:: 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;