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: p2 `1 > e and
A3: p in P and
A4: p `1 = e ; :: thesis:

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 object 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;
reconsider s2 = xs as Real by A10;
A13: s2 <= 1 by A10, BORSUK_1:40, XXREAL_1:1;
for q being Point of (TOP-REAL 2) st q = f . 1 holds
q `1 <> e by A2, A7;
then A14: 1 in { s where s is Real : ( 1 >= 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 : ( 1 >= s & s >= s2 & ( for q being Point of (TOP-REAL 2) st q = f . s holds
q `1 <> e ) ) } c= REAL

let x be object ; :: according to TARSKI:def 3 :: thesis:
assume x in { s where s is Real : ( 1 >= s & s >= s2 & ( for q being Point of (TOP-REAL 2) st q = f . s holds
q `1 <> e ) ) } ; :: thesis:
then consider s being Real such that
A15: ( s = x & 1 >= s & s >= s2 & ( for q being Point of (TOP-REAL 2) st q = f . s holds
q `1 <> e ) ) ;
s in REAL by XREAL_0:def 1;
hence x in REAL by A15; :: thesis:

end;

then reconsider R = { s where s is Real : ( 1 >= 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;
reconsider s0 = lower_bound R as Real ;
A16: for s being Real st s in R holds
s > s2

proof

let s be Real; :: thesis:
assume s in R ; :: thesis:
then A17: ex s3 being Real st
( s3 = s & 1 >= 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 A17, XXREAL_0:1; :: thesis:

end;

then for s being Real st s in R holds
s >= s2 ;
then A18: s0 >= s2 by SEQ_4:43;
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 A19: pro1 is continuous by JGRAPH_2:29;
reconsider h = pro1 * f as Function of I[01],R^1 ;
A20: dom h = [.0,1.] by BORSUK_1:40, FUNCT_2:def 1;
for s being ExtReal st s in R holds
s >= s2 by A16;
then s2 is LowerBound of R by XXREAL_2:def 2;
then A21: R is bounded_below by XXREAL_2:def 9;
A22: 0 <= s2 by A10, BORSUK_1:40, XXREAL_1:1;
R c= [.0,1.]

proof

let x be object ; :: according to TARSKI:def 3 :: thesis:
assume x in R ; :: thesis:
then ex s being Real st
( s = x & 1 >= s & s >= s2 & ( for q being Point of (TOP-REAL 2) st q = f . s holds
q `1 <> e ) ) ;
hence x in [.0,1.] by A22, XXREAL_1:1; :: thesis:

end;

then A23: 1 >= s0 by A14, BORSUK_1:40, BORSUK_4:26;
then s0 in dom f by A12, A22, A18, 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) ;
A24: LE p,p9,P,p1,p2 by A1, A5, A6, A7, A11, A22, A13, A23, A18, JORDAN5C:8;
A25: rng f = P by A9, PRE_TOPC:def 5;
A26: for p8 being Point of (TOP-REAL 2) st LE p8,p9,P,p1,p2 & LE p,p8,P,p1,p2 holds
p8 `1 = e

proof

let p8 be Point of (TOP-REAL 2); :: thesis:
assume that
A27: LE p8,p9,P,p1,p2 and
A28: LE p,p8,P,p1,p2 ; :: thesis:
A29: p8 in P by A27, JORDAN5C:def 3;
then consider x8 being object such that
A30: x8 in dom f and
A31: p8 = f . x8 by A25, FUNCT_1:def 3;
reconsider s8 = x8 as Element of REAL by A12, A30, BORSUK_1:40;
A32: s8 <= 1 by A30, BORSUK_1:40, XXREAL_1:1;
0 <= s8 by A30, BORSUK_1:40, XXREAL_1:1;
then A33: s8 >= s2 by A5, A6, A7, A11, A13, A28, A31, A32, JORDAN5C:def 3;
A34: s0 >= s8 by A5, A6, A7, A22, A23, A18, A27, A31, A32, JORDAN5C:def 3;

now :: thesis:

reconsider s8n = s8 as Point of RealSpace by METRIC_1:def 13;
reconsider s8m = s8 as Point of (Closed-Interval-MSpace (0,1)) by A30, BORSUK_1:40, TOPMETR:10;
reconsider ee = |.((p8 `1) - e).| / 2 as Real ;
reconsider w = p8 `1 as Element of RealSpace by METRIC_1:def 13, XREAL_0:def 1;
reconsider B = Ball (w,ee) as Subset of R^1 by METRIC_1:def 13, TOPMETR:17;
A35: 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 A36: p8 `1 <> e ; :: thesis:
then (p8 `1) - e <> 0 ;
then |.((p8 `1) - e).| > 0 by COMPLEX1:47;
then A37: w in Ball (w,ee) by GOBOARD6:1, XREAL_1:139;
A38: ( h " B is open & I[01] = TopSpaceMetr (Closed-Interval-MSpace (0,1)) ) by A8, A19, TOPMETR:20, TOPMETR:def 6, TOPMETR:def 7, UNIFORM1:2;
h . s8 = pro1 . (f . s8) by A20, A30, BORSUK_1:40, FUNCT_1:12
.= proj1 . p8 by A29, A31, FUNCT_1:49
.= p8 `1 by PSCOMP_1:def 5 ;
then s8 in h " B by A20, A30, A37, BORSUK_1:40, FUNCT_1:def 7;
then consider r0 being Real such that
A39: r0 > 0 and
A40: Ball (s8m,r0) c= h " B by A38, TOPMETR:15;
reconsider r0 = r0 as Real ;
reconsider r01 = min ((s8 - s2),r0) 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 A41: Ball (s8m,r01) = ].(s8 - r01),(s8 + r01).[ /\ [.0,1.] by TOPMETR:9;
s8 > s2 by A4, A11, A31, A33, A36, XXREAL_0:1;
then s8 - s2 > 0 by XREAL_1:50;
then A42: r01 > 0 by A39, XXREAL_0:21;
then A43: (r01 - (r01 / 2)) + (r01 / 2) > 0 + (r01 / 2) by XREAL_1:6;
then A44: s8 - r01 < s8 - (r01 / 2) by XREAL_1:10;
A45: r01 / 2 > 0 by A42, XREAL_1:139;
then A46: s8 + (- (r01 / 2)) < s8 + (r01 / 2) by XREAL_1:8;
s8 + (r01 / 2) < s8 + r01 by A43, XREAL_1:8;
then s8 - (r01 / 2) < s8 + r01 by A46, XXREAL_0:2;
then A47: s8 - (r01 / 2) in ].(s8 - r01),(s8 + r01).[ by A44, XXREAL_1:4;
A48: s8 - (r01 / 2) > s8 - r01 by A43, XREAL_1:10;
Ball (s8m,r01) c= Ball (s8m,r0) by PCOMPS_1:1, XXREAL_0:17;
then A49: ].(s8 - r01),(s8 + r01).[ /\ [.0,1.] c= h " B by A40, A41;
reconsider s70 = s8 - (r01 / 2) as Real ;
s8 - s2 >= r01 by XXREAL_0:17;
then - (s8 - s2) <= - r01 by XREAL_1:24;
then A50: (s2 - s8) + s8 <= (- r01) + s8 by XREAL_1:7;
- (- (r01 / 2)) > 0 by A42, XREAL_1:139;
then - (r01 / 2) < 0 ;
then A51: s8 + 0 > s8 + (- (r01 / 2)) by XREAL_1:8;
then A52: 1 >= s70 by A32, XXREAL_0:2;
1 - 0 > s8 - (r01 / 2) by A32, A45, XREAL_1:15;
then s8 - (r01 / 2) in [.0,1.] by A22, A50, A48, XXREAL_1:1;
then A53: s8 - (r01 / 2) in ].(s8 - r01),(s8 + r01).[ /\ [.0,1.] by A47, XBOOLE_0:def 4;
then A54: h . (s8 - (r01 / 2)) in B by A49, FUNCT_1:def 7;
A55: s8 - (r01 / 2) in dom h by A49, 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 in [.0,1.] by A22, A50, A44, A52, 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 A35, A54, XXREAL_1:4;
A62: (p8 `1) - ee < p7 `1 by A35, A54, A60, XXREAL_1:4;

now :: thesis:

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

now :: thesis:

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;

s70 >= s2 by A50, A44, XXREAL_0:2;
then consider s7 being Real such that
A68: s8 > s7 and
A69: ( 1 >= s7 & s7 >= s2 & ( for p7 being Point of (TOP-REAL 2) st p7 = f . s7 holds
p7 `1 <> e ) ) by A51, A52, A56;
s7 in R by A69;
then s7 >= s0 by A21, SEQ_4:def 2;
hence contradiction by A34, A68, XXREAL_0:2; :: thesis:

end;

hence p8 `1 = e ; :: thesis:

end;

assume not p is_OSout P,p1,p2,e ; :: thesis:
then consider p4 being Point of (TOP-REAL 2) such that
A70: LE p9,p4,P,p1,p2 and
A71: p4 <> p9 and
A72: ( for p5 being Point of (TOP-REAL 2) st LE p5,p4,P,p1,p2 & LE p9,p5,P,p1,p2 holds
p5 `1 <= e or for p6 being Point of (TOP-REAL 2) st LE p6,p4,P,p1,p2 & LE p9,p6,P,p1,p2 holds
p6 `1 >= e ) by A1, A3, A4, A24, A26;
A73: p9 in P by A70, JORDAN5C:def 3;

now :: thesis:

per cases by A72;

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

A75: now :: thesis:

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

proof

let p51 be Point of (TOP-REAL 2); :: thesis:
assume ( LE p51,p4,P,p1,p2 & LE p9,p51,P,p1,p2 ) ; :: thesis:
then ( p51 `1 >= e & p51 `1 <= e ) by A74, A79;
hence p51 `1 = e by XXREAL_0:1; :: thesis:

end;

A81: now :: thesis:

assume s4 > s0 ; :: thesis:
then - (- (s4 - s0)) > 0 by XREAL_1:50;
then - (s4 - s0) < 0 ;
then A82: (s0 - s4) / 2 < 0 by XREAL_1:141;
then - ((s0 - s4) / 2) > 0 by XREAL_1:58;
then consider r being Real such that
A83: r in R and
A84: r < s0 + (- ((s0 - s4) / 2)) by A21, SEQ_4:def 2;
reconsider rss = r as Real ;
A85: ex s7 being Real st
( s7 = r & 1 >= s7 & s7 >= s2 & ( for q being Point of (TOP-REAL 2) st q = f . s7 holds
q `1 <> e ) ) by A83;
then r in [.0,1.] by A22, 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 + 0 > s4 + ((s0 - s4) / 2) by A82, XREAL_1:8;
then A86: s4 > r by A84, XXREAL_0:2;
then A87: 1 > r by A78, XXREAL_0:2;
A88: r >= s0 by A21, A83, SEQ_4:def 2;
then A89: LE p9,pss,P,p1,p2 by A1, A5, A6, A7, A22, A23, A18, A87, JORDAN5C:8;
LE pss,p4,P,p1,p2 by A1, A5, A6, A7, A22, A18, A77, A78, A88, A86, A87, JORDAN5C:8;
then pss `1 = e by A80, A89;
hence contradiction by A85; :: thesis:

end;

0 <= s4 by A76, BORSUK_1:40, XXREAL_1:1;
then s4 >= s0 by A5, A6, A7, A23, A70, A77, A78, JORDAN5C:def 3;
hence contradiction by A71, A77, A81, XXREAL_0:1; :: thesis:

end;

now :: thesis:

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

proof

let p5 be Point of (TOP-REAL 2); :: thesis:
assume that
A94: LE p5,p51,P,p1,p2 and
A95: LE p,p5,P,p1,p2 ; :: thesis:
A96: LE p5,p4,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 :: thesis:

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

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

hence p5 `1 <= e by A26, A95; :: thesis:

end;

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

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

end;

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

end;

LE p,p51,P,p1,p2 by A24, A91, JORDAN5C:13;
hence p is_Lout P,p1,p2,e by A1, A3, A4, A92, A93; :: thesis:

end;

hence ( p is_Lout P,p1,p2,e or p is_Rout P,p1,p2,e ) by A75; :: thesis:

end;

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

A100: now :: thesis:

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

proof

let p51 be Point of (TOP-REAL 2); :: thesis:
assume ( LE p51,p4,P,p1,p2 & LE p9,p51,P,p1,p2 ) ; :: thesis:
then ( p51 `1 <= e & p51 `1 >= e ) by A99, A104;
hence p51 `1 = e by XXREAL_0:1; :: thesis:

end;

A106: now :: thesis:

assume s4 > s0 ; :: thesis:
then - (- (s4 - s0)) > 0 by XREAL_1:50;
then - (s4 - s0) < 0 ;
then A107: (s0 - s4) / 2 < 0 by XREAL_1:141;
then - ((s0 - s4) / 2) > 0 by XREAL_1:58;
then consider r being Real such that
A108: r in R and
A109: r < s0 + (- ((s0 - s4) / 2)) by A21, SEQ_4:def 2;
reconsider rss = r as Real ;
A110: ex s7 being Real st
( s7 = r & 1 >= s7 & s7 >= s2 & ( for q being Point of (TOP-REAL 2) st q = f . s7 holds
q `1 <> e ) ) by A108;
then r in [.0,1.] by A22, 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 + 0 > s4 + ((s0 - s4) / 2) by A107, XREAL_1:8;
then A111: s4 > r by A109, XXREAL_0:2;
then A112: 1 > r by A103, XXREAL_0:2;
A113: r >= s0 by A21, A108, SEQ_4:def 2;
then A114: LE p9,pss,P,p1,p2 by A1, A5, A6, A7, A22, A23, A18, A112, JORDAN5C:8;
LE pss,p4,P,p1,p2 by A1, A5, A6, A7, A22, A18, A102, A103, A113, A111, A112, JORDAN5C:8;
then pss `1 = e by A105, A114;
hence contradiction by A110; :: thesis:

end;

0 <= s4 by A101, BORSUK_1:40, XXREAL_1:1;
then s4 >= s0 by A5, A6, A7, A23, A70, A102, A103, JORDAN5C:def 3;
hence contradiction by A71, A102, A106, XXREAL_0:1; :: thesis:

end;

now :: thesis:

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

proof

let p5 be Point of (TOP-REAL 2); :: thesis:
assume that
A119: LE p5,p51,P,p1,p2 and
A120: LE p,p5,P,p1,p2 ; :: thesis:
A121: LE p5,p4,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 :: thesis:

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

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

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

end;

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

hence p5 `1 >= e by A26, A120; :: thesis:

end;

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

end;

LE p,p51,P,p1,p2 by A24, A116, JORDAN5C:13;
hence p is_Rout P,p1,p2,e by A1, A3, A4, A117, A118; :: thesis:

end;

hence ( p is_Lout P,p1,p2,e or p is_Rout P,p1,p2,e ) by A100; :: thesis:

end;

hence ( p is_Lout P,p1,p2,e or p is_Rout P,p1,p2,e ) ; :: thesis:

end;

hence ( p is_Lout P,p1,p2,e or p is_Rout P,p1,p2,e or p is_OSout P,p1,p2,e ) ; :: thesis: