:: JORDAN7 semantic presentation
begin
Lm1: 2 -' 1 = 2 - 1 by XREAL_1:233
.= 1 ;
Lm2: for i, j, k being Element of NAT st i -' k <= j holds
i <= j + k
proof
let i, j, k be Element of NAT ; ::_thesis: ( i -' k <= j implies i <= j + k )
assume A1: i -' k <= j ; ::_thesis: i <= j + k
percases ( i >= k or i <= k ) ;
supposeA2: i >= k ; ::_thesis: i <= j + k
(i -' k) + k <= j + k by A1, XREAL_1:6;
hence i <= j + k by A2, XREAL_1:235; ::_thesis: verum
end;
supposeA3: i <= k ; ::_thesis: i <= j + k
k <= j + k by NAT_1:11;
hence i <= j + k by A3, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
Lm3: for i, j, k being Element of NAT st j + k <= i holds
k <= i -' j
proof
let i, j, k be Element of NAT ; ::_thesis: ( j + k <= i implies k <= i -' j )
assume A1: j + k <= i ; ::_thesis: k <= i -' j
percases ( j + k = i or j + k < i ) by A1, XXREAL_0:1;
suppose j + k = i ; ::_thesis: k <= i -' j
hence k <= i -' j by NAT_D:34; ::_thesis: verum
end;
suppose j + k < i ; ::_thesis: k <= i -' j
hence k <= i -' j by Lm2; ::_thesis: verum
end;
end;
end;
theorem Th1: :: JORDAN7:1
for P being non empty compact Subset of (TOP-REAL 2) st P is being_simple_closed_curve holds
( W-min P in Lower_Arc P & E-max P in Lower_Arc P & W-min P in Upper_Arc P & E-max P in Upper_Arc P )
proof
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P is being_simple_closed_curve implies ( W-min P in Lower_Arc P & E-max P in Lower_Arc P & W-min P in Upper_Arc P & E-max P in Upper_Arc P ) )
assume P is being_simple_closed_curve ; ::_thesis: ( W-min P in Lower_Arc P & E-max P in Lower_Arc P & W-min P in Upper_Arc P & E-max P in Upper_Arc P )
then ( Upper_Arc P is_an_arc_of W-min P, E-max P & Lower_Arc P is_an_arc_of E-max P, W-min P ) by JORDAN6:50;
hence ( W-min P in Lower_Arc P & E-max P in Lower_Arc P & W-min P in Upper_Arc P & E-max P in Upper_Arc P ) by TOPREAL1:1; ::_thesis: verum
end;
theorem Th2: :: JORDAN7:2
for P being non empty compact Subset of (TOP-REAL 2)
for q being Point of (TOP-REAL 2) st P is being_simple_closed_curve & LE q, W-min P,P holds
q = W-min P
proof
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: for q being Point of (TOP-REAL 2) st P is being_simple_closed_curve & LE q, W-min P,P holds
q = W-min P
let q be Point of (TOP-REAL 2); ::_thesis: ( P is being_simple_closed_curve & LE q, W-min P,P implies q = W-min P )
assume ( P is being_simple_closed_curve & LE q, W-min P,P ) ; ::_thesis: q = W-min P
then ( LE q, W-min P, Upper_Arc P, W-min P, E-max P & Upper_Arc P is_an_arc_of W-min P, E-max P ) by JORDAN6:def_8, JORDAN6:def_10;
hence q = W-min P by JORDAN6:54; ::_thesis: verum
end;
theorem Th3: :: JORDAN7:3
for P being non empty compact Subset of (TOP-REAL 2)
for q being Point of (TOP-REAL 2) st P is being_simple_closed_curve & q in P holds
LE W-min P,q,P
proof
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: for q being Point of (TOP-REAL 2) st P is being_simple_closed_curve & q in P holds
LE W-min P,q,P
let q be Point of (TOP-REAL 2); ::_thesis: ( P is being_simple_closed_curve & q in P implies LE W-min P,q,P )
assume that
A1: P is being_simple_closed_curve and
A2: q in P ; ::_thesis: LE W-min P,q,P
A3: q in (Upper_Arc P) \/ (Lower_Arc P) by A1, A2, JORDAN6:50;
A4: Upper_Arc P is_an_arc_of W-min P, E-max P by A1, JORDAN6:50;
A5: W-min P in Upper_Arc P by A1, Th1;
percases ( q in Upper_Arc P or q in Lower_Arc P ) by A3, XBOOLE_0:def_3;
supposeA6: q in Upper_Arc P ; ::_thesis: LE W-min P,q,P
then LE W-min P,q, Upper_Arc P, W-min P, E-max P by A4, JORDAN5C:10;
hence LE W-min P,q,P by A5, A6, JORDAN6:def_10; ::_thesis: verum
end;
supposeA7: q in Lower_Arc P ; ::_thesis: LE W-min P,q,P
percases ( not q = W-min P or q = W-min P ) ;
suppose not q = W-min P ; ::_thesis: LE W-min P,q,P
hence LE W-min P,q,P by A5, A7, JORDAN6:def_10; ::_thesis: verum
end;
supposeA8: q = W-min P ; ::_thesis: LE W-min P,q,P
then LE W-min P,q, Upper_Arc P, W-min P, E-max P by A5, JORDAN5C:9;
hence LE W-min P,q,P by A5, A8, JORDAN6:def_10; ::_thesis: verum
end;
end;
end;
end;
end;
definition
let P be non empty compact Subset of (TOP-REAL 2);
let q1, q2 be Point of (TOP-REAL 2);
func Segment (q1,q2,P) -> Subset of (TOP-REAL 2) equals :Def1: :: JORDAN7:def 1
{ p where p is Point of (TOP-REAL 2) : ( LE q1,p,P & LE p,q2,P ) } if q2 <> W-min P
otherwise { p1 where p1 is Point of (TOP-REAL 2) : ( LE q1,p1,P or ( q1 in P & p1 = W-min P ) ) } ;
correctness
coherence
( ( q2 <> W-min P implies { p where p is Point of (TOP-REAL 2) : ( LE q1,p,P & LE p,q2,P ) } is Subset of (TOP-REAL 2) ) & ( not q2 <> W-min P implies { p1 where p1 is Point of (TOP-REAL 2) : ( LE q1,p1,P or ( q1 in P & p1 = W-min P ) ) } is Subset of (TOP-REAL 2) ) );
consistency
for b1 being Subset of (TOP-REAL 2) holds verum;
proof
ex B being Subset of (TOP-REAL 2) st
( ( q2 <> W-min P implies B = { p where p is Point of (TOP-REAL 2) : ( LE q1,p,P & LE p,q2,P ) } ) & ( not q2 <> W-min P implies B = { p1 where p1 is Point of (TOP-REAL 2) : ( LE q1,p1,P or ( q1 in P & p1 = W-min P ) ) } ) )
proof
percases ( q2 <> W-min P or not q2 <> W-min P ) ;
supposeA1: q2 <> W-min P ; ::_thesis: ex B being Subset of (TOP-REAL 2) st
( ( q2 <> W-min P implies B = { p where p is Point of (TOP-REAL 2) : ( LE q1,p,P & LE p,q2,P ) } ) & ( not q2 <> W-min P implies B = { p1 where p1 is Point of (TOP-REAL 2) : ( LE q1,p1,P or ( q1 in P & p1 = W-min P ) ) } ) )
defpred S1[ Point of (TOP-REAL 2)] means ( LE q1,$1,P & LE $1,q2,P );
{ p where p is Point of (TOP-REAL 2) : S1[p] } is Subset of (TOP-REAL 2) from DOMAIN_1:sch_7();
then reconsider C = { p where p is Point of (TOP-REAL 2) : ( LE q1,p,P & LE p,q2,P ) } as Subset of (TOP-REAL 2) ;
( q2 <> W-min P implies C = { p where p is Point of (TOP-REAL 2) : ( LE q1,p,P & LE p,q2,P ) } ) ;
hence ex B being Subset of (TOP-REAL 2) st
( ( q2 <> W-min P implies B = { p where p is Point of (TOP-REAL 2) : ( LE q1,p,P & LE p,q2,P ) } ) & ( not q2 <> W-min P implies B = { p1 where p1 is Point of (TOP-REAL 2) : ( LE q1,p1,P or ( q1 in P & p1 = W-min P ) ) } ) ) by A1; ::_thesis: verum
end;
supposeA2: not q2 <> W-min P ; ::_thesis: ex B being Subset of (TOP-REAL 2) st
( ( q2 <> W-min P implies B = { p where p is Point of (TOP-REAL 2) : ( LE q1,p,P & LE p,q2,P ) } ) & ( not q2 <> W-min P implies B = { p1 where p1 is Point of (TOP-REAL 2) : ( LE q1,p1,P or ( q1 in P & p1 = W-min P ) ) } ) )
defpred S1[ Point of (TOP-REAL 2)] means ( LE q1,$1,P or ( q1 in P & $1 = W-min P ) );
{ p1 where p1 is Point of (TOP-REAL 2) : S1[p1] } is Subset of (TOP-REAL 2) from DOMAIN_1:sch_7();
then reconsider C = { p1 where p1 is Point of (TOP-REAL 2) : ( LE q1,p1,P or ( q1 in P & p1 = W-min P ) ) } as Subset of (TOP-REAL 2) ;
( not q2 <> W-min P implies C = { p1 where p1 is Point of (TOP-REAL 2) : ( LE q1,p1,P or ( q1 in P & p1 = W-min P ) ) } ) ;
hence ex B being Subset of (TOP-REAL 2) st
( ( q2 <> W-min P implies B = { p where p is Point of (TOP-REAL 2) : ( LE q1,p,P & LE p,q2,P ) } ) & ( not q2 <> W-min P implies B = { p1 where p1 is Point of (TOP-REAL 2) : ( LE q1,p1,P or ( q1 in P & p1 = W-min P ) ) } ) ) by A2; ::_thesis: verum
end;
end;
end;
hence ( ( q2 <> W-min P implies { p where p is Point of (TOP-REAL 2) : ( LE q1,p,P & LE p,q2,P ) } is Subset of (TOP-REAL 2) ) & ( not q2 <> W-min P implies { p1 where p1 is Point of (TOP-REAL 2) : ( LE q1,p1,P or ( q1 in P & p1 = W-min P ) ) } is Subset of (TOP-REAL 2) ) & ( for b1 being Subset of (TOP-REAL 2) holds verum ) ) ; ::_thesis: verum
end;
end;
:: deftheorem Def1 defines Segment JORDAN7:def_1_:_
for P being non empty compact Subset of (TOP-REAL 2)
for q1, q2 being Point of (TOP-REAL 2) holds
( ( q2 <> W-min P implies Segment (q1,q2,P) = { p where p is Point of (TOP-REAL 2) : ( LE q1,p,P & LE p,q2,P ) } ) & ( not q2 <> W-min P implies Segment (q1,q2,P) = { p1 where p1 is Point of (TOP-REAL 2) : ( LE q1,p1,P or ( q1 in P & p1 = W-min P ) ) } ) );
theorem :: JORDAN7:4
for P being non empty compact Subset of (TOP-REAL 2) st P is being_simple_closed_curve holds
( Segment ((W-min P),(E-max P),P) = Upper_Arc P & Segment ((E-max P),(W-min P),P) = Lower_Arc P )
proof
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P is being_simple_closed_curve implies ( Segment ((W-min P),(E-max P),P) = Upper_Arc P & Segment ((E-max P),(W-min P),P) = Lower_Arc P ) )
assume A1: P is being_simple_closed_curve ; ::_thesis: ( Segment ((W-min P),(E-max P),P) = Upper_Arc P & Segment ((E-max P),(W-min P),P) = Lower_Arc P )
then A2: Upper_Arc P is_an_arc_of W-min P, E-max P by JORDAN6:def_8;
A3: { p1 where p1 is Point of (TOP-REAL 2) : ( LE E-max P,p1,P or ( E-max P in P & p1 = W-min P ) ) } = Lower_Arc P
proof
A4: { p1 where p1 is Point of (TOP-REAL 2) : ( LE E-max P,p1,P or ( E-max P in P & p1 = W-min P ) ) } c= Lower_Arc P
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p1 where p1 is Point of (TOP-REAL 2) : ( LE E-max P,p1,P or ( E-max P in P & p1 = W-min P ) ) } or x in Lower_Arc P )
assume x in { p1 where p1 is Point of (TOP-REAL 2) : ( LE E-max P,p1,P or ( E-max P in P & p1 = W-min P ) ) } ; ::_thesis: x in Lower_Arc P
then consider p1 being Point of (TOP-REAL 2) such that
A5: p1 = x and
A6: ( LE E-max P,p1,P or ( E-max P in P & p1 = W-min P ) ) ;
percases ( LE E-max P,p1,P or ( E-max P in P & p1 = W-min P ) ) by A6;
supposeA7: LE E-max P,p1,P ; ::_thesis: x in Lower_Arc P
percases ( x in Lower_Arc P or ( E-max P in Upper_Arc P & p1 in Upper_Arc P & LE E-max P,p1, Upper_Arc P, W-min P, E-max P ) ) by A5, A7, JORDAN6:def_10;
suppose x in Lower_Arc P ; ::_thesis: x in Lower_Arc P
hence x in Lower_Arc P ; ::_thesis: verum
end;
supposeA8: ( E-max P in Upper_Arc P & p1 in Upper_Arc P & LE E-max P,p1, Upper_Arc P, W-min P, E-max P ) ; ::_thesis: x in Lower_Arc P
A9: Upper_Arc P is_an_arc_of W-min P, E-max P by A1, JORDAN6:50;
then LE p1, E-max P, Upper_Arc P, W-min P, E-max P by A8, JORDAN5C:10;
then A10: p1 = E-max P by A8, A9, JORDAN5C:12;
Lower_Arc P is_an_arc_of E-max P, W-min P by A1, JORDAN6:def_9;
hence x in Lower_Arc P by A5, A10, TOPREAL1:1; ::_thesis: verum
end;
end;
end;
suppose ( E-max P in P & p1 = W-min P ) ; ::_thesis: x in Lower_Arc P
then x in {(W-min P),(E-max P)} by A5, TARSKI:def_2;
then x in (Upper_Arc P) /\ (Lower_Arc P) by A1, JORDAN6:def_9;
hence x in Lower_Arc P by XBOOLE_0:def_4; ::_thesis: verum
end;
end;
end;
Lower_Arc P c= { p1 where p1 is Point of (TOP-REAL 2) : ( LE E-max P,p1,P or ( E-max P in P & p1 = W-min P ) ) }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Lower_Arc P or x in { p1 where p1 is Point of (TOP-REAL 2) : ( LE E-max P,p1,P or ( E-max P in P & p1 = W-min P ) ) } )
assume A11: x in Lower_Arc P ; ::_thesis: x in { p1 where p1 is Point of (TOP-REAL 2) : ( LE E-max P,p1,P or ( E-max P in P & p1 = W-min P ) ) }
then reconsider p2 = x as Point of (TOP-REAL 2) ;
Upper_Arc P is_an_arc_of W-min P, E-max P by A1, JORDAN6:50;
then ( ( not E-max P in P or not p2 = W-min P ) implies ( E-max P in Upper_Arc P & p2 in Lower_Arc P & not p2 = W-min P ) ) by A11, SPRECT_1:14, TOPREAL1:1;
then ( LE E-max P,p2,P or ( E-max P in P & p2 = W-min P ) ) by JORDAN6:def_10;
hence x in { p1 where p1 is Point of (TOP-REAL 2) : ( LE E-max P,p1,P or ( E-max P in P & p1 = W-min P ) ) } ; ::_thesis: verum
end;
hence { p1 where p1 is Point of (TOP-REAL 2) : ( LE E-max P,p1,P or ( E-max P in P & p1 = W-min P ) ) } = Lower_Arc P by A4, XBOOLE_0:def_10; ::_thesis: verum
end;
A12: E-max P <> W-min P by A1, TOPREAL5:19;
{ p where p is Point of (TOP-REAL 2) : ( LE W-min P,p,P & LE p, E-max P,P ) } = Upper_Arc P
proof
A13: { p where p is Point of (TOP-REAL 2) : ( LE W-min P,p,P & LE p, E-max P,P ) } c= Upper_Arc P
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of (TOP-REAL 2) : ( LE W-min P,p,P & LE p, E-max P,P ) } or x in Upper_Arc P )
assume x in { p where p is Point of (TOP-REAL 2) : ( LE W-min P,p,P & LE p, E-max P,P ) } ; ::_thesis: x in Upper_Arc P
then consider p being Point of (TOP-REAL 2) such that
A14: p = x and
LE W-min P,p,P and
A15: LE p, E-max P,P ;
percases ( ( p in Upper_Arc P & E-max P in Lower_Arc P & not E-max P = W-min P ) or ( p in Upper_Arc P & E-max P in Upper_Arc P & LE p, E-max P, Upper_Arc P, W-min P, E-max P ) or ( p in Lower_Arc P & E-max P in Lower_Arc P & not E-max P = W-min P & LE p, E-max P, Lower_Arc P, E-max P, W-min P ) ) by A15, JORDAN6:def_10;
suppose ( p in Upper_Arc P & E-max P in Lower_Arc P & not E-max P = W-min P ) ; ::_thesis: x in Upper_Arc P
hence x in Upper_Arc P by A14; ::_thesis: verum
end;
suppose ( p in Upper_Arc P & E-max P in Upper_Arc P & LE p, E-max P, Upper_Arc P, W-min P, E-max P ) ; ::_thesis: x in Upper_Arc P
hence x in Upper_Arc P by A14; ::_thesis: verum
end;
supposeA16: ( p in Lower_Arc P & E-max P in Lower_Arc P & not E-max P = W-min P & LE p, E-max P, Lower_Arc P, E-max P, W-min P ) ; ::_thesis: x in Upper_Arc P
Lower_Arc P is_an_arc_of E-max P, W-min P by A1, JORDAN6:def_9;
then p = E-max P by A16, JORDAN6:54;
hence x in Upper_Arc P by A2, A14, TOPREAL1:1; ::_thesis: verum
end;
end;
end;
Upper_Arc P c= { p where p is Point of (TOP-REAL 2) : ( LE W-min P,p,P & LE p, E-max P,P ) }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Upper_Arc P or x in { p where p is Point of (TOP-REAL 2) : ( LE W-min P,p,P & LE p, E-max P,P ) } )
assume A17: x in Upper_Arc P ; ::_thesis: x in { p where p is Point of (TOP-REAL 2) : ( LE W-min P,p,P & LE p, E-max P,P ) }
then reconsider p2 = x as Point of (TOP-REAL 2) ;
E-max P in Lower_Arc P by A1, Th1;
then A18: LE p2, E-max P,P by A12, A17, JORDAN6:def_10;
A19: W-min P in Upper_Arc P by A1, Th1;
LE W-min P,p2, Upper_Arc P, W-min P, E-max P by A2, A17, JORDAN5C:10;
then LE W-min P,p2,P by A17, A19, JORDAN6:def_10;
hence x in { p where p is Point of (TOP-REAL 2) : ( LE W-min P,p,P & LE p, E-max P,P ) } by A18; ::_thesis: verum
end;
hence { p where p is Point of (TOP-REAL 2) : ( LE W-min P,p,P & LE p, E-max P,P ) } = Upper_Arc P by A13, XBOOLE_0:def_10; ::_thesis: verum
end;
hence ( Segment ((W-min P),(E-max P),P) = Upper_Arc P & Segment ((E-max P),(W-min P),P) = Lower_Arc P ) by A12, A3, Def1; ::_thesis: verum
end;
theorem Th5: :: JORDAN7:5
for P being non empty compact Subset of (TOP-REAL 2)
for q1, q2 being Point of (TOP-REAL 2) st P is being_simple_closed_curve & LE q1,q2,P holds
( q1 in P & q2 in P )
proof
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: for q1, q2 being Point of (TOP-REAL 2) st P is being_simple_closed_curve & LE q1,q2,P holds
( q1 in P & q2 in P )
let q1, q2 be Point of (TOP-REAL 2); ::_thesis: ( P is being_simple_closed_curve & LE q1,q2,P implies ( q1 in P & q2 in P ) )
assume that
A1: P is being_simple_closed_curve and
A2: LE q1,q2,P ; ::_thesis: ( q1 in P & q2 in P )
A3: (Upper_Arc P) \/ (Lower_Arc P) = P by A1, JORDAN6:50;
percases ( ( q1 in Upper_Arc P & q2 in Lower_Arc P ) or ( q1 in Upper_Arc P & q2 in Upper_Arc P ) or ( q1 in Lower_Arc P & q2 in Lower_Arc P ) ) by A2, JORDAN6:def_10;
suppose ( q1 in Upper_Arc P & q2 in Lower_Arc P ) ; ::_thesis: ( q1 in P & q2 in P )
hence ( q1 in P & q2 in P ) by A3, XBOOLE_0:def_3; ::_thesis: verum
end;
suppose ( q1 in Upper_Arc P & q2 in Upper_Arc P ) ; ::_thesis: ( q1 in P & q2 in P )
hence ( q1 in P & q2 in P ) by A3, XBOOLE_0:def_3; ::_thesis: verum
end;
suppose ( q1 in Lower_Arc P & q2 in Lower_Arc P ) ; ::_thesis: ( q1 in P & q2 in P )
hence ( q1 in P & q2 in P ) by A3, XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
theorem Th6: :: JORDAN7:6
for P being non empty compact Subset of (TOP-REAL 2)
for q1, q2 being Point of (TOP-REAL 2) st P is being_simple_closed_curve & LE q1,q2,P holds
( q1 in Segment (q1,q2,P) & q2 in Segment (q1,q2,P) )
proof
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: for q1, q2 being Point of (TOP-REAL 2) st P is being_simple_closed_curve & LE q1,q2,P holds
( q1 in Segment (q1,q2,P) & q2 in Segment (q1,q2,P) )
let q1, q2 be Point of (TOP-REAL 2); ::_thesis: ( P is being_simple_closed_curve & LE q1,q2,P implies ( q1 in Segment (q1,q2,P) & q2 in Segment (q1,q2,P) ) )
assume that
A1: P is being_simple_closed_curve and
A2: LE q1,q2,P ; ::_thesis: ( q1 in Segment (q1,q2,P) & q2 in Segment (q1,q2,P) )
hereby ::_thesis: q2 in Segment (q1,q2,P)
percases ( q2 <> W-min P or q2 = W-min P ) ;
supposeA3: q2 <> W-min P ; ::_thesis: q1 in Segment (q1,q2,P)
q1 in P by A1, A2, Th5;
then LE q1,q1,P by A1, JORDAN6:56;
then q1 in { p where p is Point of (TOP-REAL 2) : ( LE q1,p,P & LE p,q2,P ) } by A2;
hence q1 in Segment (q1,q2,P) by A3, Def1; ::_thesis: verum
end;
supposeA4: q2 = W-min P ; ::_thesis: q1 in Segment (q1,q2,P)
q1 in P by A1, A2, Th5;
then LE q1,q1,P by A1, JORDAN6:56;
then q1 in { p1 where p1 is Point of (TOP-REAL 2) : ( LE q1,p1,P or ( q1 in P & p1 = W-min P ) ) } ;
hence q1 in Segment (q1,q2,P) by A4, Def1; ::_thesis: verum
end;
end;
end;
percases ( q2 <> W-min P or q2 = W-min P ) ;
supposeA5: q2 <> W-min P ; ::_thesis: q2 in Segment (q1,q2,P)
q2 in P by A1, A2, Th5;
then LE q2,q2,P by A1, JORDAN6:56;
then q2 in { p where p is Point of (TOP-REAL 2) : ( LE q1,p,P & LE p,q2,P ) } by A2;
hence q2 in Segment (q1,q2,P) by A5, Def1; ::_thesis: verum
end;
supposeA6: q2 = W-min P ; ::_thesis: q2 in Segment (q1,q2,P)
q2 in { p1 where p1 is Point of (TOP-REAL 2) : ( LE q1,p1,P or ( q1 in P & p1 = W-min P ) ) } by A2;
hence q2 in Segment (q1,q2,P) by A6, Def1; ::_thesis: verum
end;
end;
end;
theorem Th7: :: JORDAN7:7
for P being non empty compact Subset of (TOP-REAL 2)
for q1 being Point of (TOP-REAL 2) st q1 in P & P is being_simple_closed_curve holds
q1 in Segment (q1,(W-min P),P)
proof
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: for q1 being Point of (TOP-REAL 2) st q1 in P & P is being_simple_closed_curve holds
q1 in Segment (q1,(W-min P),P)
let q1 be Point of (TOP-REAL 2); ::_thesis: ( q1 in P & P is being_simple_closed_curve implies q1 in Segment (q1,(W-min P),P) )
assume A1: q1 in P ; ::_thesis: ( not P is being_simple_closed_curve or q1 in Segment (q1,(W-min P),P) )
assume P is being_simple_closed_curve ; ::_thesis: q1 in Segment (q1,(W-min P),P)
then LE q1,q1,P by A1, JORDAN6:56;
then q1 in { p1 where p1 is Point of (TOP-REAL 2) : ( LE q1,p1,P or ( q1 in P & p1 = W-min P ) ) } ;
hence q1 in Segment (q1,(W-min P),P) by Def1; ::_thesis: verum
end;
theorem :: JORDAN7:8
for P being non empty compact Subset of (TOP-REAL 2)
for q being Point of (TOP-REAL 2) st P is being_simple_closed_curve & q in P & q <> W-min P holds
Segment (q,q,P) = {q}
proof
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: for q being Point of (TOP-REAL 2) st P is being_simple_closed_curve & q in P & q <> W-min P holds
Segment (q,q,P) = {q}
let q be Point of (TOP-REAL 2); ::_thesis: ( P is being_simple_closed_curve & q in P & q <> W-min P implies Segment (q,q,P) = {q} )
assume that
A1: P is being_simple_closed_curve and
A2: q in P and
A3: q <> W-min P ; ::_thesis: Segment (q,q,P) = {q}
for x being set holds
( x in Segment (q,q,P) iff x = q )
proof
let x be set ; ::_thesis: ( x in Segment (q,q,P) iff x = q )
hereby ::_thesis: ( x = q implies x in Segment (q,q,P) )
assume x in Segment (q,q,P) ; ::_thesis: x = q
then x in { p where p is Point of (TOP-REAL 2) : ( LE q,p,P & LE p,q,P ) } by A3, Def1;
then ex p being Point of (TOP-REAL 2) st
( p = x & LE q,p,P & LE p,q,P ) ;
hence x = q by A1, JORDAN6:57; ::_thesis: verum
end;
assume A4: x = q ; ::_thesis: x in Segment (q,q,P)
LE q,q,P by A1, A2, JORDAN6:56;
then x in { p where p is Point of (TOP-REAL 2) : ( LE q,p,P & LE p,q,P ) } by A4;
hence x in Segment (q,q,P) by A3, Def1; ::_thesis: verum
end;
hence Segment (q,q,P) = {q} by TARSKI:def_1; ::_thesis: verum
end;
theorem :: JORDAN7:9
for P being non empty compact Subset of (TOP-REAL 2)
for q1, q2 being Point of (TOP-REAL 2) st P is being_simple_closed_curve & q1 <> W-min P & q2 <> W-min P holds
not W-min P in Segment (q1,q2,P)
proof
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: for q1, q2 being Point of (TOP-REAL 2) st P is being_simple_closed_curve & q1 <> W-min P & q2 <> W-min P holds
not W-min P in Segment (q1,q2,P)
let q1, q2 be Point of (TOP-REAL 2); ::_thesis: ( P is being_simple_closed_curve & q1 <> W-min P & q2 <> W-min P implies not W-min P in Segment (q1,q2,P) )
assume that
A1: P is being_simple_closed_curve and
A2: q1 <> W-min P and
A3: q2 <> W-min P ; ::_thesis: not W-min P in Segment (q1,q2,P)
A4: Segment (q1,q2,P) = { p where p is Point of (TOP-REAL 2) : ( LE q1,p,P & LE p,q2,P ) } by A3, Def1;
now__::_thesis:_not_W-min_P_in_Segment_(q1,q2,P)
A5: Upper_Arc P is_an_arc_of W-min P, E-max P by A1, JORDAN6:def_8;
assume W-min P in Segment (q1,q2,P) ; ::_thesis: contradiction
then consider p being Point of (TOP-REAL 2) such that
A6: p = W-min P and
A7: LE q1,p,P and
LE p,q2,P by A4;
LE q1,p, Upper_Arc P, W-min P, E-max P by A6, A7, JORDAN6:def_10;
hence contradiction by A2, A6, A5, JORDAN6:54; ::_thesis: verum
end;
hence not W-min P in Segment (q1,q2,P) ; ::_thesis: verum
end;
theorem Th10: :: JORDAN7:10
for P being non empty compact Subset of (TOP-REAL 2)
for q1, q2, q3 being Point of (TOP-REAL 2) st P is being_simple_closed_curve & LE q1,q2,P & LE q2,q3,P & ( not q1 = q2 or not q1 = W-min P ) & ( not q2 = q3 or not q2 = W-min P ) holds
(Segment (q1,q2,P)) /\ (Segment (q2,q3,P)) = {q2}
proof
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: for q1, q2, q3 being Point of (TOP-REAL 2) st P is being_simple_closed_curve & LE q1,q2,P & LE q2,q3,P & ( not q1 = q2 or not q1 = W-min P ) & ( not q2 = q3 or not q2 = W-min P ) holds
(Segment (q1,q2,P)) /\ (Segment (q2,q3,P)) = {q2}
let q1, q2, q3 be Point of (TOP-REAL 2); ::_thesis: ( P is being_simple_closed_curve & LE q1,q2,P & LE q2,q3,P & ( not q1 = q2 or not q1 = W-min P ) & ( not q2 = q3 or not q2 = W-min P ) implies (Segment (q1,q2,P)) /\ (Segment (q2,q3,P)) = {q2} )
assume that
A1: P is being_simple_closed_curve and
A2: LE q1,q2,P and
A3: LE q2,q3,P and
A4: ( not q1 = q2 or not q1 = W-min P ) and
A5: ( not q2 = q3 or not q2 = W-min P ) ; ::_thesis: (Segment (q1,q2,P)) /\ (Segment (q2,q3,P)) = {q2}
A6: Upper_Arc P is_an_arc_of W-min P, E-max P by A1, JORDAN6:def_8;
thus (Segment (q1,q2,P)) /\ (Segment (q2,q3,P)) c= {q2} :: according to XBOOLE_0:def_10 ::_thesis: {q2} c= (Segment (q1,q2,P)) /\ (Segment (q2,q3,P))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (Segment (q1,q2,P)) /\ (Segment (q2,q3,P)) or x in {q2} )
assume A7: x in (Segment (q1,q2,P)) /\ (Segment (q2,q3,P)) ; ::_thesis: x in {q2}
then A8: x in Segment (q2,q3,P) by XBOOLE_0:def_4;
A9: x in Segment (q1,q2,P) by A7, XBOOLE_0:def_4;
now__::_thesis:_(_(_q3_<>_W-min_P_&_x_=_q2_)_or_(_q3_=_W-min_P_&_x_=_q2_)_)
percases ( q3 <> W-min P or q3 = W-min P ) ;
case q3 <> W-min P ; ::_thesis: verum
then x in { p where p is Point of (TOP-REAL 2) : ( LE q2,p,P & LE p,q3,P ) } by A8, Def1;
then A10: ex p being Point of (TOP-REAL 2) st
( p = x & LE q2,p,P & LE p,q3,P ) ;
percases ( q2 <> W-min P or q2 = W-min P ) ;
suppose q2 <> W-min P ; ::_thesis: x = q2
then x in { p2 where p2 is Point of (TOP-REAL 2) : ( LE q1,p2,P & LE p2,q2,P ) } by A9, Def1;
then ex p2 being Point of (TOP-REAL 2) st
( p2 = x & LE q1,p2,P & LE p2,q2,P ) ;
hence x = q2 by A1, A10, JORDAN6:57; ::_thesis: verum
end;
supposeA11: q2 = W-min P ; ::_thesis: x = q2
then LE q1,q2, Upper_Arc P, W-min P, E-max P by A2, JORDAN6:def_10;
hence x = q2 by A4, A6, A11, JORDAN6:54; ::_thesis: verum
end;
end;
end;
caseA12: q3 = W-min P ; ::_thesis: x = q2
then x in { p1 where p1 is Point of (TOP-REAL 2) : ( LE q2,p1,P or ( q2 in P & p1 = W-min P ) ) } by A8, Def1;
then consider p1 being Point of (TOP-REAL 2) such that
A13: p1 = x and
A14: ( LE q2,p1,P or ( q2 in P & p1 = W-min P ) ) ;
p1 in { p where p is Point of (TOP-REAL 2) : ( LE q1,p,P & LE p,q2,P ) } by A5, A9, A12, A13, Def1;
then ex p being Point of (TOP-REAL 2) st
( p = p1 & LE q1,p,P & LE p,q2,P ) ;
hence x = q2 by A1, A3, A12, A13, A14, JORDAN6:57; ::_thesis: verum
end;
end;
end;
hence x in {q2} by TARSKI:def_1; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {q2} or x in (Segment (q1,q2,P)) /\ (Segment (q2,q3,P)) )
assume x in {q2} ; ::_thesis: x in (Segment (q1,q2,P)) /\ (Segment (q2,q3,P))
then x = q2 by TARSKI:def_1;
then ( x in Segment (q1,q2,P) & x in Segment (q2,q3,P) ) by A1, A2, A3, Th6;
hence x in (Segment (q1,q2,P)) /\ (Segment (q2,q3,P)) by XBOOLE_0:def_4; ::_thesis: verum
end;
theorem Th11: :: JORDAN7:11
for P being non empty compact Subset of (TOP-REAL 2)
for q1, q2 being Point of (TOP-REAL 2) st P is being_simple_closed_curve & LE q1,q2,P & q1 <> W-min P & q2 <> W-min P holds
(Segment (q1,q2,P)) /\ (Segment (q2,(W-min P),P)) = {q2}
proof
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: for q1, q2 being Point of (TOP-REAL 2) st P is being_simple_closed_curve & LE q1,q2,P & q1 <> W-min P & q2 <> W-min P holds
(Segment (q1,q2,P)) /\ (Segment (q2,(W-min P),P)) = {q2}
let q1, q2 be Point of (TOP-REAL 2); ::_thesis: ( P is being_simple_closed_curve & LE q1,q2,P & q1 <> W-min P & q2 <> W-min P implies (Segment (q1,q2,P)) /\ (Segment (q2,(W-min P),P)) = {q2} )
set q3 = W-min P;
assume that
A1: P is being_simple_closed_curve and
A2: LE q1,q2,P and
A3: q1 <> W-min P and
A4: not q2 = W-min P ; ::_thesis: (Segment (q1,q2,P)) /\ (Segment (q2,(W-min P),P)) = {q2}
thus (Segment (q1,q2,P)) /\ (Segment (q2,(W-min P),P)) c= {q2} :: according to XBOOLE_0:def_10 ::_thesis: {q2} c= (Segment (q1,q2,P)) /\ (Segment (q2,(W-min P),P))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (Segment (q1,q2,P)) /\ (Segment (q2,(W-min P),P)) or x in {q2} )
assume A5: x in (Segment (q1,q2,P)) /\ (Segment (q2,(W-min P),P)) ; ::_thesis: x in {q2}
then x in Segment (q2,(W-min P),P) by XBOOLE_0:def_4;
then x in { p1 where p1 is Point of (TOP-REAL 2) : ( LE q2,p1,P or ( q2 in P & p1 = W-min P ) ) } by Def1;
then consider p1 being Point of (TOP-REAL 2) such that
A6: p1 = x and
A7: ( LE q2,p1,P or ( q2 in P & p1 = W-min P ) ) ;
x in Segment (q1,q2,P) by A5, XBOOLE_0:def_4;
then p1 in { p where p is Point of (TOP-REAL 2) : ( LE q1,p,P & LE p,q2,P ) } by A4, A6, Def1;
then A8: ex p being Point of (TOP-REAL 2) st
( p = p1 & LE q1,p,P & LE p,q2,P ) ;
percases ( LE q2,p1,P or ( q2 in P & p1 = W-min P ) ) by A7;
suppose LE q2,p1,P ; ::_thesis: x in {q2}
then x = q2 by A1, A6, A8, JORDAN6:57;
hence x in {q2} by TARSKI:def_1; ::_thesis: verum
end;
suppose ( q2 in P & p1 = W-min P ) ; ::_thesis: x in {q2}
hence x in {q2} by A1, A3, A8, Th2; ::_thesis: verum
end;
end;
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {q2} or x in (Segment (q1,q2,P)) /\ (Segment (q2,(W-min P),P)) )
assume x in {q2} ; ::_thesis: x in (Segment (q1,q2,P)) /\ (Segment (q2,(W-min P),P))
then A9: x = q2 by TARSKI:def_1;
q2 in P by A1, A2, Th5;
then A10: x in Segment (q2,(W-min P),P) by A1, A9, Th7;
x in Segment (q1,q2,P) by A1, A2, A9, Th6;
hence x in (Segment (q1,q2,P)) /\ (Segment (q2,(W-min P),P)) by A10, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem Th12: :: JORDAN7:12
for P being non empty compact Subset of (TOP-REAL 2)
for q1, q2 being Point of (TOP-REAL 2) st P is being_simple_closed_curve & LE q1,q2,P & q1 <> q2 & q1 <> W-min P holds
(Segment (q2,(W-min P),P)) /\ (Segment ((W-min P),q1,P)) = {(W-min P)}
proof
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: for q1, q2 being Point of (TOP-REAL 2) st P is being_simple_closed_curve & LE q1,q2,P & q1 <> q2 & q1 <> W-min P holds
(Segment (q2,(W-min P),P)) /\ (Segment ((W-min P),q1,P)) = {(W-min P)}
let q1, q2 be Point of (TOP-REAL 2); ::_thesis: ( P is being_simple_closed_curve & LE q1,q2,P & q1 <> q2 & q1 <> W-min P implies (Segment (q2,(W-min P),P)) /\ (Segment ((W-min P),q1,P)) = {(W-min P)} )
assume that
A1: P is being_simple_closed_curve and
A2: LE q1,q2,P and
A3: q1 <> q2 and
A4: q1 <> W-min P ; ::_thesis: (Segment (q2,(W-min P),P)) /\ (Segment ((W-min P),q1,P)) = {(W-min P)}
thus (Segment (q2,(W-min P),P)) /\ (Segment ((W-min P),q1,P)) c= {(W-min P)} :: according to XBOOLE_0:def_10 ::_thesis: {(W-min P)} c= (Segment (q2,(W-min P),P)) /\ (Segment ((W-min P),q1,P))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (Segment (q2,(W-min P),P)) /\ (Segment ((W-min P),q1,P)) or x in {(W-min P)} )
assume A5: x in (Segment (q2,(W-min P),P)) /\ (Segment ((W-min P),q1,P)) ; ::_thesis: x in {(W-min P)}
then x in Segment (q2,(W-min P),P) by XBOOLE_0:def_4;
then x in { p1 where p1 is Point of (TOP-REAL 2) : ( LE q2,p1,P or ( q2 in P & p1 = W-min P ) ) } by Def1;
then consider p1 being Point of (TOP-REAL 2) such that
A6: p1 = x and
A7: ( LE q2,p1,P or ( q2 in P & p1 = W-min P ) ) ;
A8: x in Segment ((W-min P),q1,P) by A5, XBOOLE_0:def_4;
now__::_thesis:_(_(_LE_q2,p1,P_&_contradiction_)_or_(_q2_in_P_&_p1_=_W-min_P_&_x_=_W-min_P_)_)
percases ( LE q2,p1,P or ( q2 in P & p1 = W-min P ) ) by A7;
caseA9: LE q2,p1,P ; ::_thesis: contradiction
x in { p where p is Point of (TOP-REAL 2) : ( LE W-min P,p,P & LE p,q1,P ) } by A4, A8, Def1;
then ex p2 being Point of (TOP-REAL 2) st
( p2 = x & LE W-min P,p2,P & LE p2,q1,P ) ;
then LE q2,q1,P by A1, A6, A9, JORDAN6:58;
hence contradiction by A1, A2, A3, JORDAN6:57; ::_thesis: verum
end;
case ( q2 in P & p1 = W-min P ) ; ::_thesis: x = W-min P
hence x = W-min P by A6; ::_thesis: verum
end;
end;
end;
hence x in {(W-min P)} by TARSKI:def_1; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(W-min P)} or x in (Segment (q2,(W-min P),P)) /\ (Segment ((W-min P),q1,P)) )
assume x in {(W-min P)} ; ::_thesis: x in (Segment (q2,(W-min P),P)) /\ (Segment ((W-min P),q1,P))
then A10: x = W-min P by TARSKI:def_1;
q2 in P by A1, A2, Th5;
then x in { p1 where p1 is Point of (TOP-REAL 2) : ( LE q2,p1,P or ( q2 in P & p1 = W-min P ) ) } by A10;
then A11: x in Segment (q2,(W-min P),P) by Def1;
q1 in P by A1, A2, Th5;
then LE W-min P,q1,P by A1, Th3;
then x in Segment ((W-min P),q1,P) by A1, A10, Th6;
hence x in (Segment (q2,(W-min P),P)) /\ (Segment ((W-min P),q1,P)) by A11, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem Th13: :: JORDAN7:13
for P being non empty compact Subset of (TOP-REAL 2)
for q1, q2, q3, q4 being Point of (TOP-REAL 2) st P is being_simple_closed_curve & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P & q1 <> q2 & q2 <> q3 holds
Segment (q1,q2,P) misses Segment (q3,q4,P)
proof
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: for q1, q2, q3, q4 being Point of (TOP-REAL 2) st P is being_simple_closed_curve & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P & q1 <> q2 & q2 <> q3 holds
Segment (q1,q2,P) misses Segment (q3,q4,P)
let q1, q2, q3, q4 be Point of (TOP-REAL 2); ::_thesis: ( P is being_simple_closed_curve & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P & q1 <> q2 & q2 <> q3 implies Segment (q1,q2,P) misses Segment (q3,q4,P) )
assume that
A1: P is being_simple_closed_curve and
A2: LE q1,q2,P and
A3: LE q2,q3,P and
A4: LE q3,q4,P and
A5: q1 <> q2 and
A6: q2 <> q3 ; ::_thesis: Segment (q1,q2,P) misses Segment (q3,q4,P)
set x = the Element of (Segment (q1,q2,P)) /\ (Segment (q3,q4,P));
assume A7: (Segment (q1,q2,P)) /\ (Segment (q3,q4,P)) <> {} ; :: according to XBOOLE_0:def_7 ::_thesis: contradiction
then A8: the Element of (Segment (q1,q2,P)) /\ (Segment (q3,q4,P)) in Segment (q1,q2,P) by XBOOLE_0:def_4;
A9: the Element of (Segment (q1,q2,P)) /\ (Segment (q3,q4,P)) in Segment (q3,q4,P) by A7, XBOOLE_0:def_4;
percases ( q4 = W-min P or q4 <> W-min P ) ;
suppose q4 = W-min P ; ::_thesis: contradiction
then q3 = W-min P by A1, A4, Th2;
hence contradiction by A1, A3, A6, Th2; ::_thesis: verum
end;
supposeA10: q4 <> W-min P ; ::_thesis: contradiction
q2 <> W-min P by A1, A2, A5, Th2;
then the Element of (Segment (q1,q2,P)) /\ (Segment (q3,q4,P)) in { p2 where p2 is Point of (TOP-REAL 2) : ( LE q1,p2,P & LE p2,q2,P ) } by A8, Def1;
then A11: ex p2 being Point of (TOP-REAL 2) st
( p2 = the Element of (Segment (q1,q2,P)) /\ (Segment (q3,q4,P)) & LE q1,p2,P & LE p2,q2,P ) ;
the Element of (Segment (q1,q2,P)) /\ (Segment (q3,q4,P)) in { p1 where p1 is Point of (TOP-REAL 2) : ( LE q3,p1,P & LE p1,q4,P ) } by A9, A10, Def1;
then ex p1 being Point of (TOP-REAL 2) st
( p1 = the Element of (Segment (q1,q2,P)) /\ (Segment (q3,q4,P)) & LE q3,p1,P & LE p1,q4,P ) ;
then LE q3,q2,P by A1, A11, JORDAN6:58;
hence contradiction by A1, A3, A6, JORDAN6:57; ::_thesis: verum
end;
end;
end;
theorem Th14: :: JORDAN7:14
for P being non empty compact Subset of (TOP-REAL 2)
for q1, q2, q3 being Point of (TOP-REAL 2) st P is being_simple_closed_curve & LE q1,q2,P & LE q2,q3,P & q1 <> W-min P & q2 <> q3 holds
Segment (q1,q2,P) misses Segment (q3,(W-min P),P)
proof
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: for q1, q2, q3 being Point of (TOP-REAL 2) st P is being_simple_closed_curve & LE q1,q2,P & LE q2,q3,P & q1 <> W-min P & q2 <> q3 holds
Segment (q1,q2,P) misses Segment (q3,(W-min P),P)
let q1, q2, q3 be Point of (TOP-REAL 2); ::_thesis: ( P is being_simple_closed_curve & LE q1,q2,P & LE q2,q3,P & q1 <> W-min P & q2 <> q3 implies Segment (q1,q2,P) misses Segment (q3,(W-min P),P) )
assume that
A1: P is being_simple_closed_curve and
A2: LE q1,q2,P and
A3: LE q2,q3,P and
A4: q1 <> W-min P and
A5: q2 <> q3 ; ::_thesis: Segment (q1,q2,P) misses Segment (q3,(W-min P),P)
set x = the Element of (Segment (q1,q2,P)) /\ (Segment (q3,(W-min P),P));
assume A6: (Segment (q1,q2,P)) /\ (Segment (q3,(W-min P),P)) <> {} ; :: according to XBOOLE_0:def_7 ::_thesis: contradiction
then A7: the Element of (Segment (q1,q2,P)) /\ (Segment (q3,(W-min P),P)) in Segment (q1,q2,P) by XBOOLE_0:def_4;
the Element of (Segment (q1,q2,P)) /\ (Segment (q3,(W-min P),P)) in Segment (q3,(W-min P),P) by A6, XBOOLE_0:def_4;
then the Element of (Segment (q1,q2,P)) /\ (Segment (q3,(W-min P),P)) in { p1 where p1 is Point of (TOP-REAL 2) : ( LE q3,p1,P or ( q3 in P & p1 = W-min P ) ) } by Def1;
then A8: ex p1 being Point of (TOP-REAL 2) st
( p1 = the Element of (Segment (q1,q2,P)) /\ (Segment (q3,(W-min P),P)) & ( LE q3,p1,P or ( q3 in P & p1 = W-min P ) ) ) ;
q2 <> W-min P by A1, A2, A4, Th2;
then the Element of (Segment (q1,q2,P)) /\ (Segment (q3,(W-min P),P)) in { p where p is Point of (TOP-REAL 2) : ( LE q1,p,P & LE p,q2,P ) } by A7, Def1;
then ex p3 being Point of (TOP-REAL 2) st
( p3 = the Element of (Segment (q1,q2,P)) /\ (Segment (q3,(W-min P),P)) & LE q1,p3,P & LE p3,q2,P ) ;
then LE q3,q2,P by A1, A4, A8, Th2, JORDAN6:58;
hence contradiction by A1, A3, A5, JORDAN6:57; ::_thesis: verum
end;
begin
theorem Th15: :: JORDAN7:15
for n being Element of NAT
for P being non empty Subset of (TOP-REAL n)
for f being Function of I[01],((TOP-REAL n) | P) st f is being_homeomorphism holds
ex g being Function of I[01],(TOP-REAL n) st
( f = g & g is continuous & g is one-to-one )
proof
let n be Element of NAT ; ::_thesis: for P being non empty Subset of (TOP-REAL n)
for f being Function of I[01],((TOP-REAL n) | P) st f is being_homeomorphism holds
ex g being Function of I[01],(TOP-REAL n) st
( f = g & g is continuous & g is one-to-one )
let P be non empty Subset of (TOP-REAL n); ::_thesis: for f being Function of I[01],((TOP-REAL n) | P) st f is being_homeomorphism holds
ex g being Function of I[01],(TOP-REAL n) st
( f = g & g is continuous & g is one-to-one )
let f be Function of I[01],((TOP-REAL n) | P); ::_thesis: ( f is being_homeomorphism implies ex g being Function of I[01],(TOP-REAL n) st
( f = g & g is continuous & g is one-to-one ) )
A1: [#] ((TOP-REAL n) | P) = P by PRE_TOPC:def_5;
the carrier of ((TOP-REAL n) | P) = [#] ((TOP-REAL n) | P)
.= P by PRE_TOPC:def_5 ;
then reconsider g1 = f as Function of I[01],(TOP-REAL n) by FUNCT_2:7;
assume A2: f is being_homeomorphism ; ::_thesis: ex g being Function of I[01],(TOP-REAL n) st
( f = g & g is continuous & g is one-to-one )
then A3: f is one-to-one by TOPS_2:def_5;
A4: ( [#] ((TOP-REAL n) | P) <> {} & f is continuous ) by A2, TOPS_2:def_5;
A5: for P2 being Subset of (TOP-REAL n) st P2 is open holds
g1 " P2 is open
proof
let P2 be Subset of (TOP-REAL n); ::_thesis: ( P2 is open implies g1 " P2 is open )
reconsider B1 = P2 /\ P as Subset of ((TOP-REAL n) | P) by A1, XBOOLE_1:17;
f " (rng f) c= f " P by A1, RELAT_1:143;
then A6: dom f c= f " P by RELAT_1:134;
assume P2 is open ; ::_thesis: g1 " P2 is open
then B1 is open by A1, TOPS_2:24;
then A7: f " B1 is open by A4, TOPS_2:43;
f " P c= dom f by RELAT_1:132;
then ( f " B1 = (f " P2) /\ (f " P) & f " P = dom f ) by A6, FUNCT_1:68, XBOOLE_0:def_10;
hence g1 " P2 is open by A7, RELAT_1:132, XBOOLE_1:28; ::_thesis: verum
end;
[#] (TOP-REAL n) <> {} ;
then g1 is continuous by A5, TOPS_2:43;
hence ex g being Function of I[01],(TOP-REAL n) st
( f = g & g is continuous & g is one-to-one ) by A3; ::_thesis: verum
end;
theorem Th16: :: JORDAN7:16
for n being Element of NAT
for P being non empty Subset of (TOP-REAL n)
for g being Function of I[01],(TOP-REAL n) st g is continuous & g is one-to-one & rng g = P holds
ex f being Function of I[01],((TOP-REAL n) | P) st
( f = g & f is being_homeomorphism )
proof
let n be Element of NAT ; ::_thesis: for P being non empty Subset of (TOP-REAL n)
for g being Function of I[01],(TOP-REAL n) st g is continuous & g is one-to-one & rng g = P holds
ex f being Function of I[01],((TOP-REAL n) | P) st
( f = g & f is being_homeomorphism )
let P be non empty Subset of (TOP-REAL n); ::_thesis: for g being Function of I[01],(TOP-REAL n) st g is continuous & g is one-to-one & rng g = P holds
ex f being Function of I[01],((TOP-REAL n) | P) st
( f = g & f is being_homeomorphism )
let g be Function of I[01],(TOP-REAL n); ::_thesis: ( g is continuous & g is one-to-one & rng g = P implies ex f being Function of I[01],((TOP-REAL n) | P) st
( f = g & f is being_homeomorphism ) )
assume that
A1: ( g is continuous & g is one-to-one ) and
A2: rng g = P ; ::_thesis: ex f being Function of I[01],((TOP-REAL n) | P) st
( f = g & f is being_homeomorphism )
the carrier of ((TOP-REAL n) | P) = [#] ((TOP-REAL n) | P) ;
then A3: the carrier of ((TOP-REAL n) | P) = P by PRE_TOPC:def_5;
then reconsider f = g as Function of I[01],((TOP-REAL n) | P) by A2, FUNCT_2:6;
take f ; ::_thesis: ( f = g & f is being_homeomorphism )
thus f = g ; ::_thesis: f is being_homeomorphism
A4: [#] ((TOP-REAL n) | P) = P by PRE_TOPC:def_5;
A5: dom f = the carrier of I[01] by FUNCT_2:def_1
.= [#] I[01] ;
A6: [#] (TOP-REAL n) <> {} ;
for P2 being Subset of ((TOP-REAL n) | P) st P2 is open holds
f " P2 is open
proof
let P2 be Subset of ((TOP-REAL n) | P); ::_thesis: ( P2 is open implies f " P2 is open )
assume P2 is open ; ::_thesis: f " P2 is open
then consider C being Subset of (TOP-REAL n) such that
A7: C is open and
A8: C /\ ([#] ((TOP-REAL n) | P)) = P2 by TOPS_2:24;
g " P = [#] I[01] by A3, A5, RELSET_1:22;
then f " P2 = (f " C) /\ ([#] I[01]) by A4, A8, FUNCT_1:68
.= f " C by XBOOLE_1:28 ;
hence f " P2 is open by A1, A6, A7, TOPS_2:43; ::_thesis: verum
end;
then A9: f is continuous by A4, TOPS_2:43;
rng f = [#] ((TOP-REAL n) | P) by A2, PRE_TOPC:def_5;
hence f is being_homeomorphism by A1, A5, A9, COMPTS_1:17; ::_thesis: verum
end;
Lm4: now__::_thesis:_for_h2_being_Element_of_NAT_holds_(h2_-_1)_-_1_<_h2
let h2 be Element of NAT ; ::_thesis: (h2 - 1) - 1 < h2
h2 < h2 + 1 by NAT_1:13;
then A1: h2 - 1 < (h2 + 1) - 1 by XREAL_1:9;
then (h2 - 1) - 1 < h2 - 1 by XREAL_1:9;
hence (h2 - 1) - 1 < h2 by A1, XXREAL_0:2; ::_thesis: verum
end;
Lm5: 0 in [.0,1.]
by XXREAL_1:1;
Lm6: 1 in [.0,1.]
by XXREAL_1:1;
theorem Th17: :: JORDAN7:17
for A being Subset of (TOP-REAL 2)
for p1, p2 being Point of (TOP-REAL 2) st A is_an_arc_of p1,p2 holds
ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = A & g . 0 = p1 & g . 1 = p2 )
proof
let A be Subset of (TOP-REAL 2); ::_thesis: for p1, p2 being Point of (TOP-REAL 2) st A is_an_arc_of p1,p2 holds
ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = A & g . 0 = p1 & g . 1 = p2 )
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( A is_an_arc_of p1,p2 implies ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = A & g . 0 = p1 & g . 1 = p2 ) )
assume A1: A is_an_arc_of p1,p2 ; ::_thesis: ex g being Function of I[01],(TOP-REAL 2) st
( g is continuous & g is one-to-one & rng g = A & g . 0 = p1 & g . 1 = p2 )
then reconsider A9 = A as non empty Subset of (TOP-REAL 2) by TOPREAL1:1;
consider f being Function of I[01],((TOP-REAL 2) | A9) such that
A2: f is being_homeomorphism and
A3: ( f . 0 = p1 & f . 1 = p2 ) by A1, TOPREAL1:def_1;
consider g being Function of I[01],(TOP-REAL 2) such that
A4: f = g and
A5: ( g is continuous & g is one-to-one ) by A2, Th15;
take g ; ::_thesis: ( g is continuous & g is one-to-one & rng g = A & g . 0 = p1 & g . 1 = p2 )
thus ( g is continuous & g is one-to-one ) by A5; ::_thesis: ( rng g = A & g . 0 = p1 & g . 1 = p2 )
rng f = [#] ((TOP-REAL 2) | A9) by A2, TOPS_2:def_5;
hence rng g = A by A4, PRE_TOPC:def_5; ::_thesis: ( g . 0 = p1 & g . 1 = p2 )
thus ( g . 0 = p1 & g . 1 = p2 ) by A3, A4; ::_thesis: verum
end;
theorem Th18: :: JORDAN7:18
for P being non empty Subset of (TOP-REAL 2)
for p1, p2, q1, q2 being Point of (TOP-REAL 2)
for g being Function of I[01],(TOP-REAL 2)
for s1, s2 being Real st P is_an_arc_of p1,p2 & g is continuous & g is one-to-one & rng g = P & g . 0 = p1 & g . 1 = p2 & g . s1 = q1 & 0 <= s1 & s1 <= 1 & g . s2 = q2 & 0 <= s2 & s2 <= 1 & s1 <= s2 holds
LE q1,q2,P,p1,p2
proof
let P be non empty Subset of (TOP-REAL 2); ::_thesis: for p1, p2, q1, q2 being Point of (TOP-REAL 2)
for g being Function of I[01],(TOP-REAL 2)
for s1, s2 being Real st P is_an_arc_of p1,p2 & g is continuous & g is one-to-one & rng g = P & g . 0 = p1 & g . 1 = p2 & g . s1 = q1 & 0 <= s1 & s1 <= 1 & g . s2 = q2 & 0 <= s2 & s2 <= 1 & s1 <= s2 holds
LE q1,q2,P,p1,p2
let p1, p2, q1, q2 be Point of (TOP-REAL 2); ::_thesis: for g being Function of I[01],(TOP-REAL 2)
for s1, s2 being Real st P is_an_arc_of p1,p2 & g is continuous & g is one-to-one & rng g = P & g . 0 = p1 & g . 1 = p2 & g . s1 = q1 & 0 <= s1 & s1 <= 1 & g . s2 = q2 & 0 <= s2 & s2 <= 1 & s1 <= s2 holds
LE q1,q2,P,p1,p2
let g be Function of I[01],(TOP-REAL 2); ::_thesis: for s1, s2 being Real st P is_an_arc_of p1,p2 & g is continuous & g is one-to-one & rng g = P & g . 0 = p1 & g . 1 = p2 & g . s1 = q1 & 0 <= s1 & s1 <= 1 & g . s2 = q2 & 0 <= s2 & s2 <= 1 & s1 <= s2 holds
LE q1,q2,P,p1,p2
let s1, s2 be Real; ::_thesis: ( P is_an_arc_of p1,p2 & g is continuous & g is one-to-one & rng g = P & g . 0 = p1 & g . 1 = p2 & g . s1 = q1 & 0 <= s1 & s1 <= 1 & g . s2 = q2 & 0 <= s2 & s2 <= 1 & s1 <= s2 implies LE q1,q2,P,p1,p2 )
assume that
A1: P is_an_arc_of p1,p2 and
A2: ( g is continuous & g is one-to-one & rng g = P ) ; ::_thesis: ( not g . 0 = p1 or not g . 1 = p2 or not g . s1 = q1 or not 0 <= s1 or not s1 <= 1 or not g . s2 = q2 or not 0 <= s2 or not s2 <= 1 or not s1 <= s2 or LE q1,q2,P,p1,p2 )
ex f being Function of I[01],((TOP-REAL 2) | P) st
( f = g & f is being_homeomorphism ) by A2, Th16;
hence ( not g . 0 = p1 or not g . 1 = p2 or not g . s1 = q1 or not 0 <= s1 or not s1 <= 1 or not g . s2 = q2 or not 0 <= s2 or not s2 <= 1 or not s1 <= s2 or LE q1,q2,P,p1,p2 ) by A1, JORDAN5C:8; ::_thesis: verum
end;
theorem Th19: :: JORDAN7:19
for P being non empty Subset of (TOP-REAL 2)
for p1, p2, q1, q2 being Point of (TOP-REAL 2)
for g being Function of I[01],(TOP-REAL 2)
for s1, s2 being Real st g is continuous & g is one-to-one & rng g = P & g . 0 = p1 & g . 1 = p2 & g . s1 = q1 & 0 <= s1 & s1 <= 1 & g . s2 = q2 & 0 <= s2 & s2 <= 1 & LE q1,q2,P,p1,p2 holds
s1 <= s2
proof
let P be non empty Subset of (TOP-REAL 2); ::_thesis: for p1, p2, q1, q2 being Point of (TOP-REAL 2)
for g being Function of I[01],(TOP-REAL 2)
for s1, s2 being Real st g is continuous & g is one-to-one & rng g = P & g . 0 = p1 & g . 1 = p2 & g . s1 = q1 & 0 <= s1 & s1 <= 1 & g . s2 = q2 & 0 <= s2 & s2 <= 1 & LE q1,q2,P,p1,p2 holds
s1 <= s2
let p1, p2, q1, q2 be Point of (TOP-REAL 2); ::_thesis: for g being Function of I[01],(TOP-REAL 2)
for s1, s2 being Real st g is continuous & g is one-to-one & rng g = P & g . 0 = p1 & g . 1 = p2 & g . s1 = q1 & 0 <= s1 & s1 <= 1 & g . s2 = q2 & 0 <= s2 & s2 <= 1 & LE q1,q2,P,p1,p2 holds
s1 <= s2
let g be Function of I[01],(TOP-REAL 2); ::_thesis: for s1, s2 being Real st g is continuous & g is one-to-one & rng g = P & g . 0 = p1 & g . 1 = p2 & g . s1 = q1 & 0 <= s1 & s1 <= 1 & g . s2 = q2 & 0 <= s2 & s2 <= 1 & LE q1,q2,P,p1,p2 holds
s1 <= s2
let s1, s2 be Real; ::_thesis: ( g is continuous & g is one-to-one & rng g = P & g . 0 = p1 & g . 1 = p2 & g . s1 = q1 & 0 <= s1 & s1 <= 1 & g . s2 = q2 & 0 <= s2 & s2 <= 1 & LE q1,q2,P,p1,p2 implies s1 <= s2 )
assume ( g is continuous & g is one-to-one & rng g = P ) ; ::_thesis: ( not g . 0 = p1 or not g . 1 = p2 or not g . s1 = q1 or not 0 <= s1 or not s1 <= 1 or not g . s2 = q2 or not 0 <= s2 or not s2 <= 1 or not LE q1,q2,P,p1,p2 or s1 <= s2 )
then ex f being Function of I[01],((TOP-REAL 2) | P) st
( f = g & f is being_homeomorphism ) by Th16;
hence ( not g . 0 = p1 or not g . 1 = p2 or not g . s1 = q1 or not 0 <= s1 or not s1 <= 1 or not g . s2 = q2 or not 0 <= s2 or not s2 <= 1 or not LE q1,q2,P,p1,p2 or s1 <= s2 ) by JORDAN5C:def_3; ::_thesis: verum
end;
theorem :: JORDAN7:20
for P being non empty compact Subset of (TOP-REAL 2)
for e being Real st P is being_simple_closed_curve & e > 0 holds
ex h being FinSequence of the carrier of (TOP-REAL 2) st
( h . 1 = W-min P & h is one-to-one & 8 <= len h & rng h c= P & ( for i being Element of NAT st 1 <= i & i < len h holds
LE h /. i,h /. (i + 1),P ) & ( for i being Element of NAT
for W being Subset of (Euclid 2) st 1 <= i & i < len h & W = Segment ((h /. i),(h /. (i + 1)),P) holds
diameter W < e ) & ( for W being Subset of (Euclid 2) st W = Segment ((h /. (len h)),(h /. 1),P) holds
diameter W < e ) & ( for i being Element of NAT st 1 <= i & i + 1 < len h holds
(Segment ((h /. i),(h /. (i + 1)),P)) /\ (Segment ((h /. (i + 1)),(h /. (i + 2)),P)) = {(h /. (i + 1))} ) & (Segment ((h /. (len h)),(h /. 1),P)) /\ (Segment ((h /. 1),(h /. 2),P)) = {(h /. 1)} & (Segment ((h /. ((len h) -' 1)),(h /. (len h)),P)) /\ (Segment ((h /. (len h)),(h /. 1),P)) = {(h /. (len h))} & Segment ((h /. ((len h) -' 1)),(h /. (len h)),P) misses Segment ((h /. 1),(h /. 2),P) & ( for i, j being Element of NAT st 1 <= i & i < j & j < len h & not i,j are_adjacent1 holds
Segment ((h /. i),(h /. (i + 1)),P) misses Segment ((h /. j),(h /. (j + 1)),P) ) & ( for i being Element of NAT st 1 < i & i + 1 < len h holds
Segment ((h /. (len h)),(h /. 1),P) misses Segment ((h /. i),(h /. (i + 1)),P) ) )
proof
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: for e being Real st P is being_simple_closed_curve & e > 0 holds
ex h being FinSequence of the carrier of (TOP-REAL 2) st
( h . 1 = W-min P & h is one-to-one & 8 <= len h & rng h c= P & ( for i being Element of NAT st 1 <= i & i < len h holds
LE h /. i,h /. (i + 1),P ) & ( for i being Element of NAT
for W being Subset of (Euclid 2) st 1 <= i & i < len h & W = Segment ((h /. i),(h /. (i + 1)),P) holds
diameter W < e ) & ( for W being Subset of (Euclid 2) st W = Segment ((h /. (len h)),(h /. 1),P) holds
diameter W < e ) & ( for i being Element of NAT st 1 <= i & i + 1 < len h holds
(Segment ((h /. i),(h /. (i + 1)),P)) /\ (Segment ((h /. (i + 1)),(h /. (i + 2)),P)) = {(h /. (i + 1))} ) & (Segment ((h /. (len h)),(h /. 1),P)) /\ (Segment ((h /. 1),(h /. 2),P)) = {(h /. 1)} & (Segment ((h /. ((len h) -' 1)),(h /. (len h)),P)) /\ (Segment ((h /. (len h)),(h /. 1),P)) = {(h /. (len h))} & Segment ((h /. ((len h) -' 1)),(h /. (len h)),P) misses Segment ((h /. 1),(h /. 2),P) & ( for i, j being Element of NAT st 1 <= i & i < j & j < len h & not i,j are_adjacent1 holds
Segment ((h /. i),(h /. (i + 1)),P) misses Segment ((h /. j),(h /. (j + 1)),P) ) & ( for i being Element of NAT st 1 < i & i + 1 < len h holds
Segment ((h /. (len h)),(h /. 1),P) misses Segment ((h /. i),(h /. (i + 1)),P) ) )
let e be Real; ::_thesis: ( P is being_simple_closed_curve & e > 0 implies ex h being FinSequence of the carrier of (TOP-REAL 2) st
( h . 1 = W-min P & h is one-to-one & 8 <= len h & rng h c= P & ( for i being Element of NAT st 1 <= i & i < len h holds
LE h /. i,h /. (i + 1),P ) & ( for i being Element of NAT
for W being Subset of (Euclid 2) st 1 <= i & i < len h & W = Segment ((h /. i),(h /. (i + 1)),P) holds
diameter W < e ) & ( for W being Subset of (Euclid 2) st W = Segment ((h /. (len h)),(h /. 1),P) holds
diameter W < e ) & ( for i being Element of NAT st 1 <= i & i + 1 < len h holds
(Segment ((h /. i),(h /. (i + 1)),P)) /\ (Segment ((h /. (i + 1)),(h /. (i + 2)),P)) = {(h /. (i + 1))} ) & (Segment ((h /. (len h)),(h /. 1),P)) /\ (Segment ((h /. 1),(h /. 2),P)) = {(h /. 1)} & (Segment ((h /. ((len h) -' 1)),(h /. (len h)),P)) /\ (Segment ((h /. (len h)),(h /. 1),P)) = {(h /. (len h))} & Segment ((h /. ((len h) -' 1)),(h /. (len h)),P) misses Segment ((h /. 1),(h /. 2),P) & ( for i, j being Element of NAT st 1 <= i & i < j & j < len h & not i,j are_adjacent1 holds
Segment ((h /. i),(h /. (i + 1)),P) misses Segment ((h /. j),(h /. (j + 1)),P) ) & ( for i being Element of NAT st 1 < i & i + 1 < len h holds
Segment ((h /. (len h)),(h /. 1),P) misses Segment ((h /. i),(h /. (i + 1)),P) ) ) )
assume that
A1: P is being_simple_closed_curve and
A2: e > 0 ; ::_thesis: ex h being FinSequence of the carrier of (TOP-REAL 2) st
( h . 1 = W-min P & h is one-to-one & 8 <= len h & rng h c= P & ( for i being Element of NAT st 1 <= i & i < len h holds
LE h /. i,h /. (i + 1),P ) & ( for i being Element of NAT
for W being Subset of (Euclid 2) st 1 <= i & i < len h & W = Segment ((h /. i),(h /. (i + 1)),P) holds
diameter W < e ) & ( for W being Subset of (Euclid 2) st W = Segment ((h /. (len h)),(h /. 1),P) holds
diameter W < e ) & ( for i being Element of NAT st 1 <= i & i + 1 < len h holds
(Segment ((h /. i),(h /. (i + 1)),P)) /\ (Segment ((h /. (i + 1)),(h /. (i + 2)),P)) = {(h /. (i + 1))} ) & (Segment ((h /. (len h)),(h /. 1),P)) /\ (Segment ((h /. 1),(h /. 2),P)) = {(h /. 1)} & (Segment ((h /. ((len h) -' 1)),(h /. (len h)),P)) /\ (Segment ((h /. (len h)),(h /. 1),P)) = {(h /. (len h))} & Segment ((h /. ((len h) -' 1)),(h /. (len h)),P) misses Segment ((h /. 1),(h /. 2),P) & ( for i, j being Element of NAT st 1 <= i & i < j & j < len h & not i,j are_adjacent1 holds
Segment ((h /. i),(h /. (i + 1)),P) misses Segment ((h /. j),(h /. (j + 1)),P) ) & ( for i being Element of NAT st 1 < i & i + 1 < len h holds
Segment ((h /. (len h)),(h /. 1),P) misses Segment ((h /. i),(h /. (i + 1)),P) ) )
A3: Upper_Arc P is_an_arc_of W-min P, E-max P by A1, JORDAN6:def_8;
then consider g1 being Function of I[01],(TOP-REAL 2) such that
A4: ( g1 is continuous & g1 is one-to-one ) and
A5: rng g1 = Upper_Arc P and
A6: g1 . 0 = W-min P and
A7: g1 . 1 = E-max P by Th17;
consider h1 being FinSequence of REAL such that
A8: h1 . 1 = 0 and
A9: h1 . (len h1) = 1 and
A10: 5 <= len h1 and
A11: rng h1 c= the carrier of I[01] and
A12: h1 is increasing and
A13: for i being Element of NAT
for Q being Subset of I[01]
for W being Subset of (Euclid 2) st 1 <= i & i < len h1 & Q = [.(h1 /. i),(h1 /. (i + 1)).] & W = g1 .: Q holds
diameter W < e by A2, A4, UNIFORM1:13;
h1 is FinSequence of the carrier of I[01] by A11, FINSEQ_1:def_4;
then reconsider h11 = g1 * h1 as FinSequence of the carrier of (TOP-REAL 2) by FINSEQ_2:32;
A14: 2 < len h1 by A10, XXREAL_0:2;
then A15: 2 in dom h1 by FINSEQ_3:25;
A16: 1 <= len h1 by A10, XXREAL_0:2;
then A17: 1 in dom h1 by FINSEQ_3:25;
A18: 1 + 1 in dom h1 by A14, FINSEQ_3:25;
then A19: h1 . (1 + 1) in rng h1 by FUNCT_1:def_3;
A20: h11 is one-to-one by A4, A12;
A21: Lower_Arc P is_an_arc_of E-max P, W-min P by A1, JORDAN6:def_9;
then consider g2 being Function of I[01],(TOP-REAL 2) such that
A22: ( g2 is continuous & g2 is one-to-one ) and
A23: rng g2 = Lower_Arc P and
A24: g2 . 0 = E-max P and
A25: g2 . 1 = W-min P by Th17;
consider h2 being FinSequence of REAL such that
A26: h2 . 1 = 0 and
A27: h2 . (len h2) = 1 and
A28: 5 <= len h2 and
A29: rng h2 c= the carrier of I[01] and
A30: h2 is increasing and
A31: for i being Element of NAT
for Q being Subset of I[01]
for W being Subset of (Euclid 2) st 1 <= i & i < len h2 & Q = [.(h2 /. i),(h2 /. (i + 1)).] & W = g2 .: Q holds
diameter W < e by A2, A22, UNIFORM1:13;
h2 is FinSequence of the carrier of I[01] by A29, FINSEQ_1:def_4;
then reconsider h21 = g2 * h2 as FinSequence of the carrier of (TOP-REAL 2) by FINSEQ_2:32;
A32: h21 is one-to-one by A22, A30;
A33: 1 <= len h2 by A28, XXREAL_0:2;
then A34: len h2 in dom h2 by FINSEQ_3:25;
then A35: h21 . (len h2) = W-min P by A25, A27, FUNCT_1:13;
reconsider h0 = h11 ^ (mid (h21,2,((len h21) -' 1))) as FinSequence of the carrier of (TOP-REAL 2) ;
A36: len h0 = (len h11) + (len (mid (h21,2,((len h21) -' 1)))) by FINSEQ_1:22;
set i = (len h0) -' 1;
take h0 ; ::_thesis: ( h0 . 1 = W-min P & h0 is one-to-one & 8 <= len h0 & rng h0 c= P & ( for i being Element of NAT st 1 <= i & i < len h0 holds
LE h0 /. i,h0 /. (i + 1),P ) & ( for i being Element of NAT
for W being Subset of (Euclid 2) st 1 <= i & i < len h0 & W = Segment ((h0 /. i),(h0 /. (i + 1)),P) holds
diameter W < e ) & ( for W being Subset of (Euclid 2) st W = Segment ((h0 /. (len h0)),(h0 /. 1),P) holds
diameter W < e ) & ( for i being Element of NAT st 1 <= i & i + 1 < len h0 holds
(Segment ((h0 /. i),(h0 /. (i + 1)),P)) /\ (Segment ((h0 /. (i + 1)),(h0 /. (i + 2)),P)) = {(h0 /. (i + 1))} ) & (Segment ((h0 /. (len h0)),(h0 /. 1),P)) /\ (Segment ((h0 /. 1),(h0 /. 2),P)) = {(h0 /. 1)} & (Segment ((h0 /. ((len h0) -' 1)),(h0 /. (len h0)),P)) /\ (Segment ((h0 /. (len h0)),(h0 /. 1),P)) = {(h0 /. (len h0))} & Segment ((h0 /. ((len h0) -' 1)),(h0 /. (len h0)),P) misses Segment ((h0 /. 1),(h0 /. 2),P) & ( for i, j being Element of NAT st 1 <= i & i < j & j < len h0 & not i,j are_adjacent1 holds
Segment ((h0 /. i),(h0 /. (i + 1)),P) misses Segment ((h0 /. j),(h0 /. (j + 1)),P) ) & ( for i being Element of NAT st 1 < i & i + 1 < len h0 holds
Segment ((h0 /. (len h0)),(h0 /. 1),P) misses Segment ((h0 /. i),(h0 /. (i + 1)),P) ) )
A37: rng h1 c= dom g1 by A11, FUNCT_2:def_1;
then A38: dom h1 = dom h11 by RELAT_1:27;
then A39: len h1 = len h11 by FINSEQ_3:29;
then A40: h0 . 2 = h11 . 2 by A14, FINSEQ_1:64;
A41: h0 . (1 + 1) = h11 . (1 + 1) by A39, A14, FINSEQ_1:64;
then A42: h0 . (1 + 1) = g1 . (h1 . (1 + 1)) by A18, FUNCT_1:13;
set k = (((len h0) -' (len h11)) + 2) -' 1;
0 + 2 <= ((len h0) -' (len h11)) + 2 by XREAL_1:6;
then A43: 2 -' 1 <= (((len h0) -' (len h11)) + 2) -' 1 by NAT_D:42;
A44: 0 in dom g1 by Lm5, BORSUK_1:40, FUNCT_2:def_1;
A45: len h1 in dom h1 by A16, FINSEQ_3:25;
dom g2 = the carrier of I[01] by FUNCT_2:def_1;
then A46: dom h2 = dom h21 by A29, RELAT_1:27;
then A47: len h2 = len h21 by FINSEQ_3:29;
then A48: 2 <= len h21 by A28, XXREAL_0:2;
len h21 <= (len h21) + 1 by NAT_1:12;
then A49: (len h21) - 1 <= ((len h21) + 1) - 1 by XREAL_1:9;
then A50: (len h21) -' 1 <= len h21 by A28, A47, XREAL_0:def_2;
2 <= len h21 by A28, A47, XXREAL_0:2;
then A51: (1 + 1) - 1 <= (len h21) - 1 by XREAL_1:9;
then A52: (len h21) -' 1 = (len h21) - 1 by XREAL_0:def_2;
3 < len h21 by A28, A47, XXREAL_0:2;
then A53: (2 + 1) - 1 < (len h21) - 1 by XREAL_1:9;
then A54: 2 < (len h21) -' 1 by A51, NAT_D:39;
then A55: ((len h21) -' 1) -' 2 = ((len h21) -' 1) - 2 by XREAL_1:233;
A56: 1 <= (len h21) -' 1 by A51, XREAL_0:def_2;
then A57: len (mid (h21,2,((len h21) -' 1))) = (((len h21) -' 1) -' 2) + 1 by A48, A50, A54, FINSEQ_6:118;
(3 + 2) - 2 <= (len h2) - 2 by A28, XREAL_1:9;
then A58: 5 + 3 <= (len h1) + ((len h2) - 2) by A10, XREAL_1:7;
then A59: len h0 > (1 + 1) + 1 by A39, A47, A36, A52, A55, A57, XXREAL_0:2;
then A60: (len h0) -' 1 > 1 + 1 by Lm2;
then A61: 1 < (len h0) -' 1 by XXREAL_0:2;
A62: (3 + 2) - 2 <= (len h2) - 2 by A28, XREAL_1:9;
then A63: 1 <= (len h2) - 2 by XXREAL_0:2;
then A64: (len h1) + 1 <= len h0 by A39, A47, A36, A52, A55, A57, XREAL_1:7;
then A65: len h0 > len h1 by NAT_1:13;
then A66: 1 < len h0 by A16, XXREAL_0:2;
then A67: 1 in dom h0 by FINSEQ_3:25;
then A68: h0 /. 1 = h0 . 1 by PARTFUN1:def_6;
A69: dom g1 = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1;
then A70: 1 in dom (g1 * h1) by A8, A17, Lm5, FUNCT_1:11;
then A71: h11 . 1 = W-min P by A6, A8, FUNCT_1:12;
hence A72: h0 . 1 = W-min P by A70, FINSEQ_1:def_7; ::_thesis: ( h0 is one-to-one & 8 <= len h0 & rng h0 c= P & ( for i being Element of NAT st 1 <= i & i < len h0 holds
LE h0 /. i,h0 /. (i + 1),P ) & ( for i being Element of NAT
for W being Subset of (Euclid 2) st 1 <= i & i < len h0 & W = Segment ((h0 /. i),(h0 /. (i + 1)),P) holds
diameter W < e ) & ( for W being Subset of (Euclid 2) st W = Segment ((h0 /. (len h0)),(h0 /. 1),P) holds
diameter W < e ) & ( for i being Element of NAT st 1 <= i & i + 1 < len h0 holds
(Segment ((h0 /. i),(h0 /. (i + 1)),P)) /\ (Segment ((h0 /. (i + 1)),(h0 /. (i + 2)),P)) = {(h0 /. (i + 1))} ) & (Segment ((h0 /. (len h0)),(h0 /. 1),P)) /\ (Segment ((h0 /. 1),(h0 /. 2),P)) = {(h0 /. 1)} & (Segment ((h0 /. ((len h0) -' 1)),(h0 /. (len h0)),P)) /\ (Segment ((h0 /. (len h0)),(h0 /. 1),P)) = {(h0 /. (len h0))} & Segment ((h0 /. ((len h0) -' 1)),(h0 /. (len h0)),P) misses Segment ((h0 /. 1),(h0 /. 2),P) & ( for i, j being Element of NAT st 1 <= i & i < j & j < len h0 & not i,j are_adjacent1 holds
Segment ((h0 /. i),(h0 /. (i + 1)),P) misses Segment ((h0 /. j),(h0 /. (j + 1)),P) ) & ( for i being Element of NAT st 1 < i & i + 1 < len h0 holds
Segment ((h0 /. (len h0)),(h0 /. 1),P) misses Segment ((h0 /. i),(h0 /. (i + 1)),P) ) )
then A73: h0 /. 1 = W-min P by A67, PARTFUN1:def_6;
A74: len h0 in dom h0 by A66, FINSEQ_3:25;
then A75: h0 /. (len h0) = h0 . (len h0) by PARTFUN1:def_6;
A76: 1 in dom h2 by A33, FINSEQ_3:25;
then A77: h21 . 1 = E-max P by A24, A26, FUNCT_1:13;
thus A78: h0 is one-to-one ::_thesis: ( 8 <= len h0 & rng h0 c= P & ( for i being Element of NAT st 1 <= i & i < len h0 holds
LE h0 /. i,h0 /. (i + 1),P ) & ( for i being Element of NAT
for W being Subset of (Euclid 2) st 1 <= i & i < len h0 & W = Segment ((h0 /. i),(h0 /. (i + 1)),P) holds
diameter W < e ) & ( for W being Subset of (Euclid 2) st W = Segment ((h0 /. (len h0)),(h0 /. 1),P) holds
diameter W < e ) & ( for i being Element of NAT st 1 <= i & i + 1 < len h0 holds
(Segment ((h0 /. i),(h0 /. (i + 1)),P)) /\ (Segment ((h0 /. (i + 1)),(h0 /. (i + 2)),P)) = {(h0 /. (i + 1))} ) & (Segment ((h0 /. (len h0)),(h0 /. 1),P)) /\ (Segment ((h0 /. 1),(h0 /. 2),P)) = {(h0 /. 1)} & (Segment ((h0 /. ((len h0) -' 1)),(h0 /. (len h0)),P)) /\ (Segment ((h0 /. (len h0)),(h0 /. 1),P)) = {(h0 /. (len h0))} & Segment ((h0 /. ((len h0) -' 1)),(h0 /. (len h0)),P) misses Segment ((h0 /. 1),(h0 /. 2),P) & ( for i, j being Element of NAT st 1 <= i & i < j & j < len h0 & not i,j are_adjacent1 holds
Segment ((h0 /. i),(h0 /. (i + 1)),P) misses Segment ((h0 /. j),(h0 /. (j + 1)),P) ) & ( for i being Element of NAT st 1 < i & i + 1 < len h0 holds
Segment ((h0 /. (len h0)),(h0 /. 1),P) misses Segment ((h0 /. i),(h0 /. (i + 1)),P) ) )
proof
let x, y be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not x in K128(h0) or not y in K128(h0) or not h0 . x = h0 . y or x = y )
assume that
A79: x in dom h0 and
A80: y in dom h0 and
A81: h0 . x = h0 . y ; ::_thesis: x = y
reconsider nx = x, ny = y as Element of NAT by A79, A80;
A82: 1 <= nx by A79, FINSEQ_3:25;
A83: nx <= len h0 by A79, FINSEQ_3:25;
A84: 1 <= ny by A80, FINSEQ_3:25;
A85: ny <= len h0 by A80, FINSEQ_3:25;
percases ( nx <= len h1 or nx > len h1 ) ;
suppose nx <= len h1 ; ::_thesis: x = y
then A86: nx in dom h1 by A82, FINSEQ_3:25;
then A87: h1 . nx in rng h1 by FUNCT_1:def_3;
A88: h0 . nx = h11 . nx by A38, A86, FINSEQ_1:def_7
.= g1 . (h1 . nx) by A38, A86, FUNCT_1:12 ;
then A89: h0 . nx in Upper_Arc P by A5, A37, A87, FUNCT_1:def_3;
percases ( ny <= len h1 or ny > len h1 ) ;
suppose ny <= len h1 ; ::_thesis: x = y
then A90: ny in dom h1 by A84, FINSEQ_3:25;
then A91: h1 . ny in rng h1 by FUNCT_1:def_3;
h0 . ny = h11 . ny by A38, A90, FINSEQ_1:def_7
.= g1 . (h1 . ny) by A90, FUNCT_1:13 ;
then h1 . nx = h1 . ny by A4, A37, A81, A87, A88, A91, FUNCT_1:def_4;
hence x = y by A12, A86, A90, FUNCT_1:def_4; ::_thesis: verum
end;
supposeA92: ny > len h1 ; ::_thesis: x = y
A93: 0 + 2 <= (ny -' (len h11)) + 2 by XREAL_1:6;
then A94: 1 <= ((ny -' (len h11)) + 2) -' 1 by Lm1, NAT_D:42;
(len h1) + 1 <= ny by A92, NAT_1:13;
then A95: ((len h1) + 1) - (len h1) <= ny - (len h1) by XREAL_1:9;
then 1 <= ny -' (len h11) by A39, A92, XREAL_1:233;
then 1 + 1 <= (((ny -' (len h11)) + 1) + 1) - 1 by XREAL_1:6;
then A96: 2 <= ((ny -' (len h11)) + 2) -' 1 by A93, Lm1, NAT_D:39, NAT_D:42;
A97: ny - (len h11) = ny -' (len h11) by A39, A92, XREAL_1:233;
ny - (len h11) <= ((len h1) + ((len h2) - 2)) - (len h11) by A39, A47, A36, A52, A55, A57, A85, XREAL_1:9;
then A98: (ny -' (len h11)) + 2 <= ((len h2) - 2) + 2 by A39, A97, XREAL_1:6;
then ((ny -' (len h11)) + 2) -' 1 <= len h21 by A47, NAT_D:44;
then A99: ((ny -' (len h11)) + 2) -' 1 in dom h21 by A94, FINSEQ_3:25;
((ny -' (len h11)) + 2) - 1 <= (len h2) - 1 by A98, XREAL_1:9;
then A100: ((ny -' (len h11)) + 2) -' 1 <= (len h2) - 1 by A93, Lm1, NAT_D:39, NAT_D:42;
A101: ny <= (len h11) + (len (mid (h21,2,((len h21) -' 1)))) by A85, FINSEQ_1:22;
then A102: ny - (len h11) <= ((len h11) + (len (mid (h21,2,((len h21) -' 1))))) - (len h11) by XREAL_1:9;
(len h11) + 1 <= ny by A39, A92, NAT_1:13;
then A103: h0 . ny = (mid (h21,2,((len h21) -' 1))) . (ny - (len h11)) by A101, FINSEQ_1:23;
then A104: h0 . ny = h21 . (((ny -' (len h11)) + 2) -' 1) by A39, A48, A56, A50, A54, A97, A102, A95, FINSEQ_6:118;
then h0 . ny in rng h21 by A99, FUNCT_1:def_3;
then h0 . ny in rng g2 by FUNCT_1:14;
then h0 . nx in (Upper_Arc P) /\ (Lower_Arc P) by A23, A81, A89, XBOOLE_0:def_4;
then A105: h0 . nx in {(W-min P),(E-max P)} by A1, JORDAN6:50;
percases ( h0 . nx = W-min P or h0 . nx = E-max P ) by A105, TARSKI:def_2;
suppose h0 . nx = W-min P ; ::_thesis: x = y
then h21 . (((ny -' (len h11)) + 2) -' 1) = W-min P by A39, A48, A56, A50, A54, A81, A103, A97, A102, A95, FINSEQ_6:118;
then len h21 = ((ny -' (len h11)) + 2) -' 1 by A46, A47, A34, A35, A32, A99, FUNCT_1:def_4;
then (len h21) + 1 <= ((len h21) - 1) + 1 by A47, A100, XREAL_1:6;
then ((len h21) + 1) - (len h21) <= (len h21) - (len h21) by XREAL_1:9;
then ((len h21) + 1) - (len h21) <= 0 ;
hence x = y ; ::_thesis: verum
end;
suppose h0 . nx = E-max P ; ::_thesis: x = y
then 1 = ((ny -' (len h11)) + 2) -' 1 by A46, A76, A77, A32, A81, A104, A99, FUNCT_1:def_4;
hence x = y by A96; ::_thesis: verum
end;
end;
end;
end;
end;
supposeA106: nx > len h1 ; ::_thesis: x = y
then (len h1) + 1 <= nx by NAT_1:13;
then A107: ((len h1) + 1) - (len h1) <= nx - (len h1) by XREAL_1:9;
then 1 <= nx -' (len h11) by A39, A106, XREAL_1:233;
then A108: 1 + 1 <= (((nx -' (len h11)) + 1) + 1) - 1 by XREAL_1:6;
A109: nx <= (len h11) + (len (mid (h21,2,((len h21) -' 1)))) by A83, FINSEQ_1:22;
then A110: nx - (len h11) <= ((len h11) + (len (mid (h21,2,((len h21) -' 1))))) - (len h11) by XREAL_1:9;
A111: nx - (len h11) = nx -' (len h11) by A39, A106, XREAL_1:233;
nx - (len h11) <= ((len h1) + ((len h2) - 2)) - (len h11) by A39, A47, A36, A52, A55, A57, A83, XREAL_1:9;
then A112: (nx -' (len h11)) + 2 <= ((len h2) - 2) + 2 by A39, A111, XREAL_1:6;
then A113: ((nx -' (len h11)) + 2) - 1 <= (len h2) - 1 by XREAL_1:9;
A114: ((nx -' (len h11)) + 2) -' 1 <= len h21 by A47, A112, NAT_D:44;
A115: 0 + 2 <= (nx -' (len h11)) + 2 by XREAL_1:6;
then 1 <= ((nx -' (len h11)) + 2) -' 1 by Lm1, NAT_D:42;
then A116: ((nx -' (len h11)) + 2) -' 1 in dom h21 by A114, FINSEQ_3:25;
(len h11) + 1 <= nx by A39, A106, NAT_1:13;
then A117: h0 . nx = (mid (h21,2,((len h21) -' 1))) . (nx - (len h11)) by A109, FINSEQ_1:23;
then A118: h0 . nx = h21 . (((nx -' (len h11)) + 2) -' 1) by A39, A48, A56, A50, A54, A111, A110, A107, FINSEQ_6:118;
then h0 . nx in rng h21 by A116, FUNCT_1:def_3;
then A119: h0 . nx in Lower_Arc P by A23, FUNCT_1:14;
percases ( ny <= len h1 or ny > len h1 ) ;
suppose ny <= len h1 ; ::_thesis: x = y
then A120: ny in dom h1 by A84, FINSEQ_3:25;
then A121: h1 . ny in rng h1 by FUNCT_1:def_3;
h0 . ny = h11 . ny by A38, A120, FINSEQ_1:def_7
.= g1 . (h1 . ny) by A38, A120, FUNCT_1:12 ;
then h0 . ny in rng g1 by A37, A121, FUNCT_1:def_3;
then h0 . ny in (Upper_Arc P) /\ (Lower_Arc P) by A5, A81, A119, XBOOLE_0:def_4;
then A122: h0 . ny in {(W-min P),(E-max P)} by A1, JORDAN6:50;
A123: ((nx -' (len h11)) + 2) -' 1 <= (len h2) - 1 by A115, A113, Lm1, NAT_D:39, NAT_D:42;
A124: 2 <= ((nx -' (len h11)) + 2) -' 1 by A108, A115, Lm1, NAT_D:39, NAT_D:42;
percases ( h0 . ny = W-min P or h0 . ny = E-max P ) by A122, TARSKI:def_2;
suppose h0 . ny = W-min P ; ::_thesis: x = y
then len h21 = ((nx -' (len h11)) + 2) -' 1 by A46, A47, A34, A35, A32, A81, A118, A116, FUNCT_1:def_4;
then (len h21) + 1 <= ((len h21) - 1) + 1 by A47, A123, XREAL_1:6;
then ((len h21) + 1) - (len h21) <= (len h21) - (len h21) by XREAL_1:9;
then ((len h21) + 1) - (len h21) <= 0 ;
hence x = y ; ::_thesis: verum
end;
suppose h0 . ny = E-max P ; ::_thesis: x = y
then h21 . (((nx -' (len h11)) + 2) -' 1) = E-max P by A39, A48, A56, A50, A54, A81, A117, A111, A110, A107, FINSEQ_6:118;
then 1 = ((nx -' (len h11)) + 2) -' 1 by A46, A76, A77, A32, A116, FUNCT_1:def_4;
hence x = y by A124; ::_thesis: verum
end;
end;
end;
supposeA125: ny > len h1 ; ::_thesis: x = y
then A126: ny - (len h11) = ny -' (len h11) by A39, XREAL_1:233;
(len h1) + 1 <= ny by A125, NAT_1:13;
then A127: ( h0 . ny = (mid (h21,2,((len h21) -' 1))) . (ny - (len h11)) & ((len h1) + 1) - (len h1) <= ny - (len h1) ) by A39, A36, A85, FINSEQ_1:23, XREAL_1:9;
0 + 2 <= (ny -' (len h11)) + 2 by XREAL_1:6;
then A128: 1 <= ((ny -' (len h11)) + 2) -' 1 by Lm1, NAT_D:42;
ny - (len h11) <= ((len h11) + (len (mid (h21,2,((len h21) -' 1))))) - (len h11) by A36, A85, XREAL_1:9;
then A129: h0 . ny = h21 . (((ny -' (len h11)) + 2) -' 1) by A39, A48, A56, A50, A54, A126, A127, FINSEQ_6:118;
ny - (len h11) <= ((len h1) + ((len h2) - 2)) - (len h11) by A39, A47, A36, A52, A55, A57, A85, XREAL_1:9;
then (ny -' (len h11)) + 2 <= ((len h2) - 2) + 2 by A39, A126, XREAL_1:6;
then ((ny -' (len h11)) + 2) -' 1 <= len h21 by A47, NAT_D:44;
then ((ny -' (len h11)) + 2) -' 1 in dom h21 by A128, FINSEQ_3:25;
then ((nx -' (len h1)) + 2) -' 1 = ((ny -' (len h1)) + 2) -' 1 by A39, A32, A81, A118, A116, A129, FUNCT_1:def_4;
then ((nx -' (len h1)) + 2) - 1 = ((ny -' (len h1)) + 2) -' 1 by A39, A115, Lm1, NAT_D:39, NAT_D:42;
then (nx -' (len h1)) + (2 - 1) = ((ny -' (len h1)) + 2) - 1 by A39, A128, NAT_D:39;
then ((len h1) + nx) - (len h1) = (len h1) + (ny - (len h1)) by A39, A111, A126, XCMPLX_1:29;
hence x = y ; ::_thesis: verum
end;
end;
end;
end;
end;
then A130: h0 /. (len h0) <> W-min P by A16, A72, A65, A74, A75, A67, FUNCT_1:def_4;
A131: dom g2 = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1;
thus 8 <= len h0 by A38, A47, A36, A52, A55, A57, A58, FINSEQ_3:29; ::_thesis: ( rng h0 c= P & ( for i being Element of NAT st 1 <= i & i < len h0 holds
LE h0 /. i,h0 /. (i + 1),P ) & ( for i being Element of NAT
for W being Subset of (Euclid 2) st 1 <= i & i < len h0 & W = Segment ((h0 /. i),(h0 /. (i + 1)),P) holds
diameter W < e ) & ( for W being Subset of (Euclid 2) st W = Segment ((h0 /. (len h0)),(h0 /. 1),P) holds
diameter W < e ) & ( for i being Element of NAT st 1 <= i & i + 1 < len h0 holds
(Segment ((h0 /. i),(h0 /. (i + 1)),P)) /\ (Segment ((h0 /. (i + 1)),(h0 /. (i + 2)),P)) = {(h0 /. (i + 1))} ) & (Segment ((h0 /. (len h0)),(h0 /. 1),P)) /\ (Segment ((h0 /. 1),(h0 /. 2),P)) = {(h0 /. 1)} & (Segment ((h0 /. ((len h0) -' 1)),(h0 /. (len h0)),P)) /\ (Segment ((h0 /. (len h0)),(h0 /. 1),P)) = {(h0 /. (len h0))} & Segment ((h0 /. ((len h0) -' 1)),(h0 /. (len h0)),P) misses Segment ((h0 /. 1),(h0 /. 2),P) & ( for i, j being Element of NAT st 1 <= i & i < j & j < len h0 & not i,j are_adjacent1 holds
Segment ((h0 /. i),(h0 /. (i + 1)),P) misses Segment ((h0 /. j),(h0 /. (j + 1)),P) ) & ( for i being Element of NAT st 1 < i & i + 1 < len h0 holds
Segment ((h0 /. (len h0)),(h0 /. 1),P) misses Segment ((h0 /. i),(h0 /. (i + 1)),P) ) )
( rng (mid (h21,2,((len h21) -' 1))) c= rng h21 & rng (g2 * h2) c= rng g2 ) by FINSEQ_6:119, RELAT_1:26;
then ( rng (g1 * h1) c= rng g1 & rng (mid (h21,2,((len h21) -' 1))) c= rng g2 ) by RELAT_1:26, XBOOLE_1:1;
then (rng h11) \/ (rng (mid (h21,2,((len h21) -' 1)))) c= (Upper_Arc P) \/ (Lower_Arc P) by A5, A23, XBOOLE_1:13;
then rng h0 c= (Upper_Arc P) \/ (Lower_Arc P) by FINSEQ_1:31;
hence rng h0 c= P by A1, JORDAN6:def_9; ::_thesis: ( ( for i being Element of NAT st 1 <= i & i < len h0 holds
LE h0 /. i,h0 /. (i + 1),P ) & ( for i being Element of NAT
for W being Subset of (Euclid 2) st 1 <= i & i < len h0 & W = Segment ((h0 /. i),(h0 /. (i + 1)),P) holds
diameter W < e ) & ( for W being Subset of (Euclid 2) st W = Segment ((h0 /. (len h0)),(h0 /. 1),P) holds
diameter W < e ) & ( for i being Element of NAT st 1 <= i & i + 1 < len h0 holds
(Segment ((h0 /. i),(h0 /. (i + 1)),P)) /\ (Segment ((h0 /. (i + 1)),(h0 /. (i + 2)),P)) = {(h0 /. (i + 1))} ) & (Segment ((h0 /. (len h0)),(h0 /. 1),P)) /\ (Segment ((h0 /. 1),(h0 /. 2),P)) = {(h0 /. 1)} & (Segment ((h0 /. ((len h0) -' 1)),(h0 /. (len h0)),P)) /\ (Segment ((h0 /. (len h0)),(h0 /. 1),P)) = {(h0 /. (len h0))} & Segment ((h0 /. ((len h0) -' 1)),(h0 /. (len h0)),P) misses Segment ((h0 /. 1),(h0 /. 2),P) & ( for i, j being Element of NAT st 1 <= i & i < j & j < len h0 & not i,j are_adjacent1 holds
Segment ((h0 /. i),(h0 /. (i + 1)),P) misses Segment ((h0 /. j),(h0 /. (j + 1)),P) ) & ( for i being Element of NAT st 1 < i & i + 1 < len h0 holds
Segment ((h0 /. (len h0)),(h0 /. 1),P) misses Segment ((h0 /. i),(h0 /. (i + 1)),P) ) )
A132: dom g1 = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1;
thus for i being Element of NAT st 1 <= i & i < len h0 holds
LE h0 /. i,h0 /. (i + 1),P ::_thesis: ( ( for i being Element of NAT
for W being Subset of (Euclid 2) st 1 <= i & i < len h0 & W = Segment ((h0 /. i),(h0 /. (i + 1)),P) holds
diameter W < e ) & ( for W being Subset of (Euclid 2) st W = Segment ((h0 /. (len h0)),(h0 /. 1),P) holds
diameter W < e ) & ( for i being Element of NAT st 1 <= i & i + 1 < len h0 holds
(Segment ((h0 /. i),(h0 /. (i + 1)),P)) /\ (Segment ((h0 /. (i + 1)),(h0 /. (i + 2)),P)) = {(h0 /. (i + 1))} ) & (Segment ((h0 /. (len h0)),(h0 /. 1),P)) /\ (Segment ((h0 /. 1),(h0 /. 2),P)) = {(h0 /. 1)} & (Segment ((h0 /. ((len h0) -' 1)),(h0 /. (len h0)),P)) /\ (Segment ((h0 /. (len h0)),(h0 /. 1),P)) = {(h0 /. (len h0))} & Segment ((h0 /. ((len h0) -' 1)),(h0 /. (len h0)),P) misses Segment ((h0 /. 1),(h0 /. 2),P) & ( for i, j being Element of NAT st 1 <= i & i < j & j < len h0 & not i,j are_adjacent1 holds
Segment ((h0 /. i),(h0 /. (i + 1)),P) misses Segment ((h0 /. j),(h0 /. (j + 1)),P) ) & ( for i being Element of NAT st 1 < i & i + 1 < len h0 holds
Segment ((h0 /. (len h0)),(h0 /. 1),P) misses Segment ((h0 /. i),(h0 /. (i + 1)),P) ) )
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i < len h0 implies LE h0 /. i,h0 /. (i + 1),P )
assume that
A133: 1 <= i and
A134: i < len h0 ; ::_thesis: LE h0 /. i,h0 /. (i + 1),P
A135: i + 1 <= len h0 by A134, NAT_1:13;
A136: i < i + 1 by NAT_1:13;
A137: 1 < i + 1 by A133, NAT_1:13;
percases ( i < len h1 or i >= len h1 ) ;
supposeA138: i < len h1 ; ::_thesis: LE h0 /. i,h0 /. (i + 1),P
then A139: i + 1 <= len h1 by NAT_1:13;
then A140: i + 1 in dom h1 by A137, FINSEQ_3:25;
then A141: h1 . (i + 1) in rng h1 by FUNCT_1:def_3;
then A142: h1 . (i + 1) <= 1 by A11, BORSUK_1:40, XXREAL_1:1;
h0 . (i + 1) = h11 . (i + 1) by A39, A137, A139, FINSEQ_1:64;
then A143: h0 . (i + 1) = g1 . (h1 . (i + 1)) by A140, FUNCT_1:13;
then A144: h0 . (i + 1) in Upper_Arc P by A5, A132, A11, A141, BORSUK_1:40, FUNCT_1:def_3;
i in dom h0 by A133, A134, FINSEQ_3:25;
then A145: h0 /. i = h0 . i by PARTFUN1:def_6;
A146: i in dom h1 by A133, A138, FINSEQ_3:25;
then A147: h1 . i in rng h1 by FUNCT_1:def_3;
then A148: ( 0 <= h1 . i & h1 . i <= 1 ) by A11, BORSUK_1:40, XXREAL_1:1;
A149: g1 . (h1 . i) in rng g1 by A132, A11, A147, BORSUK_1:40, FUNCT_1:def_3;
h0 . i = h11 . i by A39, A133, A138, FINSEQ_1:64;
then A150: h0 . i = g1 . (h1 . i) by A146, FUNCT_1:13;
i + 1 in dom h0 by A135, A137, FINSEQ_3:25;
then A151: h0 /. (i + 1) = h0 . (i + 1) by PARTFUN1:def_6;
h1 . i < h1 . (i + 1) by A12, A136, A146, A140, SEQM_3:def_1;
then LE h0 /. i,h0 /. (i + 1), Upper_Arc P, W-min P, E-max P by A3, A4, A5, A6, A7, A150, A148, A143, A142, A145, A151, Th18;
hence LE h0 /. i,h0 /. (i + 1),P by A5, A150, A145, A151, A149, A144, JORDAN6:def_10; ::_thesis: verum
end;
supposeA152: i >= len h1 ; ::_thesis: LE h0 /. i,h0 /. (i + 1),P
percases ( i > len h1 or i = len h1 ) by A152, XXREAL_0:1;
supposeA153: i > len h1 ; ::_thesis: LE h0 /. i,h0 /. (i + 1),P
then (len h11) + 1 <= i by A39, NAT_1:13;
then A154: h0 . i = (mid (h21,2,((len h21) -' 1))) . (i - (len h11)) by A36, A134, FINSEQ_1:23;
A155: (i + 1) - (len h11) <= ((len h11) + (len (mid (h21,2,((len h21) -' 1))))) - (len h11) by A36, A135, XREAL_1:9;
i + 1 > len h11 by A39, A153, NAT_1:13;
then A156: (i + 1) - (len h11) = (i + 1) -' (len h11) by XREAL_1:233;
A157: (len h1) + 1 <= i by A153, NAT_1:13;
then A158: ((len h1) + 1) - (len h1) <= i - (len h1) by XREAL_1:9;
A159: i - (len h11) = i -' (len h11) by A39, A153, XREAL_1:233;
A160: (len h1) + 1 <= i + 1 by A157, NAT_1:13;
then A161: ((len h1) + 1) - (len h1) <= (i + 1) - (len h1) by XREAL_1:9;
then A162: 1 < ((i + 1) -' (len h11)) + (2 - 1) by A39, A156, NAT_1:13;
then A163: 0 < (((i + 1) -' (len h11)) + 2) - 1 ;
h0 . (i + 1) = (mid (h21,2,((len h21) -' 1))) . ((i + 1) - (len h11)) by A39, A36, A135, A160, FINSEQ_1:23;
then A164: h0 . (i + 1) = h21 . ((((i + 1) -' (len h11)) + 2) -' 1) by A39, A48, A56, A50, A54, A156, A155, A161, FINSEQ_6:118;
set j = ((i -' (len h11)) + 2) -' 1;
len h2 < (len h2) + 1 by NAT_1:13;
then A165: (len h2) - 1 < ((len h2) + 1) - 1 by XREAL_1:9;
A166: 0 + 2 <= (i -' (len h11)) + 2 by XREAL_1:6;
then A167: 1 <= ((i -' (len h11)) + 2) -' 1 by Lm1, NAT_D:42;
then A168: 1 < (((i -' (len h11)) + 2) -' 1) + 1 by NAT_1:13;
i - (len h11) <= ((len h1) + ((len h2) - 2)) - (len h11) by A39, A47, A36, A52, A55, A57, A134, XREAL_1:9;
then A169: (i -' (len h11)) + 2 <= ((len h2) - 2) + 2 by A39, A159, XREAL_1:6;
then ((i -' (len h11)) + 2) -' 1 <= len h21 by A47, NAT_D:44;
then A170: ((i -' (len h11)) + 2) -' 1 in dom h2 by A46, A167, FINSEQ_3:25;
i - (len h11) <= ((len h11) + (len (mid (h21,2,((len h21) -' 1))))) - (len h11) by A36, A134, XREAL_1:9;
then h0 . i = h21 . (((i -' (len h11)) + 2) -' 1) by A39, A48, A56, A50, A54, A154, A159, A158, FINSEQ_6:118;
then A171: h0 . i = g2 . (h2 . (((i -' (len h11)) + 2) -' 1)) by A170, FUNCT_1:13;
A172: h2 . (((i -' (len h11)) + 2) -' 1) in rng h2 by A170, FUNCT_1:def_3;
then A173: h0 . i in Lower_Arc P by A23, A131, A29, A171, BORSUK_1:40, FUNCT_1:def_3;
i + 1 in dom h0 by A135, A137, FINSEQ_3:25;
then A174: h0 /. (i + 1) = h0 . (i + 1) by PARTFUN1:def_6;
(((i -' (len h11)) + 2) -' 1) + 1 = ((((i -' (len h11)) + 1) + 1) - 1) + 1 by A166, Lm1, NAT_D:39, NAT_D:42
.= (i -' (len h11)) + 2 ;
then A175: (((i -' (len h11)) + 2) -' 1) + 1 in dom h2 by A169, A168, FINSEQ_3:25;
then A176: h2 . ((((i -' (len h11)) + 2) -' 1) + 1) in rng h2 by FUNCT_1:def_3;
then A177: h2 . ((((i -' (len h11)) + 2) -' 1) + 1) <= 1 by A29, BORSUK_1:40, XXREAL_1:1;
A178: (((i -' (len h11)) + 2) -' 1) + 1 = (((i - (len h11)) + 2) - 1) + 1 by A159, A166, Lm1, NAT_D:39, NAT_D:42
.= (((i + 1) -' (len h11)) + 2) -' 1 by A156, A163, XREAL_0:def_2 ;
then A179: h0 . (i + 1) = g2 . (h2 . ((((i -' (len h11)) + 2) -' 1) + 1)) by A164, A175, FUNCT_1:13;
then A180: h0 . (i + 1) in Lower_Arc P by A23, A131, A29, A176, BORSUK_1:40, FUNCT_1:def_3;
A181: (((i + 1) -' (len h11)) + 2) - 1 = (((i + 1) -' (len h11)) + 2) -' 1 by A162, XREAL_0:def_2;
(i + 1) - (len h11) <= ((len h1) + ((len h2) - 2)) - (len h11) by A39, A47, A36, A52, A55, A57, A135, XREAL_1:9;
then A182: ((i + 1) -' (len h11)) + 2 <= ((len h2) - 2) + 2 by A39, A156, XREAL_1:6;
then (((i + 1) -' (len h11)) + 2) - 1 <= (len h2) - 1 by XREAL_1:9;
then (((i + 1) -' (len h11)) + 2) -' 1 < len h2 by A181, A165, XXREAL_0:2;
then A183: (((i + 1) -' (len h11)) + 2) -' 1 in dom h2 by A178, A168, FINSEQ_3:25;
A184: now__::_thesis:_not_h0_/._(i_+_1)_=_W-min_P
assume h0 /. (i + 1) = W-min P ; ::_thesis: contradiction
then len h21 = (((i + 1) -' (len h11)) + 2) -' 1 by A46, A47, A34, A35, A32, A164, A174, A183, FUNCT_1:def_4;
then ((len h21) + 1) - (len h21) <= (len h21) - (len h21) by A47, A181, A182, XREAL_1:9;
then ((len h21) + 1) - (len h21) <= 0 ;
hence contradiction ; ::_thesis: verum
end;
((i -' (len h11)) + 2) -' 1 < (((i -' (len h11)) + 2) -' 1) + 1 by NAT_1:13;
then A185: h2 . (((i -' (len h11)) + 2) -' 1) < h2 . ((((i -' (len h11)) + 2) -' 1) + 1) by A30, A170, A175, SEQM_3:def_1;
i in dom h0 by A133, A134, FINSEQ_3:25;
then A186: h0 /. i = h0 . i by PARTFUN1:def_6;
( 0 <= h2 . (((i -' (len h11)) + 2) -' 1) & h2 . (((i -' (len h11)) + 2) -' 1) <= 1 ) by A29, A172, BORSUK_1:40, XXREAL_1:1;
then LE h0 /. i,h0 /. (i + 1), Lower_Arc P, E-max P, W-min P by A21, A22, A23, A24, A25, A171, A179, A177, A185, A186, A174, Th18;
hence LE h0 /. i,h0 /. (i + 1),P by A186, A174, A173, A180, A184, JORDAN6:def_10; ::_thesis: verum
end;
supposeA187: i = len h1 ; ::_thesis: LE h0 /. i,h0 /. (i + 1),P
then ( h0 . i = h11 . i & i in dom h1 ) by A39, A133, FINSEQ_1:64, FINSEQ_3:25;
then A188: h0 . i = E-max P by A7, A9, A187, FUNCT_1:13;
i in dom h0 by A133, A134, FINSEQ_3:25;
then h0 /. i = E-max P by A188, PARTFUN1:def_6;
then A189: h0 /. i in Upper_Arc P by A1, Th1;
i + 1 in dom h0 by A135, A137, FINSEQ_3:25;
then A190: h0 /. (i + 1) = h0 . (i + 1) by PARTFUN1:def_6;
set j = ((i -' (len h11)) + 2) -' 1;
(len h11) -' (len h11) = (len h11) - (len h11) by XREAL_1:233
.= 0 ;
then A191: ((i -' (len h11)) + 2) -' 1 = 2 - 1 by A39, A187, XREAL_1:233;
then (((i -' (len h11)) + 2) -' 1) + 1 <= len h2 by A28, XXREAL_0:2;
then A192: (((i -' (len h11)) + 2) -' 1) + 1 in dom h2 by A191, FINSEQ_3:25;
then A193: h2 . ((((i -' (len h11)) + 2) -' 1) + 1) in rng h2 by FUNCT_1:def_3;
2 <= len h21 by A28, A47, XXREAL_0:2;
then A194: 2 in dom h21 by FINSEQ_3:25;
A195: (i + 1) - (len h11) <= ((len h11) + (len (mid (h21,2,((len h21) -' 1))))) - (len h11) by A36, A135, XREAL_1:9;
A196: (((i + 1) -' (len h11)) + 2) -' 1 = (1 + 2) -' 1 by A39, A187, NAT_D:34
.= 2 by NAT_D:34 ;
( h0 . (i + 1) = (mid (h21,2,((len h21) -' 1))) . ((i + 1) - (len h11)) & (i + 1) - (len h11) = (i + 1) -' (len h11) ) by A39, A36, A135, A136, A187, FINSEQ_1:23, XREAL_1:233;
then A197: h0 . (i + 1) = h21 . ((((i + 1) -' (len h11)) + 2) -' 1) by A39, A48, A56, A50, A54, A187, A195, FINSEQ_6:118;
then h0 . (i + 1) = g2 . (h2 . ((((i -' (len h11)) + 2) -' 1) + 1)) by A191, A196, A192, FUNCT_1:13;
then A198: h0 . (i + 1) in Lower_Arc P by A23, A131, A29, A193, BORSUK_1:40, FUNCT_1:def_3;
len h21 <> (((i + 1) -' (len h11)) + 2) -' 1 by A28, A47, A196;
then h0 /. (i + 1) <> W-min P by A46, A47, A34, A35, A32, A197, A196, A190, A194, FUNCT_1:def_4;
hence LE h0 /. i,h0 /. (i + 1),P by A189, A190, A198, JORDAN6:def_10; ::_thesis: verum
end;
end;
end;
end;
end;
A199: (len h0) -' 1 < len h0 by A66, JORDAN5B:1;
then A200: ((len h0) -' 1) + 1 <= len h0 by NAT_1:13;
A201: 1 + 1 <= len h0 by A65, A14, XXREAL_0:2;
then A202: 1 <= (len h0) -' 1 by Lm3;
then A203: (len h0) -' 1 in dom h0 by A199, FINSEQ_3:25;
then A204: h0 /. ((len h0) -' 1) = h0 . ((len h0) -' 1) by PARTFUN1:def_6;
A205: 1 + 1 <= len h0 by A66, NAT_1:13;
then 1 + 1 in dom h0 by FINSEQ_3:25;
then A206: h0 /. (1 + 1) = h0 . (1 + 1) by PARTFUN1:def_6;
A207: now__::_thesis:_not_h0_/._(1_+_1)_=_h0_/._((len_h0)_-'_1)
A208: 1 + 1 in dom h1 by A14, FINSEQ_3:25;
then A209: h1 . (1 + 1) in rng h1 by FUNCT_1:def_3;
A210: h0 . (1 + 1) = h11 . (1 + 1) by A39, A14, FINSEQ_1:64;
then h0 . (1 + 1) = g1 . (h1 . (1 + 1)) by A208, FUNCT_1:13;
then A211: h0 . (1 + 1) in Upper_Arc P by A5, A132, A11, A209, BORSUK_1:40, FUNCT_1:def_3;
assume A212: h0 /. (1 + 1) = h0 /. ((len h0) -' 1) ; ::_thesis: contradiction
percases ( (len h0) -' 1 <= len h1 or (len h0) -' 1 > len h1 ) ;
suppose (len h0) -' 1 <= len h1 ; ::_thesis: contradiction
then ( h0 . ((len h0) -' 1) = h11 . ((len h0) -' 1) & (len h0) -' 1 in dom h1 ) by A39, A202, FINSEQ_1:64, FINSEQ_3:25;
hence contradiction by A38, A20, A60, A204, A206, A212, A208, A210, FUNCT_1:def_4; ::_thesis: verum
end;
supposeA213: (len h0) -' 1 > len h1 ; ::_thesis: contradiction
(len h0) -' 1 in dom h0 by A202, A199, FINSEQ_3:25;
then A214: h0 /. ((len h0) -' 1) = h0 . ((len h0) -' 1) by PARTFUN1:def_6;
A215: ((len h0) -' 1) - (len h11) = ((len h0) -' 1) -' (len h11) by A39, A213, XREAL_1:233;
((len h0) -' 1) - (len h11) <= ((len h1) + ((len h2) - 2)) - (len h11) by A39, A47, A36, A52, A55, A57, A199, XREAL_1:9;
then (((len h0) -' 1) -' (len h11)) + 2 <= ((len h2) - 2) + 2 by A39, A215, XREAL_1:6;
then A216: ((((len h0) -' 1) -' (len h11)) + 2) -' 1 <= len h21 by A47, NAT_D:44;
((len h0) -' 1) - (len h11) <= ((len h1) + ((len h2) - 2)) - (len h11) by A39, A47, A36, A52, A55, A57, A199, XREAL_1:9;
then A217: (((len h0) -' 1) -' (len h11)) + 2 <= ((len h2) - 2) + 2 by A39, A215, XREAL_1:6;
set k = ((((len h0) -' 1) -' (len h11)) + 2) -' 1;
A218: ((len h0) -' 1) - (len h11) <= ((len h11) + (len (mid (h21,2,((len h21) -' 1))))) - (len h11) by A36, A199, XREAL_1:9;
A219: 0 + 2 <= (((len h0) -' 1) -' (len h11)) + 2 by XREAL_1:6;
then A220: ((((len h0) -' 1) -' (len h11)) + 2) -' 1 = ((((len h0) -' 1) -' (len h11)) + 2) - 1 by Lm1, NAT_D:39, NAT_D:42;
1 <= ((((len h0) -' 1) -' (len h11)) + 2) -' 1 by A219, Lm1, NAT_D:42;
then A221: ((((len h0) -' 1) -' (len h11)) + 2) -' 1 in dom h2 by A46, A216, FINSEQ_3:25;
then h2 . (((((len h0) -' 1) -' (len h11)) + 2) -' 1) in rng h2 by FUNCT_1:def_3;
then A222: g2 . (h2 . (((((len h0) -' 1) -' (len h11)) + 2) -' 1)) in rng g2 by A131, A29, BORSUK_1:40, FUNCT_1:def_3;
A223: (len h1) + 1 <= (len h0) -' 1 by A213, NAT_1:13;
then ( h0 . ((len h0) -' 1) = (mid (h21,2,((len h21) -' 1))) . (((len h0) -' 1) - (len h11)) & ((len h1) + 1) - (len h1) <= ((len h0) -' 1) - (len h1) ) by A39, A36, A199, FINSEQ_1:23, XREAL_1:9;
then A224: h0 . ((len h0) -' 1) = h21 . (((((len h0) -' 1) -' (len h11)) + 2) -' 1) by A39, A48, A56, A50, A54, A215, A218, FINSEQ_6:118;
then h0 . ((len h0) -' 1) = g2 . (h2 . (((((len h0) -' 1) -' (len h11)) + 2) -' 1)) by A221, FUNCT_1:13;
then h0 . ((len h0) -' 1) in (Upper_Arc P) /\ (Lower_Arc P) by A23, A206, A212, A211, A214, A222, XBOOLE_0:def_4;
then h0 . ((len h0) -' 1) in {(W-min P),(E-max P)} by A1, JORDAN6:def_9;
then A225: ( h0 . ((len h0) -' 1) = W-min P or h0 . ((len h0) -' 1) = E-max P ) by TARSKI:def_2;
((len h11) + 1) - (len h11) <= ((len h0) -' 1) - (len h11) by A39, A223, XREAL_1:9;
then 1 <= ((len h0) -' 1) -' (len h11) by NAT_D:39;
then 1 + 2 <= (((len h0) -' 1) -' (len h11)) + 2 by XREAL_1:6;
then (1 + 2) - 1 <= ((((len h0) -' 1) -' (len h11)) + 2) - 1 by XREAL_1:9;
then A226: 1 < ((((len h0) -' 1) -' (len h11)) + 2) -' 1 by A220, XXREAL_0:2;
((((len h0) -' 1) -' (len h11)) + 2) -' 1 < (((((len h0) -' 1) -' (len h11)) + 2) - 1) + 1 by A220, NAT_1:13;
hence contradiction by A46, A76, A34, A77, A35, A32, A224, A217, A226, A221, A225, FUNCT_1:def_4; ::_thesis: verum
end;
end;
end;
A227: 1 in dom g2 by Lm6, BORSUK_1:40, FUNCT_2:def_1;
A228: ((len h2) - 1) - 1 < len h2 by Lm4;
A229: now__::_thesis:_(_(_(len_h0)_-'_1_<=_len_h1_&_LE_h0_/._(1_+_1),h0_/._((len_h0)_-'_1),P_)_or_(_(len_h0)_-'_1_>_len_h1_&_LE_h0_/._(1_+_1),h0_/._((len_h0)_-'_1),P_)_)
percases ( (len h0) -' 1 <= len h1 or (len h0) -' 1 > len h1 ) ;
caseA230: (len h0) -' 1 <= len h1 ; ::_thesis: LE h0 /. (1 + 1),h0 /. ((len h0) -' 1),P
A231: h0 /. (1 + 1) in Upper_Arc P by A5, A132, A11, A206, A42, A19, BORSUK_1:40, FUNCT_1:def_3;
A232: ( 0 <= h1 . (1 + 1) & h1 . (1 + 1) <= 1 ) by A11, A19, BORSUK_1:40, XXREAL_1:1;
A233: (len h0) -' 1 in dom h1 by A61, A230, FINSEQ_3:25;
then A234: h1 . ((len h0) -' 1) in rng h1 by FUNCT_1:def_3;
then A235: h1 . ((len h0) -' 1) <= 1 by A11, BORSUK_1:40, XXREAL_1:1;
h0 . ((len h0) -' 1) = h11 . ((len h0) -' 1) by A39, A61, A230, FINSEQ_1:64;
then A236: h0 . ((len h0) -' 1) = g1 . (h1 . ((len h0) -' 1)) by A233, FUNCT_1:13;
then A237: h0 /. ((len h0) -' 1) in Upper_Arc P by A5, A132, A11, A204, A234, BORSUK_1:40, FUNCT_1:def_3;
h1 . (1 + 1) < h1 . ((len h0) -' 1) by A12, A60, A18, A233, SEQM_3:def_1;
then LE h0 /. (1 + 1),h0 /. ((len h0) -' 1), Upper_Arc P, W-min P, E-max P by A3, A4, A5, A6, A7, A204, A206, A42, A236, A235, A232, Th18;
hence LE h0 /. (1 + 1),h0 /. ((len h0) -' 1),P by A231, A237, JORDAN6:def_10; ::_thesis: verum
end;
caseA238: (len h0) -' 1 > len h1 ; ::_thesis: LE h0 /. (1 + 1),h0 /. ((len h0) -' 1),P
1 + 1 in dom h1 by A14, FINSEQ_3:25;
then A239: h11 . (1 + 1) = g1 . (h1 . (1 + 1)) by FUNCT_1:13;
A240: ((len h0) -' 1) - (len h11) = ((len h0) -' 1) -' (len h11) by A39, A238, XREAL_1:233;
(((len h0) -' 1) + 1) - 1 <= ((len h1) + ((len h2) - 2)) - 1 by A39, A47, A36, A52, A55, A57, A200, XREAL_1:9;
then ((len h0) -' 1) - (len h11) <= ((len h1) + (((len h2) - 2) - 1)) - (len h11) by XREAL_1:9;
then (((len h0) -' 1) -' (len h11)) + 2 <= (((len h2) - 2) - 1) + 2 by A39, A240, XREAL_1:6;
then A241: ((((len h0) -' 1) -' (len h11)) + 2) - 1 <= ((len h2) - 1) - 1 by XREAL_1:9;
A242: (len h1) + 1 <= (len h0) -' 1 by A238, NAT_1:13;
then A243: ((len h1) + 1) - (len h1) <= ((len h0) -' 1) - (len h1) by XREAL_1:9;
h1 . (1 + 1) in rng h1 by A15, FUNCT_1:def_3;
then A244: g1 . (h1 . (1 + 1)) in rng g1 by A132, A11, BORSUK_1:40, FUNCT_1:def_3;
0 + 2 <= (((len h0) -' 1) -' (len h11)) + 2 by XREAL_1:6;
then A245: 2 -' 1 <= ((((len h0) -' 1) -' (len h11)) + 2) -' 1 by NAT_D:42;
set k = ((((len h0) -' 1) -' (len h11)) + 2) -' 1;
0 + 1 <= (((((len h0) -' 1) -' (len h11)) + 1) + 1) - 1 by XREAL_1:6;
then A246: ((((len h0) -' 1) -' (len h11)) + 2) -' 1 = ((((len h0) -' 1) -' (len h11)) + 2) - 1 by NAT_D:39;
A247: ((len h0) -' 1) - (len h11) <= ((len h11) + (len (mid (h21,2,((len h21) -' 1))))) - (len h11) by A36, A199, XREAL_1:9;
((len h0) -' 1) - (len h11) <= ((len h1) + ((len h2) - 2)) - (len h11) by A39, A47, A36, A52, A55, A57, A199, XREAL_1:9;
then (((len h0) -' 1) -' (len h11)) + 2 <= ((len h2) - 2) + 2 by A39, A240, XREAL_1:6;
then ((((len h0) -' 1) -' (len h11)) + 2) -' 1 <= len h21 by A47, NAT_D:44;
then A248: ((((len h0) -' 1) -' (len h11)) + 2) -' 1 in dom h21 by A245, Lm1, FINSEQ_3:25;
A249: h0 . ((len h0) -' 1) = (mid (h21,2,((len h21) -' 1))) . (((len h0) -' 1) - (len h11)) by A39, A36, A199, A242, FINSEQ_1:23;
then h0 . ((len h0) -' 1) = h21 . (((((len h0) -' 1) -' (len h11)) + 2) -' 1) by A39, A48, A56, A50, A54, A240, A247, A243, FINSEQ_6:118;
then A250: h0 . ((len h0) -' 1) = g2 . (h2 . (((((len h0) -' 1) -' (len h11)) + 2) -' 1)) by A46, A248, FUNCT_1:13;
h2 . (((((len h0) -' 1) -' (len h11)) + 2) -' 1) in rng h2 by A46, A248, FUNCT_1:def_3;
then A251: h0 . ((len h0) -' 1) in Lower_Arc P by A23, A131, A29, A250, BORSUK_1:40, FUNCT_1:def_3;
1 <= ((len h0) -' 1) - (len h11) by A38, A243, FINSEQ_3:29;
then h0 . ((len h0) -' 1) = h21 . (((((len h0) -' 1) -' (len h11)) + 2) -' 1) by A48, A56, A50, A54, A240, A247, A249, FINSEQ_6:118;
then h0 /. ((len h0) -' 1) <> W-min P by A228, A46, A34, A35, A32, A204, A246, A248, A241, FUNCT_1:def_4;
hence LE h0 /. (1 + 1),h0 /. ((len h0) -' 1),P by A5, A204, A206, A41, A239, A244, A251, JORDAN6:def_10; ::_thesis: verum
end;
end;
end;
A252: (len h0) - (len h11) = (len h0) -' (len h11) by A39, A65, XREAL_1:233;
then (((len h0) -' (len h11)) + 2) -' 1 <= len h21 by A36, A52, A55, A57, NAT_D:44;
then (((len h0) -' (len h11)) + 2) -' 1 in dom h21 by A43, Lm1, FINSEQ_3:25;
then A253: ( h21 . ((((len h0) -' (len h11)) + 2) -' 1) = g2 . (h2 . ((((len h0) -' (len h11)) + 2) -' 1)) & h2 . ((((len h0) -' (len h11)) + 2) -' 1) in rng h2 ) by A46, FUNCT_1:13, FUNCT_1:def_3;
h1 . (len h1) in dom g1 by A9, A69, XXREAL_1:1;
then A254: len h1 in dom (g1 * h1) by A45, FUNCT_1:11;
then A255: h11 . (len h1) = E-max P by A7, A9, FUNCT_1:12;
A256: for i being Element of NAT st 1 <= i & i + 1 <= len h0 holds
( LE h0 /. i,h0 /. (i + 1),P & h0 /. (i + 1) <> W-min P & h0 /. i <> h0 /. (i + 1) )
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i + 1 <= len h0 implies ( LE h0 /. i,h0 /. (i + 1),P & h0 /. (i + 1) <> W-min P & h0 /. i <> h0 /. (i + 1) ) )
assume that
A257: 1 <= i and
A258: i + 1 <= len h0 ; ::_thesis: ( LE h0 /. i,h0 /. (i + 1),P & h0 /. (i + 1) <> W-min P & h0 /. i <> h0 /. (i + 1) )
A259: i < i + 1 by NAT_1:13;
A260: 1 < i + 1 by A257, NAT_1:13;
then i + 1 in dom h0 by A258, FINSEQ_3:25;
then A261: h0 /. (i + 1) = h0 . (i + 1) by PARTFUN1:def_6;
A262: i < len h0 by A258, NAT_1:13;
then i in dom h0 by A257, FINSEQ_3:25;
then A263: h0 /. i = h0 . i by PARTFUN1:def_6;
percases ( i < len h1 or i >= len h1 ) ;
supposeA264: i < len h1 ; ::_thesis: ( LE h0 /. i,h0 /. (i + 1),P & h0 /. (i + 1) <> W-min P & h0 /. i <> h0 /. (i + 1) )
then A265: i + 1 <= len h1 by NAT_1:13;
then A266: i + 1 in dom h1 by A260, FINSEQ_3:25;
then A267: h1 . (i + 1) in rng h1 by FUNCT_1:def_3;
then A268: h1 . (i + 1) <= 1 by A11, BORSUK_1:40, XXREAL_1:1;
A269: ( i + 1 <> 1 & i + 1 <> i ) by A257, NAT_1:13;
A270: i in dom h1 by A257, A264, FINSEQ_3:25;
then A271: h1 . i in rng h1 by FUNCT_1:def_3;
then A272: ( 0 <= h1 . i & h1 . i <= 1 ) by A11, BORSUK_1:40, XXREAL_1:1;
A273: h0 . (i + 1) = h11 . (i + 1) by A39, A260, A265, FINSEQ_1:64;
then A274: h0 . (i + 1) = g1 . (h1 . (i + 1)) by A266, FUNCT_1:13;
then A275: h0 . (i + 1) in Upper_Arc P by A5, A132, A11, A267, BORSUK_1:40, FUNCT_1:def_3;
A276: h0 . i = h11 . i by A39, A257, A264, FINSEQ_1:64;
then A277: g1 . (h1 . i) = h0 /. i by A263, A270, FUNCT_1:13;
g1 . (h1 . i) in rng g1 by A132, A11, A271, BORSUK_1:40, FUNCT_1:def_3;
then A278: h0 . i in Upper_Arc P by A5, A276, A270, FUNCT_1:13;
h1 . i < h1 . (i + 1) by A12, A259, A270, A266, SEQM_3:def_1;
then LE h0 /. i,h0 /. (i + 1), Upper_Arc P, W-min P, E-max P by A3, A4, A5, A6, A7, A261, A274, A277, A272, A268, Th18;
hence ( LE h0 /. i,h0 /. (i + 1),P & h0 /. (i + 1) <> W-min P & h0 /. i <> h0 /. (i + 1) ) by A38, A17, A71, A20, A263, A261, A276, A270, A273, A266, A278, A275, A269, FUNCT_1:def_4, JORDAN6:def_10; ::_thesis: verum
end;
supposeA279: i >= len h1 ; ::_thesis: ( LE h0 /. i,h0 /. (i + 1),P & h0 /. (i + 1) <> W-min P & h0 /. i <> h0 /. (i + 1) )
percases ( i > len h1 or i = len h1 ) by A279, XXREAL_0:1;
supposeA280: i > len h1 ; ::_thesis: ( LE h0 /. i,h0 /. (i + 1),P & h0 /. (i + 1) <> W-min P & h0 /. i <> h0 /. (i + 1) )
i - (len h11) < ((len h11) + ((len h2) - 2)) - (len h11) by A47, A36, A52, A55, A57, A262, XREAL_1:9;
then A281: (i - (len h11)) + 2 < ((len h2) - 2) + 2 by XREAL_1:6;
i + 1 > len h11 by A39, A280, NAT_1:13;
then A282: (i + 1) - (len h11) = (i + 1) -' (len h11) by XREAL_1:233;
set j = ((i -' (len h11)) + 2) -' 1;
A283: (i + 1) - (len h11) <= ((len h11) + (len (mid (h21,2,((len h21) -' 1))))) - (len h11) by A36, A258, XREAL_1:9;
A284: 0 + 2 <= (i -' (len h11)) + 2 by XREAL_1:6;
then A285: (((i -' (len h11)) + 2) -' 1) + 1 = (((i -' (len h11)) + (1 + 1)) - 1) + 1 by Lm1, NAT_D:39, NAT_D:42
.= (i -' (len h11)) + (1 + 1) ;
A286: (len h1) + 1 <= i by A280, NAT_1:13;
then A287: ((len h1) + 1) - (len h1) <= i - (len h1) by XREAL_1:9;
i + 1 in dom h0 by A258, A260, FINSEQ_3:25;
then A288: h0 /. (i + 1) = h0 . (i + 1) by PARTFUN1:def_6;
A289: (len h1) + 1 <= i + 1 by A286, NAT_1:13;
then A290: ((len h1) + 1) - (len h1) <= (i + 1) - (len h1) by XREAL_1:9;
then 1 < ((i + 1) -' (len h11)) + (2 - 1) by A39, A282, NAT_1:13;
then A291: 0 < (((i + 1) -' (len h11)) + 2) - 1 ;
h0 . (i + 1) = (mid (h21,2,((len h21) -' 1))) . ((i + 1) - (len h11)) by A39, A36, A258, A289, FINSEQ_1:23;
then A292: h0 . (i + 1) = h21 . ((((i + 1) -' (len h11)) + 2) -' 1) by A39, A48, A56, A50, A54, A282, A283, A290, FINSEQ_6:118;
A293: i <= (len h11) + (len (mid (h21,2,((len h21) -' 1)))) by A262, FINSEQ_1:22;
(len h11) + 1 <= i by A39, A280, NAT_1:13;
then A294: h0 . i = (mid (h21,2,((len h21) -' 1))) . (i - (len h11)) by A293, FINSEQ_1:23;
A295: i - (len h11) = i -' (len h11) by A39, A280, XREAL_1:233;
A296: 1 <= ((i -' (len h11)) + 2) -' 1 by A284, Lm1, NAT_D:42;
then 1 < (((i -' (len h11)) + 2) -' 1) + 1 by NAT_1:13;
then A297: (((i -' (len h11)) + 2) -' 1) + 1 in dom h2 by A295, A281, A285, FINSEQ_3:25;
then A298: h2 . ((((i -' (len h11)) + 2) -' 1) + 1) in rng h2 by FUNCT_1:def_3;
then A299: h2 . ((((i -' (len h11)) + 2) -' 1) + 1) <= 1 by A29, BORSUK_1:40, XXREAL_1:1;
i - (len h11) <= ((len h1) + ((len h2) - 2)) - (len h11) by A39, A47, A36, A52, A55, A57, A262, XREAL_1:9;
then (i -' (len h11)) + 2 <= ((len h2) - 2) + 2 by A39, A295, XREAL_1:6;
then ((i -' (len h11)) + 2) -' 1 <= len h21 by A47, NAT_D:44;
then A300: ((i -' (len h11)) + 2) -' 1 in dom h2 by A46, A296, FINSEQ_3:25;
((i -' (len h11)) + 2) -' 1 < (((i -' (len h11)) + 2) -' 1) + 1 by NAT_1:13;
then A301: h2 . (((i -' (len h11)) + 2) -' 1) < h2 . ((((i -' (len h11)) + 2) -' 1) + 1) by A30, A300, A297, SEQM_3:def_1;
i in dom h0 by A257, A262, FINSEQ_3:25;
then A302: h0 /. i = h0 . i by PARTFUN1:def_6;
i - (len h11) <= ((len h11) + (len (mid (h21,2,((len h21) -' 1))))) - (len h11) by A293, XREAL_1:9;
then A303: h0 . i = h21 . (((i -' (len h11)) + 2) -' 1) by A39, A48, A56, A50, A54, A294, A295, A287, FINSEQ_6:118;
then A304: h0 . i = g2 . (h2 . (((i -' (len h11)) + 2) -' 1)) by A300, FUNCT_1:13;
A305: (((i -' (len h11)) + 2) -' 1) + 1 = (((i - (len h11)) + 2) - 1) + 1 by A295, A284, Lm1, NAT_D:39, NAT_D:42
.= (((i + 1) -' (len h11)) + 2) -' 1 by A282, A291, XREAL_0:def_2 ;
then A306: h0 /. (i + 1) <> W-min P by A46, A34, A35, A32, A295, A292, A281, A285, A297, A288, FUNCT_1:def_4;
A307: h0 . (i + 1) = g2 . (h2 . ((((i -' (len h11)) + 2) -' 1) + 1)) by A292, A305, A297, FUNCT_1:13;
then A308: h0 /. (i + 1) in Lower_Arc P by A23, A131, A29, A298, A288, BORSUK_1:40, FUNCT_1:def_3;
A309: h2 . (((i -' (len h11)) + 2) -' 1) in rng h2 by A300, FUNCT_1:def_3;
then A310: ( ((i -' (len h11)) + 2) -' 1 < (((i -' (len h11)) + 2) -' 1) + 1 & h0 /. i in Lower_Arc P ) by A23, A131, A29, A304, A302, BORSUK_1:40, FUNCT_1:def_3, NAT_1:13;
( 0 <= h2 . (((i -' (len h11)) + 2) -' 1) & h2 . (((i -' (len h11)) + 2) -' 1) <= 1 ) by A29, A309, BORSUK_1:40, XXREAL_1:1;
then LE h0 /. i,h0 /. (i + 1), Lower_Arc P, E-max P, W-min P by A21, A22, A23, A24, A25, A304, A307, A301, A302, A288, A299, Th18;
hence ( LE h0 /. i,h0 /. (i + 1),P & h0 /. (i + 1) <> W-min P & h0 /. i <> h0 /. (i + 1) ) by A46, A32, A303, A292, A305, A300, A297, A302, A288, A306, A310, A308, FUNCT_1:def_4, JORDAN6:def_10; ::_thesis: verum
end;
supposeA311: i = len h1 ; ::_thesis: ( LE h0 /. i,h0 /. (i + 1),P & h0 /. (i + 1) <> W-min P & h0 /. i <> h0 /. (i + 1) )
then A312: i - (len h11) = i -' (len h11) by A39, XREAL_1:233;
i - (len h11) <= ((len h1) + ((len h2) - 2)) - (len h11) by A39, A47, A36, A52, A55, A57, A262, XREAL_1:9;
then A313: (i -' (len h11)) + 2 <= ((len h2) - 2) + 2 by A39, A312, XREAL_1:6;
then A314: ((i -' (len h11)) + 2) -' 1 <= len h21 by A47, NAT_D:44;
set j = ((i -' (len h11)) + 2) -' 1;
A315: (((i -' (len h11)) + 2) -' 1) + 1 <> ((i -' (len h11)) + 2) -' 1 ;
A316: 0 + 2 <= (i -' (len h11)) + 2 by XREAL_1:6;
then A317: (((i -' (len h11)) + 2) -' 1) + 1 = (((i -' (len h11)) + (1 + 1)) - 1) + 1 by Lm1, NAT_D:39, NAT_D:42
.= (i -' (len h11)) + 2 ;
2 -' 1 <= ((i -' (len h11)) + 2) -' 1 by A316, NAT_D:42;
then 1 < (((i -' (len h11)) + 2) -' 1) + 1 by Lm1, NAT_1:13;
then A318: (((i -' (len h11)) + 2) -' 1) + 1 in dom h2 by A313, A317, FINSEQ_3:25;
then A319: h2 . ((((i -' (len h11)) + 2) -' 1) + 1) in rng h2 by FUNCT_1:def_3;
i + 1 <= (len h11) + (len (mid (h21,2,((len h21) -' 1)))) by A258, FINSEQ_1:22;
then A320: (i + 1) - (len h11) <= ((len h11) + (len (mid (h21,2,((len h21) -' 1))))) - (len h11) by XREAL_1:9;
A321: (i + 1) - (len h11) = (i + 1) -' (len h11) by A39, A259, A311, XREAL_1:233;
then A322: 0 < (((i + 1) -' (len h11)) + 2) - 1 by A39, A311;
h0 . (i + 1) = (mid (h21,2,((len h21) -' 1))) . ((i + 1) - (len h11)) by A39, A36, A258, A311, FINSEQ_1:23;
then A323: h0 . (i + 1) = h21 . ((((i + 1) -' (len h11)) + 2) -' 1) by A39, A48, A56, A50, A54, A311, A321, A320, FINSEQ_6:118;
A324: h0 . i = E-max P by A39, A255, A257, A311, FINSEQ_1:64;
then A325: h0 . i in Upper_Arc P by A1, Th1;
(len h1) -' (len h11) = (len h11) - (len h11) by A39, XREAL_0:def_2;
then (0 + 2) - 1 = (((len h1) -' (len h11)) + 2) - 1 ;
then A326: h0 . i = g2 . (h2 . (((i -' (len h11)) + 2) -' 1)) by A24, A26, A311, A324, NAT_D:39;
1 <= ((i -' (len h11)) + 2) -' 1 by A316, Lm1, NAT_D:42;
then A327: ((i -' (len h11)) + 2) -' 1 in dom h2 by A46, A314, FINSEQ_3:25;
then A328: h21 . (((i -' (len h11)) + 2) -' 1) = g2 . (h2 . (((i -' (len h11)) + 2) -' 1)) by FUNCT_1:13;
i in dom h0 by A257, A262, FINSEQ_3:25;
then A329: h0 /. i = h0 . i by PARTFUN1:def_6;
i + 1 in dom h0 by A258, A260, FINSEQ_3:25;
then A330: h0 /. (i + 1) = h0 . (i + 1) by PARTFUN1:def_6;
A331: (((i -' (len h11)) + 2) -' 1) + 1 = (((i - (len h11)) + 2) - 1) + 1 by A39, A311, Lm1, XREAL_0:def_2
.= (((i + 1) -' (len h11)) + 2) -' 1 by A321, A322, XREAL_0:def_2 ;
then h0 . (i + 1) = g2 . (h2 . ((((i -' (len h11)) + 2) -' 1) + 1)) by A323, A318, FUNCT_1:13;
then A332: h0 . (i + 1) in Lower_Arc P by A23, A131, A29, A319, BORSUK_1:40, FUNCT_1:def_3;
i - (len h11) < ((len h11) + ((len h2) - 2)) - (len h11) by A47, A36, A52, A55, A57, A262, XREAL_1:9;
then (i - (len h11)) + 2 < ((len h2) - 2) + 2 by XREAL_1:6;
then (((i -' (len h11)) + 2) -' 1) + 1 < len h2 by A39, A311, A317, XREAL_0:def_2;
then h0 /. (i + 1) <> W-min P by A46, A34, A35, A32, A323, A331, A318, A330, FUNCT_1:def_4;
hence ( LE h0 /. i,h0 /. (i + 1),P & h0 /. (i + 1) <> W-min P & h0 /. i <> h0 /. (i + 1) ) by A46, A32, A323, A331, A327, A328, A326, A318, A329, A330, A332, A325, A315, FUNCT_1:def_4, JORDAN6:def_10; ::_thesis: verum
end;
end;
end;
end;
end;
then A333: ( LE h0 /. 1,h0 /. (1 + 1),P & h0 /. 1 <> h0 /. (1 + 1) ) by A205;
A334: E-max P in Upper_Arc P by A1, Th1;
thus for i being Element of NAT
for W being Subset of (Euclid 2) st 1 <= i & i < len h0 & W = Segment ((h0 /. i),(h0 /. (i + 1)),P) holds
diameter W < e ::_thesis: ( ( for W being Subset of (Euclid 2) st W = Segment ((h0 /. (len h0)),(h0 /. 1),P) holds
diameter W < e ) & ( for i being Element of NAT st 1 <= i & i + 1 < len h0 holds
(Segment ((h0 /. i),(h0 /. (i + 1)),P)) /\ (Segment ((h0 /. (i + 1)),(h0 /. (i + 2)),P)) = {(h0 /. (i + 1))} ) & (Segment ((h0 /. (len h0)),(h0 /. 1),P)) /\ (Segment ((h0 /. 1),(h0 /. 2),P)) = {(h0 /. 1)} & (Segment ((h0 /. ((len h0) -' 1)),(h0 /. (len h0)),P)) /\ (Segment ((h0 /. (len h0)),(h0 /. 1),P)) = {(h0 /. (len h0))} & Segment ((h0 /. ((len h0) -' 1)),(h0 /. (len h0)),P) misses Segment ((h0 /. 1),(h0 /. 2),P) & ( for i, j being Element of NAT st 1 <= i & i < j & j < len h0 & not i,j are_adjacent1 holds
Segment ((h0 /. i),(h0 /. (i + 1)),P) misses Segment ((h0 /. j),(h0 /. (j + 1)),P) ) & ( for i being Element of NAT st 1 < i & i + 1 < len h0 holds
Segment ((h0 /. (len h0)),(h0 /. 1),P) misses Segment ((h0 /. i),(h0 /. (i + 1)),P) ) )
proof
let i be Element of NAT ; ::_thesis: for W being Subset of (Euclid 2) st 1 <= i & i < len h0 & W = Segment ((h0 /. i),(h0 /. (i + 1)),P) holds
diameter W < e
let W be Subset of (Euclid 2); ::_thesis: ( 1 <= i & i < len h0 & W = Segment ((h0 /. i),(h0 /. (i + 1)),P) implies diameter W < e )
assume that
A335: 1 <= i and
A336: i < len h0 and
A337: W = Segment ((h0 /. i),(h0 /. (i + 1)),P) ; ::_thesis: diameter W < e
A338: i + 1 <= len h0 by A336, NAT_1:13;
A339: i < i + 1 by NAT_1:13;
A340: 1 < i + 1 by A335, NAT_1:13;
percases ( i < len h1 or i > len h1 or i = len h1 ) by XXREAL_0:1;
supposeA341: i < len h1 ; ::_thesis: diameter W < e
then A342: i in dom h1 by A335, FINSEQ_3:25;
then A343: h1 . i in rng h1 by FUNCT_1:def_3;
then A344: h1 . i <= 1 by A11, BORSUK_1:40, XXREAL_1:1;
A345: 0 <= h1 . i by A11, A343, BORSUK_1:40, XXREAL_1:1;
A346: h1 /. i = h1 . i by A335, A341, FINSEQ_4:15;
A347: h11 . i = g1 . (h1 . i) by A342, FUNCT_1:13;
then A348: h0 . i = g1 . (h1 . i) by A39, A335, A341, FINSEQ_1:64;
then A349: h0 . i in Upper_Arc P by A5, A132, A11, A343, BORSUK_1:40, FUNCT_1:def_3;
i in dom h0 by A335, A336, FINSEQ_3:25;
then A350: h0 /. i = h0 . i by PARTFUN1:def_6;
i + 1 in dom h0 by A338, A340, FINSEQ_3:25;
then A351: h0 /. (i + 1) = h0 . (i + 1) by PARTFUN1:def_6;
A352: i + 1 <= len h1 by A341, NAT_1:13;
then A353: i + 1 in dom h1 by A340, FINSEQ_3:25;
then A354: h1 . i < h1 . (i + 1) by A12, A339, A342, SEQM_3:def_1;
A355: h1 /. (i + 1) = h1 . (i + 1) by A340, A352, FINSEQ_4:15;
A356: h1 . (i + 1) in rng h1 by A353, FUNCT_1:def_3;
then reconsider Q1 = [.(h1 /. i),(h1 /. (i + 1)).] as Subset of I[01] by A11, A343, A346, A355, BORSUK_1:40, XXREAL_2:def_12;
A357: h0 . i = h11 . i by A39, A335, A341, FINSEQ_1:64;
A358: h0 . (i + 1) = h11 . (i + 1) by A39, A340, A352, FINSEQ_1:64;
then A359: h0 . (i + 1) = g1 . (h1 . (i + 1)) by A353, FUNCT_1:13;
then A360: h0 . (i + 1) in Upper_Arc P by A5, A132, A11, A356, BORSUK_1:40, FUNCT_1:def_3;
A361: Segment ((h0 /. i),(h0 /. (i + 1)),P) c= g1 .: [.(h1 /. i),(h1 /. (i + 1)).]
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Segment ((h0 /. i),(h0 /. (i + 1)),P) or x in g1 .: [.(h1 /. i),(h1 /. (i + 1)).] )
A362: h0 /. (i + 1) <> W-min P by A38, A17, A71, A20, A340, A358, A353, A351, FUNCT_1:def_4;
assume x in Segment ((h0 /. i),(h0 /. (i + 1)),P) ; ::_thesis: x in g1 .: [.(h1 /. i),(h1 /. (i + 1)).]
then x in { p where p is Point of (TOP-REAL 2) : ( LE h0 /. i,p,P & LE p,h0 /. (i + 1),P ) } by A362, Def1;
then consider p being Point of (TOP-REAL 2) such that
A363: p = x and
A364: LE h0 /. i,p,P and
A365: LE p,h0 /. (i + 1),P ;
A366: ( ( h0 /. i in Upper_Arc P & p in Lower_Arc P & not p = W-min P ) or ( h0 /. i in Upper_Arc P & p in Upper_Arc P & LE h0 /. i,p, Upper_Arc P, W-min P, E-max P ) or ( h0 /. i in Lower_Arc P & p in Lower_Arc P & not p = W-min P & LE h0 /. i,p, Lower_Arc P, E-max P, W-min P ) ) by A364, JORDAN6:def_10;
A367: ( ( p in Upper_Arc P & h0 /. (i + 1) in Lower_Arc P ) or ( p in Upper_Arc P & h0 /. (i + 1) in Upper_Arc P & LE p,h0 /. (i + 1), Upper_Arc P, W-min P, E-max P ) or ( p in Lower_Arc P & h0 /. (i + 1) in Lower_Arc P & LE p,h0 /. (i + 1), Lower_Arc P, E-max P, W-min P ) ) by A365, JORDAN6:def_10;
now__::_thesis:_ex_z_being_set_st_
(_z_in_dom_g1_&_z_in_[.(h1_/._i),(h1_/._(i_+_1)).]_&_x_=_g1_._z_)
percases ( i + 1 < len h1 or i + 1 = len h1 ) by A352, XXREAL_0:1;
suppose i + 1 < len h1 ; ::_thesis: ex z being set st
( z in dom g1 & z in [.(h1 /. i),(h1 /. (i + 1)).] & x = g1 . z )
then A368: h0 /. (i + 1) <> E-max P by A38, A45, A255, A20, A358, A353, A351, FUNCT_1:def_4;
A369: now__::_thesis:_not_h0_/._(i_+_1)_in_Lower_Arc_P
assume h0 /. (i + 1) in Lower_Arc P ; ::_thesis: contradiction
then h0 /. (i + 1) in (Upper_Arc P) /\ (Lower_Arc P) by A351, A360, XBOOLE_0:def_4;
then h0 /. (i + 1) in {(W-min P),(E-max P)} by A1, JORDAN6:def_9;
hence contradiction by A362, A368, TARSKI:def_2; ::_thesis: verum
end;
then A370: LE p,h0 /. (i + 1), Upper_Arc P, W-min P, E-max P by A365, JORDAN6:def_10;
then A371: p <> E-max P by A3, A368, JORDAN6:55;
A372: p in Upper_Arc P by A365, A369, JORDAN6:def_10;
percases ( i > 1 or i = 1 ) by A335, XXREAL_0:1;
suppose i > 1 ; ::_thesis: ex z being set st
( z in dom g1 & z in [.(h1 /. i),(h1 /. (i + 1)).] & x = g1 . z )
then A373: h0 /. i <> W-min P by A38, A17, A71, A20, A342, A347, A348, A350, FUNCT_1:def_4;
A374: h11 . i <> E-max P by A38, A45, A255, A20, A341, A342, FUNCT_1:def_4;
now__::_thesis:_not_h0_/._i_in_Lower_Arc_P
assume h0 /. i in Lower_Arc P ; ::_thesis: contradiction
then h0 /. i in (Upper_Arc P) /\ (Lower_Arc P) by A350, A349, XBOOLE_0:def_4;
then h0 /. i in {(W-min P),(E-max P)} by A1, JORDAN6:def_9;
hence contradiction by A357, A350, A373, A374, TARSKI:def_2; ::_thesis: verum
end;
then A375: ( ( h0 /. i in Upper_Arc P & p in Lower_Arc P & not p = W-min P ) or ( h0 /. i in Upper_Arc P & p in Upper_Arc P & LE h0 /. i,p, Upper_Arc P, W-min P, E-max P ) ) by A364, JORDAN6:def_10;
then A376: p <> W-min P by A3, A373, JORDAN6:54;
A377: now__::_thesis:_not_p_in_Lower_Arc_P
assume p in Lower_Arc P ; ::_thesis: contradiction
then p in (Upper_Arc P) /\ (Lower_Arc P) by A372, XBOOLE_0:def_4;
then p in {(W-min P),(E-max P)} by A1, JORDAN6:def_9;
hence contradiction by A371, A376, TARSKI:def_2; ::_thesis: verum
end;
then consider z being set such that
A378: z in dom g1 and
A379: p = g1 . z by A5, A375, FUNCT_1:def_3;
reconsider rz = z as Real by A132, A378;
A380: rz <= 1 by A378, BORSUK_1:40, XXREAL_1:1;
h1 . (i + 1) in rng h1 by A353, FUNCT_1:def_3;
then A381: ( 0 <= h1 /. (i + 1) & h1 /. (i + 1) <= 1 ) by A11, A355, BORSUK_1:40, XXREAL_1:1;
take z = z; ::_thesis: ( z in dom g1 & z in [.(h1 /. i),(h1 /. (i + 1)).] & x = g1 . z )
thus z in dom g1 by A378; ::_thesis: ( z in [.(h1 /. i),(h1 /. (i + 1)).] & x = g1 . z )
A382: ( g1 . (h1 /. i) = h0 /. i & h1 /. i <= 1 ) by A335, A341, A357, A347, A344, A350, FINSEQ_4:15;
g1 . (h1 /. (i + 1)) = h0 /. (i + 1) by A340, A352, A359, A351, FINSEQ_4:15;
then A383: rz <= h1 /. (i + 1) by A4, A5, A6, A7, A370, A379, A381, A380, Th19;
0 <= rz by A378, BORSUK_1:40, XXREAL_1:1;
then h1 /. i <= rz by A4, A5, A6, A7, A375, A377, A379, A382, A380, Th19;
hence z in [.(h1 /. i),(h1 /. (i + 1)).] by A383, XXREAL_1:1; ::_thesis: x = g1 . z
thus x = g1 . z by A363, A379; ::_thesis: verum
end;
supposeA384: i = 1 ; ::_thesis: ex y being set st
( y in dom g1 & y in [.(h1 /. i),(h1 /. (i + 1)).] & x = g1 . y )
now__::_thesis:_(_(_p_<>_W-min_P_&_ex_rz_being_Real_st_
(_rz_in_dom_g1_&_rz_in_[.(h1_/._1),(h1_/._(1_+_1)).]_&_x_=_g1_._rz_)_)_or_(_p_=_W-min_P_&_0_in_[.(h1_/._1),(h1_/._(1_+_1)).]_&_x_=_g1_._0_)_)
percases ( p <> W-min P or p = W-min P ) ;
caseA385: p <> W-min P ; ::_thesis: ex rz being Real st
( rz in dom g1 & rz in [.(h1 /. 1),(h1 /. (1 + 1)).] & x = g1 . rz )
now__::_thesis:_not_p_in_Lower_Arc_P
assume p in Lower_Arc P ; ::_thesis: contradiction
then p in (Upper_Arc P) /\ (Lower_Arc P) by A372, XBOOLE_0:def_4;
then p in {(W-min P),(E-max P)} by A1, JORDAN6:def_9;
hence contradiction by A371, A385, TARSKI:def_2; ::_thesis: verum
end;
then consider z being set such that
A386: z in dom g1 and
A387: p = g1 . z by A5, A366, FUNCT_1:def_3;
reconsider rz = z as Real by A132, A386;
A388: h1 /. 1 <= rz by A8, A346, A384, A386, BORSUK_1:40, XXREAL_1:1;
h1 . (1 + 1) in rng h1 by A353, A384, FUNCT_1:def_3;
then A389: ( 0 <= h1 /. (1 + 1) & h1 /. (1 + 1) <= 1 ) by A11, A355, A384, BORSUK_1:40, XXREAL_1:1;
A390: g1 . (h1 /. (1 + 1)) = h0 /. (1 + 1) by A352, A359, A351, A384, FINSEQ_4:15;
take rz = rz; ::_thesis: ( rz in dom g1 & rz in [.(h1 /. 1),(h1 /. (1 + 1)).] & x = g1 . rz )
rz <= 1 by A386, BORSUK_1:40, XXREAL_1:1;
then rz <= h1 /. (1 + 1) by A4, A5, A6, A7, A370, A384, A387, A390, A389, Th19;
hence ( rz in dom g1 & rz in [.(h1 /. 1),(h1 /. (1 + 1)).] & x = g1 . rz ) by A363, A386, A387, A388, XXREAL_1:1; ::_thesis: verum
end;
caseA391: p = W-min P ; ::_thesis: ( 0 in [.(h1 /. 1),(h1 /. (1 + 1)).] & x = g1 . 0 )
thus 0 in [.(h1 /. 1),(h1 /. (1 + 1)).] by A8, A354, A346, A355, A384, XXREAL_1:1; ::_thesis: x = g1 . 0
thus x = g1 . 0 by A6, A363, A391; ::_thesis: verum
end;
end;
end;
hence ex y being set st
( y in dom g1 & y in [.(h1 /. i),(h1 /. (i + 1)).] & x = g1 . y ) by A44, A384; ::_thesis: verum
end;
end;
end;
supposeA392: i + 1 = len h1 ; ::_thesis: ex y being set st
( y in dom g1 & y in [.(h1 /. i),(h1 /. (i + 1)).] & x = g1 . y )
then A393: h0 /. (i + 1) = E-max P by A7, A9, A45, A358, A351, FUNCT_1:13;
A394: now__::_thesis:_(_p_in_Lower_Arc_P_implies_p_in_Upper_Arc_P_)
assume that
A395: p in Lower_Arc P and
A396: not p in Upper_Arc P ; ::_thesis: contradiction
LE h0 /. (i + 1),p, Lower_Arc P, E-max P, W-min P by A21, A393, A395, JORDAN5C:10;
hence contradiction by A334, A21, A367, A393, A396, JORDAN5C:12; ::_thesis: verum
end;
( p in Upper_Arc P or p in Lower_Arc P ) by A364, JORDAN6:def_10;
then A397: LE p,h0 /. (i + 1), Upper_Arc P, W-min P, E-max P by A3, A393, A394, JORDAN5C:10;
percases ( p <> E-max P or p = E-max P ) ;
supposeA398: p <> E-max P ; ::_thesis: ex y being set st
( y in dom g1 & y in [.(h1 /. i),(h1 /. (i + 1)).] & x = g1 . y )
now__::_thesis:_(_(_p_<>_W-min_P_&_ex_rz_being_Real_st_
(_rz_in_dom_g1_&_rz_in_[.(h1_/._i),(h1_/._(i_+_1)).]_&_x_=_g1_._rz_)_)_or_(_p_=_W-min_P_&_ex_y_being_set_st_
(_y_in_dom_g1_&_y_in_[.(h1_/._i),(h1_/._(i_+_1)).]_&_x_=_g1_._y_)_)_)
percases ( p <> W-min P or p = W-min P ) ;
caseA399: p <> W-min P ; ::_thesis: ex rz being Real st
( rz in dom g1 & rz in [.(h1 /. i),(h1 /. (i + 1)).] & x = g1 . rz )
A400: now__::_thesis:_not_p_in_Lower_Arc_P
assume p in Lower_Arc P ; ::_thesis: contradiction
then p in (Upper_Arc P) /\ (Lower_Arc P) by A394, XBOOLE_0:def_4;
then p in {(W-min P),(E-max P)} by A1, JORDAN6:def_9;
hence contradiction by A398, A399, TARSKI:def_2; ::_thesis: verum
end;
then consider z being set such that
A401: z in dom g1 and
A402: p = g1 . z by A5, A366, FUNCT_1:def_3;
reconsider rz = z as Real by A132, A401;
A403: rz <= 1 by A401, BORSUK_1:40, XXREAL_1:1;
h1 . (i + 1) in rng h1 by A353, FUNCT_1:def_3;
then A404: ( 0 <= h1 /. (i + 1) & h1 /. (i + 1) <= 1 ) by A11, A355, BORSUK_1:40, XXREAL_1:1;
g1 . (h1 /. (i + 1)) = h0 /. (i + 1) by A340, A352, A359, A351, FINSEQ_4:15;
then A405: rz <= h1 /. (i + 1) by A4, A5, A6, A7, A397, A402, A404, A403, Th19;
take rz = rz; ::_thesis: ( rz in dom g1 & rz in [.(h1 /. i),(h1 /. (i + 1)).] & x = g1 . rz )
0 <= rz by A401, BORSUK_1:40, XXREAL_1:1;
then h1 /. i <= rz by A4, A5, A6, A7, A357, A347, A344, A350, A346, A366, A400, A402, A403, Th19;
hence ( rz in dom g1 & rz in [.(h1 /. i),(h1 /. (i + 1)).] & x = g1 . rz ) by A363, A401, A402, A405, XXREAL_1:1; ::_thesis: verum
end;
caseA406: p = W-min P ; ::_thesis: ex y being set st
( y in dom g1 & y in [.(h1 /. i),(h1 /. (i + 1)).] & x = g1 . y )
then h11 . i = W-min P by A3, A357, A350, A366, JORDAN6:54;
then i = 1 by A38, A17, A71, A20, A342, FUNCT_1:def_4;
then 0 in [.(h1 /. i),(h1 /. (i + 1)).] by A8, A354, A346, A355, XXREAL_1:1;
hence ex y being set st
( y in dom g1 & y in [.(h1 /. i),(h1 /. (i + 1)).] & x = g1 . y ) by A6, A44, A363, A406; ::_thesis: verum
end;
end;
end;
hence ex y being set st
( y in dom g1 & y in [.(h1 /. i),(h1 /. (i + 1)).] & x = g1 . y ) ; ::_thesis: verum
end;
supposeA407: p = E-max P ; ::_thesis: ex y being set st
( y in dom g1 & y in [.(h1 /. i),(h1 /. (i + 1)).] & x = g1 . y )
1 in [.(h1 /. i),(h1 /. (i + 1)).] by A9, A354, A346, A355, A392, XXREAL_1:1;
hence ex y being set st
( y in dom g1 & y in [.(h1 /. i),(h1 /. (i + 1)).] & x = g1 . y ) by A7, A69, A363, A407, Lm6; ::_thesis: verum
end;
end;
end;
end;
end;
hence x in g1 .: [.(h1 /. i),(h1 /. (i + 1)).] by FUNCT_1:def_6; ::_thesis: verum
end;
A408: h1 . (i + 1) <= 1 by A11, A356, BORSUK_1:40, XXREAL_1:1;
g1 .: [.(h1 /. i),(h1 /. (i + 1)).] c= Segment ((h0 /. i),(h0 /. (i + 1)),P)
proof
A409: ( g1 . (h1 /. i) = h0 /. i & 0 <= h1 /. i ) by A335, A341, A348, A345, A350, FINSEQ_4:15;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in g1 .: [.(h1 /. i),(h1 /. (i + 1)).] or x in Segment ((h0 /. i),(h0 /. (i + 1)),P) )
assume x in g1 .: [.(h1 /. i),(h1 /. (i + 1)).] ; ::_thesis: x in Segment ((h0 /. i),(h0 /. (i + 1)),P)
then consider y being set such that
A410: y in dom g1 and
A411: y in [.(h1 /. i),(h1 /. (i + 1)).] and
A412: x = g1 . y by FUNCT_1:def_6;
reconsider sy = y as Real by A411;
A413: sy <= 1 by A410, BORSUK_1:40, XXREAL_1:1;
A414: x in Upper_Arc P by A5, A410, A412, FUNCT_1:def_3;
then reconsider p1 = x as Point of (TOP-REAL 2) ;
A415: h1 /. i <= 1 by A335, A341, A344, FINSEQ_4:15;
h1 /. i <= sy by A411, XXREAL_1:1;
then LE h0 /. i,p1, Upper_Arc P, W-min P, E-max P by A3, A4, A5, A6, A7, A412, A409, A415, A413, Th18;
then A416: LE h0 /. i,p1,P by A350, A349, A414, JORDAN6:def_10;
( sy <= h1 /. (i + 1) & 0 <= sy ) by A410, A411, BORSUK_1:40, XXREAL_1:1;
then LE p1,h0 /. (i + 1), Upper_Arc P, W-min P, E-max P by A3, A4, A5, A6, A7, A359, A408, A351, A355, A412, A413, Th18;
then LE p1,h0 /. (i + 1),P by A351, A360, A414, JORDAN6:def_10;
then A417: x in { p where p is Point of (TOP-REAL 2) : ( LE h0 /. i,p,P & LE p,h0 /. (i + 1),P ) } by A416;
not h0 /. (i + 1) = W-min P by A38, A17, A71, A20, A340, A358, A353, A351, FUNCT_1:def_4;
hence x in Segment ((h0 /. i),(h0 /. (i + 1)),P) by A417, Def1; ::_thesis: verum
end;
then W = g1 .: Q1 by A337, A361, XBOOLE_0:def_10;
hence diameter W < e by A13, A335, A341; ::_thesis: verum
end;
supposeA418: i > len h1 ; ::_thesis: diameter W < e
i - (len h11) < ((len h11) + ((len h2) - 2)) - (len h11) by A47, A36, A52, A55, A57, A336, XREAL_1:9;
then A419: (i - (len h11)) + 2 < ((len h2) - 2) + 2 by XREAL_1:6;
A420: (len h1) + 1 <= i by A418, NAT_1:13;
then A421: ((len h1) + 1) - (len h1) <= i - (len h1) by XREAL_1:9;
A422: i - (len h11) = i -' (len h11) by A39, A418, XREAL_1:233;
i - (len h11) <= ((len h1) + ((len h2) - 2)) - (len h11) by A39, A47, A36, A52, A55, A57, A336, XREAL_1:9;
then A423: (i -' (len h11)) + 2 <= ((len h2) - 2) + 2 by A39, A422, XREAL_1:6;
i + 1 > len h11 by A39, A418, NAT_1:13;
then A424: (i + 1) - (len h11) = (i + 1) -' (len h11) by XREAL_1:233;
i + 1 in dom h0 by A338, A340, FINSEQ_3:25;
then A425: h0 /. (i + 1) = h0 . (i + 1) by PARTFUN1:def_6;
A426: i <= (len h11) + (len (mid (h21,2,((len h21) -' 1)))) by A336, FINSEQ_1:22;
(len h11) + 1 <= i by A39, A418, NAT_1:13;
then A427: h0 . i = (mid (h21,2,((len h21) -' 1))) . (i - (len h11)) by A426, FINSEQ_1:23;
A428: (i + 1) - (len h11) <= ((len h11) + (len (mid (h21,2,((len h21) -' 1))))) - (len h11) by A36, A338, XREAL_1:9;
i in dom h0 by A335, A336, FINSEQ_3:25;
then A429: h0 /. i = h0 . i by PARTFUN1:def_6;
set j = ((i -' (len h11)) + 2) -' 1;
len h2 < (len h2) + 1 by NAT_1:13;
then A430: (len h2) - 1 < ((len h2) + 1) - 1 by XREAL_1:9;
A431: 0 + 2 <= (i -' (len h11)) + 2 by XREAL_1:6;
then A432: (((i -' (len h11)) + 2) -' 1) + 1 = ((((i -' (len h11)) + 1) + 1) - 1) + 1 by Lm1, NAT_D:39, NAT_D:42
.= (i -' (len h11)) + (1 + 1) ;
A433: 1 <= ((i -' (len h11)) + 2) -' 1 by A431, Lm1, NAT_D:42;
then A434: 1 < (((i -' (len h11)) + 2) -' 1) + 1 by NAT_1:13;
then A435: (((i -' (len h11)) + 2) -' 1) + 1 in dom h2 by A423, A432, FINSEQ_3:25;
then A436: h2 . ((((i -' (len h11)) + 2) -' 1) + 1) in rng h2 by FUNCT_1:def_3;
then A437: h2 . ((((i -' (len h11)) + 2) -' 1) + 1) <= 1 by A29, BORSUK_1:40, XXREAL_1:1;
A438: h2 /. ((((i -' (len h11)) + 2) -' 1) + 1) = h2 . ((((i -' (len h11)) + 2) -' 1) + 1) by A423, A432, A434, FINSEQ_4:15;
((i -' (len h11)) + 2) -' 1 <= len h21 by A47, A423, NAT_D:44;
then A439: ((i -' (len h11)) + 2) -' 1 in dom h2 by A46, A433, FINSEQ_3:25;
i - (len h11) <= ((len h11) + (len (mid (h21,2,((len h21) -' 1))))) - (len h11) by A36, A336, XREAL_1:9;
then A440: h0 . i = h21 . (((i -' (len h11)) + 2) -' 1) by A39, A48, A56, A50, A54, A427, A422, A421, FINSEQ_6:118;
then A441: h0 . i = g2 . (h2 . (((i -' (len h11)) + 2) -' 1)) by A439, FUNCT_1:13;
A442: h2 . (((i -' (len h11)) + 2) -' 1) in rng h2 by A439, FUNCT_1:def_3;
then g2 . (h2 . (((i -' (len h11)) + 2) -' 1)) in rng g2 by A131, A29, BORSUK_1:40, FUNCT_1:def_3;
then A443: h0 . i in Lower_Arc P by A23, A440, A439, FUNCT_1:13;
((i -' (len h11)) + 2) - 1 <= (len h2) - 1 by A423, XREAL_1:9;
then ((i -' (len h11)) + 2) - 1 < len h2 by A430, XXREAL_0:2;
then A444: ((i -' (len h11)) + 2) -' 1 < len h2 by A431, Lm1, NAT_D:39, NAT_D:42;
then A445: h2 /. (((i -' (len h11)) + 2) -' 1) = h2 . (((i -' (len h11)) + 2) -' 1) by A431, Lm1, FINSEQ_4:15, NAT_D:42;
then reconsider Q1 = [.(h2 /. (((i -' (len h11)) + 2) -' 1)),(h2 /. ((((i -' (len h11)) + 2) -' 1) + 1)).] as Subset of I[01] by A29, A442, A436, A438, BORSUK_1:40, XXREAL_2:def_12;
A446: ( 0 <= h2 . (((i -' (len h11)) + 2) -' 1) & h2 . (((i -' (len h11)) + 2) -' 1) <= 1 ) by A29, A442, BORSUK_1:40, XXREAL_1:1;
A447: ((i -' (len h11)) + 2) -' 1 = ((i -' (len h11)) + 2) - 1 by A431, Lm1, NAT_D:39, NAT_D:42;
A448: (len h1) + 1 <= i + 1 by A420, NAT_1:13;
then A449: ((len h1) + 1) - (len h1) <= (i + 1) - (len h1) by XREAL_1:9;
then 1 < ((i + 1) -' (len h11)) + (2 - 1) by A39, A424, NAT_1:13;
then 0 < (((i + 1) -' (len h11)) + 2) - 1 ;
then A450: (((i -' (len h11)) + 2) -' 1) + 1 = (((i + 1) -' (len h11)) + 2) -' 1 by A422, A424, A447, XREAL_0:def_2;
h0 . (i + 1) = (mid (h21,2,((len h21) -' 1))) . ((i + 1) - (len h11)) by A39, A36, A338, A448, FINSEQ_1:23;
then A451: h0 . (i + 1) = h21 . ((((i + 1) -' (len h11)) + 2) -' 1) by A39, A48, A56, A50, A54, A424, A428, A449, FINSEQ_6:118;
then h0 . (i + 1) = g2 . (h2 . ((((i -' (len h11)) + 2) -' 1) + 1)) by A450, A435, FUNCT_1:13;
then A452: h0 . (i + 1) in Lower_Arc P by A23, A131, A29, A436, BORSUK_1:40, FUNCT_1:def_3;
(len h1) + 1 <= i by A418, NAT_1:13;
then ((len h11) + 1) - (len h11) <= i - (len h11) by A39, XREAL_1:9;
then 1 <= i -' (len h11) by NAT_D:39;
then 1 + 2 <= (i -' (len h11)) + 2 by XREAL_1:6;
then (1 + 2) - 1 <= ((i -' (len h11)) + 2) - 1 by XREAL_1:9;
then A453: 1 < ((i -' (len h11)) + 2) -' 1 by A447, XXREAL_0:2;
A454: Segment ((h0 /. i),(h0 /. (i + 1)),P) c= g2 .: [.(h2 /. (((i -' (len h11)) + 2) -' 1)),(h2 /. ((((i -' (len h11)) + 2) -' 1) + 1)).]
proof
h2 . ((((i -' (len h11)) + 2) -' 1) + 1) in rng h2 by A435, FUNCT_1:def_3;
then A455: ( 0 <= h2 /. ((((i -' (len h11)) + 2) -' 1) + 1) & h2 /. ((((i -' (len h11)) + 2) -' 1) + 1) <= 1 ) by A29, A438, BORSUK_1:40, XXREAL_1:1;
A456: g2 . (h2 /. ((((i -' (len h11)) + 2) -' 1) + 1)) = h0 /. (i + 1) by A451, A450, A435, A425, A438, FUNCT_1:13;
((i -' (len h11)) + 2) -' 1 < len h2 by A423, A432, NAT_1:13;
then A457: h0 /. i <> W-min P by A46, A34, A35, A32, A440, A439, A429, FUNCT_1:def_4;
A458: h2 /. (((i -' (len h11)) + 2) -' 1) <= 1 by A29, A442, A445, BORSUK_1:40, XXREAL_1:1;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Segment ((h0 /. i),(h0 /. (i + 1)),P) or x in g2 .: [.(h2 /. (((i -' (len h11)) + 2) -' 1)),(h2 /. ((((i -' (len h11)) + 2) -' 1) + 1)).] )
assume A459: x in Segment ((h0 /. i),(h0 /. (i + 1)),P) ; ::_thesis: x in g2 .: [.(h2 /. (((i -' (len h11)) + 2) -' 1)),(h2 /. ((((i -' (len h11)) + 2) -' 1) + 1)).]
h0 /. (i + 1) <> W-min P by A46, A34, A35, A32, A422, A451, A419, A432, A450, A435, A425, FUNCT_1:def_4;
then x in { p where p is Point of (TOP-REAL 2) : ( LE h0 /. i,p,P & LE p,h0 /. (i + 1),P ) } by A459, Def1;
then consider p being Point of (TOP-REAL 2) such that
A460: p = x and
A461: LE h0 /. i,p,P and
A462: LE p,h0 /. (i + 1),P ;
A463: ( ( h0 /. i in Upper_Arc P & p in Lower_Arc P & not p = W-min P ) or ( h0 /. i in Upper_Arc P & p in Upper_Arc P & LE h0 /. i,p, Upper_Arc P, W-min P, E-max P ) or ( h0 /. i in Lower_Arc P & p in Lower_Arc P & not p = W-min P & LE h0 /. i,p, Lower_Arc P, E-max P, W-min P ) ) by A461, JORDAN6:def_10;
A464: h21 . (((i -' (len h11)) + 2) -' 1) <> E-max P by A46, A76, A77, A32, A453, A439, FUNCT_1:def_4;
A465: now__::_thesis:_not_h0_/._i_in_Upper_Arc_P
assume h0 /. i in Upper_Arc P ; ::_thesis: contradiction
then h0 /. i in (Upper_Arc P) /\ (Lower_Arc P) by A429, A443, XBOOLE_0:def_4;
then h0 /. i in {(W-min P),(E-max P)} by A1, JORDAN6:def_9;
hence contradiction by A440, A429, A457, A464, TARSKI:def_2; ::_thesis: verum
end;
then A466: LE h0 /. i,p, Lower_Arc P, E-max P, W-min P by A461, JORDAN6:def_10;
A467: h0 /. i <> E-max P by A46, A76, A77, A32, A440, A453, A439, A429, FUNCT_1:def_4;
A468: now__::_thesis:_not_p_in_Upper_Arc_P
assume p in Upper_Arc P ; ::_thesis: contradiction
then p in (Upper_Arc P) /\ (Lower_Arc P) by A463, A465, XBOOLE_0:def_4;
then p in {(W-min P),(E-max P)} by A1, JORDAN6:def_9;
then ( p = W-min P or p = E-max P ) by TARSKI:def_2;
hence contradiction by A21, A463, A465, A467, JORDAN6:54; ::_thesis: verum
end;
A469: ( h0 /. (i + 1) <> E-max P & h21 . ((((i -' (len h11)) + 2) -' 1) + 1) <> W-min P ) by A46, A76, A34, A77, A35, A32, A422, A451, A419, A432, A450, A434, A435, A425, FUNCT_1:def_4;
now__::_thesis:_not_h0_/._(i_+_1)_in_Upper_Arc_P
assume h0 /. (i + 1) in Upper_Arc P ; ::_thesis: contradiction
then h0 /. (i + 1) in (Upper_Arc P) /\ (Lower_Arc P) by A425, A452, XBOOLE_0:def_4;
then h0 /. (i + 1) in {(W-min P),(E-max P)} by A1, JORDAN6:def_9;
hence contradiction by A451, A450, A425, A469, TARSKI:def_2; ::_thesis: verum
end;
then ( ( p in Lower_Arc P & h0 /. (i + 1) in Lower_Arc P & not h0 /. (i + 1) = W-min P & LE p,h0 /. (i + 1), Lower_Arc P, E-max P, W-min P ) or ( p in Upper_Arc P & h0 /. (i + 1) in Lower_Arc P & not h0 /. (i + 1) = W-min P ) ) by A462, JORDAN6:def_10;
then consider z being set such that
A470: z in dom g2 and
A471: p = g2 . z by A23, A468, FUNCT_1:def_3;
reconsider rz = z as Real by A131, A470;
A472: rz <= 1 by A470, BORSUK_1:40, XXREAL_1:1;
LE p,h0 /. (i + 1), Lower_Arc P, E-max P, W-min P by A462, A468, JORDAN6:def_10;
then A473: rz <= h2 /. ((((i -' (len h11)) + 2) -' 1) + 1) by A22, A23, A24, A25, A471, A456, A472, A455, Th19;
0 <= rz by A470, BORSUK_1:40, XXREAL_1:1;
then h2 /. (((i -' (len h11)) + 2) -' 1) <= rz by A22, A23, A24, A25, A441, A429, A445, A466, A471, A458, A472, Th19;
then rz in [.(h2 /. (((i -' (len h11)) + 2) -' 1)),(h2 /. ((((i -' (len h11)) + 2) -' 1) + 1)).] by A473, XXREAL_1:1;
hence x in g2 .: [.(h2 /. (((i -' (len h11)) + 2) -' 1)),(h2 /. ((((i -' (len h11)) + 2) -' 1) + 1)).] by A460, A470, A471, FUNCT_1:def_6; ::_thesis: verum
end;
A474: g2 . (h2 . ((((i -' (len h11)) + 2) -' 1) + 1)) = h0 /. (i + 1) by A451, A450, A435, A425, FUNCT_1:13;
g2 .: [.(h2 /. (((i -' (len h11)) + 2) -' 1)),(h2 /. ((((i -' (len h11)) + 2) -' 1) + 1)).] c= Segment ((h0 /. i),(h0 /. (i + 1)),P)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in g2 .: [.(h2 /. (((i -' (len h11)) + 2) -' 1)),(h2 /. ((((i -' (len h11)) + 2) -' 1) + 1)).] or x in Segment ((h0 /. i),(h0 /. (i + 1)),P) )
assume x in g2 .: [.(h2 /. (((i -' (len h11)) + 2) -' 1)),(h2 /. ((((i -' (len h11)) + 2) -' 1) + 1)).] ; ::_thesis: x in Segment ((h0 /. i),(h0 /. (i + 1)),P)
then consider y being set such that
A475: y in dom g2 and
A476: y in [.(h2 /. (((i -' (len h11)) + 2) -' 1)),(h2 /. ((((i -' (len h11)) + 2) -' 1) + 1)).] and
A477: x = g2 . y by FUNCT_1:def_6;
reconsider sy = y as Real by A476;
A478: sy <= h2 /. ((((i -' (len h11)) + 2) -' 1) + 1) by A476, XXREAL_1:1;
A479: x in Lower_Arc P by A23, A475, A477, FUNCT_1:def_3;
then reconsider p1 = x as Point of (TOP-REAL 2) ;
A480: h2 . ((((i -' (len h11)) + 2) -' 1) + 1) <> 1 by A27, A30, A34, A422, A419, A432, A435, FUNCT_1:def_4;
A481: now__::_thesis:_not_p1_=_W-min_P
assume p1 = W-min P ; ::_thesis: contradiction
then 1 = sy by A22, A25, A227, A475, A477, FUNCT_1:def_4;
hence contradiction by A437, A438, A478, A480, XXREAL_0:1; ::_thesis: verum
end;
A482: sy <= 1 by A475, BORSUK_1:40, XXREAL_1:1;
h2 /. (((i -' (len h11)) + 2) -' 1) <= sy by A476, XXREAL_1:1;
then LE h0 /. i,p1, Lower_Arc P, E-max P, W-min P by A21, A22, A23, A24, A25, A441, A429, A446, A445, A477, A482, Th18;
then A483: LE h0 /. i,p1,P by A429, A443, A479, A481, JORDAN6:def_10;
A484: h0 /. (i + 1) <> W-min P by A46, A34, A35, A32, A422, A451, A419, A432, A450, A435, A425, FUNCT_1:def_4;
0 <= sy by A475, BORSUK_1:40, XXREAL_1:1;
then LE p1,h0 /. (i + 1), Lower_Arc P, E-max P, W-min P by A21, A22, A23, A24, A25, A474, A437, A438, A477, A478, A482, Th18;
then LE p1,h0 /. (i + 1),P by A425, A452, A479, A484, JORDAN6:def_10;
then x in { p where p is Point of (TOP-REAL 2) : ( LE h0 /. i,p,P & LE p,h0 /. (i + 1),P ) } by A483;
hence x in Segment ((h0 /. i),(h0 /. (i + 1)),P) by A484, Def1; ::_thesis: verum
end;
then W = g2 .: Q1 by A337, A454, XBOOLE_0:def_10;
hence diameter W < e by A31, A433, A444; ::_thesis: verum
end;
supposeA485: i = len h1 ; ::_thesis: diameter W < e
A486: (i + 1) - (len h11) <= ((len h11) + (len (mid (h21,2,((len h21) -' 1))))) - (len h11) by A36, A338, XREAL_1:9;
then 1 <= (i + 1) -' (len h11) by A39, A339, A485, XREAL_1:233;
then 1 < ((i + 1) -' (len h11)) + (2 - 1) by NAT_1:13;
then A487: 0 < (((i + 1) -' (len h11)) + 2) - 1 ;
i in dom h0 by A335, A336, FINSEQ_3:25;
then A488: h0 /. i = h0 . i by PARTFUN1:def_6;
A489: h0 . i = E-max P by A39, A255, A335, A485, FINSEQ_1:64;
set j = ((i -' (len h11)) + 2) -' 1;
A490: 0 + 2 <= (i -' (len h11)) + 2 by XREAL_1:6;
then A491: (((i -' (len h11)) + 2) -' 1) + 1 = ((((i -' (len h11)) + 1) + 1) - 1) + 1 by Lm1, NAT_D:39, NAT_D:42
.= (i -' (len h11)) + (1 + 1) ;
A492: (len h1) -' (len h11) = (len h11) - (len h11) by A39, XREAL_0:def_2;
then A493: (0 + 2) - 1 = (((len h1) -' (len h11)) + 2) - 1 ;
then A494: h0 . i = g2 . (h2 . (((i -' (len h11)) + 2) -' 1)) by A24, A26, A485, A489, NAT_D:39;
A495: i - (len h11) = i -' (len h11) by A39, A485, XREAL_1:233;
i - (len h11) <= ((len h1) + ((len h2) - 2)) - (len h11) by A39, A47, A36, A52, A55, A57, A336, XREAL_1:9;
then A496: (i -' (len h11)) + 2 <= ((len h2) - 2) + 2 by A39, A495, XREAL_1:6;
i - (len h11) < ((len h11) + ((len h2) - 2)) - (len h11) by A47, A36, A52, A55, A57, A336, XREAL_1:9;
then A497: (i - (len h11)) + 2 < ((len h2) - 2) + 2 by XREAL_1:6;
then A498: (((i -' (len h11)) + 2) -' 1) + 1 < len h2 by A39, A485, A491, XREAL_0:def_2;
A499: h0 . (i + 1) = (mid (h21,2,((len h21) -' 1))) . ((i + 1) - (len h11)) by A39, A36, A338, A485, FINSEQ_1:23;
len h2 < (len h2) + 1 by NAT_1:13;
then A500: (len h2) - 1 < ((len h2) + 1) - 1 by XREAL_1:9;
i + 1 in dom h0 by A338, A340, FINSEQ_3:25;
then A501: h0 /. (i + 1) = h0 . (i + 1) by PARTFUN1:def_6;
A502: 1 <= ((i -' (len h11)) + 2) -' 1 by A490, Lm1, NAT_D:42;
then A503: 1 < (((i -' (len h11)) + 2) -' 1) + 1 by NAT_1:13;
then A504: (((i -' (len h11)) + 2) -' 1) + 1 in dom h2 by A496, A491, FINSEQ_3:25;
then A505: h2 . ((((i -' (len h11)) + 2) -' 1) + 1) in rng h2 by FUNCT_1:def_3;
then A506: h2 . ((((i -' (len h11)) + 2) -' 1) + 1) <= 1 by A29, BORSUK_1:40, XXREAL_1:1;
((i -' (len h11)) + 2) -' 1 <= len h21 by A47, A496, NAT_D:44;
then A507: ((i -' (len h11)) + 2) -' 1 in dom h2 by A46, A502, FINSEQ_3:25;
then A508: h2 . (((i -' (len h11)) + 2) -' 1) in rng h2 by FUNCT_1:def_3;
then g2 . (h2 . (((i -' (len h11)) + 2) -' 1)) in rng g2 by A131, A29, BORSUK_1:40, FUNCT_1:def_3;
then A509: h0 . i in Lower_Arc P by A23, A24, A26, A485, A489, A493, NAT_D:39;
A510: h2 /. ((((i -' (len h11)) + 2) -' 1) + 1) = h2 . ((((i -' (len h11)) + 2) -' 1) + 1) by A496, A491, A503, FINSEQ_4:15;
((i -' (len h11)) + 2) - 1 <= (len h2) - 1 by A496, XREAL_1:9;
then A511: ((i -' (len h11)) + 2) - 1 < len h2 by A500, XXREAL_0:2;
then A512: ((i -' (len h11)) + 2) -' 1 < len h2 by A490, Lm1, NAT_D:39, NAT_D:42;
then h2 /. (((i -' (len h11)) + 2) -' 1) = h2 . (((i -' (len h11)) + 2) -' 1) by A490, Lm1, FINSEQ_4:15, NAT_D:42;
then reconsider Q1 = [.(h2 /. (((i -' (len h11)) + 2) -' 1)),(h2 /. ((((i -' (len h11)) + 2) -' 1) + 1)).] as Subset of I[01] by A29, A508, A505, A510, BORSUK_1:40, XXREAL_2:def_12;
A513: (i + 1) - (len h11) = (i + 1) -' (len h11) by A39, A339, A485, XREAL_1:233;
(((i -' (len h11)) + 2) -' 1) + 1 = (((i - (len h11)) + 2) - 1) + 1 by A39, A485, Lm1, XREAL_0:def_2
.= (((i + 1) -' (len h11)) + 2) -' 1 by A513, A487, XREAL_0:def_2 ;
then A514: h0 . (i + 1) = h21 . ((((i -' (len h11)) + 2) -' 1) + 1) by A39, A48, A56, A50, A54, A485, A499, A513, A486, FINSEQ_6:118;
then A515: h0 . (i + 1) = g2 . (h2 . ((((i -' (len h11)) + 2) -' 1) + 1)) by A504, FUNCT_1:13;
then A516: h0 . (i + 1) in Lower_Arc P by A23, A131, A29, A505, BORSUK_1:40, FUNCT_1:def_3;
A517: h21 . (((i -' (len h11)) + 2) -' 1) = g2 . (h2 . (((i -' (len h11)) + 2) -' 1)) by A507, FUNCT_1:13;
A518: Segment ((h0 /. i),(h0 /. (i + 1)),P) c= g2 .: [.(h2 /. (((i -' (len h11)) + 2) -' 1)),(h2 /. ((((i -' (len h11)) + 2) -' 1) + 1)).]
proof
(((i -' (len h11)) + 2) -' 1) + 1 < len h2 by A39, A485, A497, A491, XREAL_0:def_2;
then ((i -' (len h11)) + 2) -' 1 < len h2 by NAT_1:13;
then A519: h0 /. i <> W-min P by A46, A34, A35, A32, A507, A517, A494, A488, FUNCT_1:def_4;
A520: g2 . (h2 /. ((((i -' (len h11)) + 2) -' 1) + 1)) = h0 /. (i + 1) by A496, A491, A503, A515, A501, FINSEQ_4:15;
h2 . ((((i -' (len h11)) + 2) -' 1) + 1) in rng h2 by A504, FUNCT_1:def_3;
then A521: ( 0 <= h2 /. ((((i -' (len h11)) + 2) -' 1) + 1) & h2 /. ((((i -' (len h11)) + 2) -' 1) + 1) <= 1 ) by A29, A510, BORSUK_1:40, XXREAL_1:1;
A522: h0 /. (i + 1) in Lower_Arc P by A23, A131, A29, A515, A505, A501, BORSUK_1:40, FUNCT_1:def_3;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Segment ((h0 /. i),(h0 /. (i + 1)),P) or x in g2 .: [.(h2 /. (((i -' (len h11)) + 2) -' 1)),(h2 /. ((((i -' (len h11)) + 2) -' 1) + 1)).] )
assume A523: x in Segment ((h0 /. i),(h0 /. (i + 1)),P) ; ::_thesis: x in g2 .: [.(h2 /. (((i -' (len h11)) + 2) -' 1)),(h2 /. ((((i -' (len h11)) + 2) -' 1) + 1)).]
h0 /. (i + 1) <> W-min P by A46, A34, A35, A32, A498, A514, A504, A501, FUNCT_1:def_4;
then x in { p where p is Point of (TOP-REAL 2) : ( LE h0 /. i,p,P & LE p,h0 /. (i + 1),P ) } by A523, Def1;
then consider p being Point of (TOP-REAL 2) such that
A524: p = x and
A525: LE h0 /. i,p,P and
A526: LE p,h0 /. (i + 1),P ;
A527: ( ( h0 /. i in Upper_Arc P & p in Lower_Arc P & not p = W-min P ) or ( h0 /. i in Upper_Arc P & p in Upper_Arc P & LE h0 /. i,p, Upper_Arc P, W-min P, E-max P ) or ( h0 /. i in Lower_Arc P & p in Lower_Arc P & not p = W-min P & LE h0 /. i,p, Lower_Arc P, E-max P, W-min P ) ) by A525, JORDAN6:def_10;
dom (g1 * h1) c= dom h0 by FINSEQ_1:26;
then A528: h0 /. i = E-max P by A254, A485, A489, PARTFUN1:def_6;
A529: now__::_thesis:_p_in_Lower_Arc_P
assume A530: not p in Lower_Arc P ; ::_thesis: contradiction
then p = E-max P by A3, A527, A528, JORDAN6:55;
hence contradiction by A1, A530, Th1; ::_thesis: verum
end;
A531: now__::_thesis:_(_p_in_Upper_Arc_P_implies_p_=_E-max_P_)
assume p in Upper_Arc P ; ::_thesis: p = E-max P
then p in (Upper_Arc P) /\ (Lower_Arc P) by A529, XBOOLE_0:def_4;
then A532: p in {(W-min P),(E-max P)} by A1, JORDAN6:def_9;
p <> W-min P by A3, A527, A519, JORDAN6:54;
hence p = E-max P by A532, TARSKI:def_2; ::_thesis: verum
end;
then p in rng g2 by A1, A23, A525, Th1, JORDAN6:def_10;
then consider z being set such that
A533: z in dom g2 and
A534: p = g2 . z by FUNCT_1:def_3;
reconsider rz = z as Real by A131, A533;
0 <= rz by A533, BORSUK_1:40, XXREAL_1:1;
then A535: h2 /. (((i -' (len h11)) + 2) -' 1) <= rz by A26, A485, A511, A492, Lm1, FINSEQ_4:15;
A536: not h0 /. (i + 1) = E-max P by A46, A76, A77, A32, A503, A514, A504, A501, FUNCT_1:def_4;
now__::_thesis:_not_h0_/._(i_+_1)_in_Upper_Arc_P
assume h0 /. (i + 1) in Upper_Arc P ; ::_thesis: contradiction
then h0 /. (i + 1) in (Upper_Arc P) /\ (Lower_Arc P) by A522, XBOOLE_0:def_4;
then h0 /. (i + 1) in {(W-min P),(E-max P)} by A1, JORDAN6:def_9;
then h21 . ((((i -' (len h11)) + 2) -' 1) + 1) = W-min P by A514, A501, A536, TARSKI:def_2;
hence contradiction by A46, A34, A35, A32, A498, A504, FUNCT_1:def_4; ::_thesis: verum
end;
then ( ( p in Lower_Arc P & h0 /. (i + 1) in Lower_Arc P & not h0 /. (i + 1) = W-min P & LE p,h0 /. (i + 1), Lower_Arc P, E-max P, W-min P ) or ( p in Upper_Arc P & h0 /. (i + 1) in Lower_Arc P & not h0 /. (i + 1) = W-min P ) ) by A526, JORDAN6:def_10;
then A537: LE p,h0 /. (i + 1), Lower_Arc P, E-max P, W-min P by A21, A531, JORDAN5C:10;
rz <= 1 by A533, BORSUK_1:40, XXREAL_1:1;
then rz <= h2 /. ((((i -' (len h11)) + 2) -' 1) + 1) by A22, A23, A24, A25, A537, A534, A520, A521, Th19;
then rz in [.(h2 /. (((i -' (len h11)) + 2) -' 1)),(h2 /. ((((i -' (len h11)) + 2) -' 1) + 1)).] by A535, XXREAL_1:1;
hence x in g2 .: [.(h2 /. (((i -' (len h11)) + 2) -' 1)),(h2 /. ((((i -' (len h11)) + 2) -' 1) + 1)).] by A524, A533, A534, FUNCT_1:def_6; ::_thesis: verum
end;
A538: g2 . (h2 . ((((i -' (len h11)) + 2) -' 1) + 1)) = h0 /. (i + 1) by A514, A504, A501, FUNCT_1:13;
g2 .: [.(h2 /. (((i -' (len h11)) + 2) -' 1)),(h2 /. ((((i -' (len h11)) + 2) -' 1) + 1)).] c= Segment ((h0 /. i),(h0 /. (i + 1)),P)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in g2 .: [.(h2 /. (((i -' (len h11)) + 2) -' 1)),(h2 /. ((((i -' (len h11)) + 2) -' 1) + 1)).] or x in Segment ((h0 /. i),(h0 /. (i + 1)),P) )
assume x in g2 .: [.(h2 /. (((i -' (len h11)) + 2) -' 1)),(h2 /. ((((i -' (len h11)) + 2) -' 1) + 1)).] ; ::_thesis: x in Segment ((h0 /. i),(h0 /. (i + 1)),P)
then consider y being set such that
A539: y in dom g2 and
A540: y in [.(h2 /. (((i -' (len h11)) + 2) -' 1)),(h2 /. ((((i -' (len h11)) + 2) -' 1) + 1)).] and
A541: x = g2 . y by FUNCT_1:def_6;
reconsider sy = y as Real by A540;
A542: sy <= h2 /. ((((i -' (len h11)) + 2) -' 1) + 1) by A540, XXREAL_1:1;
A543: x in Lower_Arc P by A23, A539, A541, FUNCT_1:def_3;
then reconsider p1 = x as Point of (TOP-REAL 2) ;
A544: h2 . ((((i -' (len h11)) + 2) -' 1) + 1) <> 1 by A27, A30, A34, A498, A504, FUNCT_1:def_4;
A545: now__::_thesis:_not_p1_=_W-min_P
assume p1 = W-min P ; ::_thesis: contradiction
then 1 = sy by A22, A25, A227, A539, A541, FUNCT_1:def_4;
hence contradiction by A506, A510, A542, A544, XXREAL_0:1; ::_thesis: verum
end;
A546: ( 0 <= sy & sy <= 1 ) by A539, BORSUK_1:40, XXREAL_1:1;
then LE h0 /. i,p1, Lower_Arc P, E-max P, W-min P by A21, A22, A23, A24, A25, A489, A488, A541, Th18;
then A547: LE h0 /. i,p1,P by A488, A509, A543, A545, JORDAN6:def_10;
A548: h0 /. (i + 1) <> W-min P by A46, A34, A35, A32, A498, A514, A504, A501, FUNCT_1:def_4;
LE p1,h0 /. (i + 1), Lower_Arc P, E-max P, W-min P by A21, A22, A23, A24, A25, A538, A506, A510, A541, A542, A546, Th18;
then LE p1,h0 /. (i + 1),P by A501, A516, A543, A548, JORDAN6:def_10;
then x in { p where p is Point of (TOP-REAL 2) : ( LE h0 /. i,p,P & LE p,h0 /. (i + 1),P ) } by A547;
hence x in Segment ((h0 /. i),(h0 /. (i + 1)),P) by A548, Def1; ::_thesis: verum
end;
then W = g2 .: Q1 by A337, A518, XBOOLE_0:def_10;
hence diameter W < e by A31, A502, A512; ::_thesis: verum
end;
end;
end;
A549: len h0 = (len h1) + ((len h2) - 2) by A38, A47, A36, A52, A55, A57, FINSEQ_3:29;
thus for W being Subset of (Euclid 2) st W = Segment ((h0 /. (len h0)),(h0 /. 1),P) holds
diameter W < e ::_thesis: ( ( for i being Element of NAT st 1 <= i & i + 1 < len h0 holds
(Segment ((h0 /. i),(h0 /. (i + 1)),P)) /\ (Segment ((h0 /. (i + 1)),(h0 /. (i + 2)),P)) = {(h0 /. (i + 1))} ) & (Segment ((h0 /. (len h0)),(h0 /. 1),P)) /\ (Segment ((h0 /. 1),(h0 /. 2),P)) = {(h0 /. 1)} & (Segment ((h0 /. ((len h0) -' 1)),(h0 /. (len h0)),P)) /\ (Segment ((h0 /. (len h0)),(h0 /. 1),P)) = {(h0 /. (len h0))} & Segment ((h0 /. ((len h0) -' 1)),(h0 /. (len h0)),P) misses Segment ((h0 /. 1),(h0 /. 2),P) & ( for i, j being Element of NAT st 1 <= i & i < j & j < len h0 & not i,j are_adjacent1 holds
Segment ((h0 /. i),(h0 /. (i + 1)),P) misses Segment ((h0 /. j),(h0 /. (j + 1)),P) ) & ( for i being Element of NAT st 1 < i & i + 1 < len h0 holds
Segment ((h0 /. (len h0)),(h0 /. 1),P) misses Segment ((h0 /. i),(h0 /. (i + 1)),P) ) )
proof
set i = len h0;
let W be Subset of (Euclid 2); ::_thesis: ( W = Segment ((h0 /. (len h0)),(h0 /. 1),P) implies diameter W < e )
set j = (((len h0) -' (len h11)) + 2) -' 1;
A550: 0 + 2 <= ((len h0) -' (len h11)) + 2 by XREAL_1:6;
then A551: 1 <= (((len h0) -' (len h11)) + 2) -' 1 by Lm1, NAT_D:42;
((len h11) + 1) - (len h11) <= (len h0) - (len h11) by A47, A36, A52, A55, A57, A62, XXREAL_0:2;
then 1 <= (len h0) -' (len h11) by NAT_D:39;
then 1 + 2 <= ((len h0) -' (len h11)) + 2 by XREAL_1:6;
then A552: (1 + 2) - 1 <= (((len h0) -' (len h11)) + 2) - 1 by XREAL_1:9;
len h0 <= (len h11) + (len (mid (h21,2,((len h21) -' 1)))) by FINSEQ_1:22;
then A553: h0 . (len h0) = (mid (h21,2,((len h21) -' 1))) . ((len h0) - (len h11)) by A39, A64, FINSEQ_1:23;
A554: (len h0) - (len h11) = (len h0) -' (len h11) by A39, A65, XREAL_1:233;
then (((len h0) -' (len h11)) + 2) -' 1 <= len h21 by A36, A52, A55, A57, NAT_D:44;
then A555: (((len h0) -' (len h11)) + 2) -' 1 in dom h2 by A46, A551, FINSEQ_3:25;
then A556: h2 . ((((len h0) -' (len h11)) + 2) -' 1) in rng h2 by FUNCT_1:def_3;
( (len h0) - (len h11) <= ((len h11) + (len (mid (h21,2,((len h21) -' 1))))) - (len h11) & ((len h1) + 1) - (len h1) <= (len h0) - (len h1) ) by A549, A62, FINSEQ_1:22, XXREAL_0:2;
then A557: h0 . (len h0) = h21 . ((((len h0) -' (len h11)) + 2) -' 1) by A39, A48, A56, A50, A54, A553, A554, FINSEQ_6:118;
then A558: h0 . (len h0) = g2 . (h2 . ((((len h0) -' (len h11)) + 2) -' 1)) by A555, FUNCT_1:13;
then A559: h0 . (len h0) in Lower_Arc P by A23, A131, A29, A556, BORSUK_1:40, FUNCT_1:def_3;
A560: 2 -' 1 <= (((len h0) -' (len h11)) + 2) -' 1 by A550, NAT_D:42;
then A561: 1 < ((((len h0) -' (len h11)) + 2) -' 1) + 1 by Lm1, NAT_1:13;
then A562: h2 /. (((((len h0) -' (len h11)) + 2) -' 1) + 1) = h2 . (((((len h0) -' (len h11)) + 2) -' 1) + 1) by A47, A36, A52, A55, A57, A554, FINSEQ_4:15;
len h2 < (len h2) + 1 by NAT_1:13;
then A563: (len h2) - 1 < ((len h2) + 1) - 1 by XREAL_1:9;
then A564: h2 /. ((((len h0) -' (len h11)) + 2) -' 1) = h2 . ((((len h0) -' (len h11)) + 2) -' 1) by A47, A36, A52, A55, A57, A554, A560, Lm1, FINSEQ_4:15;
((((len h0) -' (len h11)) + 2) -' 1) + 1 in dom h2 by A46, A36, A52, A55, A57, A554, A561, FINSEQ_3:25;
then h2 . (((((len h0) -' (len h11)) + 2) -' 1) + 1) in rng h2 by FUNCT_1:def_3;
then reconsider Q1 = [.(h2 /. ((((len h0) -' (len h11)) + 2) -' 1)),(h2 /. (((((len h0) -' (len h11)) + 2) -' 1) + 1)).] as Subset of I[01] by A29, A556, A564, A562, BORSUK_1:40, XXREAL_2:def_12;
len h0 in dom h0 by A66, FINSEQ_3:25;
then A565: h0 /. (len h0) = h0 . (len h0) by PARTFUN1:def_6;
(((len h0) -' (len h11)) + 2) -' 1 = (((len h0) -' (len h11)) + 2) - 1 by A550, Lm1, NAT_D:39, NAT_D:42;
then A566: 1 < (((len h0) -' (len h11)) + 2) -' 1 by A552, XXREAL_0:2;
A567: now__::_thesis:_not_h0_._(len_h0)_in_Upper_Arc_P
assume h0 . (len h0) in Upper_Arc P ; ::_thesis: contradiction
then h0 . (len h0) in (Upper_Arc P) /\ (Lower_Arc P) by A559, XBOOLE_0:def_4;
then h0 . (len h0) in {(W-min P),(E-max P)} by A1, JORDAN6:def_9;
then ( h0 . (len h0) = W-min P or h0 . (len h0) = E-max P ) by TARSKI:def_2;
hence contradiction by A46, A47, A76, A34, A77, A35, A36, A52, A55, A57, A32, A554, A557, A563, A566, A555, FUNCT_1:def_4; ::_thesis: verum
end;
A568: h2 . ((((len h0) -' (len h11)) + 2) -' 1) <= 1 by A29, A556, BORSUK_1:40, XXREAL_1:1;
A569: Segment ((h0 /. (len h0)),(h0 /. 1),P) c= g2 .: [.(h2 /. ((((len h0) -' (len h11)) + 2) -' 1)),(h2 /. (((((len h0) -' (len h11)) + 2) -' 1) + 1)).]
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Segment ((h0 /. (len h0)),(h0 /. 1),P) or x in g2 .: [.(h2 /. ((((len h0) -' (len h11)) + 2) -' 1)),(h2 /. (((((len h0) -' (len h11)) + 2) -' 1) + 1)).] )
assume A570: x in Segment ((h0 /. (len h0)),(h0 /. 1),P) ; ::_thesis: x in g2 .: [.(h2 /. ((((len h0) -' (len h11)) + 2) -' 1)),(h2 /. (((((len h0) -' (len h11)) + 2) -' 1) + 1)).]
h0 /. 1 = W-min P by A72, A67, PARTFUN1:def_6;
then A571: x in { p where p is Point of (TOP-REAL 2) : ( LE h0 /. (len h0),p,P or ( h0 /. (len h0) in P & p = W-min P ) ) } by A570, Def1;
A572: ((((len h0) -' (len h11)) + 2) -' 1) + 1 in dom h2 by A46, A36, A52, A55, A57, A554, A561, FINSEQ_3:25;
( (((len h0) -' (len h11)) + 2) -' 1 < ((((len h0) -' (len h11)) + 2) -' 1) + 1 & (((len h0) -' (len h11)) + 2) -' 1 in dom h2 ) by A47, A36, A52, A55, A57, A554, A560, A563, Lm1, FINSEQ_3:25;
then h2 . ((((len h0) -' (len h11)) + 2) -' 1) < h2 . (((((len h0) -' (len h11)) + 2) -' 1) + 1) by A30, A572, SEQM_3:def_1;
then A573: h2 /. (((((len h0) -' (len h11)) + 2) -' 1) + 1) in [.(h2 /. ((((len h0) -' (len h11)) + 2) -' 1)),(h2 /. (((((len h0) -' (len h11)) + 2) -' 1) + 1)).] by A564, A562, XXREAL_1:1;
consider p being Point of (TOP-REAL 2) such that
A574: p = x and
A575: ( LE h0 /. (len h0),p,P or ( h0 /. (len h0) in P & p = W-min P ) ) by A571;
A576: h2 /. (((((len h0) -' (len h11)) + 2) -' 1) + 1) = 1 by A27, A47, A34, A36, A52, A55, A57, A554, PARTFUN1:def_6;
now__::_thesis:_ex_z_being_set_st_
(_z_in_dom_g2_&_z_in_[.(h2_/._((((len_h0)_-'_(len_h11))_+_2)_-'_1)),(h2_/._(((((len_h0)_-'_(len_h11))_+_2)_-'_1)_+_1)).]_&_x_=_g2_._z_)
percases ( ( LE h0 /. (len h0),p,P & ( p <> W-min P or not h0 /. (len h0) in P ) ) or ( h0 /. (len h0) in P & p = W-min P ) ) by A575;
supposeA577: ( LE h0 /. (len h0),p,P & ( p <> W-min P or not h0 /. (len h0) in P ) ) ; ::_thesis: ex z being set st
( z in dom g2 & z in [.(h2 /. ((((len h0) -' (len h11)) + 2) -' 1)),(h2 /. (((((len h0) -' (len h11)) + 2) -' 1) + 1)).] & x = g2 . z )
then p in Lower_Arc P by A567, A565, JORDAN6:def_10;
then consider z being set such that
A578: z in dom g2 and
A579: p = g2 . z by A23, FUNCT_1:def_3;
take z = z; ::_thesis: ( z in dom g2 & z in [.(h2 /. ((((len h0) -' (len h11)) + 2) -' 1)),(h2 /. (((((len h0) -' (len h11)) + 2) -' 1) + 1)).] & x = g2 . z )
thus z in dom g2 by A578; ::_thesis: ( z in [.(h2 /. ((((len h0) -' (len h11)) + 2) -' 1)),(h2 /. (((((len h0) -' (len h11)) + 2) -' 1) + 1)).] & x = g2 . z )
reconsider rz = z as Real by A131, A578;
A580: rz <= 1 by A578, BORSUK_1:40, XXREAL_1:1;
then A581: rz <= h2 /. (((((len h0) -' (len h11)) + 2) -' 1) + 1) by A27, A47, A36, A52, A55, A57, A554, A561, FINSEQ_4:15;
A582: LE h0 /. (len h0),p, Lower_Arc P, E-max P, W-min P by A567, A565, A577, JORDAN6:def_10;
0 <= rz by A578, BORSUK_1:40, XXREAL_1:1;
then h2 /. ((((len h0) -' (len h11)) + 2) -' 1) <= rz by A22, A23, A24, A25, A558, A568, A565, A564, A582, A579, A580, Th19;
hence ( z in [.(h2 /. ((((len h0) -' (len h11)) + 2) -' 1)),(h2 /. (((((len h0) -' (len h11)) + 2) -' 1) + 1)).] & x = g2 . z ) by A574, A579, A581, XXREAL_1:1; ::_thesis: verum
end;
suppose ( h0 /. (len h0) in P & p = W-min P ) ; ::_thesis: ex y being set st
( y in dom g2 & y in [.(h2 /. ((((len h0) -' (len h11)) + 2) -' 1)),(h2 /. (((((len h0) -' (len h11)) + 2) -' 1) + 1)).] & x = g2 . y )
hence ex y being set st
( y in dom g2 & y in [.(h2 /. ((((len h0) -' (len h11)) + 2) -' 1)),(h2 /. (((((len h0) -' (len h11)) + 2) -' 1) + 1)).] & x = g2 . y ) by A25, A227, A574, A573, A576; ::_thesis: verum
end;
end;
end;
hence x in g2 .: [.(h2 /. ((((len h0) -' (len h11)) + 2) -' 1)),(h2 /. (((((len h0) -' (len h11)) + 2) -' 1) + 1)).] by FUNCT_1:def_6; ::_thesis: verum
end;
A583: ( 0 <= h2 . ((((len h0) -' (len h11)) + 2) -' 1) & h2 . ((((len h0) -' (len h11)) + 2) -' 1) <= 1 ) by A29, A556, BORSUK_1:40, XXREAL_1:1;
A584: g2 .: [.(h2 /. ((((len h0) -' (len h11)) + 2) -' 1)),(h2 /. (((((len h0) -' (len h11)) + 2) -' 1) + 1)).] c= Segment ((h0 /. (len h0)),(h0 /. 1),P)
proof
A585: (Upper_Arc P) \/ (Lower_Arc P) = P by A1, JORDAN6:def_9;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in g2 .: [.(h2 /. ((((len h0) -' (len h11)) + 2) -' 1)),(h2 /. (((((len h0) -' (len h11)) + 2) -' 1) + 1)).] or x in Segment ((h0 /. (len h0)),(h0 /. 1),P) )
assume x in g2 .: [.(h2 /. ((((len h0) -' (len h11)) + 2) -' 1)),(h2 /. (((((len h0) -' (len h11)) + 2) -' 1) + 1)).] ; ::_thesis: x in Segment ((h0 /. (len h0)),(h0 /. 1),P)
then consider y being set such that
A586: y in dom g2 and
A587: y in [.(h2 /. ((((len h0) -' (len h11)) + 2) -' 1)),(h2 /. (((((len h0) -' (len h11)) + 2) -' 1) + 1)).] and
A588: x = g2 . y by FUNCT_1:def_6;
reconsider sy = y as Real by A587;
A589: x in Lower_Arc P by A23, A586, A588, FUNCT_1:def_3;
then reconsider p1 = x as Point of (TOP-REAL 2) ;
( h2 /. ((((len h0) -' (len h11)) + 2) -' 1) <= sy & sy <= 1 ) by A586, A587, BORSUK_1:40, XXREAL_1:1;
then A590: LE h0 /. (len h0),p1, Lower_Arc P, E-max P, W-min P by A21, A22, A23, A24, A25, A558, A565, A583, A564, A588, Th18;
now__::_thesis:_(_(_p1_=_W-min_P_&_(_LE_h0_/._(len_h0),p1,P_or_(_h0_/._(len_h0)_in_P_&_p1_=_W-min_P_)_)_)_or_(_p1_<>_W-min_P_&_(_LE_h0_/._(len_h0),p1,P_or_(_h0_/._(len_h0)_in_P_&_p1_=_W-min_P_)_)_)_)
percases ( p1 = W-min P or p1 <> W-min P ) ;
case p1 = W-min P ; ::_thesis: ( LE h0 /. (len h0),p1,P or ( h0 /. (len h0) in P & p1 = W-min P ) )
hence ( LE h0 /. (len h0),p1,P or ( h0 /. (len h0) in P & p1 = W-min P ) ) by A559, A565, A585, XBOOLE_0:def_3; ::_thesis: verum
end;
case p1 <> W-min P ; ::_thesis: ( LE h0 /. (len h0),p1,P or ( h0 /. (len h0) in P & p1 = W-min P ) )
hence ( LE h0 /. (len h0),p1,P or ( h0 /. (len h0) in P & p1 = W-min P ) ) by A559, A565, A589, A590, JORDAN6:def_10; ::_thesis: verum
end;
end;
end;
then A591: x in { p where p is Point of (TOP-REAL 2) : ( LE h0 /. (len h0),p,P or ( h0 /. (len h0) in P & p = W-min P ) ) } ;
h0 /. 1 = W-min P by A72, A67, PARTFUN1:def_6;
hence x in Segment ((h0 /. (len h0)),(h0 /. 1),P) by A591, Def1; ::_thesis: verum
end;
assume W = Segment ((h0 /. (len h0)),(h0 /. 1),P) ; ::_thesis: diameter W < e
then W = g2 .: Q1 by A569, A584, XBOOLE_0:def_10;
hence diameter W < e by A31, A47, A36, A52, A55, A57, A554, A560, A563, Lm1; ::_thesis: verum
end;
A592: for i being Element of NAT st 1 <= i & i + 1 < len h0 holds
LE h0 /. (i + 1),h0 /. (i + 2),P
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i + 1 < len h0 implies LE h0 /. (i + 1),h0 /. (i + 2),P )
assume that
A593: 1 <= i and
A594: i + 1 < len h0 ; ::_thesis: LE h0 /. (i + 1),h0 /. (i + 2),P
A595: (i + 1) + 1 <= len h0 by A594, NAT_1:13;
A596: i + 1 < (i + 1) + 1 by NAT_1:13;
A597: 1 < i + 1 by A593, NAT_1:13;
then A598: 1 < (i + 1) + 1 by NAT_1:13;
percases ( i + 1 < len h1 or i + 1 >= len h1 ) ;
supposeA599: i + 1 < len h1 ; ::_thesis: LE h0 /. (i + 1),h0 /. (i + 2),P
then A600: i + 1 in dom h1 by A597, FINSEQ_3:25;
then A601: h1 . (i + 1) in rng h1 by FUNCT_1:def_3;
then A602: ( 0 <= h1 . (i + 1) & h1 . (i + 1) <= 1 ) by A11, BORSUK_1:40, XXREAL_1:1;
A603: 1 < (i + 1) + 1 by A597, NAT_1:13;
then (i + 1) + 1 in dom h0 by A595, FINSEQ_3:25;
then A604: h0 /. ((i + 1) + 1) = h0 . ((i + 1) + 1) by PARTFUN1:def_6;
A605: (i + 1) + 1 <= len h1 by A599, NAT_1:13;
then A606: (i + 1) + 1 in dom h1 by A603, FINSEQ_3:25;
then A607: h1 . ((i + 1) + 1) in rng h1 by FUNCT_1:def_3;
then A608: h1 . ((i + 1) + 1) <= 1 by A11, BORSUK_1:40, XXREAL_1:1;
h0 . ((i + 1) + 1) = h11 . ((i + 1) + 1) by A39, A605, A603, FINSEQ_1:64;
then A609: h0 . ((i + 1) + 1) = g1 . (h1 . ((i + 1) + 1)) by A606, FUNCT_1:13;
then A610: h0 /. ((i + 1) + 1) in Upper_Arc P by A5, A132, A11, A607, A604, BORSUK_1:40, FUNCT_1:def_3;
i + 1 in dom h0 by A594, A597, FINSEQ_3:25;
then A611: h0 /. (i + 1) = h0 . (i + 1) by PARTFUN1:def_6;
A612: h0 . (i + 1) = h11 . (i + 1) by A39, A597, A599, FINSEQ_1:64;
then A613: h0 . (i + 1) = g1 . (h1 . (i + 1)) by A600, FUNCT_1:13;
g1 . (h1 . (i + 1)) in rng g1 by A132, A11, A601, BORSUK_1:40, FUNCT_1:def_3;
then A614: h0 /. (i + 1) in Upper_Arc P by A5, A612, A600, A611, FUNCT_1:13;
h1 . (i + 1) < h1 . ((i + 1) + 1) by A12, A596, A600, A606, SEQM_3:def_1;
then LE h0 /. (i + 1),h0 /. ((i + 1) + 1), Upper_Arc P, W-min P, E-max P by A3, A4, A5, A6, A7, A613, A602, A609, A608, A611, A604, Th18;
hence LE h0 /. (i + 1),h0 /. (i + 2),P by A614, A610, JORDAN6:def_10; ::_thesis: verum
end;
supposeA615: i + 1 >= len h1 ; ::_thesis: LE h0 /. (i + 1),h0 /. (i + 2),P
percases ( i + 1 > len h1 or i + 1 = len h1 ) by A615, XXREAL_0:1;
supposeA616: i + 1 > len h1 ; ::_thesis: LE h0 /. (i + 1),h0 /. (i + 2),P
set j = (((i + 1) -' (len h11)) + 2) -' 1;
A617: ((i + 1) + 1) - (len h11) <= ((len h11) + (len (mid (h21,2,((len h21) -' 1))))) - (len h11) by A36, A595, XREAL_1:9;
A618: 0 + 2 <= ((i + 1) -' (len h11)) + 2 by XREAL_1:6;
then A619: 1 <= (((i + 1) -' (len h11)) + 2) -' 1 by Lm1, NAT_D:42;
A620: ((((i + 1) -' (len h11)) + 2) -' 1) + 1 = ((((i + 1) -' (len h11)) + (1 + 1)) - 1) + 1 by A618, Lm1, NAT_D:39, NAT_D:42
.= ((i + 1) -' (len h11)) + 2 ;
A621: (len h1) + 1 <= i + 1 by A616, NAT_1:13;
then A622: ((len h1) + 1) - (len h1) <= (i + 1) - (len h1) by XREAL_1:9;
A623: (i + 1) - (len h11) = (i + 1) -' (len h11) by A39, A616, XREAL_1:233;
(i + 1) + 1 > len h11 by A39, A616, NAT_1:13;
then A624: ((i + 1) + 1) - (len h11) = ((i + 1) + 1) -' (len h11) by XREAL_1:233;
A625: (len h1) + 1 <= (i + 1) + 1 by A621, NAT_1:13;
then A626: ((len h1) + 1) - (len h1) <= ((i + 1) + 1) - (len h1) by XREAL_1:9;
then 1 < (((i + 1) + 1) -' (len h11)) + (2 - 1) by A39, A624, NAT_1:13;
then A627: 0 < ((((i + 1) + 1) -' (len h11)) + 2) - 1 ;
(((i + 1) -' (len h11)) + 2) -' 1 = (((i + 1) -' (len h11)) + 2) - 1 by A618, Lm1, NAT_D:39, NAT_D:42;
then A628: ((((i + 1) -' (len h11)) + 2) -' 1) + 1 = ((((i + 1) + 1) -' (len h11)) + 2) -' 1 by A623, A624, A627, XREAL_0:def_2;
(i + 1) - (len h11) <= ((len h1) + ((len h2) - 2)) - (len h11) by A39, A47, A36, A52, A55, A57, A594, XREAL_1:9;
then A629: ((i + 1) -' (len h11)) + 2 <= ((len h2) - 2) + 2 by A39, A623, XREAL_1:6;
then (((i + 1) -' (len h11)) + 2) -' 1 <= len h21 by A47, NAT_D:44;
then A630: (((i + 1) -' (len h11)) + 2) -' 1 in dom h2 by A46, A619, FINSEQ_3:25;
2 -' 1 <= (((i + 1) -' (len h11)) + 2) -' 1 by A618, NAT_D:42;
then 1 < ((((i + 1) -' (len h11)) + 2) -' 1) + 1 by Lm1, NAT_1:13;
then A631: ((((i + 1) -' (len h11)) + 2) -' 1) + 1 in dom h2 by A629, A620, FINSEQ_3:25;
then A632: h2 . (((((i + 1) -' (len h11)) + 2) -' 1) + 1) in rng h2 by FUNCT_1:def_3;
then A633: h2 . (((((i + 1) -' (len h11)) + 2) -' 1) + 1) <= 1 by A29, BORSUK_1:40, XXREAL_1:1;
(((i + 1) -' (len h11)) + 2) -' 1 < ((((i + 1) -' (len h11)) + 2) -' 1) + 1 by NAT_1:13;
then A634: h2 . ((((i + 1) -' (len h11)) + 2) -' 1) < h2 . (((((i + 1) -' (len h11)) + 2) -' 1) + 1) by A30, A630, A631, SEQM_3:def_1;
A635: i + 1 <= (len h11) + (len (mid (h21,2,((len h21) -' 1)))) by A594, FINSEQ_1:22;
(len h11) + 1 <= i + 1 by A39, A616, NAT_1:13;
then A636: h0 . (i + 1) = (mid (h21,2,((len h21) -' 1))) . ((i + 1) - (len h11)) by A635, FINSEQ_1:23;
(i + 1) - (len h11) <= ((len h11) + (len (mid (h21,2,((len h21) -' 1))))) - (len h11) by A635, XREAL_1:9;
then h0 . (i + 1) = h21 . ((((i + 1) -' (len h11)) + 2) -' 1) by A39, A48, A56, A50, A54, A636, A623, A622, FINSEQ_6:118;
then A637: h0 . (i + 1) = g2 . (h2 . ((((i + 1) -' (len h11)) + 2) -' 1)) by A630, FUNCT_1:13;
A638: h2 . ((((i + 1) -' (len h11)) + 2) -' 1) in rng h2 by A630, FUNCT_1:def_3;
then A639: h0 . (i + 1) in Lower_Arc P by A23, A131, A29, A637, BORSUK_1:40, FUNCT_1:def_3;
(i + 1) + 1 in dom h0 by A595, A598, FINSEQ_3:25;
then A640: h0 /. ((i + 1) + 1) = h0 . ((i + 1) + 1) by PARTFUN1:def_6;
h0 . ((i + 1) + 1) = (mid (h21,2,((len h21) -' 1))) . (((i + 1) + 1) - (len h11)) by A39, A36, A595, A625, FINSEQ_1:23;
then A641: h0 . ((i + 1) + 1) = h21 . (((((i + 1) + 1) -' (len h11)) + 2) -' 1) by A39, A48, A56, A50, A54, A624, A617, A626, FINSEQ_6:118;
then A642: h0 . ((i + 1) + 1) = g2 . (h2 . (((((i + 1) -' (len h11)) + 2) -' 1) + 1)) by A628, A631, FUNCT_1:13;
then A643: h0 . ((i + 1) + 1) in Lower_Arc P by A23, A131, A29, A632, BORSUK_1:40, FUNCT_1:def_3;
(i + 1) - (len h11) < ((len h11) + ((len h2) - 2)) - (len h11) by A47, A36, A52, A55, A57, A594, XREAL_1:9;
then ((i + 1) - (len h11)) + 2 < ((len h2) - 2) + 2 by XREAL_1:6;
then A644: h0 /. ((i + 1) + 1) <> W-min P by A46, A34, A35, A32, A623, A641, A620, A628, A631, A640, FUNCT_1:def_4;
i + 1 in dom h0 by A594, A597, FINSEQ_3:25;
then A645: h0 /. (i + 1) = h0 . (i + 1) by PARTFUN1:def_6;
( 0 <= h2 . ((((i + 1) -' (len h11)) + 2) -' 1) & h2 . ((((i + 1) -' (len h11)) + 2) -' 1) <= 1 ) by A29, A638, BORSUK_1:40, XXREAL_1:1;
then LE h0 /. (i + 1),h0 /. ((i + 1) + 1), Lower_Arc P, E-max P, W-min P by A21, A22, A23, A24, A25, A637, A642, A634, A645, A640, A633, Th18;
hence LE h0 /. (i + 1),h0 /. (i + 2),P by A645, A640, A639, A643, A644, JORDAN6:def_10; ::_thesis: verum
end;
supposeA646: i + 1 = len h1 ; ::_thesis: LE h0 /. (i + 1),h0 /. (i + 2),P
then (len h1) + 1 <= (len h11) + (len (mid (h21,2,((len h21) -' 1)))) by A595, FINSEQ_1:22;
then A647: ((i + 1) + 1) - (len h11) <= ((len h11) + (len (mid (h21,2,((len h21) -' 1))))) - (len h11) by A646, XREAL_1:9;
then 1 <= ((i + 1) + 1) -' (len h11) by A39, A596, A646, XREAL_1:233;
then 1 < (((i + 1) + 1) -' (len h11)) + (2 - 1) by NAT_1:13;
then A648: 0 < ((((i + 1) + 1) -' (len h11)) + 2) - 1 ;
A649: ((i + 1) + 1) - (len h11) = ((i + 1) + 1) -' (len h11) by A39, A596, A646, XREAL_1:233;
len h1 in dom h0 by A594, A597, A646, FINSEQ_3:25;
then A650: h0 /. (len h1) = h0 . (len h1) by PARTFUN1:def_6;
set j = (((i + 1) -' (len h11)) + 2) -' 1;
A651: 0 + 2 <= ((i + 1) -' (len h11)) + 2 by XREAL_1:6;
then A652: ((((i + 1) -' (len h11)) + 2) -' 1) + 1 = ((((i + 1) -' (len h11)) + (1 + 1)) - 1) + 1 by Lm1, NAT_D:39, NAT_D:42
.= ((i + 1) -' (len h11)) + (1 + 1) ;
2 -' 1 <= (((i + 1) -' (len h11)) + 2) -' 1 by A651, NAT_D:42;
then A653: 1 < ((((i + 1) -' (len h11)) + 2) -' 1) + 1 by Lm1, NAT_1:13;
( (len h1) - (len h11) = (len h1) -' (len h11) & (i + 1) - (len h11) <= ((len h1) + ((len h2) - 2)) - (len h11) ) by A39, A47, A36, A52, A55, A57, A594, XREAL_1:9, XREAL_1:233;
then ((i + 1) -' (len h11)) + 2 <= ((len h2) - 2) + 2 by A39, A646, XREAL_1:6;
then A654: ((((i + 1) -' (len h11)) + 2) -' 1) + 1 in dom h2 by A652, A653, FINSEQ_3:25;
then A655: h2 . (((((i + 1) -' (len h11)) + 2) -' 1) + 1) in rng h2 by FUNCT_1:def_3;
h0 . (len h1) = E-max P by A39, A255, A597, A646, FINSEQ_1:64;
then A656: h0 . (i + 1) in Upper_Arc P by A1, A646, Th1;
(i + 1) + 1 in dom h0 by A595, A598, FINSEQ_3:25;
then A657: h0 /. ((i + 1) + 1) = h0 . ((i + 1) + 1) by PARTFUN1:def_6;
h0 . ((i + 1) + 1) = (mid (h21,2,((len h21) -' 1))) . (((i + 1) + 1) - (len h11)) by A39, A36, A595, A646, FINSEQ_1:23;
then A658: h0 . ((i + 1) + 1) = h21 . (((((i + 1) + 1) -' (len h11)) + 2) -' 1) by A39, A48, A56, A50, A54, A646, A649, A647, FINSEQ_6:118;
A659: ((((i + 1) -' (len h11)) + 2) -' 1) + 1 = ((((i + 1) - (len h11)) + 2) - 1) + 1 by A39, A646, Lm1, XREAL_0:def_2
.= ((((i + 1) + 1) -' (len h11)) + 2) -' 1 by A649, A648, XREAL_0:def_2 ;
then h0 . ((i + 1) + 1) = g2 . (h2 . (((((i + 1) -' (len h11)) + 2) -' 1) + 1)) by A658, A654, FUNCT_1:13;
then A660: h0 . ((i + 1) + 1) in Lower_Arc P by A23, A131, A29, A655, BORSUK_1:40, FUNCT_1:def_3;
(i + 1) - (len h11) < ((len h11) + ((len h2) - 2)) - (len h11) by A47, A36, A52, A55, A57, A594, XREAL_1:9;
then ((i + 1) - (len h11)) + 2 < ((len h2) - 2) + 2 by XREAL_1:6;
then ((((i + 1) -' (len h11)) + 2) -' 1) + 1 < len h2 by A39, A646, A652, XREAL_0:def_2;
then h0 /. ((i + 1) + 1) <> W-min P by A46, A34, A35, A32, A658, A659, A654, A657, FUNCT_1:def_4;
hence LE h0 /. (i + 1),h0 /. (i + 2),P by A646, A650, A657, A660, A656, JORDAN6:def_10; ::_thesis: verum
end;
end;
end;
end;
end;
thus for i being Element of NAT st 1 <= i & i + 1 < len h0 holds
(Segment ((h0 /. i),(h0 /. (i + 1)),P)) /\ (Segment ((h0 /. (i + 1)),(h0 /. (i + 2)),P)) = {(h0 /. (i + 1))} ::_thesis: ( (Segment ((h0 /. (len h0)),(h0 /. 1),P)) /\ (Segment ((h0 /. 1),(h0 /. 2),P)) = {(h0 /. 1)} & (Segment ((h0 /. ((len h0) -' 1)),(h0 /. (len h0)),P)) /\ (Segment ((h0 /. (len h0)),(h0 /. 1),P)) = {(h0 /. (len h0))} & Segment ((h0 /. ((len h0) -' 1)),(h0 /. (len h0)),P) misses Segment ((h0 /. 1),(h0 /. 2),P) & ( for i, j being Element of NAT st 1 <= i & i < j & j < len h0 & not i,j are_adjacent1 holds
Segment ((h0 /. i),(h0 /. (i + 1)),P) misses Segment ((h0 /. j),(h0 /. (j + 1)),P) ) & ( for i being Element of NAT st 1 < i & i + 1 < len h0 holds
Segment ((h0 /. (len h0)),(h0 /. 1),P) misses Segment ((h0 /. i),(h0 /. (i + 1)),P) ) )
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i + 1 < len h0 implies (Segment ((h0 /. i),(h0 /. (i + 1)),P)) /\ (Segment ((h0 /. (i + 1)),(h0 /. (i + 2)),P)) = {(h0 /. (i + 1))} )
assume A661: ( 1 <= i & i + 1 < len h0 ) ; ::_thesis: (Segment ((h0 /. i),(h0 /. (i + 1)),P)) /\ (Segment ((h0 /. (i + 1)),(h0 /. (i + 2)),P)) = {(h0 /. (i + 1))}
then A662: ( LE h0 /. i,h0 /. (i + 1),P & h0 /. (i + 1) <> W-min P ) by A256;
( h0 /. i <> h0 /. (i + 1) & LE h0 /. (i + 1),h0 /. (i + 2),P ) by A592, A256, A661;
hence (Segment ((h0 /. i),(h0 /. (i + 1)),P)) /\ (Segment ((h0 /. (i + 1)),(h0 /. (i + 2)),P)) = {(h0 /. (i + 1))} by A1, A662, Th10; ::_thesis: verum
end;
A663: 2 in dom h0 by A201, FINSEQ_3:25;
(len h0) -' 1 <> 1 by A59, Lm2;
then A664: h0 /. ((len h0) -' 1) <> h0 /. 1 by A67, A78, A203, PARTFUN2:10;
A665: len h1 in dom h1 by A16, FINSEQ_3:25;
thus (Segment ((h0 /. (len h0)),(h0 /. 1),P)) /\ (Segment ((h0 /. 1),(h0 /. 2),P)) = {(h0 /. 1)} ::_thesis: ( (Segment ((h0 /. ((len h0) -' 1)),(h0 /. (len h0)),P)) /\ (Segment ((h0 /. (len h0)),(h0 /. 1),P)) = {(h0 /. (len h0))} & Segment ((h0 /. ((len h0) -' 1)),(h0 /. (len h0)),P) misses Segment ((h0 /. 1),(h0 /. 2),P) & ( for i, j being Element of NAT st 1 <= i & i < j & j < len h0 & not i,j are_adjacent1 holds
Segment ((h0 /. i),(h0 /. (i + 1)),P) misses Segment ((h0 /. j),(h0 /. (j + 1)),P) ) & ( for i being Element of NAT st 1 < i & i + 1 < len h0 holds
Segment ((h0 /. (len h0)),(h0 /. 1),P) misses Segment ((h0 /. i),(h0 /. (i + 1)),P) ) )
proof
defpred S1[ Element of NAT ] means ( $1 + 2 <= len h0 implies LE h0 /. 2,h0 /. ($1 + 2),P );
set j = (((len h0) -' (len h11)) + 2) -' 1;
A666: (len h0) - (len h11) <= ((len h11) + (len (mid (h21,2,((len h21) -' 1))))) - (len h11) by FINSEQ_1:22;
A667: h0 /. 2 = h0 . 2 by A663, PARTFUN1:def_6;
A668: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A669: ( k + 2 <= len h0 implies LE h0 /. 2,h0 /. (k + 2),P ) ; ::_thesis: S1[k + 1]
now__::_thesis:_(_(k_+_1)_+_2_<=_len_h0_implies_LE_h0_/._2,h0_/._((k_+_1)_+_2),P_)
A670: (k + 1) + 1 = k + 2 ;
A671: (k + 1) + 2 = (k + 2) + 1 ;
assume A672: (k + 1) + 2 <= len h0 ; ::_thesis: LE h0 /. 2,h0 /. ((k + 1) + 2),P
then k + 2 < len h0 by A671, NAT_1:13;
then LE h0 /. (k + 2),h0 /. ((k + 2) + 1),P by A592, A671, A670, NAT_1:11;
hence LE h0 /. 2,h0 /. ((k + 1) + 2),P by A1, A669, A672, JORDAN6:58, NAT_1:13; ::_thesis: verum
end;
hence S1[k + 1] ; ::_thesis: verum
end;
(len h0) -' 2 = (len h0) - 2 by A65, A14, XREAL_1:233, XXREAL_0:2;
then A673: ((len h0) -' 2) + 2 = len h0 ;
0 + 2 <= ((len h0) -' (len h11)) + 2 by XREAL_1:6;
then A674: 1 <= (((len h0) -' (len h11)) + 2) -' 1 by Lm1, NAT_D:42;
(((len h0) -' (len h11)) + 2) -' 1 <= len h21 by A36, A52, A55, A57, A252, NAT_D:44;
then A675: (((len h0) -' (len h11)) + 2) -' 1 in dom h2 by A46, A674, FINSEQ_3:25;
( h0 . 2 = g1 . (h1 . 2) & h1 . 2 in rng h1 ) by A15, A40, FUNCT_1:13, FUNCT_1:def_3;
then A676: h0 /. 2 in Upper_Arc P by A5, A132, A11, A667, BORSUK_1:40, FUNCT_1:def_3;
(Upper_Arc P) \/ (Lower_Arc P) = P by A1, JORDAN6:50;
then h0 /. 2 in P by A676, XBOOLE_0:def_3;
then A677: S1[ 0 ] by A1, JORDAN6:56;
A678: for i being Element of NAT holds S1[i] from NAT_1:sch_1(A677, A668);
A679: h11 . 2 <> W-min P by A38, A17, A71, A15, A20, FUNCT_1:def_4;
( ((len h1) + 1) - (len h1) <= (len h0) - (len h1) & h0 . (len h0) = (mid (h21,2,((len h21) -' 1))) . ((len h0) - (len h11)) ) by A39, A36, A549, A62, A64, FINSEQ_1:23, XXREAL_0:2;
then h0 . (len h0) = h21 . ((((len h0) -' (len h11)) + 2) -' 1) by A39, A48, A56, A50, A54, A252, A666, FINSEQ_6:118;
then A680: h0 . (len h0) = g2 . (h2 . ((((len h0) -' (len h11)) + 2) -' 1)) by A675, FUNCT_1:13;
A681: now__::_thesis:_not_h0_/._2_=_h0_/._(len_h0)
h2 . ((((len h0) -' (len h11)) + 2) -' 1) in rng h2 by A675, FUNCT_1:def_3;
then A682: g2 . (h2 . ((((len h0) -' (len h11)) + 2) -' 1)) in rng g2 by A131, A29, BORSUK_1:40, FUNCT_1:def_3;
assume h0 /. 2 = h0 /. (len h0) ; ::_thesis: contradiction
then h0 /. 2 in (Upper_Arc P) /\ (Lower_Arc P) by A23, A75, A676, A680, A682, XBOOLE_0:def_4;
then h0 /. 2 in {(W-min P),(E-max P)} by A1, JORDAN6:def_9;
then h11 . 2 = E-max P by A40, A679, A667, TARSKI:def_2;
hence contradiction by A38, A665, A255, A14, A15, A20, FUNCT_1:def_4; ::_thesis: verum
end;
h0 /. 2 <> W-min P by A663, A40, A679, PARTFUN1:def_6;
hence (Segment ((h0 /. (len h0)),(h0 /. 1),P)) /\ (Segment ((h0 /. 1),(h0 /. 2),P)) = {(h0 /. 1)} by A1, A73, A681, A678, A673, Th12; ::_thesis: verum
end;
A683: ((len h0) -' 1) + 1 = len h0 by A16, A65, XREAL_1:235, XXREAL_0:2;
then ( LE h0 /. ((len h0) -' 1),h0 /. (((len h0) -' 1) + 1),P & h0 /. (((len h0) -' 1) + 1) <> W-min P ) by A256, A202;
hence (Segment ((h0 /. ((len h0) -' 1)),(h0 /. (len h0)),P)) /\ (Segment ((h0 /. (len h0)),(h0 /. 1),P)) = {(h0 /. (len h0))} by A1, A73, A683, A664, Th11; ::_thesis: ( Segment ((h0 /. ((len h0) -' 1)),(h0 /. (len h0)),P) misses Segment ((h0 /. 1),(h0 /. 2),P) & ( for i, j being Element of NAT st 1 <= i & i < j & j < len h0 & not i,j are_adjacent1 holds
Segment ((h0 /. i),(h0 /. (i + 1)),P) misses Segment ((h0 /. j),(h0 /. (j + 1)),P) ) & ( for i being Element of NAT st 1 < i & i + 1 < len h0 holds
Segment ((h0 /. (len h0)),(h0 /. 1),P) misses Segment ((h0 /. i),(h0 /. (i + 1)),P) ) )
LE h0 /. ((len h0) -' 1),h0 /. (((len h0) -' 1) + 1),P by A256, A202, A200;
hence Segment ((h0 /. ((len h0) -' 1)),(h0 /. (len h0)),P) misses Segment ((h0 /. 1),(h0 /. 2),P) by A1, A683, A333, A229, A207, Th13; ::_thesis: ( ( for i, j being Element of NAT st 1 <= i & i < j & j < len h0 & not i,j are_adjacent1 holds
Segment ((h0 /. i),(h0 /. (i + 1)),P) misses Segment ((h0 /. j),(h0 /. (j + 1)),P) ) & ( for i being Element of NAT st 1 < i & i + 1 < len h0 holds
Segment ((h0 /. (len h0)),(h0 /. 1),P) misses Segment ((h0 /. i),(h0 /. (i + 1)),P) ) )
thus for i, j being Element of NAT st 1 <= i & i < j & j < len h0 & not i,j are_adjacent1 holds
Segment ((h0 /. i),(h0 /. (i + 1)),P) misses Segment ((h0 /. j),(h0 /. (j + 1)),P) ::_thesis: for i being Element of NAT st 1 < i & i + 1 < len h0 holds
Segment ((h0 /. (len h0)),(h0 /. 1),P) misses Segment ((h0 /. i),(h0 /. (i + 1)),P)
proof
let i, j be Element of NAT ; ::_thesis: ( 1 <= i & i < j & j < len h0 & not i,j are_adjacent1 implies Segment ((h0 /. i),(h0 /. (i + 1)),P) misses Segment ((h0 /. j),(h0 /. (j + 1)),P) )
assume that
A684: 1 <= i and
A685: i < j and
A686: j < len h0 and
A687: not i,j are_adjacent1 ; ::_thesis: Segment ((h0 /. i),(h0 /. (i + 1)),P) misses Segment ((h0 /. j),(h0 /. (j + 1)),P)
A688: 1 < j by A684, A685, XXREAL_0:2;
i < len h0 by A685, A686, XXREAL_0:2;
then A689: i + 1 <= len h0 by NAT_1:13;
then A690: ( LE h0 /. i,h0 /. (i + 1),P & h0 /. i <> h0 /. (i + 1) ) by A256, A684;
A691: i + 1 <= j by A685, NAT_1:13;
then A692: i + 1 < len h0 by A686, XXREAL_0:2;
A693: not j = i + 1 by A687, GOBRD10:def_1;
then A694: i + 1 < j by A691, XXREAL_0:1;
A695: now__::_thesis:_not_h0_/._(i_+_1)_=_h0_/._j
assume A696: h0 /. (i + 1) = h0 /. j ; ::_thesis: contradiction
percases ( i + 1 <= len h1 or i + 1 > len h1 ) ;
supposeA697: i + 1 <= len h1 ; ::_thesis: contradiction
A698: 1 < i + 1 by A684, NAT_1:13;
then A699: i + 1 in dom h1 by A697, FINSEQ_3:25;
then A700: h1 . (i + 1) in rng h1 by FUNCT_1:def_3;
i + 1 in dom h0 by A689, A698, FINSEQ_3:25;
then A701: h0 /. (i + 1) = h0 . (i + 1) by PARTFUN1:def_6;
A702: h0 . (i + 1) = h11 . (i + 1) by A39, A697, A698, FINSEQ_1:64;
then h0 . (i + 1) = g1 . (h1 . (i + 1)) by A699, FUNCT_1:13;
then A703: h0 . (i + 1) in Upper_Arc P by A5, A132, A11, A700, BORSUK_1:40, FUNCT_1:def_3;
percases ( j <= len h1 or j > len h1 ) ;
supposeA704: j <= len h1 ; ::_thesis: contradiction
j in dom h0 by A686, A688, FINSEQ_3:25;
then A705: h0 /. j = h0 . j by PARTFUN1:def_6;
( h0 . j = h11 . j & j in dom h1 ) by A39, A688, A704, FINSEQ_1:64, FINSEQ_3:25;
hence contradiction by A38, A20, A693, A696, A699, A701, A702, A705, FUNCT_1:def_4; ::_thesis: verum
end;
supposeA706: j > len h1 ; ::_thesis: contradiction
j in dom h0 by A686, A688, FINSEQ_3:25;
then A707: h0 /. j = h0 . j by PARTFUN1:def_6;
A708: j - (len h11) = j -' (len h11) by A39, A706, XREAL_1:233;
j - (len h11) <= ((len h1) + ((len h2) - 2)) - (len h11) by A39, A47, A36, A52, A55, A57, A686, XREAL_1:9;
then (j -' (len h11)) + 2 <= ((len h2) - 2) + 2 by A39, A708, XREAL_1:6;
then A709: ((j -' (len h11)) + 2) -' 1 <= len h21 by A47, NAT_D:44;
j - (len h11) <= ((len h1) + ((len h2) - 2)) - (len h11) by A39, A47, A36, A52, A55, A57, A686, XREAL_1:9;
then A710: (j -' (len h11)) + 2 <= ((len h2) - 2) + 2 by A39, A708, XREAL_1:6;
set k = ((j -' (len h11)) + 2) -' 1;
j <= (len h11) + (len (mid (h21,2,((len h21) -' 1)))) by A686, FINSEQ_1:22;
then A711: j - (len h11) <= ((len h11) + (len (mid (h21,2,((len h21) -' 1))))) - (len h11) by XREAL_1:9;
A712: 0 + 2 <= (j -' (len h11)) + 2 by XREAL_1:6;
then A713: ((j -' (len h11)) + 2) -' 1 = ((j -' (len h11)) + 2) - 1 by Lm1, NAT_D:39, NAT_D:42;
1 <= ((j -' (len h11)) + 2) -' 1 by A712, Lm1, NAT_D:42;
then A714: ((j -' (len h11)) + 2) -' 1 in dom h2 by A46, A709, FINSEQ_3:25;
then h2 . (((j -' (len h11)) + 2) -' 1) in rng h2 by FUNCT_1:def_3;
then A715: g2 . (h2 . (((j -' (len h11)) + 2) -' 1)) in rng g2 by A131, A29, BORSUK_1:40, FUNCT_1:def_3;
A716: (len h1) + 1 <= j by A706, NAT_1:13;
then ( h0 . j = (mid (h21,2,((len h21) -' 1))) . (j - (len h11)) & ((len h1) + 1) - (len h1) <= j - (len h1) ) by A39, A36, A686, FINSEQ_1:23, XREAL_1:9;
then A717: h0 . j = h21 . (((j -' (len h11)) + 2) -' 1) by A39, A48, A56, A50, A54, A708, A711, FINSEQ_6:118;
then h0 . j = g2 . (h2 . (((j -' (len h11)) + 2) -' 1)) by A714, FUNCT_1:13;
then h0 . j in (Upper_Arc P) /\ (Lower_Arc P) by A23, A696, A701, A703, A707, A715, XBOOLE_0:def_4;
then A718: h0 . j in {(W-min P),(E-max P)} by A1, JORDAN6:def_9;
((len h11) + 1) - (len h11) <= j - (len h11) by A39, A716, XREAL_1:9;
then 1 <= j -' (len h11) by NAT_D:39;
then 1 + 2 <= (j -' (len h11)) + 2 by XREAL_1:6;
then (1 + 2) - 1 <= ((j -' (len h11)) + 2) - 1 by XREAL_1:9;
then 1 < ((j -' (len h11)) + 2) -' 1 by A713, XXREAL_0:2;
then A719: h0 . j <> E-max P by A46, A76, A77, A32, A717, A714, FUNCT_1:def_4;
((j -' (len h11)) + 2) -' 1 < (((j -' (len h11)) + 2) -' 1) + 1 by NAT_1:13;
then h0 . j <> W-min P by A46, A34, A35, A32, A717, A713, A710, A714, FUNCT_1:def_4;
hence contradiction by A718, A719, TARSKI:def_2; ::_thesis: verum
end;
end;
end;
supposeA720: i + 1 > len h1 ; ::_thesis: contradiction
then A721: j > len h1 by A691, XXREAL_0:2;
then A722: (len h1) + 1 <= j by NAT_1:13;
then A723: ((len h1) + 1) - (len h1) <= j - (len h1) by XREAL_1:9;
((len h11) + 1) - (len h11) <= j - (len h11) by A39, A722, XREAL_1:9;
then A724: j -' (len h11) = j - (len h11) by NAT_D:39;
A725: (len h1) + 1 <= i + 1 by A720, NAT_1:13;
then ((len h11) + 1) - (len h11) <= (i + 1) - (len h11) by A39, XREAL_1:9;
then (i + 1) -' (len h11) = (i + 1) - (len h11) by NAT_D:39;
then (i + 1) -' (len h11) < j -' (len h11) by A694, A724, XREAL_1:9;
then A726: ((i + 1) -' (len h11)) + 2 < (j -' (len h11)) + 2 by XREAL_1:6;
set k = ((j -' (len h11)) + 2) -' 1;
set j0 = (((i + 1) -' (len h11)) + 2) -' 1;
A727: j <= (len h11) + (len (mid (h21,2,((len h21) -' 1)))) by A686, FINSEQ_1:22;
A728: (i + 1) - (len h11) <= ((len h11) + (len (mid (h21,2,((len h21) -' 1))))) - (len h11) by A36, A689, XREAL_1:9;
A729: j - (len h11) = j -' (len h11) by A39, A691, A720, XREAL_1:233, XXREAL_0:2;
j - (len h11) <= ((len h1) + ((len h2) - 2)) - (len h11) by A39, A47, A36, A52, A55, A57, A686, XREAL_1:9;
then (j -' (len h11)) + 2 <= ((len h2) - 2) + 2 by A39, A729, XREAL_1:6;
then A730: ((j -' (len h11)) + 2) -' 1 <= len h21 by A47, NAT_D:44;
A731: 0 + 2 <= (j -' (len h11)) + 2 by XREAL_1:6;
then A732: ((j -' (len h11)) + 2) -' 1 = ((j -' (len h11)) + 2) - 1 by Lm1, NAT_D:39, NAT_D:42;
1 <= ((j -' (len h11)) + 2) -' 1 by A731, Lm1, NAT_D:42;
then A733: ((j -' (len h11)) + 2) -' 1 in dom h2 by A46, A730, FINSEQ_3:25;
A734: (i + 1) - (len h11) = (i + 1) -' (len h11) by A39, A720, XREAL_1:233;
(len h11) + 1 <= j by A39, A721, NAT_1:13;
then A735: h0 . j = (mid (h21,2,((len h21) -' 1))) . (j - (len h11)) by A727, FINSEQ_1:23;
j - (len h11) <= ((len h11) + (len (mid (h21,2,((len h21) -' 1))))) - (len h11) by A36, A686, XREAL_1:9;
then A736: h0 . j = h21 . (((j -' (len h11)) + 2) -' 1) by A39, A48, A56, A50, A54, A735, A729, A723, FINSEQ_6:118;
1 <= i + 1 by A684, NAT_1:13;
then i + 1 in dom h0 by A689, FINSEQ_3:25;
then A737: h0 /. (i + 1) = h0 . (i + 1) by PARTFUN1:def_6;
j in dom h0 by A686, A688, FINSEQ_3:25;
then A738: h0 /. j = h0 . j by PARTFUN1:def_6;
(i + 1) - (len h11) <= ((len h1) + ((len h2) - 2)) - (len h11) by A39, A47, A36, A52, A55, A57, A689, XREAL_1:9;
then ((i + 1) -' (len h11)) + 2 <= ((len h2) - 2) + 2 by A39, A734, XREAL_1:6;
then A739: (((i + 1) -' (len h11)) + 2) -' 1 <= len h21 by A47, NAT_D:44;
A740: 0 + 2 <= ((i + 1) -' (len h11)) + 2 by XREAL_1:6;
then 1 <= (((i + 1) -' (len h11)) + 2) -' 1 by Lm1, NAT_D:42;
then A741: (((i + 1) -' (len h11)) + 2) -' 1 in dom h2 by A46, A739, FINSEQ_3:25;
(((i + 1) -' (len h11)) + 2) -' 1 = (((i + 1) -' (len h11)) + 2) - 1 by A740, Lm1, NAT_D:39, NAT_D:42;
then A742: (((i + 1) -' (len h11)) + 2) -' 1 < ((j -' (len h11)) + 2) -' 1 by A732, A726, XREAL_1:9;
( h0 . (i + 1) = (mid (h21,2,((len h21) -' 1))) . ((i + 1) - (len h11)) & ((len h1) + 1) - (len h1) <= (i + 1) - (len h1) ) by A39, A36, A689, A725, FINSEQ_1:23, XREAL_1:9;
then h0 . (i + 1) = h21 . ((((i + 1) -' (len h11)) + 2) -' 1) by A39, A48, A56, A50, A54, A734, A728, FINSEQ_6:118;
hence contradiction by A46, A32, A696, A741, A737, A736, A742, A733, A738, FUNCT_1:def_4; ::_thesis: verum
end;
end;
end;
A743: j + 1 <= len h0 by A686, NAT_1:13;
A744: 1 < i + 1 by A684, NAT_1:13;
A745: 1 <= i + 1 by A684, NAT_1:13;
A746: i + 1 < len h0 by A686, A691, XXREAL_0:2;
A747: LE h0 /. (i + 1),h0 /. j,P
proof
percases ( i + 1 <= len h1 or i + 1 > len h1 ) ;
supposeA748: i + 1 <= len h1 ; ::_thesis: LE h0 /. (i + 1),h0 /. j,P
percases ( j <= len h1 or j > len h1 ) ;
supposeA749: j <= len h1 ; ::_thesis: LE h0 /. (i + 1),h0 /. j,P
A750: 1 < j by A694, A745, XXREAL_0:2;
then A751: j in dom h1 by A749, FINSEQ_3:25;
then A752: h1 . j in rng h1 by FUNCT_1:def_3;
then A753: h1 . j <= 1 by A11, BORSUK_1:40, XXREAL_1:1;
j in dom h0 by A686, A750, FINSEQ_3:25;
then A754: h0 /. j = h0 . j by PARTFUN1:def_6;
h0 . j = h11 . j by A39, A749, A750, FINSEQ_1:64;
then A755: g1 . (h1 . j) = h0 /. j by A751, A754, FUNCT_1:13;
then A756: h0 /. j in Upper_Arc P by A5, A132, A11, A752, BORSUK_1:40, FUNCT_1:def_3;
i + 1 in dom h0 by A745, A692, FINSEQ_3:25;
then A757: h0 /. (i + 1) = h0 . (i + 1) by PARTFUN1:def_6;
A758: Upper_Arc P is_an_arc_of W-min P, E-max P by A1, JORDAN6:def_8;
A759: i + 1 in dom h1 by A745, A748, FINSEQ_3:25;
then A760: h1 . (i + 1) in rng h1 by FUNCT_1:def_3;
then A761: ( 0 <= h1 . (i + 1) & h1 . (i + 1) <= 1 ) by A11, BORSUK_1:40, XXREAL_1:1;
h0 . (i + 1) = h11 . (i + 1) by A39, A745, A748, FINSEQ_1:64;
then A762: g1 . (h1 . (i + 1)) = h0 /. (i + 1) by A759, A757, FUNCT_1:13;
then A763: h0 /. (i + 1) in Upper_Arc P by A5, A132, A11, A760, BORSUK_1:40, FUNCT_1:def_3;
h1 . (i + 1) <= h1 . j by A12, A694, A759, A751, SEQM_3:def_1;
then LE h0 /. (i + 1),h0 /. j, Upper_Arc P, W-min P, E-max P by A4, A5, A6, A7, A758, A762, A761, A755, A753, Th18;
hence LE h0 /. (i + 1),h0 /. j,P by A763, A756, JORDAN6:def_10; ::_thesis: verum
end;
supposeA764: j > len h1 ; ::_thesis: LE h0 /. (i + 1),h0 /. j,P
set k = ((j -' (len h11)) + 2) -' 1;
0 + 2 <= (j -' (len h11)) + 2 by XREAL_1:6;
then A765: 2 -' 1 <= ((j -' (len h11)) + 2) -' 1 by NAT_D:42;
A766: j - (len h11) = j -' (len h11) by A39, A764, XREAL_1:233;
j - (len h11) <= ((len h1) + ((len h2) - 2)) - (len h11) by A39, A47, A36, A52, A55, A57, A686, XREAL_1:9;
then (j -' (len h11)) + 2 <= ((len h2) - 2) + 2 by A39, A766, XREAL_1:6;
then ((j -' (len h11)) + 2) -' 1 <= len h21 by A47, NAT_D:44;
then A767: ((j -' (len h11)) + 2) -' 1 in dom h21 by A765, Lm1, FINSEQ_3:25;
(j + 1) - 1 <= ((len h1) + ((len h2) - 2)) - 1 by A39, A47, A36, A52, A55, A57, A743, XREAL_1:9;
then j - (len h11) <= ((len h1) + (((len h2) - 2) - 1)) - (len h11) by XREAL_1:9;
then (j -' (len h11)) + 2 <= (((len h2) - 2) - 1) + 2 by A39, A766, XREAL_1:6;
then A768: ((j -' (len h11)) + 2) - 1 <= ((len h2) - 1) - 1 by XREAL_1:9;
A769: h0 . (i + 1) = h11 . (i + 1) by A39, A744, A748, FINSEQ_1:64;
i + 1 in dom h1 by A744, A748, FINSEQ_3:25;
then h1 . (i + 1) in rng h1 by FUNCT_1:def_3;
then A770: g1 . (h1 . (i + 1)) in rng g1 by A132, A11, BORSUK_1:40, FUNCT_1:def_3;
0 + 1 <= (((j -' (len h11)) + 1) + 1) - 1 by XREAL_1:6;
then A771: ((j -' (len h11)) + 2) -' 1 = ((j -' (len h11)) + 2) - 1 by NAT_D:39;
(len h1) + 1 <= j by A764, NAT_1:13;
then A772: ( h0 . j = (mid (h21,2,((len h21) -' 1))) . (j - (len h11)) & ((len h1) + 1) - (len h1) <= j - (len h1) ) by A39, A36, A686, FINSEQ_1:23, XREAL_1:9;
A773: j - (len h11) <= ((len h11) + (len (mid (h21,2,((len h21) -' 1))))) - (len h11) by A36, A686, XREAL_1:9;
then h0 . j = h21 . (((j -' (len h11)) + 2) -' 1) by A39, A48, A56, A50, A54, A766, A772, FINSEQ_6:118;
then A774: h0 . j = g2 . (h2 . (((j -' (len h11)) + 2) -' 1)) by A46, A767, FUNCT_1:13;
j - (len h11) = j -' (len h11) by A39, A764, XREAL_1:233;
then A775: h0 . j = h21 . (((j -' (len h11)) + 2) -' 1) by A39, A48, A56, A50, A54, A773, A772, FINSEQ_6:118;
j in dom h0 by A686, A688, FINSEQ_3:25;
then A776: h0 /. j = h0 . j by PARTFUN1:def_6;
h2 . (((j -' (len h11)) + 2) -' 1) in rng h2 by A46, A767, FUNCT_1:def_3;
then A777: h0 . j in Lower_Arc P by A23, A131, A29, A774, BORSUK_1:40, FUNCT_1:def_3;
i + 1 in Seg (len h1) by A745, A748, FINSEQ_1:1;
then i + 1 in dom h1 by FINSEQ_1:def_3;
then A778: h11 . (i + 1) = g1 . (h1 . (i + 1)) by FUNCT_1:13;
i + 1 in dom h0 by A745, A692, FINSEQ_3:25;
then A779: h0 /. (i + 1) = h0 . (i + 1) by PARTFUN1:def_6;
((len h2) - 1) - 1 < len h2 by Lm4;
then h0 /. j <> W-min P by A46, A34, A35, A32, A771, A767, A768, A775, A776, FUNCT_1:def_4;
hence LE h0 /. (i + 1),h0 /. j,P by A5, A769, A778, A779, A776, A770, A777, JORDAN6:def_10; ::_thesis: verum
end;
end;
end;
supposeA780: i + 1 > len h1 ; ::_thesis: LE h0 /. (i + 1),h0 /. j,P
set j0 = (((i + 1) -' (len h11)) + 2) -' 1;
set k = ((j -' (len h11)) + 2) -' 1;
A781: 0 + 2 <= ((i + 1) -' (len h11)) + 2 by XREAL_1:6;
then A782: 1 <= (((i + 1) -' (len h11)) + 2) -' 1 by Lm1, NAT_D:42;
A783: j - (len h11) = j -' (len h11) by A39, A691, A780, XREAL_1:233, XXREAL_0:2;
len h1 < j by A691, A780, XXREAL_0:2;
then A784: (len h11) + 1 <= j by A39, NAT_1:13;
then A785: ((len h1) + 1) - (len h1) <= j - (len h1) by A39, XREAL_1:9;
j <= (len h11) + (len (mid (h21,2,((len h21) -' 1)))) by A686, FINSEQ_1:22;
then A786: h0 . j = (mid (h21,2,((len h21) -' 1))) . (j - (len h11)) by A784, FINSEQ_1:23;
A787: (i + 1) - (len h11) < ((len h11) + (len (mid (h21,2,((len h21) -' 1))))) - (len h11) by A36, A746, XREAL_1:9;
then j - (len h11) <= len (mid (h21,2,((len h21) -' 1))) by A36, A686, XREAL_1:9;
then A788: h0 . j = h21 . (((j -' (len h11)) + 2) -' 1) by A39, A48, A56, A50, A54, A783, A786, A785, FINSEQ_6:118;
A789: (i + 1) - (len h11) = (i + 1) -' (len h11) by A39, A780, XREAL_1:233;
then ((i + 1) -' (len h11)) + 2 <= ((len h2) - 2) + 2 by A47, A52, A55, A57, A787, XREAL_1:6;
then (((i + 1) -' (len h11)) + 2) -' 1 <= len h21 by A47, NAT_D:44;
then A790: (((i + 1) -' (len h11)) + 2) -' 1 in dom h2 by A46, A782, FINSEQ_3:25;
then A791: h2 . ((((i + 1) -' (len h11)) + 2) -' 1) in rng h2 by FUNCT_1:def_3;
then A792: ( 0 <= h2 . ((((i + 1) -' (len h11)) + 2) -' 1) & h2 . ((((i + 1) -' (len h11)) + 2) -' 1) <= 1 ) by A29, BORSUK_1:40, XXREAL_1:1;
A793: g2 . (h2 . ((((i + 1) -' (len h11)) + 2) -' 1)) in rng g2 by A131, A29, A791, BORSUK_1:40, FUNCT_1:def_3;
A794: j - (len h11) = j -' (len h11) by A39, A691, A780, XREAL_1:233, XXREAL_0:2;
(j + 1) - 1 <= ((len h1) + ((len h2) - 2)) - 1 by A39, A47, A36, A52, A55, A57, A743, XREAL_1:9;
then j - (len h11) <= ((len h1) + (((len h2) - 2) - 1)) - (len h11) by XREAL_1:9;
then (j -' (len h11)) + 2 <= (((len h2) - 2) - 1) + 2 by A39, A794, XREAL_1:6;
then A795: ((j -' (len h11)) + 2) - 1 <= ((len h2) - 1) - 1 by XREAL_1:9;
0 + 1 <= (((j -' (len h11)) + 1) + 1) - 1 by XREAL_1:6;
then A796: ((j -' (len h11)) + 2) -' 1 = ((j -' (len h11)) + 2) - 1 by NAT_D:39;
j - (len h11) <= ((len h1) + ((len h2) - 2)) - (len h11) by A39, A47, A36, A52, A55, A57, A686, XREAL_1:9;
then (j -' (len h11)) + 2 <= ((len h2) - 2) + 2 by A39, A794, XREAL_1:6;
then A797: ((j -' (len h11)) + 2) -' 1 <= len h21 by A47, NAT_D:44;
0 + 2 <= (j -' (len h11)) + 2 by XREAL_1:6;
then 2 -' 1 <= ((j -' (len h11)) + 2) -' 1 by NAT_D:42;
then A798: ((j -' (len h11)) + 2) -' 1 in dom h21 by A797, Lm1, FINSEQ_3:25;
then A799: h2 . (((j -' (len h11)) + 2) -' 1) in rng h2 by A46, FUNCT_1:def_3;
then A800: h2 . (((j -' (len h11)) + 2) -' 1) <= 1 by A29, BORSUK_1:40, XXREAL_1:1;
j - (len h11) <= ((len h11) + (len (mid (h21,2,((len h21) -' 1))))) - (len h11) by A36, A686, XREAL_1:9;
then A801: h0 . j = h21 . (((j -' (len h11)) + 2) -' 1) by A39, A48, A56, A50, A54, A794, A786, A785, FINSEQ_6:118;
then h0 . j = g2 . (h2 . (((j -' (len h11)) + 2) -' 1)) by A46, A798, FUNCT_1:13;
then A802: h0 . j in Lower_Arc P by A23, A131, A29, A799, BORSUK_1:40, FUNCT_1:def_3;
A803: (((i + 1) -' (len h11)) + 2) -' 1 = (((i + 1) -' (len h11)) + 2) - 1 by A781, Lm1, NAT_D:39, NAT_D:42;
A804: i + 1 < (len h11) + (len (mid (h21,2,((len h21) -' 1)))) by A746, FINSEQ_1:22;
(i + 1) - (len h11) < j - (len h11) by A694, XREAL_1:9;
then ((i + 1) -' (len h11)) + 2 < (j -' (len h11)) + 2 by A783, A789, XREAL_1:6;
then (((i + 1) -' (len h11)) + 2) -' 1 < ((j -' (len h11)) + 2) -' 1 by A796, A803, XREAL_1:9;
then A805: h2 . ((((i + 1) -' (len h11)) + 2) -' 1) < h2 . (((j -' (len h11)) + 2) -' 1) by A30, A46, A798, A790, SEQM_3:def_1;
i + 1 in dom h0 by A745, A692, FINSEQ_3:25;
then A806: h0 /. (i + 1) = h0 . (i + 1) by PARTFUN1:def_6;
(len h1) + 1 <= i + 1 by A780, NAT_1:13;
then A807: ((len h1) + 1) - (len h1) <= (i + 1) - (len h1) by XREAL_1:9;
then A808: (i + 1) -' (len h11) = (i + 1) - (len h11) by A39, NAT_D:39;
j in dom h0 by A686, A688, FINSEQ_3:25;
then A809: h0 /. j = h0 . j by PARTFUN1:def_6;
(len h11) + 1 <= i + 1 by A39, A780, NAT_1:13;
then h0 . (i + 1) = (mid (h21,2,((len h21) -' 1))) . ((i + 1) - (len h11)) by A804, FINSEQ_1:23
.= h21 . ((((i + 1) -' (len h11)) + 2) -' 1) by A39, A48, A56, A50, A54, A787, A807, A808, FINSEQ_6:118 ;
then A810: h0 . (i + 1) = g2 . (h2 . ((((i + 1) -' (len h11)) + 2) -' 1)) by A790, FUNCT_1:13;
((len h2) - 1) - 1 < len h2 by Lm4;
then A811: h0 /. j <> W-min P by A46, A34, A35, A32, A796, A798, A795, A788, A809, FUNCT_1:def_4;
h21 . (((j -' (len h11)) + 2) -' 1) = g2 . (h2 . (((j -' (len h11)) + 2) -' 1)) by A46, A798, FUNCT_1:13;
then LE h0 /. (i + 1),h0 /. j, Lower_Arc P, E-max P, W-min P by A21, A22, A23, A24, A25, A801, A800, A810, A792, A805, A806, A809, Th18;
hence LE h0 /. (i + 1),h0 /. j,P by A23, A810, A806, A809, A793, A802, A811, JORDAN6:def_10; ::_thesis: verum
end;
end;
end;
LE h0 /. j,h0 /. (j + 1),P by A256, A688, A743;
hence Segment ((h0 /. i),(h0 /. (i + 1)),P) misses Segment ((h0 /. j),(h0 /. (j + 1)),P) by A1, A690, A747, A695, Th13; ::_thesis: verum
end;
let i be Element of NAT ; ::_thesis: ( 1 < i & i + 1 < len h0 implies Segment ((h0 /. (len h0)),(h0 /. 1),P) misses Segment ((h0 /. i),(h0 /. (i + 1)),P) )
assume that
A812: 1 < i and
A813: i + 1 < len h0 ; ::_thesis: Segment ((h0 /. (len h0)),(h0 /. 1),P) misses Segment ((h0 /. i),(h0 /. (i + 1)),P)
A814: 1 < i + 1 by A812, NAT_1:13;
then A815: i + 1 in dom h0 by A813, FINSEQ_3:25;
A816: 1 <= (len h0) - (len h1) by A549, A62, XXREAL_0:2;
A817: now__::_thesis:_not_h0_/._(i_+_1)_=_h0_/._(len_h0)
assume A818: h0 /. (i + 1) = h0 /. (len h0) ; ::_thesis: contradiction
percases ( i + 1 <= len h1 or i + 1 > len h1 ) ;
supposeA819: i + 1 <= len h1 ; ::_thesis: contradiction
then A820: i + 1 in dom h1 by A814, FINSEQ_3:25;
h0 . (i + 1) = h11 . (i + 1) by A39, A814, A819, FINSEQ_1:64;
then A821: h0 . (i + 1) = g1 . (h1 . (i + 1)) by A820, FUNCT_1:13;
h1 . (i + 1) in rng h1 by A820, FUNCT_1:def_3;
then A822: h0 . (i + 1) in Upper_Arc P by A5, A132, A11, A821, BORSUK_1:40, FUNCT_1:def_3;
i + 1 in dom h0 by A813, A814, FINSEQ_3:25;
then A823: h0 /. (i + 1) = h0 . (i + 1) by PARTFUN1:def_6;
1 + 2 <= ((len h0) -' (len h11)) + 2 by A47, A36, A52, A55, A57, A63, A252, XREAL_1:6;
then A824: (1 + 2) - 1 <= (((len h0) -' (len h11)) + 2) - 1 by XREAL_1:9;
set k = (((len h0) -' (len h11)) + 2) -' 1;
A825: 0 + 2 <= ((len h0) -' (len h11)) + 2 by XREAL_1:6;
then A826: 2 -' 1 <= (((len h0) -' (len h11)) + 2) -' 1 by NAT_D:42;
(((len h0) -' (len h11)) + 2) -' 1 <= len h21 by A36, A52, A55, A57, A252, NAT_D:44;
then A827: (((len h0) -' (len h11)) + 2) -' 1 in dom h2 by A46, A826, Lm1, FINSEQ_3:25;
then h2 . ((((len h0) -' (len h11)) + 2) -' 1) in rng h2 by FUNCT_1:def_3;
then A828: g2 . (h2 . ((((len h0) -' (len h11)) + 2) -' 1)) in rng g2 by A131, A29, BORSUK_1:40, FUNCT_1:def_3;
h0 . (len h0) = (mid (h21,2,((len h21) -' 1))) . ((len h0) - (len h11)) by A39, A36, A64, FINSEQ_1:23;
then A829: h0 . (len h0) = h21 . ((((len h0) -' (len h11)) + 2) -' 1) by A39, A36, A48, A56, A50, A54, A816, A252, FINSEQ_6:118;
then h0 . (len h0) = g2 . (h2 . ((((len h0) -' (len h11)) + 2) -' 1)) by A827, FUNCT_1:13;
then h0 . (len h0) in (Upper_Arc P) /\ (Lower_Arc P) by A23, A75, A818, A823, A822, A828, XBOOLE_0:def_4;
then A830: h0 . (len h0) in {(W-min P),(E-max P)} by A1, JORDAN6:def_9;
(((len h0) -' (len h11)) + 2) -' 1 = (((len h0) -' (len h11)) + 2) - 1 by A825, Lm1, NAT_D:39, NAT_D:42;
then 1 < (((len h0) -' (len h11)) + 2) -' 1 by A824, XXREAL_0:2;
then A831: h0 . (len h0) <> E-max P by A46, A76, A77, A32, A829, A827, FUNCT_1:def_4;
(((len h0) -' (len h11)) + 2) -' 1 < ((((len h0) -' (len h11)) + 2) -' 1) + 1 by NAT_1:13;
then h0 . (len h0) <> W-min P by A46, A47, A34, A35, A36, A52, A55, A57, A252, A32, A829, A827, FUNCT_1:def_4;
hence contradiction by A830, A831, TARSKI:def_2; ::_thesis: verum
end;
supposeA832: i + 1 > len h1 ; ::_thesis: contradiction
set k = (((len h0) -' (len h11)) + 2) -' 1;
set j0 = (((i + 1) -' (len h11)) + 2) -' 1;
A833: 0 + 2 <= ((len h0) -' (len h11)) + 2 by XREAL_1:6;
then A834: 2 -' 1 <= (((len h0) -' (len h11)) + 2) -' 1 by NAT_D:42;
(((len h0) -' (len h11)) + 2) -' 1 <= len h21 by A36, A52, A55, A57, A252, NAT_D:44;
then A835: (((len h0) -' (len h11)) + 2) -' 1 in dom h2 by A46, A834, Lm1, FINSEQ_3:25;
i + 1 <= (len h11) + (len (mid (h21,2,((len h21) -' 1)))) by A813, FINSEQ_1:22;
then A836: (i + 1) - (len h11) <= ((len h11) + (len (mid (h21,2,((len h21) -' 1))))) - (len h11) by XREAL_1:9;
A837: (len h1) + 1 <= i + 1 by A832, NAT_1:13;
then ((len h11) + 1) - (len h11) <= (i + 1) - (len h11) by A39, XREAL_1:9;
then A838: (i + 1) -' (len h11) = (i + 1) - (len h11) by NAT_D:39;
(len h0) -' (len h11) = (len h0) - (len h11) by A36, A57, XREAL_0:def_2;
then (i + 1) -' (len h11) < (len h0) -' (len h11) by A813, A838, XREAL_1:9;
then A839: ((i + 1) -' (len h11)) + 2 < ((len h0) -' (len h11)) + 2 by XREAL_1:6;
1 <= i + 1 by A812, NAT_1:13;
then i + 1 in dom h0 by A813, FINSEQ_3:25;
then A840: h0 /. (i + 1) = h0 . (i + 1) by PARTFUN1:def_6;
A841: (((len h0) -' (len h11)) + 2) -' 1 = (((len h0) -' (len h11)) + 2) - 1 by A833, Lm1, NAT_D:39, NAT_D:42;
A842: (i + 1) - (len h11) = (i + 1) -' (len h11) by A39, A832, XREAL_1:233;
(i + 1) - (len h11) <= ((len h1) + ((len h2) - 2)) - (len h11) by A39, A47, A36, A52, A55, A57, A813, XREAL_1:9;
then ((i + 1) -' (len h11)) + 2 <= ((len h2) - 2) + 2 by A39, A842, XREAL_1:6;
then A843: (((i + 1) -' (len h11)) + 2) -' 1 <= len h21 by A47, NAT_D:44;
A844: 0 + 2 <= ((i + 1) -' (len h11)) + 2 by XREAL_1:6;
then 2 -' 1 <= (((i + 1) -' (len h11)) + 2) -' 1 by NAT_D:42;
then A845: (((i + 1) -' (len h11)) + 2) -' 1 in dom h2 by A46, A843, Lm1, FINSEQ_3:25;
( h0 . (i + 1) = (mid (h21,2,((len h21) -' 1))) . ((i + 1) - (len h11)) & ((len h1) + 1) - (len h1) <= (i + 1) - (len h1) ) by A39, A36, A813, A837, FINSEQ_1:23, XREAL_1:9;
then A846: h0 . (i + 1) = h21 . ((((i + 1) -' (len h11)) + 2) -' 1) by A39, A48, A56, A50, A54, A842, A836, FINSEQ_6:118;
(((i + 1) -' (len h11)) + 2) -' 1 = (((i + 1) -' (len h11)) + 2) - 1 by A844, Lm1, NAT_D:39, NAT_D:42;
then A847: (((i + 1) -' (len h11)) + 2) -' 1 < (((len h0) -' (len h11)) + 2) -' 1 by A841, A839, XREAL_1:9;
h0 . (len h0) = (mid (h21,2,((len h21) -' 1))) . ((len h0) - (len h11)) by A39, A36, A64, FINSEQ_1:23;
then h0 . (len h0) = h21 . ((((len h0) -' (len h11)) + 2) -' 1) by A47, A36, A51, A52, A49, A48, A53, A55, A57, A63, A252, FINSEQ_6:118;
hence contradiction by A46, A75, A32, A818, A846, A845, A840, A847, A835, FUNCT_1:def_4; ::_thesis: verum
end;
end;
end;
h0 . (len h0) = (mid (h21,2,((len h21) -' 1))) . ((len h0) - (len h11)) by A39, A36, A64, FINSEQ_1:23;
then A848: h0 . (len h0) = h21 . ((((len h0) -' (len h11)) + 2) -' 1) by A39, A36, A48, A56, A50, A54, A816, A252, FINSEQ_6:118;
then A849: h0 . (len h0) in Lower_Arc P by A23, A131, A29, A253, BORSUK_1:40, FUNCT_1:def_3;
A850: LE h0 /. (i + 1),h0 /. (len h0),P
proof
percases ( i + 1 <= len h1 or i + 1 > len h1 ) ;
supposeA851: i + 1 <= len h1 ; ::_thesis: LE h0 /. (i + 1),h0 /. (len h0),P
then i + 1 in dom h1 by A814, FINSEQ_3:25;
then h1 . (i + 1) in rng h1 by FUNCT_1:def_3;
then A852: g1 . (h1 . (i + 1)) in rng g1 by A132, A11, BORSUK_1:40, FUNCT_1:def_3;
A853: h0 /. (i + 1) = h0 . (i + 1) by A815, PARTFUN1:def_6;
i + 1 in dom h1 by A814, A851, FINSEQ_3:25;
then A854: h11 . (i + 1) = g1 . (h1 . (i + 1)) by FUNCT_1:13;
h0 . (i + 1) = h11 . (i + 1) by A39, A814, A851, FINSEQ_1:64;
hence LE h0 /. (i + 1),h0 /. (len h0),P by A5, A75, A130, A849, A854, A853, A852, JORDAN6:def_10; ::_thesis: verum
end;
supposeA855: i + 1 > len h1 ; ::_thesis: LE h0 /. (i + 1),h0 /. (len h0),P
then (len h1) + 1 <= i + 1 by NAT_1:13;
then A856: ((len h1) + 1) - (len h1) <= (i + 1) - (len h1) by XREAL_1:9;
then A857: (i + 1) -' (len h11) = (i + 1) - (len h11) by A39, NAT_D:39;
A858: (i + 1) - (len h11) < ((len h11) + (len (mid (h21,2,((len h21) -' 1))))) - (len h11) by A36, A813, XREAL_1:9;
A859: i + 1 < (len h11) + (len (mid (h21,2,((len h21) -' 1)))) by A813, FINSEQ_1:22;
(len h11) + 1 <= i + 1 by A39, A855, NAT_1:13;
then A860: h0 . (i + 1) = (mid (h21,2,((len h21) -' 1))) . ((i + 1) - (len h11)) by A859, FINSEQ_1:23
.= h21 . ((((i + 1) -' (len h11)) + 2) -' 1) by A39, A48, A56, A50, A54, A858, A856, A857, FINSEQ_6:118 ;
set j0 = (((i + 1) -' (len h11)) + 2) -' 1;
set k = (((len h0) -' (len h11)) + 2) -' 1;
0 + 1 <= ((((len h0) -' (len h11)) + 1) + 1) - 1 by XREAL_1:6;
then A861: (((len h0) -' (len h11)) + 2) -' 1 = (((len h0) -' (len h11)) + 2) - 1 by NAT_D:39;
A862: (((len h0) -' (len h11)) + 2) -' 1 <= len h21 by A36, A52, A55, A57, A252, NAT_D:44;
then A863: (((len h0) -' (len h11)) + 2) -' 1 in dom h21 by A43, Lm1, FINSEQ_3:25;
then h2 . ((((len h0) -' (len h11)) + 2) -' 1) in rng h2 by A46, FUNCT_1:def_3;
then A864: h2 . ((((len h0) -' (len h11)) + 2) -' 1) <= 1 by A29, BORSUK_1:40, XXREAL_1:1;
(((len h0) -' (len h11)) + 2) -' 1 in dom h21 by A43, A862, Lm1, FINSEQ_3:25;
then A865: h21 . ((((len h0) -' (len h11)) + 2) -' 1) = g2 . (h2 . ((((len h0) -' (len h11)) + 2) -' 1)) by A46, FUNCT_1:13;
A866: (i + 1) - (len h11) = (i + 1) -' (len h11) by A39, A855, XREAL_1:233;
(i + 1) - (len h11) <= ((len h11) + ((len h2) - 2)) - (len h11) by A47, A36, A52, A55, A57, A813, XREAL_1:9;
then ((i + 1) -' (len h11)) + 2 <= ((len h2) - 2) + 2 by A866, XREAL_1:6;
then A867: (((i + 1) -' (len h11)) + 2) -' 1 <= len h21 by A47, NAT_D:44;
h0 . (len h0) in Lower_Arc P by A23, A131, A29, A848, A253, BORSUK_1:40, FUNCT_1:def_3;
then A868: h0 /. (len h0) in Lower_Arc P by A74, PARTFUN1:def_6;
A869: 0 + 2 <= ((i + 1) -' (len h11)) + 2 by XREAL_1:6;
then 2 -' 1 <= (((i + 1) -' (len h11)) + 2) -' 1 by NAT_D:42;
then A870: (((i + 1) -' (len h11)) + 2) -' 1 in dom h2 by A46, A867, Lm1, FINSEQ_3:25;
then A871: h2 . ((((i + 1) -' (len h11)) + 2) -' 1) in rng h2 by FUNCT_1:def_3;
then A872: ( 0 <= h2 . ((((i + 1) -' (len h11)) + 2) -' 1) & h2 . ((((i + 1) -' (len h11)) + 2) -' 1) <= 1 ) by A29, BORSUK_1:40, XXREAL_1:1;
(i + 1) - (len h11) < (len h0) - (len h11) by A813, XREAL_1:9;
then A873: ((i + 1) -' (len h11)) + 2 < ((len h0) -' (len h11)) + 2 by A252, A866, XREAL_1:6;
h0 . (len h0) = (mid (h21,2,((len h21) -' 1))) . ((len h0) - (len h11)) by A39, A36, A64, FINSEQ_1:23;
then A874: h0 . (len h0) = h21 . ((((len h0) -' (len h11)) + 2) -' 1) by A39, A36, A48, A56, A50, A54, A816, A252, FINSEQ_6:118;
A875: h0 /. (i + 1) = h0 . (i + 1) by A815, PARTFUN1:def_6;
g2 . (h2 . ((((i + 1) -' (len h11)) + 2) -' 1)) in rng g2 by A131, A29, A871, BORSUK_1:40, FUNCT_1:def_3;
then A876: h0 . (i + 1) in Lower_Arc P by A23, A860, A870, FUNCT_1:13;
(((i + 1) -' (len h11)) + 2) -' 1 = (((i + 1) -' (len h11)) + 2) - 1 by A869, Lm1, NAT_D:39, NAT_D:42;
then (((i + 1) -' (len h11)) + 2) -' 1 < (((len h0) -' (len h11)) + 2) -' 1 by A861, A873, XREAL_1:9;
then A877: h2 . ((((i + 1) -' (len h11)) + 2) -' 1) < h2 . ((((len h0) -' (len h11)) + 2) -' 1) by A30, A46, A863, A870, SEQM_3:def_1;
h0 . (i + 1) = g2 . (h2 . ((((i + 1) -' (len h11)) + 2) -' 1)) by A860, A870, FUNCT_1:13;
then LE h0 /. (i + 1),h0 /. (len h0), Lower_Arc P, E-max P, W-min P by A21, A22, A23, A24, A25, A75, A874, A865, A864, A872, A877, A875, Th18;
hence LE h0 /. (i + 1),h0 /. (len h0),P by A130, A875, A876, A868, JORDAN6:def_10; ::_thesis: verum
end;
end;
end;
i < len h0 by A813, NAT_1:13;
then A878: i in dom h0 by A812, FINSEQ_3:25;
then h0 /. i = h0 . i by PARTFUN1:def_6;
then A879: h0 /. i <> W-min P by A72, A67, A78, A812, A878, FUNCT_1:def_4;
LE h0 /. i,h0 /. (i + 1),P by A256, A812, A813;
hence Segment ((h0 /. (len h0)),(h0 /. 1),P) misses Segment ((h0 /. i),(h0 /. (i + 1)),P) by A1, A72, A68, A879, A850, A817, Th14; ::_thesis: verum
end;