A146: 1 <= len hb by A24, XXREAL_0:2;
then A147: len hb in dom hb by FINSEQ_3:25;
A148: 1 <= len hb by A24, XXREAL_0:2;
then A149: h2 . (len h2) = h2 /. (len h2) by A104, FINSEQ_4:15;
A150: dom hb = Seg (len hb) by FINSEQ_1:def_3;
A151: for i being Element of NAT st i in dom hb holds
h2 /. i = g1 . (hb . i)
proof
let i be Element of NAT ; ::_thesis: ( i in dom hb implies h2 /. i = g1 . (hb . i) )
assume A152: i in dom hb ; ::_thesis: h2 /. i = g1 . (hb . i)
then i in dom h29 by A104, FINSEQ_3:29;
then A153: h2 . i = g1 . (hb . i) by A105;
( 1 <= i & i <= len hb ) by A150, A152, FINSEQ_1:1;
hence h2 /. i = g1 . (hb . i) by A104, A153, FINSEQ_4:15; ::_thesis: verum
end;
A154: f2 . 1 = f2 /. 1 by A102, FINSEQ_4:15;
A155: 1 <= len h by A30, XXREAL_0:2;
then 1 in dom h19 by A34, FINSEQ_3:25;
then A156: h1 /. 1 = f1 . (h . 1) by A35, A43;
len h in dom h19 by A34, A155, FINSEQ_3:25;
then A157: f2 /. 1 <> f2 /. (len f2) by A1, A14, A15, A16, A28, A29, A34, A35, A96, A97, A43, A154, A103, A156, JORDAN6:37;
5 <= len f2 by A30, A34, A98, XXREAL_0:2;
then A158: 2 <= len f2 by XXREAL_0:2;
A159: h1 . (len h1) = h1 /. (len h1) by A42, FINSEQ_4:15;
then A160: (X_axis f2) . (len f2) = (h1 /. (len h1)) `1 by A97, A102, A103, Th41
.= (X_axis h1) . (len h1) by A42, Th41 ;
A161: h2 . 1 = h2 /. 1 by A148, A104, FINSEQ_4:15;
A162: len h2 >= 1 by A24, A104, XXREAL_0:2;
A163: for i being Element of NAT st i in dom h2 holds
( (h2 /. 1) `2 <= (h2 /. i) `2 & (h2 /. i) `2 <= (h2 /. (len h2)) `2 )
proof
let i be Element of NAT ; ::_thesis: ( i in dom h2 implies ( (h2 /. 1) `2 <= (h2 /. i) `2 & (h2 /. i) `2 <= (h2 /. (len h2)) `2 ) )
assume i in dom h2 ; ::_thesis: ( (h2 /. 1) `2 <= (h2 /. i) `2 & (h2 /. i) `2 <= (h2 /. (len h2)) `2 )
then i in Seg (len h2) by FINSEQ_1:def_3;
then i in dom hb by A104, FINSEQ_1:def_3;
then A164: ( h2 /. i = g1 . (hb . i) & hb . i in rng hb ) by A151, FUNCT_1:def_3;
dom g1 = [#] I[01] by A7, A18, TOPS_2:def_5
.= the carrier of I[01] ;
then A165: h2 /. i in rng g by A18, A25, A164, FUNCT_1:def_3;
1 in Seg (len hb) by A104, A162, FINSEQ_1:1;
then 1 in dom hb by FINSEQ_1:def_3;
then A166: h2 /. 1 = g1 . (hb . 1) by A151;
len hb in Seg (len hb) by A104, A162, FINSEQ_1:1;
then len hb in dom hb by FINSEQ_1:def_3;
then A167: h2 /. (len h2) = g1 . (hb . (len hb)) by A104, A151;
rng g = [#] ((TOP-REAL 2) | Q) by A7, TOPS_2:def_5
.= Q by PRE_TOPC:def_5 ;
hence ( (h2 /. 1) `2 <= (h2 /. i) `2 & (h2 /. i) `2 <= (h2 /. (len h2)) `2 ) by A6, A8, A9, A18, A22, A23, A166, A167, A165; ::_thesis: verum
end;
for i being Element of NAT st i in dom (Y_axis h2) holds
( (Y_axis h2) . 1 <= (Y_axis h2) . i & (Y_axis h2) . i <= (Y_axis h2) . (len h2) )
proof
let i be Element of NAT ; ::_thesis: ( i in dom (Y_axis h2) implies ( (Y_axis h2) . 1 <= (Y_axis h2) . i & (Y_axis h2) . i <= (Y_axis h2) . (len h2) ) )
A168: ( (Y_axis h2) . 1 = (h2 /. 1) `2 & (Y_axis h2) . (len h2) = (h2 /. (len h2)) `2 ) by A162, Th42;
assume i in dom (Y_axis h2) ; ::_thesis: ( (Y_axis h2) . 1 <= (Y_axis h2) . i & (Y_axis h2) . i <= (Y_axis h2) . (len h2) )
then i in Seg (len (Y_axis h2)) by FINSEQ_1:def_3;
then i in Seg (len h2) by A162, Th42;
then A169: i in dom h2 by FINSEQ_1:def_3;
then (Y_axis h2) . i = (h2 /. i) `2 by Th43;
hence ( (Y_axis h2) . 1 <= (Y_axis h2) . i & (Y_axis h2) . i <= (Y_axis h2) . (len h2) ) by A163, A169, A168; ::_thesis: verum
end;
then A170: Y_axis h2 lies_between (Y_axis h2) . 1,(Y_axis h2) . (len h2) by GOBOARD4:def_2;
A171: for i being Element of NAT st i in dom h2 holds
( p1 `1 <= (h2 /. i) `1 & (h2 /. i) `1 <= p2 `1 )
proof
let i be Element of NAT ; ::_thesis: ( i in dom h2 implies ( p1 `1 <= (h2 /. i) `1 & (h2 /. i) `1 <= p2 `1 ) )
A172: rng g = [#] ((TOP-REAL 2) | Q) by A7, TOPS_2:def_5
.= Q by PRE_TOPC:def_5 ;
assume i in dom h2 ; ::_thesis: ( p1 `1 <= (h2 /. i) `1 & (h2 /. i) `1 <= p2 `1 )
then i in Seg (len h2) by FINSEQ_1:def_3;
then i in dom hb by A104, FINSEQ_1:def_3;
then A173: ( h2 /. i = g1 . (hb . i) & hb . i in rng hb ) by A151, FUNCT_1:def_3;
dom g1 = [#] I[01] by A7, A18, TOPS_2:def_5
.= the carrier of I[01] ;
then h2 /. i in rng g by A18, A25, A173, FUNCT_1:def_3;
hence ( p1 `1 <= (h2 /. i) `1 & (h2 /. i) `1 <= p2 `1 ) by A4, A172; ::_thesis: verum
end;
for i being Element of NAT st i in dom (X_axis h2) holds
( p1 `1 <= (X_axis h2) . i & (X_axis h2) . i <= p2 `1 )
proof
let i be Element of NAT ; ::_thesis: ( i in dom (X_axis h2) implies ( p1 `1 <= (X_axis h2) . i & (X_axis h2) . i <= p2 `1 ) )
assume i in dom (X_axis h2) ; ::_thesis: ( p1 `1 <= (X_axis h2) . i & (X_axis h2) . i <= p2 `1 )
then i in Seg (len (X_axis h2)) by FINSEQ_1:def_3;
then i in Seg (len h2) by A162, Th41;
then A174: i in dom h2 by FINSEQ_1:def_3;
then (X_axis h2) . i = (h2 /. i) `1 by Th43;
hence ( p1 `1 <= (X_axis h2) . i & (X_axis h2) . i <= p2 `1 ) by A171, A174; ::_thesis: verum
end;
then X_axis h2 lies_between p1 `1 ,p2 `1 by GOBOARD4:def_2;
then consider g2 being FinSequence of (TOP-REAL 2) such that
A175: g2 is special and
A176: g2 . 1 = h2 . 1 and
A177: g2 . (len g2) = h2 . (len h2) and
A178: len g2 >= len h2 and
A179: ( Y_axis g2 lies_between (Y_axis h2) . 1,(Y_axis h2) . (len h2) & X_axis g2 lies_between p1 `1 ,p2 `1 ) and
A180: for j being Element of NAT st j in dom g2 holds
ex k being Element of NAT st
( k in dom h2 & |.((g2 /. j) - (h2 /. k)).| < (min_dist_min (P9,Q9)) / 5 ) and
A181: for j being Element of NAT st 1 <= j & j + 1 <= len g2 holds
|.((g2 /. j) - (g2 /. (j + 1))).| < (min_dist_min (P9,Q9)) / 5 by A21, A162, A170, A112, Th40;
5 <= len g2 by A24, A104, A178, XXREAL_0:2;
then A182: 2 <= len g2 by XXREAL_0:2;
A183: len g2 >= 1 by A162, A178, XXREAL_0:2;
then g2 . 1 = g2 /. 1 by FINSEQ_4:15;
then A184: (Y_axis g2) . 1 = (h2 /. 1) `2 by A176, A183, A161, Th42
.= (Y_axis h2) . 1 by A162, Th42 ;
1 in dom hb by A146, FINSEQ_3:25;
then h2 /. 1 = g1 . (hb . 1) by A151;
then A185: g2 . 1 <> g2 . (len g2) by A2, A8, A9, A18, A22, A23, A104, A151, A176, A177, A161, A149, A147, JORDAN6:37;
len hb in dom hb by A148, FINSEQ_3:25;
then A186: g2 . (len g2) = q2 by A9, A18, A23, A104, A151, A177, A149;
g2 /. (len g2) = g2 . (len g2) by A183, FINSEQ_4:15;
then A187: (Y_axis g2) . (len g2) = q2 `2 by A183, A186, Th42;
1 in dom hb by A148, FINSEQ_3:25;
then A188: h2 /. 1 = q1 by A8, A18, A22, A151;
A189: rng g = [#] ((TOP-REAL 2) | Q) by A7, TOPS_2:def_5
.= Q by PRE_TOPC:def_5 ;
len h in dom h19 by A34, A42, FINSEQ_3:25;
then h1 /. (len h1) = p2 by A15, A16, A29, A34, A35, A159;
then A190: (X_axis f2) . (len f2) = p2 `1 by A97, A102, A159, A103, Th41;
1 in dom h19 by A42, FINSEQ_3:25;
then h1 . 1 = f1 . (h . 1) by A35;
then A191: (X_axis f2) . 1 = p1 `1 by A14, A16, A28, A96, A102, A154, Th41;
g2 . (len g2) = g2 /. (len g2) by A183, FINSEQ_4:15;
then A192: (Y_axis g2) . (len g2) = (h2 /. (len h2)) `2 by A177, A183, A149, Th42
.= (Y_axis h2) . (len h2) by A162, Th42 ;
g2 /. 1 = g2 . 1 by A183, FINSEQ_4:15;
then A193: (Y_axis g2) . 1 = q1 `2 by A176, A183, A188, A161, Th42;
(X_axis f2) . 1 = (h1 /. 1) `1 by A96, A102, A43, A154, Th41
.= (X_axis h1) . 1 by A42, Th41 ;
then L~ f2 meets L~ g2 by A95, A99, A175, A179, A154, A103, A191, A190, A193, A187, A160, A184, A192, A158, A182, A157, A185, Th26;
then consider s being set such that
A194: s in L~ f2 and
A195: s in L~ g2 by XBOOLE_0:3;
reconsider ps = s as Point of (TOP-REAL 2) by A194;
ps in union { (LSeg (g2,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len g2 ) } by A195, TOPREAL1:def_4;
then consider y being set such that
A196: ( ps in y & y in { (LSeg (g2,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len g2 ) } ) by TARSKI:def_4;
ps in union { (LSeg (f2,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len f2 ) } by A194, TOPREAL1:def_4;
then consider x being set such that
A197: ( ps in x & x in { (LSeg (f2,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len f2 ) } ) by TARSKI:def_4;
consider i being Element of NAT such that
A198: x = LSeg (f2,i) and
A199: 1 <= i and
A200: i + 1 <= len f2 by A197;
LSeg (f2,i) = LSeg ((f2 /. i),(f2 /. (i + 1))) by A199, A200, TOPREAL1:def_3;
then A201: |.(ps - (f2 /. i)).| <= |.((f2 /. i) - (f2 /. (i + 1))).| by A197, A198, Th36;
i < len f2 by A200, NAT_1:13;
then i in dom f2 by A199, FINSEQ_3:25;
then consider k being Element of NAT such that
A202: k in dom h1 and
A203: |.((f2 /. i) - (h1 /. k)).| < (min_dist_min (P9,Q9)) / 5 by A100;
k in dom h by A34, A202, FINSEQ_3:29;
then A204: h . k in rng h by FUNCT_1:def_3;
reconsider p11 = h1 /. k as Point of (TOP-REAL 2) ;
|.((f2 /. i) - (f2 /. (i + 1))).| < (min_dist_min (P9,Q9)) / 5 by A101, A199, A200;
then |.(ps - (f2 /. i)).| < (min_dist_min (P9,Q9)) / 5 by A201, XXREAL_0:2;
then ( |.(ps - (h1 /. k)).| <= |.(ps - (f2 /. i)).| + |.((f2 /. i) - (h1 /. k)).| & |.(ps - (f2 /. i)).| + |.((f2 /. i) - (h1 /. k)).| < ((min_dist_min (P9,Q9)) / 5) + ((min_dist_min (P9,Q9)) / 5) ) by A203, TOPRNS_1:34, XREAL_1:8;
then |.(ps - (h1 /. k)).| < ((min_dist_min (P9,Q9)) / 5) + ((min_dist_min (P9,Q9)) / 5) by XXREAL_0:2;
then A205: |.(p11 - ps).| < ((min_dist_min (P9,Q9)) / 5) + ((min_dist_min (P9,Q9)) / 5) by TOPRNS_1:27;
k in Seg (len h1) by A202, FINSEQ_1:def_3;
then ( 1 <= k & k <= len h1 ) by FINSEQ_1:1;
then h1 . k = h1 /. k by FINSEQ_4:15;
then A206: h1 /. k = f1 . (h . k) by A35, A202;
consider j being Element of NAT such that
A207: y = LSeg (g2,j) and
A208: 1 <= j and
A209: j + 1 <= len g2 by A196;
LSeg (g2,j) = LSeg ((g2 /. j),(g2 /. (j + 1))) by A208, A209, TOPREAL1:def_3;
then A210: |.(ps - (g2 /. j)).| <= |.((g2 /. j) - (g2 /. (j + 1))).| by A196, A207, Th36;
j < len g2 by A209, NAT_1:13;
then j in Seg (len g2) by A208, FINSEQ_1:1;
then j in dom g2 by FINSEQ_1:def_3;
then consider k9 being Element of NAT such that
A211: k9 in dom h2 and
A212: |.((g2 /. j) - (h2 /. k9)).| < (min_dist_min (P9,Q9)) / 5 by A180;
k9 in Seg (len h2) by A211, FINSEQ_1:def_3;
then A213: k9 in dom hb by A104, FINSEQ_1:def_3;
then A214: hb . k9 in rng hb by FUNCT_1:def_3;
reconsider q11 = h2 /. k9 as Point of (TOP-REAL 2) ;
|.((g2 /. j) - (g2 /. (j + 1))).| < (min_dist_min (P9,Q9)) / 5 by A181, A208, A209;
then |.(ps - (g2 /. j)).| < (min_dist_min (P9,Q9)) / 5 by A210, XXREAL_0:2;
then ( |.(ps - (h2 /. k9)).| <= |.(ps - (g2 /. j)).| + |.((g2 /. j) - (h2 /. k9)).| & |.(ps - (g2 /. j)).| + |.((g2 /. j) - (h2 /. k9)).| < ((min_dist_min (P9,Q9)) / 5) + ((min_dist_min (P9,Q9)) / 5) ) by A212, TOPRNS_1:34, XREAL_1:8;
then |.(ps - (h2 /. k9)).| < ((min_dist_min (P9,Q9)) / 5) + ((min_dist_min (P9,Q9)) / 5) by XXREAL_0:2;
then ( |.(p11 - q11).| <= |.(p11 - ps).| + |.(ps - q11).| & |.(p11 - ps).| + |.(ps - q11).| < (((min_dist_min (P9,Q9)) / 5) + ((min_dist_min (P9,Q9)) / 5)) + (((min_dist_min (P9,Q9)) / 5) + ((min_dist_min (P9,Q9)) / 5)) ) by A205, TOPRNS_1:34, XREAL_1:8;
then A215: |.(p11 - q11).| < ((((min_dist_min (P9,Q9)) / 5) + ((min_dist_min (P9,Q9)) / 5)) + ((min_dist_min (P9,Q9)) / 5)) + ((min_dist_min (P9,Q9)) / 5) by XXREAL_0:2;
h2 /. k9 = g1 . (hb . k9) by A151, A213;
then A216: h2 /. k9 in rng g by A18, A25, A214, A36, FUNCT_1:def_3;
reconsider x1 = p11, x2 = q11 as Point of (Euclid 2) by EUCLID:22;
dom f1 = [#] I[01] by A13, A16, TOPS_2:def_5
.= the carrier of I[01] ;
then h1 /. k in P by A16, A31, A206, A204, A111, FUNCT_1:def_3;
then min_dist_min (P9,Q9) <= dist (x1,x2) by A11, A12, A216, A189, WEIERSTR:34;
then min_dist_min (P9,Q9) <= |.(p11 - q11).| by Th28;
then min_dist_min (P9,Q9) < 4 * ((min_dist_min (P9,Q9)) / 5) by A215, XXREAL_0:2;
then (4 * ((min_dist_min (P9,Q9)) / 5)) - (5 * ((min_dist_min (P9,Q9)) / 5)) > 0 by XREAL_1:50;
then ((4 - 5) * ((min_dist_min (P9,Q9)) / 5)) / ((min_dist_min (P9,Q9)) / 5) > 0 by A21, XREAL_1:139;
then 4 - 5 > 0 by A20;
hence contradiction ; ::_thesis: verum
end;
theorem Th45: :: 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
proof
let X, Y be non empty TopSpace; ::_thesis: 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
let f be Function of X,Y; ::_thesis: 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
let P be non empty Subset of Y; ::_thesis: for f1 being Function of X,(Y | P) st f = f1 & f is continuous holds
f1 is continuous
let f1 be Function of X,(Y | P); ::_thesis: ( f = f1 & f is continuous implies f1 is continuous )
assume that
A1: f = f1 and
A2: f is continuous ; ::_thesis: f1 is continuous
A3: [#] Y <> {} ;
A4: for P1 being Subset of (Y | P) st P1 is open holds
f1 " P1 is open
proof
let P1 be Subset of (Y | P); ::_thesis: ( P1 is open implies f1 " P1 is open )
assume P1 is open ; ::_thesis: f1 " P1 is open
then P1 in the topology of (Y | P) by PRE_TOPC:def_2;
then consider Q being Subset of Y such that
A5: Q in the topology of Y and
A6: P1 = Q /\ ([#] (Y | P)) by PRE_TOPC:def_4;
reconsider Q = Q as Subset of Y ;
A7: f " Q = f1 " ((rng f1) /\ Q) by A1, RELAT_1:133;
A8: [#] (Y | P) = P by PRE_TOPC:def_5;
then (rng f1) /\ Q c= P /\ Q by XBOOLE_1:26;
then A9: f1 " ((rng f1) /\ Q) c= f1 " P1 by A6, A8, RELAT_1:143;
Q is open by A5, PRE_TOPC:def_2;
then A10: f " Q is open by A3, A2, TOPS_2:43;
f1 " P1 c= f " Q by A1, A6, RELAT_1:143, XBOOLE_1:17;
hence f1 " P1 is open by A10, A7, A9, XBOOLE_0:def_10; ::_thesis: verum
end;
[#] (Y | P) <> {} ;
hence f1 is continuous by A4, TOPS_2:43; ::_thesis: verum
end;
theorem Th46: :: 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 )
proof
let X, Y be non empty TopSpace; ::_thesis: 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 )
let f be Function of X,Y; ::_thesis: 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 )
let P be non empty Subset of Y; ::_thesis: ( X is compact & Y is T_2 & f is continuous & f is one-to-one & P = rng f implies ex f1 being Function of X,(Y | P) st
( f = f1 & f1 is being_homeomorphism ) )
assume that
A1: X is compact and
A2: Y is T_2 and
A3: ( f is continuous & f is one-to-one ) and
A4: P = rng f ; ::_thesis: ex f1 being Function of X,(Y | P) st
( f = f1 & f1 is being_homeomorphism )
( the carrier of (Y | P) = P & dom f = the carrier of X ) by FUNCT_2:def_1, PRE_TOPC:8;
then reconsider f2 = f as Function of X,(Y | P) by A4, FUNCT_2:1;
A5: ( dom f2 = [#] X & f2 is continuous ) by A3, Th45, FUNCT_2:def_1;
( rng f2 = [#] (Y | P) & Y | P is T_2 ) by A2, A4, PRE_TOPC:def_5, TOPMETR:2;
hence ex f1 being Function of X,(Y | P) st
( f = f1 & f1 is being_homeomorphism ) by A1, A3, A5, COMPTS_1:17; ::_thesis: verum
end;
theorem :: JGRAPH_1:47
for f, g being Function of I[01],(TOP-REAL 2)
for a, b, c, d being real number
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
let f, g be Function of I[01],(TOP-REAL 2); ::_thesis: for a, b, c, d being real number
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
let a, b, c, d be real number ; ::_thesis: 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
let O, I be Point of I[01]; ::_thesis: ( 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 ) ) implies rng f meets rng g )
assume that
A1: ( O = 0 & I = 1 ) and
A2: ( f is continuous & f is one-to-one ) and
A3: ( g is continuous & g is one-to-one ) and
A4: (f . O) `1 = a and
A5: (f . I) `1 = b and
A6: (g . O) `2 = c and
A7: (g . I) `2 = d and
A8: 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 ) ; ::_thesis: rng f meets rng g
reconsider P = rng f as non empty Subset of (TOP-REAL 2) ;
A9: I[01] is compact by HEINE:4, TOPMETR:20;
then consider f1 being Function of I[01],((TOP-REAL 2) | P) such that
A10: f = f1 and
A11: f1 is being_homeomorphism by A2, Th46;
reconsider Q = rng g as non empty Subset of (TOP-REAL 2) ;
consider g1 being Function of I[01],((TOP-REAL 2) | Q) such that
A12: g = g1 and
A13: g1 is being_homeomorphism by A3, A9, Th46;
reconsider q2 = g1 . I as Point of (TOP-REAL 2) by A7, A12;
reconsider q1 = g1 . O as Point of (TOP-REAL 2) by A6, A12;
A14: Q is_an_arc_of q1,q2 by A1, A13, TOPREAL1:def_1;
reconsider p2 = f1 . I as Point of (TOP-REAL 2) by A5, A10;
reconsider p1 = f1 . O as Point of (TOP-REAL 2) by A4, A10;
A15: for p being Point of (TOP-REAL 2) st p in P holds
( p1 `1 <= p `1 & p `1 <= p2 `1 )
proof
let p be Point of (TOP-REAL 2); ::_thesis: ( p in P implies ( p1 `1 <= p `1 & p `1 <= p2 `1 ) )
assume p in P ; ::_thesis: ( p1 `1 <= p `1 & p `1 <= p2 `1 )
then ex x being set st
( x in dom f1 & p = f1 . x ) by A10, FUNCT_1:def_3;
hence ( p1 `1 <= p `1 & p `1 <= p2 `1 ) by A4, A5, A8, A10; ::_thesis: verum
end;
A16: for p being Point of (TOP-REAL 2) st p in Q holds
( p1 `1 <= p `1 & p `1 <= p2 `1 )
proof
let p be Point of (TOP-REAL 2); ::_thesis: ( p in Q implies ( p1 `1 <= p `1 & p `1 <= p2 `1 ) )
assume p in Q ; ::_thesis: ( p1 `1 <= p `1 & p `1 <= p2 `1 )
then ex x being set st
( x in dom g1 & p = g1 . x ) by A12, FUNCT_1:def_3;
hence ( p1 `1 <= p `1 & p `1 <= p2 `1 ) by A4, A5, A8, A10, A12; ::_thesis: verum
end;
A17: for p being Point of (TOP-REAL 2) st p in Q holds
( q1 `2 <= p `2 & p `2 <= q2 `2 )
proof
let p be Point of (TOP-REAL 2); ::_thesis: ( p in Q implies ( q1 `2 <= p `2 & p `2 <= q2 `2 ) )
assume p in Q ; ::_thesis: ( q1 `2 <= p `2 & p `2 <= q2 `2 )
then ex x being set st
( x in dom g1 & p = g1 . x ) by A12, FUNCT_1:def_3;
hence ( q1 `2 <= p `2 & p `2 <= q2 `2 ) by A6, A7, A8, A12; ::_thesis: verum
end;
A18: for p being Point of (TOP-REAL 2) st p in P holds
( q1 `2 <= p `2 & p `2 <= q2 `2 )
proof
let p be Point of (TOP-REAL 2); ::_thesis: ( p in P implies ( q1 `2 <= p `2 & p `2 <= q2 `2 ) )
assume p in P ; ::_thesis: ( q1 `2 <= p `2 & p `2 <= q2 `2 )
then ex x being set st
( x in dom f1 & p = f1 . x ) by A10, FUNCT_1:def_3;
hence ( q1 `2 <= p `2 & p `2 <= q2 `2 ) by A6, A7, A8, A10, A12; ::_thesis: verum
end;
P is_an_arc_of p1,p2 by A1, A11, TOPREAL1:def_1;
hence rng f meets rng g by A14, A15, A16, A18, A17, Th44; ::_thesis: verum
end;