let P be non empty Subset of (TOP-REAL 2); :: thesis:
let p1, p2, p be Point of (TOP-REAL 2); :: thesis:
let e be Real; :: thesis:
assume that
A1: P is_an_arc_of p1,p2 and
A2: p1 `1 < e and
A3: p in P and
A4: p `1 = e ; :: thesis:

now

reconsider pr1a = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:17;
reconsider pro1 = pr1a | P as Function of ((TOP-REAL 2) | P),R^1 by PRE_TOPC:9;
consider f being Function of I[01],((TOP-REAL 2) | P) such that
A5: f is being_homeomorphism and
A6: f . 0 = p1 and
A7: f . 1 = p2 by A1, TOPREAL1:def 1;
A8: f is continuous by A5, TOPS_2:def 5;
A9: rng f = [#] ((TOP-REAL 2) | P) by A5, TOPS_2:def 5;
then p in rng f by A3, PRE_TOPC:def 5;
then consider xs being set such that
A10: xs in dom f and
A11: p = f . xs by FUNCT_1:def 3;
A12: dom f = [#] I[01] by A5, TOPS_2:def 5;
then reconsider s2 = xs as Real by A10, BORSUK_1:40;
A13: 0 <= s2 by A10, BORSUK_1:40, XXREAL_1:1;
for q being Point of (TOP-REAL 2) st q = f . 0 holds
q `1 <> e by A2, A6;
then A14: 0 in { s where s is Real : ( 0 <= s & s <= s2 & ( for q being Point of (TOP-REAL 2) st q = f . s holds
q `1 <> e ) ) } by A13;
{ s where s is Real : ( 0 <= s & s <= s2 & ( for q being Point of (TOP-REAL 2) st q = f . s holds
q `1 <> e ) ) } c= REAL

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume x in { s where s is Real : ( 0 <= s & s <= s2 & ( for q being Point of (TOP-REAL 2) st q = f . s holds
q `1 <> e ) ) } ; :: thesis:
then ex s being Real st
( s = x & 0 <= s & s <= s2 & ( for q being Point of (TOP-REAL 2) st q = f . s holds
q `1 <> e ) ) ;
hence x in REAL ; :: thesis:

end;

then reconsider R = { s where s is Real : ( 0 <= s & s <= s2 & ( for q being Point of (TOP-REAL 2) st q = f . s holds
q `1 <> e ) ) } as non empty Subset of REAL by A14;
A15: s2 <= 1 by A10, BORSUK_1:40, XXREAL_1:1;
R c= [.0,1.]

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume x in R ; :: thesis:
then consider s being Real such that
A16: ( s = x & 0 <= s ) and
A17: s <= s2 and
for q being Point of (TOP-REAL 2) st q = f . s holds
q `1 <> e ;
s <= 1 by A15, A17, XXREAL_0:2;
hence x in [.0,1.] by A16, XXREAL_1:1; :: thesis:

end;

then reconsider R99 = R as Subset of I[01] by BORSUK_1:40;
reconsider s0 = upper_bound R as Real ;
A18: for s being real number st s in R holds
s < s2

proof

let s be real number ; :: thesis:
assume s in R ; :: thesis:
then A19: ex s3 being Real st
( s3 = s & 0 <= s3 & s3 <= s2 & ( for q being Point of (TOP-REAL 2) st q = f . s3 holds
q `1 <> e ) ) ;
then s <> s2 by A4, A11;
hence s < s2 by A19, XXREAL_0:1; :: thesis:

end;

then for s being real number st s in R holds
s <= s2 ;
then A20: s0 <= s2 by SEQ_4:45;
then A21: s0 <= 1 by A15, XXREAL_0:2;
R99 = R ;
then A22: 0 <= s0 by A14, BORSUK_4:26;
then s0 in dom f by A12, A21, BORSUK_1:40, XXREAL_1:1;
then f . s0 in rng f by FUNCT_1:def 3;
then f . s0 in P by A9, PRE_TOPC:def 5;
then reconsider p9 = f . s0 as Point of (TOP-REAL 2) ;
A23: LE p9,p,P,p1,p2 by A1, A5, A6, A7, A11, A15, A22, A20, A21, JORDAN5C:8;
for p7 being Point of ((TOP-REAL 2) | P) holds pro1 . p7 = proj1 . p7

proof

let p7 be Point of ((TOP-REAL 2) | P); :: thesis:
the carrier of ((TOP-REAL 2) | P) = P by PRE_TOPC:8;
hence pro1 . p7 = proj1 . p7 by FUNCT_1:49; :: thesis:

end;

then A24: pro1 is continuous by JGRAPH_2:29;
reconsider h = pro1 * f as Function of I[01],R^1 ;
A25: dom h = [.0,1.] by BORSUK_1:40, FUNCT_2:def 1;
for s being ext-real number st s in R holds
s <= s2 by A18;
then s2 is UpperBound of R by XXREAL_2:def 1;
then A26: R is bounded_above by XXREAL_2:def 10;
A27: rng f = P by A9, PRE_TOPC:def 5;
A28: for p8 being Point of (TOP-REAL 2) st LE p9,p8,P,p1,p2 & LE p8,p,P,p1,p2 holds
p8 `1 = e

proof

let p8 be Point of (TOP-REAL 2); :: thesis:
assume that
A29: LE p9,p8,P,p1,p2 and
A30: LE p8,p,P,p1,p2 ; :: thesis:
A31: p8 in P by A29, JORDAN5C:def 3;
then consider x8 being set such that
A32: x8 in dom f and
A33: p8 = f . x8 by A27, FUNCT_1:def 3;
reconsider s8 = x8 as Real by A12, A32, BORSUK_1:40;
A34: s8 <= 1 by A32, BORSUK_1:40, XXREAL_1:1;
then A35: s8 <= s2 by A5, A6, A7, A11, A13, A15, A30, A33, JORDAN5C:def 3;
A36: 0 <= s8 by A32, BORSUK_1:40, XXREAL_1:1;
then A37: s0 <= s8 by A5, A6, A7, A21, A29, A33, A34, JORDAN5C:def 3;

now

reconsider s8n = s8 as Point of RealSpace by METRIC_1:def 13;
reconsider s8m = s8 as Point of (Closed-Interval-MSpace (0,1)) by A32, BORSUK_1:40, TOPMETR:10;
reconsider ee = (abs ((p8 `1) - e)) / 2 as Real ;
reconsider w = p8 `1 as Element of RealSpace by METRIC_1:def 13;
reconsider B = Ball (w,ee) as Subset of R^1 by METRIC_1:def 13, TOPMETR:17;
A38: B = { s7 where s7 is Real : ( (p8 `1) - ee < s7 & s7 < (p8 `1) + ee ) } by JORDAN2B:17
.= ].((p8 `1) - ee),((p8 `1) + ee).[ by RCOMP_1:def 2 ;
assume A39: p8 `1 <> e ; :: thesis:
then (p8 `1) - e <> 0 ;
then abs ((p8 `1) - e) > 0 by COMPLEX1:47;
then A40: w in Ball (w,ee) by GOBOARD6:1, XREAL_1:139;
A41: ( h " B is open & I[01] = TopSpaceMetr (Closed-Interval-MSpace (0,1)) ) by A8, A24, TOPMETR:20, TOPMETR:def 6, TOPMETR:def 7, UNIFORM1:2;
h . s8 = pro1 . (f . s8) by A25, A32, BORSUK_1:40, FUNCT_1:12
.= proj1 . p8 by A31, A33, FUNCT_1:49
.= p8 `1 by PSCOMP_1:def 5 ;
then s8 in h " B by A25, A32, A40, BORSUK_1:40, FUNCT_1:def 7;
then consider r0 being real number such that
A42: r0 > 0 and
A43: Ball (s8m,r0) c= h " B by A41, TOPMETR:15;
reconsider r0 = r0 as Real by XREAL_0:def 1;
reconsider r01 = min ((s2 - s8),r0) as Real ;
s8 < s2 by A4, A11, A33, A35, A39, XXREAL_0:1;
then s2 - s8 > 0 by XREAL_1:50;
then A44: r01 > 0 by A42, XXREAL_0:21;
then A45: (r01 - (r01 / 2)) + (r01 / 2) > 0 + (r01 / 2) by XREAL_1:6;
then A46: s8 + (r01 / 2) < s8 + r01 by XREAL_1:6;
reconsider s70 = s8 + (r01 / 2) as Real ;
( the carrier of (Closed-Interval-MSpace (0,1)) = [.0,1.] & Ball (s8n,r01) = ].(s8 - r01),(s8 + r01).[ ) by FRECHET:7, TOPMETR:10;
then A47: Ball (s8m,r01) = ].(s8 - r01),(s8 + r01).[ /\ [.0,1.] by TOPMETR:9;
s2 - s8 >= r01 by XXREAL_0:17;
then A48: (s2 - s8) + s8 >= r01 + s8 by XREAL_1:7;
then A49: s70 <= s2 by A46, XXREAL_0:2;
s8 + r01 <= 1 by A15, A48, XXREAL_0:2;
then s8 + (r01 / 2) < 1 by A46, XXREAL_0:2;
then A50: s8 + (r01 / 2) in [.0,1.] by A36, A44, XXREAL_1:1;
Ball (s8m,r01) c= Ball (s8m,r0) by PCOMPS_1:1, XXREAL_0:17;
then A51: ].(s8 - r01),(s8 + r01).[ /\ [.0,1.] c= h " B by A43, A47, XBOOLE_1:1;
s8 + 0 < s8 + ((r01 / 2) + r01) by A44, XREAL_1:6;
then (s8 - r01) + r01 < (s8 + (r01 / 2)) + r01 ;
then A52: s8 - r01 < s8 + (r01 / 2) by XREAL_1:6;
s8 + (r01 / 2) < s8 + r01 by A45, XREAL_1:6;
then s8 + (r01 / 2) in ].(s8 - r01),(s8 + r01).[ by A52, XXREAL_1:4;
then A53: s8 + (r01 / 2) in ].(s8 - r01),(s8 + r01).[ /\ [.0,1.] by A50, XBOOLE_0:def 4;
then A54: h . (s8 + (r01 / 2)) in B by A51, FUNCT_1:def 7;
A55: s8 + (r01 / 2) in dom h by A51, A53, FUNCT_1:def 7;
A56: for p7 being Point of (TOP-REAL 2) st p7 = f . s70 holds
p7 `1 <> e

proof

let p7 be Point of (TOP-REAL 2); :: thesis:
assume A57: p7 = f . s70 ; :: thesis:
s70 <= 1 by A15, A49, XXREAL_0:2;
then s70 in [.0,1.] by A36, A44, XXREAL_1:1;
then A58: p7 in rng f by A12, A57, BORSUK_1:40, FUNCT_1:def 3;
A59: rng f = [#] ((TOP-REAL 2) | P) by A5, TOPS_2:def 5
.= P by PRE_TOPC:def 5 ;
A60: h . s70 = pro1 . (f . s70) by A55, FUNCT_1:12
.= pr1a . p7 by A57, A58, A59, FUNCT_1:49
.= p7 `1 by PSCOMP_1:def 5 ;
then A61: p7 `1 < (p8 `1) + ee by A38, A54, XXREAL_1:4;
A62: (p8 `1) - ee < p7 `1 by A38, A54, A60, XXREAL_1:4;

now

assume A63: p7 `1 = e ; :: thesis:

now

per cases ) - e >= 0 or (p8 `1) - e < 0 ) ;

case A64: (p8 `1) - e >= 0 ; :: thesis:

then (p8 `1) - (((p8 `1) - e) / 2) < e by A62, A63, ABSVALUE:def 1;
then ((p8 `1) / 2) + (e / 2) < (e / 2) + (e / 2) ;
then (p8 `1) / 2 < e / 2 by XREAL_1:7;
then A65: ((p8 `1) / 2) - (e / 2) < (e / 2) - (e / 2) by XREAL_1:14;
((p8 `1) - e) / 2 >= 0 / 2 by A64;
hence contradiction by A65; :: thesis:

end;

case A66: (p8 `1) - e < 0 ; :: thesis:

then e < (p8 `1) + ((- ((p8 `1) - e)) / 2) by A61, A63, ABSVALUE:def 1;
then ((p8 `1) / 2) + (e / 2) > (e / 2) + (e / 2) ;
then (p8 `1) / 2 > e / 2 by XREAL_1:7;
then A67: ((p8 `1) / 2) - (e / 2) > (e / 2) - (e / 2) by XREAL_1:14;
((p8 `1) - e) / 2 <= 0 / 2 by A66;
hence contradiction by A67; :: thesis:

end;

hence contradiction ; :: thesis:

end;

hence p7 `1 <> e ; :: thesis:

end;

s8 < s70 by A44, XREAL_1:29, XREAL_1:139;
then consider s7 being Real such that
A68: s8 < s7 and
A69: ( 0 <= s7 & s7 <= s2 & ( for p7 being Point of (TOP-REAL 2) st p7 = f . s7 holds
p7 `1 <> e ) ) by A36, A49, A56;
s7 in R by A69;
then s7 <= s0 by A26, SEQ_4:def 1;
hence contradiction by A37, A68, XXREAL_0:2; :: thesis:

end;

hence p8 `1 = e ; :: thesis:

end;

assume not p is_OSin P,p1,p2,e ; :: thesis:
then consider p4 being Point of (TOP-REAL 2) such that
A70: LE p4,p9,P,p1,p2 and
A71: p4 <> p9 and
A72: ( for p5 being Point of (TOP-REAL 2) st LE p4,p5,P,p1,p2 & LE p5,p9,P,p1,p2 holds
p5 `1 <= e or for p6 being Point of (TOP-REAL 2) st LE p4,p6,P,p1,p2 & LE p6,p9,P,p1,p2 holds
p6 `1 >= e ) by A1, A3, A4, A23, A28, Def5;
A73: p9 in P by A70, JORDAN5C:def 3;

now

per cases by A72;

case A74: for p5 being Point of (TOP-REAL 2) st LE p4,p5,P,p1,p2 & LE p5,p9,P,p1,p2 holds
p5 `1 <= e ; :: thesis:

A75: now

p4 in P by A70, JORDAN5C:def 3;
then p4 in rng f by A9, PRE_TOPC:def 5;
then consider xs4 being set such that
A76: xs4 in dom f and
A77: p4 = f . xs4 by FUNCT_1:def 3;
reconsider s4 = xs4 as Real by A12, A76, BORSUK_1:40;
A78: 0 <= s4 by A76, BORSUK_1:40, XXREAL_1:1;
A79: s4 <= 1 by A76, BORSUK_1:40, XXREAL_1:1;
assume A80: for p51 being Point of (TOP-REAL 2) holds
( not LE p4,p51,P,p1,p2 or not LE p51,p9,P,p1,p2 or not p51 `1 < e ) ; :: thesis:
A81: for p51 being Point of (TOP-REAL 2) st LE p4,p51,P,p1,p2 & LE p51,p9,P,p1,p2 holds
p51 `1 = e

proof

let p51 be Point of (TOP-REAL 2); :: thesis:
assume ( LE p4,p51,P,p1,p2 & LE p51,p9,P,p1,p2 ) ; :: thesis:
then ( p51 `1 >= e & p51 `1 <= e ) by A74, A80;
hence p51 `1 = e by XXREAL_0:1; :: thesis:

end;

A82: now

assume s4 < s0 ; :: thesis:
then A83: s0 - s4 > 0 by XREAL_1:50;
then A84: s4 < s4 + ((s0 - s4) / 2) by XREAL_1:29, XREAL_1:139;
(s0 - s4) / 2 > 0 by A83, XREAL_1:139;
then consider r being real number such that
A85: r in R and
A86: s0 - ((s0 - s4) / 2) < r by A26, SEQ_4:def 1;
reconsider rss = r as Real by XREAL_0:def 1;
A87: ex s7 being Real st
( s7 = r & 0 <= s7 & s7 <= s2 & ( for q being Point of (TOP-REAL 2) st q = f . s7 holds
q `1 <> e ) ) by A85;
then A88: r <= 1 by A15, XXREAL_0:2;
then r in [.0,1.] by A78, A84, A86, XXREAL_1:1;
then f . rss in rng f by A12, BORSUK_1:40, FUNCT_1:def 3;
then f . rss in P by A9, PRE_TOPC:def 5;
then reconsider pss = f . rss as Point of (TOP-REAL 2) ;
s4 < r by A84, A86, XXREAL_0:2;
then A89: LE p4,pss,P,p1,p2 by A1, A5, A6, A7, A77, A78, A79, A88, JORDAN5C:8;
r <= s0 by A26, A85, SEQ_4:def 1;
then LE pss,p9,P,p1,p2 by A1, A5, A6, A7, A21, A78, A84, A86, A88, JORDAN5C:8;
then pss `1 = e by A81, A89;
hence contradiction by A87; :: thesis:

end;

s4 <= s0 by A5, A6, A7, A22, A21, A70, A77, A79, JORDAN5C:def 3;
hence contradiction by A71, A77, A82, XXREAL_0:1; :: thesis:

end;

now

assume ex p51 being Point of (TOP-REAL 2) st
( LE p4,p51,P,p1,p2 & LE p51,p9,P,p1,p2 & p51 `1 < e ) ; :: thesis:
then consider p51 being Point of (TOP-REAL 2) such that
A90: LE p4,p51,P,p1,p2 and
A91: LE p51,p9,P,p1,p2 and
A92: p51 `1 < e ;
A93: for p5 being Point of (TOP-REAL 2) st LE p51,p5,P,p1,p2 & LE p5,p,P,p1,p2 holds
p5 `1 <= e

proof

let p5 be Point of (TOP-REAL 2); :: thesis:
assume that
A94: LE p51,p5,P,p1,p2 and
A95: LE p5,p,P,p1,p2 ; :: thesis:
A96: LE p4,p5,P,p1,p2 by A90, A94, JORDAN5C:13;
A97: p5 in P by A94, JORDAN5C:def 3;
then A98: ( p5 = p9 implies LE p9,p5,P,p1,p2 ) by JORDAN5C:9;

now

per cases by A1, A73, A97, A98, JORDAN5C:14;

case LE p5,p9,P,p1,p2 ; :: thesis:

hence p5 `1 <= e by A74, A96; :: thesis:

end;

case LE p9,p5,P,p1,p2 ; :: thesis:

hence p5 `1 <= e by A28, A95; :: thesis:

end;

hence p5 `1 <= e ; :: thesis:

end;

LE p51,p,P,p1,p2 by A23, A91, JORDAN5C:13;
hence p is_Lin P,p1,p2,e by A1, A3, A4, A92, A93, Def1; :: thesis:

end;

hence ( p is_Lin P,p1,p2,e or p is_Rin P,p1,p2,e ) by A75; :: thesis:

end;

case A99: for p6 being Point of (TOP-REAL 2) st LE p4,p6,P,p1,p2 & LE p6,p9,P,p1,p2 holds
p6 `1 >= e ; :: thesis:

A100: now

p4 in P by A70, JORDAN5C:def 3;
then p4 in rng f by A9, PRE_TOPC:def 5;
then consider xs4 being set such that
A101: xs4 in dom f and
A102: p4 = f . xs4 by FUNCT_1:def 3;
reconsider s4 = xs4 as Real by A12, A101, BORSUK_1:40;
A103: 0 <= s4 by A101, BORSUK_1:40, XXREAL_1:1;
A104: s4 <= 1 by A101, BORSUK_1:40, XXREAL_1:1;
assume A105: for p51 being Point of (TOP-REAL 2) holds
( not LE p4,p51,P,p1,p2 or not LE p51,p9,P,p1,p2 or not p51 `1 > e ) ; :: thesis:
A106: for p51 being Point of (TOP-REAL 2) st LE p4,p51,P,p1,p2 & LE p51,p9,P,p1,p2 holds
p51 `1 = e

proof

let p51 be Point of (TOP-REAL 2); :: thesis:
assume ( LE p4,p51,P,p1,p2 & LE p51,p9,P,p1,p2 ) ; :: thesis:
then ( p51 `1 <= e & p51 `1 >= e ) by A99, A105;
hence p51 `1 = e by XXREAL_0:1; :: thesis:

end;

A107: now

assume s4 < s0 ; :: thesis:
then A108: s0 - s4 > 0 by XREAL_1:50;
then A109: s4 < s4 + ((s0 - s4) / 2) by XREAL_1:29, XREAL_1:139;
(s0 - s4) / 2 > 0 by A108, XREAL_1:139;
then consider r being real number such that
A110: r in R and
A111: s0 - ((s0 - s4) / 2) < r by A26, SEQ_4:def 1;
reconsider rss = r as Real by XREAL_0:def 1;
A112: ex s7 being Real st
( s7 = r & 0 <= s7 & s7 <= s2 & ( for q being Point of (TOP-REAL 2) st q = f . s7 holds
q `1 <> e ) ) by A110;
then A113: r <= 1 by A15, XXREAL_0:2;
then r in [.0,1.] by A103, A109, A111, XXREAL_1:1;
then f . rss in rng f by A12, BORSUK_1:40, FUNCT_1:def 3;
then f . rss in P by A9, PRE_TOPC:def 5;
then reconsider pss = f . rss as Point of (TOP-REAL 2) ;
s4 < r by A109, A111, XXREAL_0:2;
then A114: LE p4,pss,P,p1,p2 by A1, A5, A6, A7, A102, A103, A104, A113, JORDAN5C:8;
r <= s0 by A26, A110, SEQ_4:def 1;
then LE pss,p9,P,p1,p2 by A1, A5, A6, A7, A21, A103, A109, A111, A113, JORDAN5C:8;
then pss `1 = e by A106, A114;
hence contradiction by A112; :: thesis:

end;

s4 <= s0 by A5, A6, A7, A22, A21, A70, A102, A104, JORDAN5C:def 3;
hence contradiction by A71, A102, A107, XXREAL_0:1; :: thesis:

end;

now

assume ex p51 being Point of (TOP-REAL 2) st
( LE p4,p51,P,p1,p2 & LE p51,p9,P,p1,p2 & p51 `1 > e ) ; :: thesis:
then consider p51 being Point of (TOP-REAL 2) such that
A115: LE p4,p51,P,p1,p2 and
A116: LE p51,p9,P,p1,p2 and
A117: p51 `1 > e ;
A118: for p5 being Point of (TOP-REAL 2) st LE p51,p5,P,p1,p2 & LE p5,p,P,p1,p2 holds
p5 `1 >= e

proof

let p5 be Point of (TOP-REAL 2); :: thesis:
assume that
A119: LE p51,p5,P,p1,p2 and
A120: LE p5,p,P,p1,p2 ; :: thesis:
A121: LE p4,p5,P,p1,p2 by A115, A119, JORDAN5C:13;
A122: p5 in P by A119, JORDAN5C:def 3;
then A123: ( p5 = p9 implies LE p9,p5,P,p1,p2 ) by JORDAN5C:9;

now

per cases by A1, A73, A122, A123, JORDAN5C:14;

case LE p5,p9,P,p1,p2 ; :: thesis:

hence p5 `1 >= e by A99, A121; :: thesis:

end;

case LE p9,p5,P,p1,p2 ; :: thesis:

hence p5 `1 >= e by A28, A120; :: thesis:

end;

hence p5 `1 >= e ; :: thesis:

end;

LE p51,p,P,p1,p2 by A23, A116, JORDAN5C:13;
hence p is_Rin P,p1,p2,e by A1, A3, A4, A117, A118, Def2; :: thesis:

end;

hence ( p is_Lin P,p1,p2,e or p is_Rin P,p1,p2,e ) by A100; :: thesis:

end;

hence ( p is_Lin P,p1,p2,e or p is_Rin P,p1,p2,e ) ; :: thesis:

end;

hence ( p is_Lin P,p1,p2,e or p is_Rin P,p1,p2,e or p is_OSin P,p1,p2,e ) ; :: thesis: