:: JORDAN20 semantic presentation
begin
theorem Th1: :: JORDAN20:1
for P being Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & p in P holds
Segment (P,p1,p2,p,p) = {p}
proof
let P be Subset of (TOP-REAL 2); ::_thesis: for p1, p2, p being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & p in P holds
Segment (P,p1,p2,p,p) = {p}
let p1, p2, p be Point of (TOP-REAL 2); ::_thesis: ( P is_an_arc_of p1,p2 & p in P implies Segment (P,p1,p2,p,p) = {p} )
assume that
A1: P is_an_arc_of p1,p2 and
A2: p in P ; ::_thesis: Segment (P,p1,p2,p,p) = {p}
A3: Segment (P,p1,p2,p,p) = { q where q is Point of (TOP-REAL 2) : ( LE p,q,P,p1,p2 & LE q,p,P,p1,p2 ) } by JORDAN6:26;
A4: {p} c= Segment (P,p1,p2,p,p)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {p} or x in Segment (P,p1,p2,p,p) )
assume x in {p} ; ::_thesis: x in Segment (P,p1,p2,p,p)
then A5: x = p by TARSKI:def_1;
LE p,p,P,p1,p2 by A2, JORDAN5C:9;
hence x in Segment (P,p1,p2,p,p) by A3, A5; ::_thesis: verum
end;
Segment (P,p1,p2,p,p) c= {p}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Segment (P,p1,p2,p,p) or x in {p} )
assume x in Segment (P,p1,p2,p,p) ; ::_thesis: x in {p}
then consider q being Point of (TOP-REAL 2) such that
A6: x = q and
A7: ( LE p,q,P,p1,p2 & LE q,p,P,p1,p2 ) by A3;
p = q by A1, A7, JORDAN5C:12;
hence x in {p} by A6, TARSKI:def_1; ::_thesis: verum
end;
hence Segment (P,p1,p2,p,p) = {p} by A4, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th2: :: JORDAN20:2
for p1, p2, p being Point of (TOP-REAL 2)
for a being Real st p in LSeg (p1,p2) & p1 `1 <= a & p2 `1 <= a holds
p `1 <= a
proof
let p1, p2, p be Point of (TOP-REAL 2); ::_thesis: for a being Real st p in LSeg (p1,p2) & p1 `1 <= a & p2 `1 <= a holds
p `1 <= a
let a be Real; ::_thesis: ( p in LSeg (p1,p2) & p1 `1 <= a & p2 `1 <= a implies p `1 <= a )
assume that
A1: p in LSeg (p1,p2) and
A2: p1 `1 <= a and
A3: p2 `1 <= a ; ::_thesis: p `1 <= a
consider r being Real such that
A4: p = ((1 - r) * p1) + (r * p2) and
A5: 0 <= r and
A6: r <= 1 by A1;
A7: p `1 = (((1 - r) * p1) `1) + ((r * p2) `1) by A4, TOPREAL3:2
.= (((1 - r) * p1) `1) + (r * (p2 `1)) by TOPREAL3:4
.= ((1 - r) * (p1 `1)) + (r * (p2 `1)) by TOPREAL3:4 ;
1 - r >= 0 by A6, XREAL_1:48;
then A8: (1 - r) * (p1 `1) <= (1 - r) * a by A2, XREAL_1:64;
A9: ((1 - r) * a) + (r * a) = a ;
r * (p2 `1) <= r * a by A3, A5, XREAL_1:64;
hence p `1 <= a by A7, A8, A9, XREAL_1:7; ::_thesis: verum
end;
theorem Th3: :: JORDAN20:3
for p1, p2, p being Point of (TOP-REAL 2)
for a being Real st p in LSeg (p1,p2) & p1 `1 >= a & p2 `1 >= a holds
p `1 >= a
proof
let p1, p2, p be Point of (TOP-REAL 2); ::_thesis: for a being Real st p in LSeg (p1,p2) & p1 `1 >= a & p2 `1 >= a holds
p `1 >= a
let a be Real; ::_thesis: ( p in LSeg (p1,p2) & p1 `1 >= a & p2 `1 >= a implies p `1 >= a )
assume that
A1: p in LSeg (p1,p2) and
A2: p1 `1 >= a and
A3: p2 `1 >= a ; ::_thesis: p `1 >= a
consider r being Real such that
A4: p = ((1 - r) * p1) + (r * p2) and
A5: 0 <= r and
A6: r <= 1 by A1;
A7: p `1 = (((1 - r) * p1) `1) + ((r * p2) `1) by A4, TOPREAL3:2
.= (((1 - r) * p1) `1) + (r * (p2 `1)) by TOPREAL3:4
.= ((1 - r) * (p1 `1)) + (r * (p2 `1)) by TOPREAL3:4 ;
1 - r >= 0 by A6, XREAL_1:48;
then A8: (1 - r) * (p1 `1) >= (1 - r) * a by A2, XREAL_1:64;
A9: ((1 - r) * a) + (r * a) = a ;
r * (p2 `1) >= r * a by A3, A5, XREAL_1:64;
hence p `1 >= a by A7, A8, A9, XREAL_1:7; ::_thesis: verum
end;
theorem :: JORDAN20:4
for p1, p2, p being Point of (TOP-REAL 2)
for a being Real st p in LSeg (p1,p2) & p1 `1 < a & p2 `1 < a holds
p `1 < a
proof
let p1, p2, p be Point of (TOP-REAL 2); ::_thesis: for a being Real st p in LSeg (p1,p2) & p1 `1 < a & p2 `1 < a holds
p `1 < a
let a be Real; ::_thesis: ( p in LSeg (p1,p2) & p1 `1 < a & p2 `1 < a implies p `1 < a )
assume that
A1: p in LSeg (p1,p2) and
A2: p1 `1 < a and
A3: p2 `1 < a ; ::_thesis: p `1 < a
consider r being Real such that
A4: p = ((1 - r) * p1) + (r * p2) and
A5: 0 <= r and
A6: r <= 1 by A1;
A7: p `1 = (((1 - r) * p1) `1) + ((r * p2) `1) by A4, TOPREAL3:2
.= (((1 - r) * p1) `1) + (r * (p2 `1)) by TOPREAL3:4
.= ((1 - r) * (p1 `1)) + (r * (p2 `1)) by TOPREAL3:4 ;
percases ( 0 = r or 0 <> r ) ;
suppose 0 = r ; ::_thesis: p `1 < a
then p = p1 + (0 * p2) by A4, EUCLID:29
.= p1 + (0. (TOP-REAL 2)) by EUCLID:29
.= p1 by EUCLID:27 ;
hence p `1 < a by A2; ::_thesis: verum
end;
supposeA8: 0 <> r ; ::_thesis: p `1 < a
A9: ((1 - r) * a) + (r * a) = a ;
1 - r >= 0 by A6, XREAL_1:48;
then A10: (1 - r) * (p1 `1) <= (1 - r) * a by A2, XREAL_1:64;
r * (p2 `1) < r * a by A3, A5, A8, XREAL_1:68;
hence p `1 < a by A7, A10, A9, XREAL_1:8; ::_thesis: verum
end;
end;
end;
theorem :: JORDAN20:5
for p1, p2, p being Point of (TOP-REAL 2)
for a being Real st p in LSeg (p1,p2) & p1 `1 > a & p2 `1 > a holds
p `1 > a
proof
let p1, p2, p be Point of (TOP-REAL 2); ::_thesis: for a being Real st p in LSeg (p1,p2) & p1 `1 > a & p2 `1 > a holds
p `1 > a
let a be Real; ::_thesis: ( p in LSeg (p1,p2) & p1 `1 > a & p2 `1 > a implies p `1 > a )
assume that
A1: p in LSeg (p1,p2) and
A2: p1 `1 > a and
A3: p2 `1 > a ; ::_thesis: p `1 > a
consider r being Real such that
A4: p = ((1 - r) * p1) + (r * p2) and
A5: 0 <= r and
A6: r <= 1 by A1;
A7: p `1 = (((1 - r) * p1) `1) + ((r * p2) `1) by A4, TOPREAL3:2
.= (((1 - r) * p1) `1) + (r * (p2 `1)) by TOPREAL3:4
.= ((1 - r) * (p1 `1)) + (r * (p2 `1)) by TOPREAL3:4 ;
percases ( 0 = r or 0 <> r ) ;
suppose 0 = r ; ::_thesis: p `1 > a
then p = p1 + (0 * p2) by A4, EUCLID:29
.= p1 + (0. (TOP-REAL 2)) by EUCLID:29
.= p1 by EUCLID:27 ;
hence p `1 > a by A2; ::_thesis: verum
end;
supposeA8: 0 <> r ; ::_thesis: p `1 > a
A9: ((1 - r) * a) + (r * a) = a ;
1 - r >= 0 by A6, XREAL_1:48;
then A10: (1 - r) * (p1 `1) >= (1 - r) * a by A2, XREAL_1:64;
r * (p2 `1) > r * a by A3, A5, A8, XREAL_1:68;
hence p `1 > a by A7, A10, A9, XREAL_1:8; ::_thesis: verum
end;
end;
end;
theorem Th6: :: JORDAN20:6
for j being Element of NAT
for f being S-Sequence_in_R2
for p, q being Point of (TOP-REAL 2) st 1 <= j & j < len f & p in LSeg (f,j) & q in LSeg (f,j) & (f /. j) `2 = (f /. (j + 1)) `2 & (f /. j) `1 > (f /. (j + 1)) `1 & LE p,q, L~ f,f /. 1,f /. (len f) holds
p `1 >= q `1
proof
let j be Element of NAT ; ::_thesis: for f being S-Sequence_in_R2
for p, q being Point of (TOP-REAL 2) st 1 <= j & j < len f & p in LSeg (f,j) & q in LSeg (f,j) & (f /. j) `2 = (f /. (j + 1)) `2 & (f /. j) `1 > (f /. (j + 1)) `1 & LE p,q, L~ f,f /. 1,f /. (len f) holds
p `1 >= q `1
let f be S-Sequence_in_R2; ::_thesis: for p, q being Point of (TOP-REAL 2) st 1 <= j & j < len f & p in LSeg (f,j) & q in LSeg (f,j) & (f /. j) `2 = (f /. (j + 1)) `2 & (f /. j) `1 > (f /. (j + 1)) `1 & LE p,q, L~ f,f /. 1,f /. (len f) holds
p `1 >= q `1
let p, q be Point of (TOP-REAL 2); ::_thesis: ( 1 <= j & j < len f & p in LSeg (f,j) & q in LSeg (f,j) & (f /. j) `2 = (f /. (j + 1)) `2 & (f /. j) `1 > (f /. (j + 1)) `1 & LE p,q, L~ f,f /. 1,f /. (len f) implies p `1 >= q `1 )
assume that
A1: 1 <= j and
A2: j < len f and
A3: p in LSeg (f,j) and
A4: q in LSeg (f,j) and
A5: (f /. j) `2 = (f /. (j + 1)) `2 and
A6: (f /. j) `1 > (f /. (j + 1)) `1 and
A7: LE p,q, L~ f,f /. 1,f /. (len f) ; ::_thesis: p `1 >= q `1
j + 1 <= len f by A2, NAT_1:13;
then A8: LSeg (f,j) = LSeg ((f /. j),(f /. (j + 1))) by A1, TOPREAL1:def_3;
percases ( p `1 <> (f /. j) `1 or p `1 = (f /. j) `1 ) ;
supposeA9: p `1 <> (f /. j) `1 ; ::_thesis: p `1 >= q `1
(f /. j) `1 >= p `1 by A3, A6, A8, TOPREAL1:3;
then (f /. j) `1 > p `1 by A9, XXREAL_0:1;
then A10: ((f /. j) `1) - (p `1) > 0 by XREAL_1:50;
now__::_thesis:_not_p_`1_<_q_`1
reconsider a = (((f /. j) `1) - (q `1)) / (((f /. j) `1) - (p `1)) as Real ;
A11: 1 - a = ((((f /. j) `1) - (p `1)) / (((f /. j) `1) - (p `1))) - ((((f /. j) `1) - (q `1)) / (((f /. j) `1) - (p `1))) by A10, XCMPLX_1:60
.= ((((f /. j) `1) - (p `1)) - (((f /. j) `1) - (q `1))) / (((f /. j) `1) - (p `1)) by XCMPLX_1:120
.= ((q `1) - (p `1)) / (((f /. j) `1) - (p `1)) ;
A12: (((1 - a) * (f /. j)) + (a * p)) `1 = (((1 - a) * (f /. j)) `1) + ((a * p) `1) by TOPREAL3:2
.= ((1 - a) * ((f /. j) `1)) + ((a * p) `1) by TOPREAL3:4
.= ((1 * ((f /. j) `1)) - (a * ((f /. j) `1))) + (a * (p `1)) by TOPREAL3:4
.= ((f /. j) `1) - (a * (((f /. j) `1) - (p `1)))
.= ((f /. j) `1) - (((f /. j) `1) - (q `1)) by A10, XCMPLX_1:87
.= q `1 ;
(f /. j) `1 >= q `1 by A4, A6, A8, TOPREAL1:3;
then A13: ((f /. j) `1) - (q `1) >= 0 by XREAL_1:48;
A14: p `2 = (f /. j) `2 by A3, A5, A8, GOBOARD7:6;
(((1 - a) * (f /. j)) + (a * p)) `2 = (((1 - a) * (f /. j)) `2) + ((a * p) `2) by TOPREAL3:2
.= ((1 - a) * ((f /. j) `2)) + ((a * p) `2) by TOPREAL3:4
.= ((1 * ((f /. j) `2)) - (a * ((f /. j) `2))) + (a * (p `2)) by TOPREAL3:4
.= q `2 by A4, A5, A8, A14, GOBOARD7:6 ;
then A15: q = ((1 - a) * (f /. j)) + (a * p) by A12, TOPREAL3:6;
assume A16: p `1 < q `1 ; ::_thesis: contradiction
then (q `1) - (p `1) > 0 by XREAL_1:50;
then (1 - a) + a >= 0 + a by A10, A11, XREAL_1:7;
then q in LSeg ((f /. j),p) by A10, A13, A15;
then LE q,p, L~ f,f /. 1,f /. (len f) by A1, A2, A3, SPRECT_3:23;
hence contradiction by A7, A16, JORDAN5C:12, TOPREAL1:25; ::_thesis: verum
end;
hence p `1 >= q `1 ; ::_thesis: verum
end;
suppose p `1 = (f /. j) `1 ; ::_thesis: p `1 >= q `1
hence p `1 >= q `1 by A4, A6, A8, TOPREAL1:3; ::_thesis: verum
end;
end;
end;
theorem Th7: :: JORDAN20:7
for j being Element of NAT
for f being S-Sequence_in_R2
for p, q being Point of (TOP-REAL 2) st 1 <= j & j < len f & p in LSeg (f,j) & q in LSeg (f,j) & (f /. j) `2 = (f /. (j + 1)) `2 & (f /. j) `1 < (f /. (j + 1)) `1 & LE p,q, L~ f,f /. 1,f /. (len f) holds
p `1 <= q `1
proof
let j be Element of NAT ; ::_thesis: for f being S-Sequence_in_R2
for p, q being Point of (TOP-REAL 2) st 1 <= j & j < len f & p in LSeg (f,j) & q in LSeg (f,j) & (f /. j) `2 = (f /. (j + 1)) `2 & (f /. j) `1 < (f /. (j + 1)) `1 & LE p,q, L~ f,f /. 1,f /. (len f) holds
p `1 <= q `1
let f be S-Sequence_in_R2; ::_thesis: for p, q being Point of (TOP-REAL 2) st 1 <= j & j < len f & p in LSeg (f,j) & q in LSeg (f,j) & (f /. j) `2 = (f /. (j + 1)) `2 & (f /. j) `1 < (f /. (j + 1)) `1 & LE p,q, L~ f,f /. 1,f /. (len f) holds
p `1 <= q `1
let p, q be Point of (TOP-REAL 2); ::_thesis: ( 1 <= j & j < len f & p in LSeg (f,j) & q in LSeg (f,j) & (f /. j) `2 = (f /. (j + 1)) `2 & (f /. j) `1 < (f /. (j + 1)) `1 & LE p,q, L~ f,f /. 1,f /. (len f) implies p `1 <= q `1 )
assume that
A1: 1 <= j and
A2: j < len f and
A3: p in LSeg (f,j) and
A4: q in LSeg (f,j) and
A5: (f /. j) `2 = (f /. (j + 1)) `2 and
A6: (f /. j) `1 < (f /. (j + 1)) `1 and
A7: LE p,q, L~ f,f /. 1,f /. (len f) ; ::_thesis: p `1 <= q `1
j + 1 <= len f by A2, NAT_1:13;
then A8: LSeg (f,j) = LSeg ((f /. j),(f /. (j + 1))) by A1, TOPREAL1:def_3;
percases ( p `1 <> (f /. j) `1 or p `1 = (f /. j) `1 ) ;
supposeA9: p `1 <> (f /. j) `1 ; ::_thesis: p `1 <= q `1
(f /. j) `1 <= p `1 by A3, A6, A8, TOPREAL1:3;
then (f /. j) `1 < p `1 by A9, XXREAL_0:1;
then A10: ((f /. j) `1) - (p `1) < 0 by XREAL_1:49;
now__::_thesis:_not_p_`1_>_q_`1
reconsider a = (((f /. j) `1) - (q `1)) / (((f /. j) `1) - (p `1)) as Real ;
A11: 1 - a = ((((f /. j) `1) - (p `1)) / (((f /. j) `1) - (p `1))) - ((((f /. j) `1) - (q `1)) / (((f /. j) `1) - (p `1))) by A10, XCMPLX_1:60
.= ((((f /. j) `1) - (p `1)) - (((f /. j) `1) - (q `1))) / (((f /. j) `1) - (p `1)) by XCMPLX_1:120
.= ((q `1) - (p `1)) / (((f /. j) `1) - (p `1)) ;
A12: (((1 - a) * (f /. j)) + (a * p)) `1 = (((1 - a) * (f /. j)) `1) + ((a * p) `1) by TOPREAL3:2
.= ((1 - a) * ((f /. j) `1)) + ((a * p) `1) by TOPREAL3:4
.= ((1 * ((f /. j) `1)) - (a * ((f /. j) `1))) + (a * (p `1)) by TOPREAL3:4
.= ((f /. j) `1) - (a * (((f /. j) `1) - (p `1)))
.= ((f /. j) `1) - (((f /. j) `1) - (q `1)) by A10, XCMPLX_1:87
.= q `1 ;
(f /. j) `1 <= q `1 by A4, A6, A8, TOPREAL1:3;
then A13: ((f /. j) `1) - (q `1) <= 0 by XREAL_1:47;
A14: p `2 = (f /. j) `2 by A3, A5, A8, GOBOARD7:6;
(((1 - a) * (f /. j)) + (a * p)) `2 = (((1 - a) * (f /. j)) `2) + ((a * p) `2) by TOPREAL3:2
.= ((1 - a) * ((f /. j) `2)) + ((a * p) `2) by TOPREAL3:4
.= ((1 * ((f /. j) `2)) - (a * ((f /. j) `2))) + (a * (p `2)) by TOPREAL3:4
.= q `2 by A4, A5, A8, A14, GOBOARD7:6 ;
then A15: q = ((1 - a) * (f /. j)) + (a * p) by A12, TOPREAL3:6;
assume A16: p `1 > q `1 ; ::_thesis: contradiction
then (q `1) - (p `1) < 0 by XREAL_1:49;
then (1 - a) + a >= 0 + a by A10, A11, XREAL_1:7;
then q in LSeg ((f /. j),p) by A10, A13, A15;
then LE q,p, L~ f,f /. 1,f /. (len f) by A1, A2, A3, SPRECT_3:23;
hence contradiction by A7, A16, JORDAN5C:12, TOPREAL1:25; ::_thesis: verum
end;
hence p `1 <= q `1 ; ::_thesis: verum
end;
suppose p `1 = (f /. j) `1 ; ::_thesis: p `1 <= q `1
hence p `1 <= q `1 by A4, A6, A8, TOPREAL1:3; ::_thesis: verum
end;
end;
end;
definition
let P be Subset of (TOP-REAL 2);
let p1, p2, p be Point of (TOP-REAL 2);
let e be Real;
predp is_Lin P,p1,p2,e means :Def1: :: JORDAN20:def 1
( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p4 being Point of (TOP-REAL 2) st
( p4 `1 < e & LE p4,p,P,p1,p2 & ( for p5 being Point of (TOP-REAL 2) st LE p4,p5,P,p1,p2 & LE p5,p,P,p1,p2 holds
p5 `1 <= e ) ) );
predp is_Rin P,p1,p2,e means :Def2: :: JORDAN20:def 2
( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p4 being Point of (TOP-REAL 2) st
( p4 `1 > e & LE p4,p,P,p1,p2 & ( for p5 being Point of (TOP-REAL 2) st LE p4,p5,P,p1,p2 & LE p5,p,P,p1,p2 holds
p5 `1 >= e ) ) );
predp is_Lout P,p1,p2,e means :Def3: :: JORDAN20:def 3
( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p4 being Point of (TOP-REAL 2) st
( p4 `1 < e & LE p,p4,P,p1,p2 & ( for p5 being Point of (TOP-REAL 2) st LE p5,p4,P,p1,p2 & LE p,p5,P,p1,p2 holds
p5 `1 <= e ) ) );
predp is_Rout P,p1,p2,e means :Def4: :: JORDAN20:def 4
( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p4 being Point of (TOP-REAL 2) st
( p4 `1 > e & LE p,p4,P,p1,p2 & ( for p5 being Point of (TOP-REAL 2) st LE p5,p4,P,p1,p2 & LE p,p5,P,p1,p2 holds
p5 `1 >= e ) ) );
predp is_OSin P,p1,p2,e means :Def5: :: JORDAN20:def 5
( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p7 being Point of (TOP-REAL 2) st
( LE p7,p,P,p1,p2 & ( for p8 being Point of (TOP-REAL 2) st LE p7,p8,P,p1,p2 & LE p8,p,P,p1,p2 holds
p8 `1 = e ) & ( for p4 being Point of (TOP-REAL 2) st LE p4,p7,P,p1,p2 & p4 <> p7 holds
( ex p5 being Point of (TOP-REAL 2) st
( LE p4,p5,P,p1,p2 & LE p5,p7,P,p1,p2 & p5 `1 > e ) & ex p6 being Point of (TOP-REAL 2) st
( LE p4,p6,P,p1,p2 & LE p6,p7,P,p1,p2 & p6 `1 < e ) ) ) ) );
predp is_OSout P,p1,p2,e means :Def6: :: JORDAN20:def 6
( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p7 being Point of (TOP-REAL 2) st
( LE p,p7,P,p1,p2 & ( for p8 being Point of (TOP-REAL 2) st LE p8,p7,P,p1,p2 & LE p,p8,P,p1,p2 holds
p8 `1 = e ) & ( for p4 being Point of (TOP-REAL 2) st LE p7,p4,P,p1,p2 & p4 <> p7 holds
( ex p5 being Point of (TOP-REAL 2) st
( LE p5,p4,P,p1,p2 & LE p7,p5,P,p1,p2 & p5 `1 > e ) & ex p6 being Point of (TOP-REAL 2) st
( LE p6,p4,P,p1,p2 & LE p7,p6,P,p1,p2 & p6 `1 < e ) ) ) ) );
correctness
;
end;
:: deftheorem Def1 defines is_Lin JORDAN20:def_1_:_
for P being Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2)
for e being Real holds
( p is_Lin P,p1,p2,e iff ( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p4 being Point of (TOP-REAL 2) st
( p4 `1 < e & LE p4,p,P,p1,p2 & ( for p5 being Point of (TOP-REAL 2) st LE p4,p5,P,p1,p2 & LE p5,p,P,p1,p2 holds
p5 `1 <= e ) ) ) );
:: deftheorem Def2 defines is_Rin JORDAN20:def_2_:_
for P being Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2)
for e being Real holds
( p is_Rin P,p1,p2,e iff ( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p4 being Point of (TOP-REAL 2) st
( p4 `1 > e & LE p4,p,P,p1,p2 & ( for p5 being Point of (TOP-REAL 2) st LE p4,p5,P,p1,p2 & LE p5,p,P,p1,p2 holds
p5 `1 >= e ) ) ) );
:: deftheorem Def3 defines is_Lout JORDAN20:def_3_:_
for P being Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2)
for e being Real holds
( p is_Lout P,p1,p2,e iff ( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p4 being Point of (TOP-REAL 2) st
( p4 `1 < e & LE p,p4,P,p1,p2 & ( for p5 being Point of (TOP-REAL 2) st LE p5,p4,P,p1,p2 & LE p,p5,P,p1,p2 holds
p5 `1 <= e ) ) ) );
:: deftheorem Def4 defines is_Rout JORDAN20:def_4_:_
for P being Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2)
for e being Real holds
( p is_Rout P,p1,p2,e iff ( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p4 being Point of (TOP-REAL 2) st
( p4 `1 > e & LE p,p4,P,p1,p2 & ( for p5 being Point of (TOP-REAL 2) st LE p5,p4,P,p1,p2 & LE p,p5,P,p1,p2 holds
p5 `1 >= e ) ) ) );
:: deftheorem Def5 defines is_OSin JORDAN20:def_5_:_
for P being Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2)
for e being Real holds
( p is_OSin P,p1,p2,e iff ( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p7 being Point of (TOP-REAL 2) st
( LE p7,p,P,p1,p2 & ( for p8 being Point of (TOP-REAL 2) st LE p7,p8,P,p1,p2 & LE p8,p,P,p1,p2 holds
p8 `1 = e ) & ( for p4 being Point of (TOP-REAL 2) st LE p4,p7,P,p1,p2 & p4 <> p7 holds
( ex p5 being Point of (TOP-REAL 2) st
( LE p4,p5,P,p1,p2 & LE p5,p7,P,p1,p2 & p5 `1 > e ) & ex p6 being Point of (TOP-REAL 2) st
( LE p4,p6,P,p1,p2 & LE p6,p7,P,p1,p2 & p6 `1 < e ) ) ) ) ) );
:: deftheorem Def6 defines is_OSout JORDAN20:def_6_:_
for P being Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2)
for e being Real holds
( p is_OSout P,p1,p2,e iff ( P is_an_arc_of p1,p2 & p in P & p `1 = e & ex p7 being Point of (TOP-REAL 2) st
( LE p,p7,P,p1,p2 & ( for p8 being Point of (TOP-REAL 2) st LE p8,p7,P,p1,p2 & LE p,p8,P,p1,p2 holds
p8 `1 = e ) & ( for p4 being Point of (TOP-REAL 2) st LE p7,p4,P,p1,p2 & p4 <> p7 holds
( ex p5 being Point of (TOP-REAL 2) st
( LE p5,p4,P,p1,p2 & LE p7,p5,P,p1,p2 & p5 `1 > e ) & ex p6 being Point of (TOP-REAL 2) st
( LE p6,p4,P,p1,p2 & LE p7,p6,P,p1,p2 & p6 `1 < e ) ) ) ) ) );
theorem :: JORDAN20:8
for P being Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2)
for e being Real st P is_an_arc_of p1,p2 & p1 `1 <= e & p2 `1 >= e holds
ex p3 being Point of (TOP-REAL 2) st
( p3 in P & p3 `1 = e )
proof
let P be Subset of (TOP-REAL 2); ::_thesis: for p1, p2, p being Point of (TOP-REAL 2)
for e being Real st P is_an_arc_of p1,p2 & p1 `1 <= e & p2 `1 >= e holds
ex p3 being Point of (TOP-REAL 2) st
( p3 in P & p3 `1 = e )
let p1, p2, p be Point of (TOP-REAL 2); ::_thesis: for e being Real st P is_an_arc_of p1,p2 & p1 `1 <= e & p2 `1 >= e holds
ex p3 being Point of (TOP-REAL 2) st
( p3 in P & p3 `1 = e )
let e be Real; ::_thesis: ( P is_an_arc_of p1,p2 & p1 `1 <= e & p2 `1 >= e implies ex p3 being Point of (TOP-REAL 2) st
( p3 in P & p3 `1 = e ) )
set x = the Element of P /\ (Vertical_Line e);
assume ( P is_an_arc_of p1,p2 & p1 `1 <= e & p2 `1 >= e ) ; ::_thesis: ex p3 being Point of (TOP-REAL 2) st
( p3 in P & p3 `1 = e )
then P meets Vertical_Line e by JORDAN6:49;
then A1: P /\ (Vertical_Line e) <> {} by XBOOLE_0:def_7;
then the Element of P /\ (Vertical_Line e) in Vertical_Line e by XBOOLE_0:def_4;
then the Element of P /\ (Vertical_Line e) in { p3 where p3 is Point of (TOP-REAL 2) : p3 `1 = e } by JORDAN6:def_6;
then A2: ex p4 being Point of (TOP-REAL 2) st
( p4 = the Element of P /\ (Vertical_Line e) & p4 `1 = e ) ;
the Element of P /\ (Vertical_Line e) in P by A1, XBOOLE_0:def_4;
hence ex p3 being Point of (TOP-REAL 2) st
( p3 in P & p3 `1 = e ) by A2; ::_thesis: verum
end;
theorem :: JORDAN20:9
for P being non empty Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2)
for e being Real st P is_an_arc_of p1,p2 & p1 `1 < e & p in P & p `1 = e & not p is_Lin P,p1,p2,e & not p is_Rin P,p1,p2,e holds
p is_OSin P,p1,p2,e
proof
let P be non empty Subset of (TOP-REAL 2); ::_thesis: for p1, p2, p being Point of (TOP-REAL 2)
for e being Real st P is_an_arc_of p1,p2 & p1 `1 < e & p in P & p `1 = e & not p is_Lin P,p1,p2,e & not p is_Rin P,p1,p2,e holds
p is_OSin P,p1,p2,e
let p1, p2, p be Point of (TOP-REAL 2); ::_thesis: for e being Real st P is_an_arc_of p1,p2 & p1 `1 < e & p in P & p `1 = e & not p is_Lin P,p1,p2,e & not p is_Rin P,p1,p2,e holds
p is_OSin P,p1,p2,e
let e be Real; ::_thesis: ( P is_an_arc_of p1,p2 & p1 `1 < e & p in P & p `1 = e & not p is_Lin P,p1,p2,e & not p is_Rin P,p1,p2,e implies p is_OSin P,p1,p2,e )
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: ( p is_Lin P,p1,p2,e or p is_Rin P,p1,p2,e or p is_OSin P,p1,p2,e )
now__::_thesis:_(_p_is_OSin_P,p1,p2,e_or_p_is_Lin_P,p1,p2,e_or_p_is_Rin_P,p1,p2,e_)
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: ( not 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 ) ) } or x in REAL )
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: x in REAL
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: verum
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: ( not x in R or x in [.0,1.] )
assume x in R ; ::_thesis: x in [.0,1.]
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: verum
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: ( s in R implies s < s2 )
assume s in R ; ::_thesis: s < s2
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: verum
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: pro1 . p7 = proj1 . p7
the carrier of ((TOP-REAL 2) | P) = P by PRE_TOPC:8;
hence pro1 . p7 = proj1 . p7 by FUNCT_1:49; ::_thesis: verum
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: ( LE p9,p8,P,p1,p2 & LE p8,p,P,p1,p2 implies p8 `1 = e )
assume that
A29: LE p9,p8,P,p1,p2 and
A30: LE p8,p,P,p1,p2 ; ::_thesis: p8 `1 = e
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__::_thesis:_not_p8_`1_<>_e
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: contradiction
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: ( p7 = f . s70 implies p7 `1 <> e )
assume A57: p7 = f . s70 ; ::_thesis: p7 `1 <> e
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__::_thesis:_not_p7_`1_=_e
assume A63: p7 `1 = e ; ::_thesis: contradiction
now__::_thesis:_(_(_(p8_`1)_-_e_>=_0_&_contradiction_)_or_(_(p8_`1)_-_e_<_0_&_contradiction_)_)
percases ( (p8 `1) - e >= 0 or (p8 `1) - e < 0 ) ;
caseA64: (p8 `1) - e >= 0 ; ::_thesis: contradiction
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: verum
end;
caseA66: (p8 `1) - e < 0 ; ::_thesis: contradiction
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: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
hence p7 `1 <> e ; ::_thesis: verum
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: verum
end;
hence p8 `1 = e ; ::_thesis: verum
end;
assume not p is_OSin P,p1,p2,e ; ::_thesis: ( p is_Lin P,p1,p2,e or p is_Rin P,p1,p2,e )
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__::_thesis:_(_(_(_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_)_&_(_p_is_Lin_P,p1,p2,e_or_p_is_Rin_P,p1,p2,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_)_&_(_p_is_Lin_P,p1,p2,e_or_p_is_Rin_P,p1,p2,e_)_)_)
percases ( 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 A72;
caseA74: 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: ( p is_Lin P,p1,p2,e or p is_Rin P,p1,p2,e )
A75: now__::_thesis:_ex_p51_being_Point_of_(TOP-REAL_2)_st_
(_LE_p4,p51,P,p1,p2_&_LE_p51,p9,P,p1,p2_&_p51_`1_<_e_)
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: contradiction
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: ( LE p4,p51,P,p1,p2 & LE p51,p9,P,p1,p2 implies p51 `1 = e )
assume ( LE p4,p51,P,p1,p2 & LE p51,p9,P,p1,p2 ) ; ::_thesis: p51 `1 = e
then ( p51 `1 >= e & p51 `1 <= e ) by A74, A80;
hence p51 `1 = e by XXREAL_0:1; ::_thesis: verum
end;
A82: now__::_thesis:_not_s4_<_s0
assume s4 < s0 ; ::_thesis: contradiction
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: verum
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: verum
end;
now__::_thesis:_(_ex_p51_being_Point_of_(TOP-REAL_2)_st_
(_LE_p4,p51,P,p1,p2_&_LE_p51,p9,P,p1,p2_&_p51_`1_<_e_)_implies_p_is_Lin_P,p1,p2,e_)
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: p is_Lin P,p1,p2,e
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: ( LE p51,p5,P,p1,p2 & LE p5,p,P,p1,p2 implies p5 `1 <= e )
assume that
A94: LE p51,p5,P,p1,p2 and
A95: LE p5,p,P,p1,p2 ; ::_thesis: p5 `1 <= e
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__::_thesis:_(_(_LE_p5,p9,P,p1,p2_&_p5_`1_<=_e_)_or_(_LE_p9,p5,P,p1,p2_&_p5_`1_<=_e_)_)
percases ( LE p5,p9,P,p1,p2 or LE p9,p5,P,p1,p2 ) by A1, A73, A97, A98, JORDAN5C:14;
case LE p5,p9,P,p1,p2 ; ::_thesis: p5 `1 <= e
hence p5 `1 <= e by A74, A96; ::_thesis: verum
end;
case LE p9,p5,P,p1,p2 ; ::_thesis: p5 `1 <= e
hence p5 `1 <= e by A28, A95; ::_thesis: verum
end;
end;
end;
hence p5 `1 <= e ; ::_thesis: verum
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: verum
end;
hence ( p is_Lin P,p1,p2,e or p is_Rin P,p1,p2,e ) by A75; ::_thesis: verum
end;
caseA99: 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: ( p is_Lin P,p1,p2,e or p is_Rin P,p1,p2,e )
A100: now__::_thesis:_ex_p51_being_Point_of_(TOP-REAL_2)_st_
(_LE_p4,p51,P,p1,p2_&_LE_p51,p9,P,p1,p2_&_p51_`1_>_e_)
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: contradiction
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: ( LE p4,p51,P,p1,p2 & LE p51,p9,P,p1,p2 implies p51 `1 = e )
assume ( LE p4,p51,P,p1,p2 & LE p51,p9,P,p1,p2 ) ; ::_thesis: p51 `1 = e
then ( p51 `1 <= e & p51 `1 >= e ) by A99, A105;
hence p51 `1 = e by XXREAL_0:1; ::_thesis: verum
end;
A107: now__::_thesis:_not_s4_<_s0
assume s4 < s0 ; ::_thesis: contradiction
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: verum
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: verum
end;
now__::_thesis:_(_ex_p51_being_Point_of_(TOP-REAL_2)_st_
(_LE_p4,p51,P,p1,p2_&_LE_p51,p9,P,p1,p2_&_p51_`1_>_e_)_implies_p_is_Rin_P,p1,p2,e_)
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: p is_Rin P,p1,p2,e
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: ( LE p51,p5,P,p1,p2 & LE p5,p,P,p1,p2 implies p5 `1 >= e )
assume that
A119: LE p51,p5,P,p1,p2 and
A120: LE p5,p,P,p1,p2 ; ::_thesis: p5 `1 >= e
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__::_thesis:_(_(_LE_p5,p9,P,p1,p2_&_p5_`1_>=_e_)_or_(_LE_p9,p5,P,p1,p2_&_p5_`1_>=_e_)_)
percases ( LE p5,p9,P,p1,p2 or LE p9,p5,P,p1,p2 ) by A1, A73, A122, A123, JORDAN5C:14;
case LE p5,p9,P,p1,p2 ; ::_thesis: p5 `1 >= e
hence p5 `1 >= e by A99, A121; ::_thesis: verum
end;
case LE p9,p5,P,p1,p2 ; ::_thesis: p5 `1 >= e
hence p5 `1 >= e by A28, A120; ::_thesis: verum
end;
end;
end;
hence p5 `1 >= e ; ::_thesis: verum
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: verum
end;
hence ( p is_Lin P,p1,p2,e or p is_Rin P,p1,p2,e ) by A100; ::_thesis: verum
end;
end;
end;
hence ( p is_Lin P,p1,p2,e or p is_Rin P,p1,p2,e ) ; ::_thesis: verum
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: verum
end;
theorem :: JORDAN20:10
for P being non empty Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2)
for e being Real st P is_an_arc_of p1,p2 & p2 `1 > e & p in P & p `1 = e & not p is_Lout P,p1,p2,e & not p is_Rout P,p1,p2,e holds
p is_OSout P,p1,p2,e
proof
let P be non empty Subset of (TOP-REAL 2); ::_thesis: for p1, p2, p being Point of (TOP-REAL 2)
for e being Real st P is_an_arc_of p1,p2 & p2 `1 > e & p in P & p `1 = e & not p is_Lout P,p1,p2,e & not p is_Rout P,p1,p2,e holds
p is_OSout P,p1,p2,e
let p1, p2, p be Point of (TOP-REAL 2); ::_thesis: for e being Real st P is_an_arc_of p1,p2 & p2 `1 > e & p in P & p `1 = e & not p is_Lout P,p1,p2,e & not p is_Rout P,p1,p2,e holds
p is_OSout P,p1,p2,e
let e be Real; ::_thesis: ( P is_an_arc_of p1,p2 & p2 `1 > e & p in P & p `1 = e & not p is_Lout P,p1,p2,e & not p is_Rout P,p1,p2,e implies p is_OSout P,p1,p2,e )
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: ( p is_Lout P,p1,p2,e or p is_Rout P,p1,p2,e or p is_OSout P,p1,p2,e )
now__::_thesis:_(_p_is_OSout_P,p1,p2,e_or_p_is_Lout_P,p1,p2,e_or_p_is_Rout_P,p1,p2,e_)
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: 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
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not 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 ) ) } or x in REAL )
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: x in REAL
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 REAL ; ::_thesis: verum
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 ;
A15: for s being real number st s in R holds
s > s2
proof
let s be real number ; ::_thesis: ( s in R implies s > s2 )
assume s in R ; ::_thesis: s > s2
then A16: 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 A16, XXREAL_0:1; ::_thesis: verum
end;
then for s being real number st s in R holds
s >= s2 ;
then A17: 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: pro1 . p7 = proj1 . p7
the carrier of ((TOP-REAL 2) | P) = P by PRE_TOPC:8;
hence pro1 . p7 = proj1 . p7 by FUNCT_1:49; ::_thesis: verum
end;
then A18: pro1 is continuous by JGRAPH_2:29;
reconsider h = pro1 * f as Function of I[01],R^1 ;
A19: 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 A15;
then s2 is LowerBound of R by XXREAL_2:def_2;
then A20: R is bounded_below by XXREAL_2:def_9;
A21: 0 <= s2 by A10, BORSUK_1:40, XXREAL_1:1;
R c= [.0,1.]
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in R or x in [.0,1.] )
assume x in R ; ::_thesis: x in [.0,1.]
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 A21, XXREAL_1:1; ::_thesis: verum
end;
then A22: 1 >= s0 by A14, BORSUK_1:40, BORSUK_4:26;
then s0 in dom f by A12, A21, A17, 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 p,p9,P,p1,p2 by A1, A5, A6, A7, A11, A21, A13, A22, A17, JORDAN5C:8;
A24: rng f = P by A9, PRE_TOPC:def_5;
A25: 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: ( LE p8,p9,P,p1,p2 & LE p,p8,P,p1,p2 implies p8 `1 = e )
assume that
A26: LE p8,p9,P,p1,p2 and
A27: LE p,p8,P,p1,p2 ; ::_thesis: p8 `1 = e
A28: p8 in P by A26, JORDAN5C:def_3;
then consider x8 being set such that
A29: x8 in dom f and
A30: p8 = f . x8 by A24, FUNCT_1:def_3;
reconsider s8 = x8 as Real by A12, A29, BORSUK_1:40;
A31: s8 <= 1 by A29, BORSUK_1:40, XXREAL_1:1;
0 <= s8 by A29, BORSUK_1:40, XXREAL_1:1;
then A32: s8 >= s2 by A5, A6, A7, A11, A13, A27, A30, A31, JORDAN5C:def_3;
A33: s0 >= s8 by A5, A6, A7, A21, A22, A17, A26, A30, A31, JORDAN5C:def_3;
now__::_thesis:_not_p8_`1_<>_e
reconsider s8n = s8 as Point of RealSpace by METRIC_1:def_13;
reconsider s8m = s8 as Point of (Closed-Interval-MSpace (0,1)) by A29, 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;
A34: 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 A35: p8 `1 <> e ; ::_thesis: contradiction
then (p8 `1) - e <> 0 ;
then abs ((p8 `1) - e) > 0 by COMPLEX1:47;
then A36: w in Ball (w,ee) by GOBOARD6:1, XREAL_1:139;
A37: ( h " B is open & I[01] = TopSpaceMetr (Closed-Interval-MSpace (0,1)) ) by A8, A18, TOPMETR:20, TOPMETR:def_6, TOPMETR:def_7, UNIFORM1:2;
h . s8 = pro1 . (f . s8) by A19, A29, BORSUK_1:40, FUNCT_1:12
.= proj1 . p8 by A28, A30, FUNCT_1:49
.= p8 `1 by PSCOMP_1:def_5 ;
then s8 in h " B by A19, A29, A36, BORSUK_1:40, FUNCT_1:def_7;
then consider r0 being real number such that
A38: r0 > 0 and
A39: Ball (s8m,r0) c= h " B by A37, TOPMETR:15;
reconsider r0 = r0 as Real by XREAL_0:def_1;
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 A40: Ball (s8m,r01) = ].(s8 - r01),(s8 + r01).[ /\ [.0,1.] by TOPMETR:9;
s8 > s2 by A4, A11, A30, A32, A35, XXREAL_0:1;
then s8 - s2 > 0 by XREAL_1:50;
then A41: r01 > 0 by A38, XXREAL_0:21;
then A42: (r01 - (r01 / 2)) + (r01 / 2) > 0 + (r01 / 2) by XREAL_1:6;
then A43: s8 - r01 < s8 - (r01 / 2) by XREAL_1:10;
A44: r01 / 2 > 0 by A41, XREAL_1:139;
then A45: s8 + (- (r01 / 2)) < s8 + (r01 / 2) by XREAL_1:8;
s8 + (r01 / 2) < s8 + r01 by A42, XREAL_1:8;
then s8 - (r01 / 2) < s8 + r01 by A45, XXREAL_0:2;
then A46: s8 - (r01 / 2) in ].(s8 - r01),(s8 + r01).[ by A43, XXREAL_1:4;
A47: s8 - (r01 / 2) > s8 - r01 by A42, XREAL_1:10;
Ball (s8m,r01) c= Ball (s8m,r0) by PCOMPS_1:1, XXREAL_0:17;
then A48: ].(s8 - r01),(s8 + r01).[ /\ [.0,1.] c= h " B by A39, A40, XBOOLE_1:1;
reconsider s70 = s8 - (r01 / 2) as Real ;
s8 - s2 >= r01 by XXREAL_0:17;
then - (s8 - s2) <= - r01 by XREAL_1:24;
then A49: (s2 - s8) + s8 <= (- r01) + s8 by XREAL_1:7;
- (- (r01 / 2)) > 0 by A41, XREAL_1:139;
then - (r01 / 2) < 0 ;
then A50: s8 + 0 > s8 + (- (r01 / 2)) by XREAL_1:8;
then A51: 1 >= s70 by A31, XXREAL_0:2;
1 - 0 > s8 - (r01 / 2) by A31, A44, XREAL_1:15;
then s8 - (r01 / 2) in [.0,1.] by A21, A49, A47, XXREAL_1:1;
then A52: s8 - (r01 / 2) in ].(s8 - r01),(s8 + r01).[ /\ [.0,1.] by A46, XBOOLE_0:def_4;
then A53: h . (s8 - (r01 / 2)) in B by A48, FUNCT_1:def_7;
A54: s8 - (r01 / 2) in dom h by A48, A52, FUNCT_1:def_7;
A55: 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: ( p7 = f . s70 implies p7 `1 <> e )
assume A56: p7 = f . s70 ; ::_thesis: p7 `1 <> e
s70 in [.0,1.] by A21, A49, A43, A51, XXREAL_1:1;
then A57: p7 in rng f by A12, A56, BORSUK_1:40, FUNCT_1:def_3;
A58: rng f = [#] ((TOP-REAL 2) | P) by A5, TOPS_2:def_5
.= P by PRE_TOPC:def_5 ;
A59: h . s70 = pro1 . (f . s70) by A54, FUNCT_1:12
.= pr1a . p7 by A56, A57, A58, FUNCT_1:49
.= p7 `1 by PSCOMP_1:def_5 ;
then A60: p7 `1 < (p8 `1) + ee by A34, A53, XXREAL_1:4;
A61: (p8 `1) - ee < p7 `1 by A34, A53, A59, XXREAL_1:4;
now__::_thesis:_not_p7_`1_=_e
assume A62: p7 `1 = e ; ::_thesis: contradiction
now__::_thesis:_(_(_(p8_`1)_-_e_>=_0_&_contradiction_)_or_(_(p8_`1)_-_e_<_0_&_contradiction_)_)
percases ( (p8 `1) - e >= 0 or (p8 `1) - e < 0 ) ;
caseA63: (p8 `1) - e >= 0 ; ::_thesis: contradiction
then (p8 `1) - (((p8 `1) - e) / 2) < e by A61, A62, ABSVALUE:def_1;
then ((p8 `1) / 2) + (e / 2) < (e / 2) + (e / 2) ;
then (p8 `1) / 2 < e / 2 by XREAL_1:7;
then A64: ((p8 `1) / 2) - (e / 2) < (e / 2) - (e / 2) by XREAL_1:14;
((p8 `1) - e) / 2 >= 0 / 2 by A63;
hence contradiction by A64; ::_thesis: verum
end;
caseA65: (p8 `1) - e < 0 ; ::_thesis: contradiction
then e < (p8 `1) + ((- ((p8 `1) - e)) / 2) by A60, A62, ABSVALUE:def_1;
then ((p8 `1) / 2) + (e / 2) > (e / 2) + (e / 2) ;
then (p8 `1) / 2 > e / 2 by XREAL_1:7;
then A66: ((p8 `1) / 2) - (e / 2) > (e / 2) - (e / 2) by XREAL_1:14;
((p8 `1) - e) / 2 <= 0 / 2 by A65;
hence contradiction by A66; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
hence p7 `1 <> e ; ::_thesis: verum
end;
s70 >= s2 by A49, A43, XXREAL_0:2;
then consider s7 being Real such that
A67: s8 > s7 and
A68: ( 1 >= s7 & s7 >= s2 & ( for p7 being Point of (TOP-REAL 2) st p7 = f . s7 holds
p7 `1 <> e ) ) by A50, A51, A55;
s7 in R by A68;
then s7 >= s0 by A20, SEQ_4:def_2;
hence contradiction by A33, A67, XXREAL_0:2; ::_thesis: verum
end;
hence p8 `1 = e ; ::_thesis: verum
end;
assume not p is_OSout P,p1,p2,e ; ::_thesis: ( p is_Lout P,p1,p2,e or p is_Rout P,p1,p2,e )
then consider p4 being Point of (TOP-REAL 2) such that
A69: LE p9,p4,P,p1,p2 and
A70: p4 <> p9 and
A71: ( 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, A23, A25, Def6;
A72: p9 in P by A69, JORDAN5C:def_3;
now__::_thesis:_(_(_(_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_)_&_(_p_is_Lout_P,p1,p2,e_or_p_is_Rout_P,p1,p2,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_)_&_(_p_is_Lout_P,p1,p2,e_or_p_is_Rout_P,p1,p2,e_)_)_)
percases ( 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 A71;
caseA73: 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: ( p is_Lout P,p1,p2,e or p is_Rout P,p1,p2,e )
A74: now__::_thesis:_ex_p51_being_Point_of_(TOP-REAL_2)_st_
(_LE_p51,p4,P,p1,p2_&_LE_p9,p51,P,p1,p2_&_p51_`1_<_e_)
p4 in P by A69, JORDAN5C:def_3;
then p4 in rng f by A9, PRE_TOPC:def_5;
then consider xs4 being set such that
A75: xs4 in dom f and
A76: p4 = f . xs4 by FUNCT_1:def_3;
reconsider s4 = xs4 as Real by A12, A75, BORSUK_1:40;
A77: s4 <= 1 by A75, BORSUK_1:40, XXREAL_1:1;
assume A78: 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: contradiction
A79: 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: ( LE p51,p4,P,p1,p2 & LE p9,p51,P,p1,p2 implies p51 `1 = e )
assume ( LE p51,p4,P,p1,p2 & LE p9,p51,P,p1,p2 ) ; ::_thesis: p51 `1 = e
then ( p51 `1 >= e & p51 `1 <= e ) by A73, A78;
hence p51 `1 = e by XXREAL_0:1; ::_thesis: verum
end;
A80: now__::_thesis:_not_s4_>_s0
assume s4 > s0 ; ::_thesis: contradiction
then - (- (s4 - s0)) > 0 by XREAL_1:50;
then - (s4 - s0) < 0 ;
then A81: (s0 - s4) / 2 < 0 by XREAL_1:141;
then - ((s0 - s4) / 2) > 0 by XREAL_1:58;
then consider r being real number such that
A82: r in R and
A83: r < s0 + (- ((s0 - s4) / 2)) by A20, SEQ_4:def_2;
reconsider rss = r as Real by XREAL_0:def_1;
A84: 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 A82;
then r in [.0,1.] by A21, 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 A81, XREAL_1:8;
then A85: s4 > r by A83, XXREAL_0:2;
then A86: 1 > r by A77, XXREAL_0:2;
A87: r >= s0 by A20, A82, SEQ_4:def_2;
then A88: LE p9,pss,P,p1,p2 by A1, A5, A6, A7, A21, A22, A17, A86, JORDAN5C:8;
LE pss,p4,P,p1,p2 by A1, A5, A6, A7, A21, A17, A76, A77, A87, A85, A86, JORDAN5C:8;
then pss `1 = e by A79, A88;
hence contradiction by A84; ::_thesis: verum
end;
0 <= s4 by A75, BORSUK_1:40, XXREAL_1:1;
then s4 >= s0 by A5, A6, A7, A22, A69, A76, A77, JORDAN5C:def_3;
hence contradiction by A70, A76, A80, XXREAL_0:1; ::_thesis: verum
end;
now__::_thesis:_(_ex_p51_being_Point_of_(TOP-REAL_2)_st_
(_LE_p51,p4,P,p1,p2_&_LE_p9,p51,P,p1,p2_&_p51_`1_<_e_)_implies_p_is_Lout_P,p1,p2,e_)
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: p is_Lout P,p1,p2,e
then consider p51 being Point of (TOP-REAL 2) such that
A89: LE p51,p4,P,p1,p2 and
A90: LE p9,p51,P,p1,p2 and
A91: p51 `1 < e ;
A92: 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: ( LE p5,p51,P,p1,p2 & LE p,p5,P,p1,p2 implies p5 `1 <= e )
assume that
A93: LE p5,p51,P,p1,p2 and
A94: LE p,p5,P,p1,p2 ; ::_thesis: p5 `1 <= e
A95: LE p5,p4,P,p1,p2 by A89, A93, JORDAN5C:13;
A96: p5 in P by A93, JORDAN5C:def_3;
then A97: ( p5 = p9 implies LE p9,p5,P,p1,p2 ) by JORDAN5C:9;
now__::_thesis:_(_(_LE_p5,p9,P,p1,p2_&_p5_`1_<=_e_)_or_(_LE_p9,p5,P,p1,p2_&_p5_`1_<=_e_)_)
percases ( LE p5,p9,P,p1,p2 or LE p9,p5,P,p1,p2 ) by A1, A72, A96, A97, JORDAN5C:14;
case LE p5,p9,P,p1,p2 ; ::_thesis: p5 `1 <= e
hence p5 `1 <= e by A25, A94; ::_thesis: verum
end;
case LE p9,p5,P,p1,p2 ; ::_thesis: p5 `1 <= e
hence p5 `1 <= e by A73, A95; ::_thesis: verum
end;
end;
end;
hence p5 `1 <= e ; ::_thesis: verum
end;
LE p,p51,P,p1,p2 by A23, A90, JORDAN5C:13;
hence p is_Lout P,p1,p2,e by A1, A3, A4, A91, A92, Def3; ::_thesis: verum
end;
hence ( p is_Lout P,p1,p2,e or p is_Rout P,p1,p2,e ) by A74; ::_thesis: verum
end;
caseA98: 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: ( p is_Lout P,p1,p2,e or p is_Rout P,p1,p2,e )
A99: now__::_thesis:_ex_p51_being_Point_of_(TOP-REAL_2)_st_
(_LE_p51,p4,P,p1,p2_&_LE_p9,p51,P,p1,p2_&_p51_`1_>_e_)
p4 in P by A69, JORDAN5C:def_3;
then p4 in rng f by A9, PRE_TOPC:def_5;
then consider xs4 being set such that
A100: xs4 in dom f and
A101: p4 = f . xs4 by FUNCT_1:def_3;
reconsider s4 = xs4 as Real by A12, A100, BORSUK_1:40;
A102: s4 <= 1 by A100, BORSUK_1:40, XXREAL_1:1;
assume A103: 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: contradiction
A104: 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: ( LE p51,p4,P,p1,p2 & LE p9,p51,P,p1,p2 implies p51 `1 = e )
assume ( LE p51,p4,P,p1,p2 & LE p9,p51,P,p1,p2 ) ; ::_thesis: p51 `1 = e
then ( p51 `1 <= e & p51 `1 >= e ) by A98, A103;
hence p51 `1 = e by XXREAL_0:1; ::_thesis: verum
end;
A105: now__::_thesis:_not_s4_>_s0
assume s4 > s0 ; ::_thesis: contradiction
then - (- (s4 - s0)) > 0 by XREAL_1:50;
then - (s4 - s0) < 0 ;
then A106: (s0 - s4) / 2 < 0 by XREAL_1:141;
then - ((s0 - s4) / 2) > 0 by XREAL_1:58;
then consider r being real number such that
A107: r in R and
A108: r < s0 + (- ((s0 - s4) / 2)) by A20, SEQ_4:def_2;
reconsider rss = r as Real by XREAL_0:def_1;
A109: 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 A107;
then r in [.0,1.] by A21, 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 A106, XREAL_1:8;
then A110: s4 > r by A108, XXREAL_0:2;
then A111: 1 > r by A102, XXREAL_0:2;
A112: r >= s0 by A20, A107, SEQ_4:def_2;
then A113: LE p9,pss,P,p1,p2 by A1, A5, A6, A7, A21, A22, A17, A111, JORDAN5C:8;
LE pss,p4,P,p1,p2 by A1, A5, A6, A7, A21, A17, A101, A102, A112, A110, A111, JORDAN5C:8;
then pss `1 = e by A104, A113;
hence contradiction by A109; ::_thesis: verum
end;
0 <= s4 by A100, BORSUK_1:40, XXREAL_1:1;
then s4 >= s0 by A5, A6, A7, A22, A69, A101, A102, JORDAN5C:def_3;
hence contradiction by A70, A101, A105, XXREAL_0:1; ::_thesis: verum
end;
now__::_thesis:_(_ex_p51_being_Point_of_(TOP-REAL_2)_st_
(_LE_p51,p4,P,p1,p2_&_LE_p9,p51,P,p1,p2_&_p51_`1_>_e_)_implies_p_is_Rout_P,p1,p2,e_)
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: p is_Rout P,p1,p2,e
then consider p51 being Point of (TOP-REAL 2) such that
A114: LE p51,p4,P,p1,p2 and
A115: LE p9,p51,P,p1,p2 and
A116: p51 `1 > e ;
A117: 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: ( LE p5,p51,P,p1,p2 & LE p,p5,P,p1,p2 implies p5 `1 >= e )
assume that
A118: LE p5,p51,P,p1,p2 and
A119: LE p,p5,P,p1,p2 ; ::_thesis: p5 `1 >= e
A120: LE p5,p4,P,p1,p2 by A114, A118, JORDAN5C:13;
A121: p5 in P by A118, JORDAN5C:def_3;
then A122: ( p5 = p9 implies LE p9,p5,P,p1,p2 ) by JORDAN5C:9;
now__::_thesis:_(_(_LE_p9,p5,P,p1,p2_&_p5_`1_>=_e_)_or_(_LE_p5,p9,P,p1,p2_&_p5_`1_>=_e_)_)
percases ( LE p9,p5,P,p1,p2 or LE p5,p9,P,p1,p2 ) by A1, A72, A121, A122, JORDAN5C:14;
case LE p9,p5,P,p1,p2 ; ::_thesis: p5 `1 >= e
hence p5 `1 >= e by A98, A120; ::_thesis: verum
end;
case LE p5,p9,P,p1,p2 ; ::_thesis: p5 `1 >= e
hence p5 `1 >= e by A25, A119; ::_thesis: verum
end;
end;
end;
hence p5 `1 >= e ; ::_thesis: verum
end;
LE p,p51,P,p1,p2 by A23, A115, JORDAN5C:13;
hence p is_Rout P,p1,p2,e by A1, A3, A4, A116, A117, Def4; ::_thesis: verum
end;
hence ( p is_Lout P,p1,p2,e or p is_Rout P,p1,p2,e ) by A99; ::_thesis: verum
end;
end;
end;
hence ( p is_Lout P,p1,p2,e or p is_Rout P,p1,p2,e ) ; ::_thesis: verum
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: verum
end;
theorem Th11: :: JORDAN20:11
for P being Subset of I[01]
for s being Real st P = [.0,s.[ holds
P is open
proof
A1: [#] I[01] = [.0,1.] by TOPMETR:18, TOPMETR:20;
let P be Subset of I[01]; ::_thesis: for s being Real st P = [.0,s.[ holds
P is open
let s be Real; ::_thesis: ( P = [.0,s.[ implies P is open )
assume A2: P = [.0,s.[ ; ::_thesis: P is open
percases ( s in [.0,1.] or not s in [.0,1.] ) ;
supposeA3: s in [.0,1.] ; ::_thesis: P is open
reconsider T = [.0,1.] as Subset of R^1 by TOPMETR:17;
0 in [.0,1.] by XXREAL_1:1;
then ( [.0,s.[ c= [.0,s.] & [.0,s.] c= [.0,1.] ) by A3, XXREAL_1:24, XXREAL_2:def_12;
then [.0,s.[ c= [.0,1.] by XBOOLE_1:1;
then P c= [#] (R^1 | T) by A2, PRE_TOPC:def_5;
then reconsider P2 = P as Subset of (R^1 | T) ;
reconsider Q = ].(- 1),s.[ as Subset of R^1 by TOPMETR:17;
A4: s <= 1 by A3, XXREAL_1:1;
A5: [.0,s.[ c= ].(- 1),s.[ /\ [.0,1.]
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in [.0,s.[ or x in ].(- 1),s.[ /\ [.0,1.] )
assume A6: x in [.0,s.[ ; ::_thesis: x in ].(- 1),s.[ /\ [.0,1.]
then reconsider sx = x as Real ;
A7: 0 <= sx by A6, XXREAL_1:3;
A8: sx < s by A6, XXREAL_1:3;
then sx <= 1 by A4, XXREAL_0:2;
then A9: x in [.0,1.] by A7, XXREAL_1:1;
- 1 < sx by A7, XXREAL_0:2;
then x in ].(- 1),s.[ by A8, XXREAL_1:4;
hence x in ].(- 1),s.[ /\ [.0,1.] by A9, XBOOLE_0:def_4; ::_thesis: verum
end;
].(- 1),s.[ /\ [.0,1.] c= [.0,s.[
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ].(- 1),s.[ /\ [.0,1.] or x in [.0,s.[ )
assume A10: x in ].(- 1),s.[ /\ [.0,1.] ; ::_thesis: x in [.0,s.[
then reconsider sx = x as Real ;
x in [.0,1.] by A10, XBOOLE_0:def_4;
then A11: 0 <= sx by XXREAL_1:1;
x in ].(- 1),s.[ by A10, XBOOLE_0:def_4;
then sx < s by XXREAL_1:4;
hence x in [.0,s.[ by A11, XXREAL_1:3; ::_thesis: verum
end;
then [.0,s.[ = ].(- 1),s.[ /\ [.0,1.] by A5, XBOOLE_0:def_10;
then A12: P2 = Q /\ ([#] (R^1 | T)) by A2, PRE_TOPC:def_5;
( Q is open & Closed-Interval-TSpace (0,1) = R^1 | T ) by JORDAN6:35, TOPMETR:19;
hence P is open by A12, TOPMETR:20, TOPS_2:24; ::_thesis: verum
end;
supposeA13: not s in [.0,1.] ; ::_thesis: P is open
now__::_thesis:_(_(_s_<_0_&_P_is_open_)_or_(_s_>_1_&_contradiction_)_)
percases ( s < 0 or s > 1 ) by A13, XXREAL_1:1;
case s < 0 ; ::_thesis: P is open
then [.0,s.[ = {} by XXREAL_1:27;
then P in the topology of I[01] by A2, PRE_TOPC:1;
hence P is open by PRE_TOPC:def_2; ::_thesis: verum
end;
caseA14: s > 1 ; ::_thesis: contradiction
A15: for r being Real st 0 <= r & r < s holds
r <= 1
proof
let r be Real; ::_thesis: ( 0 <= r & r < s implies r <= 1 )
assume ( 0 <= r & r < s ) ; ::_thesis: r <= 1
then r in [.0,s.[ by XXREAL_1:3;
hence r <= 1 by A2, A1, XXREAL_1:1; ::_thesis: verum
end;
consider t being real number such that
A16: 1 < t and
A17: t < s by A14, XREAL_1:5;
reconsider t = t as Real by XREAL_0:def_1;
0 < t by A16;
hence contradiction by A16, A17, A15; ::_thesis: verum
end;
end;
end;
hence P is open ; ::_thesis: verum
end;
end;
end;
theorem Th12: :: JORDAN20:12
for P being Subset of I[01]
for s being Real st P = ].s,1.] holds
P is open
proof
A1: [#] I[01] = [.0,1.] by TOPMETR:18, TOPMETR:20;
let P be Subset of I[01]; ::_thesis: for s being Real st P = ].s,1.] holds
P is open
let s be Real; ::_thesis: ( P = ].s,1.] implies P is open )
assume A2: P = ].s,1.] ; ::_thesis: P is open
percases ( s in [.0,1.] or not s in [.0,1.] ) ;
supposeA3: s in [.0,1.] ; ::_thesis: P is open
reconsider T = [.0,1.] as Subset of R^1 by TOPMETR:17;
1 in [.0,1.] by XXREAL_1:1;
then ( ].s,1.] c= [.s,1.] & [.s,1.] c= [.0,1.] ) by A3, XXREAL_1:23, XXREAL_2:def_12;
then ].s,1.] c= [.0,1.] by XBOOLE_1:1;
then P c= [#] (R^1 | T) by A2, PRE_TOPC:def_5;
then reconsider P2 = P as Subset of (R^1 | T) ;
reconsider Q = ].s,2.[ as Subset of R^1 by TOPMETR:17;
A4: 0 <= s by A3, XXREAL_1:1;
A5: ].s,1.] c= ].s,2.[ /\ [.0,1.]
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ].s,1.] or x in ].s,2.[ /\ [.0,1.] )
assume A6: x in ].s,1.] ; ::_thesis: x in ].s,2.[ /\ [.0,1.]
then reconsider sx = x as Real ;
A7: s < sx by A6, XXREAL_1:2;
A8: sx <= 1 by A6, XXREAL_1:2;
then 2 > sx by XXREAL_0:2;
then A9: x in ].s,2.[ by A7, XXREAL_1:4;
x in [.0,1.] by A4, A7, A8, XXREAL_1:1;
hence x in ].s,2.[ /\ [.0,1.] by A9, XBOOLE_0:def_4; ::_thesis: verum
end;
].s,2.[ /\ [.0,1.] c= ].s,1.]
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ].s,2.[ /\ [.0,1.] or x in ].s,1.] )
assume A10: x in ].s,2.[ /\ [.0,1.] ; ::_thesis: x in ].s,1.]
then reconsider sx = x as Real ;
x in [.0,1.] by A10, XBOOLE_0:def_4;
then A11: sx <= 1 by XXREAL_1:1;
x in ].s,2.[ by A10, XBOOLE_0:def_4;
then s < sx by XXREAL_1:4;
hence x in ].s,1.] by A11, XXREAL_1:2; ::_thesis: verum
end;
then ].s,1.] = ].s,2.[ /\ [.0,1.] by A5, XBOOLE_0:def_10;
then A12: P2 = Q /\ ([#] (R^1 | T)) by A2, PRE_TOPC:def_5;
( Q is open & Closed-Interval-TSpace (0,1) = R^1 | T ) by JORDAN6:35, TOPMETR:19;
hence P is open by A12, TOPMETR:20, TOPS_2:24; ::_thesis: verum
end;
supposeA13: not s in [.0,1.] ; ::_thesis: P is open
now__::_thesis:_(_(_s_>_1_&_P_is_open_)_or_(_s_<_0_&_contradiction_)_)
percases ( s > 1 or s < 0 ) by A13, XXREAL_1:1;
case s > 1 ; ::_thesis: P is open
then ].s,1.] = {} by XXREAL_1:26;
then P in the topology of I[01] by A2, PRE_TOPC:1;
hence P is open by PRE_TOPC:def_2; ::_thesis: verum
end;
caseA14: s < 0 ; ::_thesis: contradiction
A15: for r being Real st s < r & r <= 1 holds
r >= 0
proof
let r be Real; ::_thesis: ( s < r & r <= 1 implies r >= 0 )
assume ( s < r & r <= 1 ) ; ::_thesis: r >= 0
then r in ].s,1.] by XXREAL_1:2;
hence r >= 0 by A2, A1, XXREAL_1:1; ::_thesis: verum
end;
consider t being real number such that
A16: s < t and
A17: t < 0 by A14, XREAL_1:5;
reconsider t = t as Real by XREAL_0:def_1;
t <= 1 by A17;
hence contradiction by A16, A17, A15; ::_thesis: verum
end;
end;
end;
hence P is open ; ::_thesis: verum
end;
end;
end;
theorem Th13: :: JORDAN20:13
for P being non empty Subset of (TOP-REAL 2)
for P1 being Subset of ((TOP-REAL 2) | P)
for Q being Subset of I[01]
for f being Function of I[01],((TOP-REAL 2) | P)
for s being Real st s <= 1 & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( 0 <= ss & ss < s & q0 = f . ss ) } & Q = [.0,s.[ holds
f .: Q = P1
proof
let P be non empty Subset of (TOP-REAL 2); ::_thesis: for P1 being Subset of ((TOP-REAL 2) | P)
for Q being Subset of I[01]
for f being Function of I[01],((TOP-REAL 2) | P)
for s being Real st s <= 1 & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( 0 <= ss & ss < s & q0 = f . ss ) } & Q = [.0,s.[ holds
f .: Q = P1
let P1 be Subset of ((TOP-REAL 2) | P); ::_thesis: for Q being Subset of I[01]
for f being Function of I[01],((TOP-REAL 2) | P)
for s being Real st s <= 1 & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( 0 <= ss & ss < s & q0 = f . ss ) } & Q = [.0,s.[ holds
f .: Q = P1
let Q be Subset of I[01]; ::_thesis: for f being Function of I[01],((TOP-REAL 2) | P)
for s being Real st s <= 1 & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( 0 <= ss & ss < s & q0 = f . ss ) } & Q = [.0,s.[ holds
f .: Q = P1
let f be Function of I[01],((TOP-REAL 2) | P); ::_thesis: for s being Real st s <= 1 & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( 0 <= ss & ss < s & q0 = f . ss ) } & Q = [.0,s.[ holds
f .: Q = P1
let s be Real; ::_thesis: ( s <= 1 & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( 0 <= ss & ss < s & q0 = f . ss ) } & Q = [.0,s.[ implies f .: Q = P1 )
assume that
A1: s <= 1 and
A2: P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( 0 <= ss & ss < s & q0 = f . ss ) } and
A3: Q = [.0,s.[ ; ::_thesis: f .: Q = P1
A4: the carrier of ((TOP-REAL 2) | P) = P by PRE_TOPC:8;
A5: f .: Q c= P1
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in f .: Q or y in P1 )
assume y in f .: Q ; ::_thesis: y in P1
then consider z being set such that
A6: z in dom f and
A7: z in Q and
A8: f . z = y by FUNCT_1:def_6;
dom f = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1;
then reconsider ss = z as Real by A6;
y in rng f by A6, A8, FUNCT_1:def_3;
then y in P by A4;
then reconsider q = y as Point of (TOP-REAL 2) ;
( 0 <= ss & ss < s ) by A3, A7, XXREAL_1:3;
then ex ss being Real st
( 0 <= ss & ss < s & q = f . ss ) by A8;
hence y in P1 by A2; ::_thesis: verum
end;
P1 c= f .: Q
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in P1 or x in f .: Q )
A9: dom f = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1;
assume x in P1 ; ::_thesis: x in f .: Q
then consider q0 being Point of (TOP-REAL 2) such that
A10: q0 = x and
A11: ex ss being Real st
( 0 <= ss & ss < s & q0 = f . ss ) by A2;
consider ss being Real such that
A12: 0 <= ss and
A13: ss < s and
A14: q0 = f . ss by A11;
ss < 1 by A1, A13, XXREAL_0:2;
then A15: ss in dom f by A12, A9, XXREAL_1:1;
ss in Q by A3, A12, A13, XXREAL_1:3;
hence x in f .: Q by A10, A14, A15, FUNCT_1:def_6; ::_thesis: verum
end;
hence f .: Q = P1 by A5, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th14: :: JORDAN20:14
for P being non empty Subset of (TOP-REAL 2)
for P1 being Subset of ((TOP-REAL 2) | P)
for Q being Subset of I[01]
for f being Function of I[01],((TOP-REAL 2) | P)
for s being Real st s >= 0 & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( s < ss & ss <= 1 & q0 = f . ss ) } & Q = ].s,1.] holds
f .: Q = P1
proof
let P be non empty Subset of (TOP-REAL 2); ::_thesis: for P1 being Subset of ((TOP-REAL 2) | P)
for Q being Subset of I[01]
for f being Function of I[01],((TOP-REAL 2) | P)
for s being Real st s >= 0 & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( s < ss & ss <= 1 & q0 = f . ss ) } & Q = ].s,1.] holds
f .: Q = P1
let P1 be Subset of ((TOP-REAL 2) | P); ::_thesis: for Q being Subset of I[01]
for f being Function of I[01],((TOP-REAL 2) | P)
for s being Real st s >= 0 & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( s < ss & ss <= 1 & q0 = f . ss ) } & Q = ].s,1.] holds
f .: Q = P1
let Q be Subset of I[01]; ::_thesis: for f being Function of I[01],((TOP-REAL 2) | P)
for s being Real st s >= 0 & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( s < ss & ss <= 1 & q0 = f . ss ) } & Q = ].s,1.] holds
f .: Q = P1
let f be Function of I[01],((TOP-REAL 2) | P); ::_thesis: for s being Real st s >= 0 & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( s < ss & ss <= 1 & q0 = f . ss ) } & Q = ].s,1.] holds
f .: Q = P1
let s be Real; ::_thesis: ( s >= 0 & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( s < ss & ss <= 1 & q0 = f . ss ) } & Q = ].s,1.] implies f .: Q = P1 )
assume that
A1: s >= 0 and
A2: P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( s < ss & ss <= 1 & q0 = f . ss ) } and
A3: Q = ].s,1.] ; ::_thesis: f .: Q = P1
A4: the carrier of ((TOP-REAL 2) | P) = P by PRE_TOPC:8;
A5: f .: Q c= P1
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in f .: Q or y in P1 )
assume y in f .: Q ; ::_thesis: y in P1
then consider z being set such that
A6: z in dom f and
A7: z in Q and
A8: f . z = y by FUNCT_1:def_6;
dom f = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1;
then reconsider ss = z as Real by A6;
y in rng f by A6, A8, FUNCT_1:def_3;
then y in P by A4;
then reconsider q = y as Point of (TOP-REAL 2) ;
( s < ss & ss <= 1 ) by A3, A7, XXREAL_1:2;
then ex ss being Real st
( s < ss & ss <= 1 & q = f . ss ) by A8;
hence y in P1 by A2; ::_thesis: verum
end;
P1 c= f .: Q
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in P1 or x in f .: Q )
assume x in P1 ; ::_thesis: x in f .: Q
then consider q0 being Point of (TOP-REAL 2) such that
A9: q0 = x and
A10: ex ss being Real st
( s < ss & ss <= 1 & q0 = f . ss ) by A2;
consider ss being Real such that
A11: ( s < ss & ss <= 1 ) and
A12: q0 = f . ss by A10;
A13: ss in Q by A3, A11, XXREAL_1:2;
dom f = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1;
then ss in dom f by A1, A11, XXREAL_1:1;
hence x in f .: Q by A9, A12, A13, FUNCT_1:def_6; ::_thesis: verum
end;
hence f .: Q = P1 by A5, XBOOLE_0:def_10; ::_thesis: verum
end;
Lm1: [#] I[01] <> {}
;
theorem Th15: :: JORDAN20:15
for P being non empty Subset of (TOP-REAL 2)
for P1 being Subset of ((TOP-REAL 2) | P)
for f being Function of I[01],((TOP-REAL 2) | P)
for s being Real st s <= 1 & f is being_homeomorphism & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( 0 <= ss & ss < s & q0 = f . ss ) } holds
P1 is open
proof
let P be non empty Subset of (TOP-REAL 2); ::_thesis: for P1 being Subset of ((TOP-REAL 2) | P)
for f being Function of I[01],((TOP-REAL 2) | P)
for s being Real st s <= 1 & f is being_homeomorphism & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( 0 <= ss & ss < s & q0 = f . ss ) } holds
P1 is open
let P1 be Subset of ((TOP-REAL 2) | P); ::_thesis: for f being Function of I[01],((TOP-REAL 2) | P)
for s being Real st s <= 1 & f is being_homeomorphism & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( 0 <= ss & ss < s & q0 = f . ss ) } holds
P1 is open
let f be Function of I[01],((TOP-REAL 2) | P); ::_thesis: for s being Real st s <= 1 & f is being_homeomorphism & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( 0 <= ss & ss < s & q0 = f . ss ) } holds
P1 is open
let s be Real; ::_thesis: ( s <= 1 & f is being_homeomorphism & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( 0 <= ss & ss < s & q0 = f . ss ) } implies P1 is open )
assume that
A1: s <= 1 and
A2: f is being_homeomorphism and
A3: P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( 0 <= ss & ss < s & q0 = f . ss ) } ; ::_thesis: P1 is open
( f is one-to-one & rng f = [#] ((TOP-REAL 2) | P) ) by A2, TOPS_2:def_5;
then A4: (f ") " = f by TOPS_2:51;
[.0,s.[ c= [.0,1.]
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in [.0,s.[ or x in [.0,1.] )
assume A5: x in [.0,s.[ ; ::_thesis: x in [.0,1.]
then reconsider sx = x as Real ;
sx < s by A5, XXREAL_1:3;
then A6: sx < 1 by A1, XXREAL_0:2;
0 <= sx by A5, XXREAL_1:3;
hence x in [.0,1.] by A6, XXREAL_1:1; ::_thesis: verum
end;
then reconsider Q = [.0,s.[ as Subset of I[01] by TOPMETR:18, TOPMETR:20;
A7: Q is open by Th11;
A8: f " is being_homeomorphism by A2, TOPS_2:56;
then A9: f " is one-to-one by TOPS_2:def_5;
rng (f ") = [#] I[01] by A8, TOPS_2:def_5;
then f " is onto by FUNCT_2:def_3;
then (f ") " = (f ") " by A9, TOPS_2:def_4;
then A10: ((f ") ") .: Q = (f ") " Q by A9, FUNCT_1:85;
( P1 = f .: Q & f " is continuous ) by A1, A2, A3, Th13, TOPS_2:def_5;
hence P1 is open by A7, A4, A10, Lm1, TOPS_2:43; ::_thesis: verum
end;
theorem Th16: :: JORDAN20:16
for P being non empty Subset of (TOP-REAL 2)
for P1 being Subset of ((TOP-REAL 2) | P)
for f being Function of I[01],((TOP-REAL 2) | P)
for s being Real st s >= 0 & f is being_homeomorphism & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( s < ss & ss <= 1 & q0 = f . ss ) } holds
P1 is open
proof
let P be non empty Subset of (TOP-REAL 2); ::_thesis: for P1 being Subset of ((TOP-REAL 2) | P)
for f being Function of I[01],((TOP-REAL 2) | P)
for s being Real st s >= 0 & f is being_homeomorphism & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( s < ss & ss <= 1 & q0 = f . ss ) } holds
P1 is open
let P1 be Subset of ((TOP-REAL 2) | P); ::_thesis: for f being Function of I[01],((TOP-REAL 2) | P)
for s being Real st s >= 0 & f is being_homeomorphism & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( s < ss & ss <= 1 & q0 = f . ss ) } holds
P1 is open
let f be Function of I[01],((TOP-REAL 2) | P); ::_thesis: for s being Real st s >= 0 & f is being_homeomorphism & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( s < ss & ss <= 1 & q0 = f . ss ) } holds
P1 is open
let s be Real; ::_thesis: ( s >= 0 & f is being_homeomorphism & P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( s < ss & ss <= 1 & q0 = f . ss ) } implies P1 is open )
assume that
A1: s >= 0 and
A2: f is being_homeomorphism and
A3: P1 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( s < ss & ss <= 1 & q0 = f . ss ) } ; ::_thesis: P1 is open
( f is one-to-one & rng f = [#] ((TOP-REAL 2) | P) ) by A2, TOPS_2:def_5;
then A4: (f ") " = f by TOPS_2:51;
].s,1.] c= [.0,1.]
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ].s,1.] or x in [.0,1.] )
assume A5: x in ].s,1.] ; ::_thesis: x in [.0,1.]
then reconsider sx = x as Real ;
( 0 < sx & sx <= 1 ) by A1, A5, XXREAL_1:2;
hence x in [.0,1.] by XXREAL_1:1; ::_thesis: verum
end;
then reconsider Q = ].s,1.] as Subset of I[01] by TOPMETR:18, TOPMETR:20;
A6: ( [#] I[01] <> {} & Q is open ) by Th12;
A7: f " is being_homeomorphism by A2, TOPS_2:56;
then A8: f " is one-to-one by TOPS_2:def_5;
rng (f ") = [#] I[01] by A7, TOPS_2:def_5;
then f " is onto by FUNCT_2:def_3;
then (f ") " = (f ") " by A8, TOPS_2:def_4;
then A9: ((f ") ") .: Q = (f ") " Q by A8, FUNCT_1:85;
( P1 = f .: Q & f " is continuous ) by A1, A2, A3, Th14, TOPS_2:def_5;
hence P1 is open by A6, A4, A9, TOPS_2:43; ::_thesis: verum
end;
theorem Th17: :: JORDAN20:17
for T being non empty TopStruct
for Q1, Q2 being Subset of T
for p1, p2 being Point of T st Q1 /\ Q2 = {} & Q1 \/ Q2 = the carrier of T & p1 in Q1 & p2 in Q2 & Q1 is open & Q2 is open holds
for P being Function of I[01],T holds
( not P is continuous or not P . 0 = p1 or not P . 1 = p2 )
proof
let T be non empty TopStruct ; ::_thesis: for Q1, Q2 being Subset of T
for p1, p2 being Point of T st Q1 /\ Q2 = {} & Q1 \/ Q2 = the carrier of T & p1 in Q1 & p2 in Q2 & Q1 is open & Q2 is open holds
for P being Function of I[01],T holds
( not P is continuous or not P . 0 = p1 or not P . 1 = p2 )
let Q1, Q2 be Subset of T; ::_thesis: for p1, p2 being Point of T st Q1 /\ Q2 = {} & Q1 \/ Q2 = the carrier of T & p1 in Q1 & p2 in Q2 & Q1 is open & Q2 is open holds
for P being Function of I[01],T holds
( not P is continuous or not P . 0 = p1 or not P . 1 = p2 )
let p1, p2 be Point of T; ::_thesis: ( Q1 /\ Q2 = {} & Q1 \/ Q2 = the carrier of T & p1 in Q1 & p2 in Q2 & Q1 is open & Q2 is open implies for P being Function of I[01],T holds
( not P is continuous or not P . 0 = p1 or not P . 1 = p2 ) )
assume that
A1: Q1 /\ Q2 = {} and
A2: Q1 \/ Q2 = the carrier of T and
A3: p1 in Q1 and
A4: p2 in Q2 and
A5: ( Q1 is open & Q2 is open ) ; ::_thesis: for P being Function of I[01],T holds
( not P is continuous or not P . 0 = p1 or not P . 1 = p2 )
assume ex P being Function of I[01],T st
( P is continuous & P . 0 = p1 & P . 1 = p2 ) ; ::_thesis: contradiction
then consider P being Function of I[01],T such that
A6: P is continuous and
A7: P . 0 = p1 and
A8: P . 1 = p2 ;
[#] T <> {} ;
then A9: ( P " Q1 is open & P " Q2 is open ) by A5, A6, TOPS_2:43;
A10: [#] I[01] = [.0,1.] by TOPMETR:18, TOPMETR:20;
then 0 in the carrier of I[01] by XXREAL_1:1;
then 0 in dom P by FUNCT_2:def_1;
then A11: ( [#] I[01] = the carrier of I[01] & P " Q1 <> {} I[01] ) by A3, A7, FUNCT_1:def_7;
(P " Q1) /\ (P " Q2) = P " (Q1 /\ Q2) by FUNCT_1:68
.= {} by A1 ;
then A12: not P " Q1 meets P " Q2 by XBOOLE_0:def_7;
1 in the carrier of I[01] by A10, XXREAL_1:1;
then 1 in dom P by FUNCT_2:def_1;
then A13: P " Q2 <> {} I[01] by A4, A8, FUNCT_1:def_7;
(P " Q1) \/ (P " Q2) = P " (Q1 \/ Q2) by RELAT_1:140
.= the carrier of I[01] by A2, FUNCT_2:40 ;
hence contradiction by A9, A11, A13, A12, CONNSP_1:11, TREAL_1:19; ::_thesis: verum
end;
theorem Th18: :: JORDAN20:18
for P being non empty Subset of (TOP-REAL 2)
for Q being Subset of ((TOP-REAL 2) | P)
for p1, p2, q being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & q in P & q <> p1 & q <> p2 & Q = P \ {q} holds
( not Q is connected & ( for R being Function of I[01],(((TOP-REAL 2) | P) | Q) holds
( not R is continuous or not R . 0 = p1 or not R . 1 = p2 ) ) )
proof
let P be non empty Subset of (TOP-REAL 2); ::_thesis: for Q being Subset of ((TOP-REAL 2) | P)
for p1, p2, q being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & q in P & q <> p1 & q <> p2 & Q = P \ {q} holds
( not Q is connected & ( for R being Function of I[01],(((TOP-REAL 2) | P) | Q) holds
( not R is continuous or not R . 0 = p1 or not R . 1 = p2 ) ) )
let Q be Subset of ((TOP-REAL 2) | P); ::_thesis: for p1, p2, q being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & q in P & q <> p1 & q <> p2 & Q = P \ {q} holds
( not Q is connected & ( for R being Function of I[01],(((TOP-REAL 2) | P) | Q) holds
( not R is continuous or not R . 0 = p1 or not R . 1 = p2 ) ) )
let p1, p2, q be Point of (TOP-REAL 2); ::_thesis: ( P is_an_arc_of p1,p2 & q in P & q <> p1 & q <> p2 & Q = P \ {q} implies ( not Q is connected & ( for R being Function of I[01],(((TOP-REAL 2) | P) | Q) holds
( not R is continuous or not R . 0 = p1 or not R . 1 = p2 ) ) ) )
assume that
A1: P is_an_arc_of p1,p2 and
A2: q in P and
A3: q <> p1 and
A4: q <> p2 and
A5: Q = P \ {q} ; ::_thesis: ( not Q is connected & ( for R being Function of I[01],(((TOP-REAL 2) | P) | Q) holds
( not R is continuous or not R . 0 = p1 or not R . 1 = p2 ) ) )
consider f being Function of I[01],((TOP-REAL 2) | P) such that
A6: f is being_homeomorphism and
A7: f . 0 = p1 and
A8: f . 1 = p2 by A1, TOPREAL1:def_1;
A9: rng f = [#] ((TOP-REAL 2) | P) by A6, TOPS_2:def_5;
A10: [#] I[01] = [.0,1.] by TOPMETR:18, TOPMETR:20;
A11: [#] ((TOP-REAL 2) | P) = P by PRE_TOPC:def_5;
then consider xs being set such that
A12: xs in dom f and
A13: f . xs = q by A2, A9, FUNCT_1:def_3;
A14: dom f = [#] I[01] by A6, TOPS_2:def_5;
then reconsider s = xs as Real by A12, A10;
{ q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( s < ss & ss <= 1 & q0 = f . ss ) } c= the carrier of ((TOP-REAL 2) | P)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( s < ss & ss <= 1 & q0 = f . ss ) } or z in the carrier of ((TOP-REAL 2) | P) )
assume z in { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( s < ss & ss <= 1 & q0 = f . ss ) } ; ::_thesis: z in the carrier of ((TOP-REAL 2) | P)
then consider q0 being Point of (TOP-REAL 2) such that
A15: q0 = z and
A16: ex ss being Real st
( s < ss & ss <= 1 & q0 = f . ss ) ;
consider ss being Real such that
A17: s < ss and
A18: ss <= 1 and
A19: q0 = f . ss by A16;
ss > 0 by A12, A10, A17, XXREAL_1:1;
then ss in dom f by A14, A10, A18, XXREAL_1:1;
then q0 in rng f by A19, FUNCT_1:def_3;
hence z in the carrier of ((TOP-REAL 2) | P) by A15; ::_thesis: verum
end;
then reconsider P29 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( s < ss & ss <= 1 & q0 = f . ss ) } as Subset of ((TOP-REAL 2) | P) ;
A20: 0 <= s by A12, A10, XXREAL_1:1;
then A21: P29 is open by A6, Th16;
A22: P29 c= Q
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in P29 or x in Q )
assume x in P29 ; ::_thesis: x in Q
then consider q00 being Point of (TOP-REAL 2) such that
A23: q00 = x and
A24: ex ss being Real st
( s < ss & ss <= 1 & q00 = f . ss ) ;
consider ss being Real such that
A25: s < ss and
A26: ss <= 1 and
A27: q00 = f . ss by A24;
ss > 0 by A12, A10, A25, XXREAL_1:1;
then A28: ss in dom f by A14, A10, A26, XXREAL_1:1;
now__::_thesis:_not_q00_=_q
assume A29: q00 = q ; ::_thesis: contradiction
f is one-to-one by A6, TOPS_2:def_5;
hence contradiction by A12, A13, A25, A27, A28, A29, FUNCT_1:def_4; ::_thesis: verum
end;
then A30: not q00 in {q} by TARSKI:def_1;
q00 in P by A9, A11, A27, A28, FUNCT_1:def_3;
hence x in Q by A5, A23, A30, XBOOLE_0:def_5; ::_thesis: verum
end;
{ q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( 0 <= ss & ss < s & q0 = f . ss ) } c= the carrier of ((TOP-REAL 2) | P)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( 0 <= ss & ss < s & q0 = f . ss ) } or z in the carrier of ((TOP-REAL 2) | P) )
assume z in { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( 0 <= ss & ss < s & q0 = f . ss ) } ; ::_thesis: z in the carrier of ((TOP-REAL 2) | P)
then consider q0 being Point of (TOP-REAL 2) such that
A31: q0 = z and
A32: ex ss being Real st
( 0 <= ss & ss < s & q0 = f . ss ) ;
consider ss being Real such that
A33: 0 <= ss and
A34: ss < s and
A35: q0 = f . ss by A32;
s <= 1 by A12, A10, XXREAL_1:1;
then ss < 1 by A34, XXREAL_0:2;
then ss in dom f by A14, A10, A33, XXREAL_1:1;
then q0 in rng f by A35, FUNCT_1:def_3;
hence z in the carrier of ((TOP-REAL 2) | P) by A31; ::_thesis: verum
end;
then reconsider P19 = { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( 0 <= ss & ss < s & q0 = f . ss ) } as Subset of ((TOP-REAL 2) | P) ;
A36: s <= 1 by A12, A10, XXREAL_1:1;
then A37: P19 is open by A6, Th15;
A38: Q c= P19 \/ P29
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Q or x in P19 \/ P29 )
assume A39: x in Q ; ::_thesis: x in P19 \/ P29
then consider xt being set such that
A40: xt in dom f and
A41: f . xt = x by A9, FUNCT_1:def_3;
reconsider t = xt as Real by A14, A10, A40;
A42: t <= 1 by A10, A40, XXREAL_1:1;
reconsider qq = x as Point of (TOP-REAL 2) by A5, A39;
not x in {q} by A5, A39, XBOOLE_0:def_5;
then A43: not x = q by TARSKI:def_1;
A44: 0 <= t by A10, A40, XXREAL_1:1;
now__::_thesis:_(_(_t_<_s_&_x_in_P19_\/_P29_)_or_(_t_>=_s_&_x_in_P19_\/_P29_)_)
percases ( t < s or t >= s ) ;
case t < s ; ::_thesis: x in P19 \/ P29
then ex ss being Real st
( 0 <= ss & ss < s & qq = f . ss ) by A41, A44;
then x in P19 ;
hence x in P19 \/ P29 by XBOOLE_0:def_3; ::_thesis: verum
end;
case t >= s ; ::_thesis: x in P19 \/ P29
then t > s by A13, A41, A43, XXREAL_0:1;
then ex ss being Real st
( s < ss & ss <= 1 & qq = f . ss ) by A41, A42;
then x in P29 ;
hence x in P19 \/ P29 by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
hence x in P19 \/ P29 ; ::_thesis: verum
end;
A45: now__::_thesis:_not_P19_meets_P29
assume P19 meets P29 ; ::_thesis: contradiction
then consider p0 being set such that
A46: p0 in P19 and
A47: p0 in P29 by XBOOLE_0:3;
consider q00 being Point of (TOP-REAL 2) such that
A48: q00 = p0 and
A49: ex ss being Real st
( 0 <= ss & ss < s & q00 = f . ss ) by A46;
consider ss1 being Real such that
A50: 0 <= ss1 and
A51: ss1 < s and
A52: q00 = f . ss1 by A49;
ss1 < 1 by A36, A51, XXREAL_0:2;
then A53: ss1 in dom f by A14, A10, A50, XXREAL_1:1;
consider q01 being Point of (TOP-REAL 2) such that
A54: q01 = p0 and
A55: ex ss being Real st
( s < ss & ss <= 1 & q01 = f . ss ) by A47;
consider ss2 being Real such that
A56: s < ss2 and
A57: ss2 <= 1 and
A58: q01 = f . ss2 by A55;
ss2 > 0 by A12, A10, A56, XXREAL_1:1;
then A59: ss2 in dom f by A14, A10, A57, XXREAL_1:1;
f is one-to-one by A6, TOPS_2:def_5;
hence contradiction by A48, A51, A52, A54, A56, A58, A53, A59, FUNCT_1:def_4; ::_thesis: verum
end;
1 > s by A4, A8, A13, A36, XXREAL_0:1;
then A60: p2 in { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( s < ss & ss <= 1 & q0 = f . ss ) } by A8;
then reconsider Q9 = Q as non empty Subset of ((TOP-REAL 2) | P) by A22;
reconsider T = ((TOP-REAL 2) | P) | Q9 as non empty TopSpace ;
A61: the carrier of T = [#] T ;
then reconsider P299 = P29 as Subset of T by A22, PRE_TOPC:def_5;
P29 /\ Q <> {} by A60, A22, XBOOLE_1:28;
then A62: P29 meets Q by XBOOLE_0:def_7;
A63: P19 c= Q
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in P19 or x in Q )
assume x in P19 ; ::_thesis: x in Q
then consider q00 being Point of (TOP-REAL 2) such that
A64: q00 = x and
A65: ex ss being Real st
( 0 <= ss & ss < s & q00 = f . ss ) ;
consider ss being Real such that
A66: 0 <= ss and
A67: ss < s and
A68: q00 = f . ss by A65;
ss < 1 by A36, A67, XXREAL_0:2;
then A69: ss in dom f by A14, A10, A66, XXREAL_1:1;
now__::_thesis:_not_q00_=_q
assume A70: q00 = q ; ::_thesis: contradiction
f is one-to-one by A6, TOPS_2:def_5;
hence contradiction by A12, A13, A67, A68, A69, A70, FUNCT_1:def_4; ::_thesis: verum
end;
then A71: not q00 in {q} by TARSKI:def_1;
q00 in P by A9, A11, A68, A69, FUNCT_1:def_3;
hence x in Q by A5, A64, A71, XBOOLE_0:def_5; ::_thesis: verum
end;
then reconsider P199 = P19 as Subset of T by A61, PRE_TOPC:def_5;
P199 = P19 /\ the carrier of T by XBOOLE_1:28;
then A72: P199 is open by A37, A61, TOPS_2:24;
s <> 0 by A3, A7, A13;
then A73: p1 in { q0 where q0 is Point of (TOP-REAL 2) : ex ss being Real st
( 0 <= ss & ss < s & q0 = f . ss ) } by A7, A20;
then P19 /\ Q <> {} by A63, XBOOLE_1:28;
then P19 meets Q by XBOOLE_0:def_7;
hence not Q is connected by A37, A21, A38, A62, A45, TOPREAL5:1; ::_thesis: for R being Function of I[01],(((TOP-REAL 2) | P) | Q) holds
( not R is continuous or not R . 0 = p1 or not R . 1 = p2 )
the carrier of T = Q by A61, PRE_TOPC:def_5;
then A74: P199 \/ P299 = the carrier of (((TOP-REAL 2) | P) | Q) by A38, XBOOLE_0:def_10;
P299 = P29 /\ the carrier of T by XBOOLE_1:28;
then A75: P299 is open by A21, A61, TOPS_2:24;
P199 /\ P299 = {} by A45, XBOOLE_0:def_7;
hence for R being Function of I[01],(((TOP-REAL 2) | P) | Q) holds
( not R is continuous or not R . 0 = p1 or not R . 1 = p2 ) by A73, A60, A72, A75, A74, Th17; ::_thesis: verum
end;
theorem Th19: :: JORDAN20:19
for P being non empty Subset of (TOP-REAL 2)
for p1, p2, q1, q2 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & q1 in P & q2 in P & not LE q1,q2,P,p1,p2 holds
LE q2,q1,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) st P is_an_arc_of p1,p2 & q1 in P & q2 in P & not LE q1,q2,P,p1,p2 holds
LE q2,q1,P,p1,p2
let p1, p2, q1, q2 be Point of (TOP-REAL 2); ::_thesis: ( P is_an_arc_of p1,p2 & q1 in P & q2 in P & not LE q1,q2,P,p1,p2 implies LE q2,q1,P,p1,p2 )
assume that
A1: P is_an_arc_of p1,p2 and
A2: q1 in P and
A3: q2 in P ; ::_thesis: ( LE q1,q2,P,p1,p2 or LE q2,q1,P,p1,p2 )
percases ( q1 <> q2 or q1 = q2 ) ;
suppose q1 <> q2 ; ::_thesis: ( LE q1,q2,P,p1,p2 or LE q2,q1,P,p1,p2 )
hence ( LE q1,q2,P,p1,p2 or LE q2,q1,P,p1,p2 ) by A1, A2, A3, JORDAN5C:14; ::_thesis: verum
end;
suppose q1 = q2 ; ::_thesis: ( LE q1,q2,P,p1,p2 or LE q2,q1,P,p1,p2 )
hence ( LE q1,q2,P,p1,p2 or LE q2,q1,P,p1,p2 ) by A2, JORDAN5C:9; ::_thesis: verum
end;
end;
end;
theorem Th20: :: JORDAN20:20
for n being Element of NAT
for p1, p2 being Point of (TOP-REAL n)
for P, P1 being non empty Subset of (TOP-REAL n) st P is_an_arc_of p1,p2 & P1 is_an_arc_of p1,p2 & P1 c= P holds
P1 = P
proof
let n be Element of NAT ; ::_thesis: for p1, p2 being Point of (TOP-REAL n)
for P, P1 being non empty Subset of (TOP-REAL n) st P is_an_arc_of p1,p2 & P1 is_an_arc_of p1,p2 & P1 c= P holds
P1 = P
let p1, p2 be Point of (TOP-REAL n); ::_thesis: for P, P1 being non empty Subset of (TOP-REAL n) st P is_an_arc_of p1,p2 & P1 is_an_arc_of p1,p2 & P1 c= P holds
P1 = P
let P, P1 be non empty Subset of (TOP-REAL n); ::_thesis: ( P is_an_arc_of p1,p2 & P1 is_an_arc_of p1,p2 & P1 c= P implies P1 = P )
assume that
A1: P is_an_arc_of p1,p2 and
A2: P1 is_an_arc_of p1,p2 and
A3: P1 c= P ; ::_thesis: P1 = P
P1 is_an_arc_of p2,p1 by A2, JORDAN5B:14;
hence P1 = P by A1, A3, TOPMETR3:14; ::_thesis: verum
end;
theorem Th21: :: JORDAN20:21
for P being non empty Subset of (TOP-REAL 2)
for p1, p2, q1 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & q1 in P & p2 <> q1 holds
Segment (P,p1,p2,q1,p2) is_an_arc_of q1,p2
proof
let P be non empty Subset of (TOP-REAL 2); ::_thesis: for p1, p2, q1 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & q1 in P & p2 <> q1 holds
Segment (P,p1,p2,q1,p2) is_an_arc_of q1,p2
let p1, p2, q1 be Point of (TOP-REAL 2); ::_thesis: ( P is_an_arc_of p1,p2 & q1 in P & p2 <> q1 implies Segment (P,p1,p2,q1,p2) is_an_arc_of q1,p2 )
assume that
A1: P is_an_arc_of p1,p2 and
A2: q1 in P and
A3: p2 <> q1 ; ::_thesis: Segment (P,p1,p2,q1,p2) is_an_arc_of q1,p2
LE q1,p2,P,p1,p2 by A1, A2, JORDAN5C:10;
hence Segment (P,p1,p2,q1,p2) is_an_arc_of q1,p2 by A1, A3, JORDAN16:21; ::_thesis: verum
end;
theorem Th22: :: JORDAN20:22
for P being non empty Subset of (TOP-REAL 2)
for p1, p2, q1, q2, q3 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & LE q1,q2,P,p1,p2 & LE q2,q3,P,p1,p2 holds
(Segment (P,p1,p2,q1,q2)) \/ (Segment (P,p1,p2,q2,q3)) = Segment (P,p1,p2,q1,q3)
proof
let P be non empty Subset of (TOP-REAL 2); ::_thesis: for p1, p2, q1, q2, q3 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & LE q1,q2,P,p1,p2 & LE q2,q3,P,p1,p2 holds
(Segment (P,p1,p2,q1,q2)) \/ (Segment (P,p1,p2,q2,q3)) = Segment (P,p1,p2,q1,q3)
let p1, p2, q1, q2, q3 be Point of (TOP-REAL 2); ::_thesis: ( P is_an_arc_of p1,p2 & LE q1,q2,P,p1,p2 & LE q2,q3,P,p1,p2 implies (Segment (P,p1,p2,q1,q2)) \/ (Segment (P,p1,p2,q2,q3)) = Segment (P,p1,p2,q1,q3) )
assume that
A1: P is_an_arc_of p1,p2 and
A2: LE q1,q2,P,p1,p2 and
A3: LE q2,q3,P,p1,p2 ; ::_thesis: (Segment (P,p1,p2,q1,q2)) \/ (Segment (P,p1,p2,q2,q3)) = Segment (P,p1,p2,q1,q3)
A4: q2 in P by A2, JORDAN5C:def_3;
A5: Segment (P,p1,p2,q1,q3) c= (Segment (P,p1,p2,q1,q2)) \/ (Segment (P,p1,p2,q2,q3))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Segment (P,p1,p2,q1,q3) or x in (Segment (P,p1,p2,q1,q2)) \/ (Segment (P,p1,p2,q2,q3)) )
assume x in Segment (P,p1,p2,q1,q3) ; ::_thesis: x in (Segment (P,p1,p2,q1,q2)) \/ (Segment (P,p1,p2,q2,q3))
then x in { p3 where p3 is Point of (TOP-REAL 2) : ( LE q1,p3,P,p1,p2 & LE p3,q3,P,p1,p2 ) } by JORDAN6:26;
then consider p3 being Point of (TOP-REAL 2) such that
A6: x = p3 and
A7: LE q1,p3,P,p1,p2 and
A8: LE p3,q3,P,p1,p2 ;
A9: p3 in P by A7, JORDAN5C:def_3;
now__::_thesis:_(_x_in_Segment_(P,p1,p2,q1,q2)_or_x_in_Segment_(P,p1,p2,q2,q3)_)
percases ( p3 = q2 or p3 <> q2 ) ;
supposeA10: p3 = q2 ; ::_thesis: ( x in Segment (P,p1,p2,q1,q2) or x in Segment (P,p1,p2,q2,q3) )
then LE p3,q2,P,p1,p2 by A4, JORDAN5C:9;
then x in { p31 where p31 is Point of (TOP-REAL 2) : ( LE q1,p31,P,p1,p2 & LE p31,q2,P,p1,p2 ) } by A2, A6, A10;
hence ( x in Segment (P,p1,p2,q1,q2) or x in Segment (P,p1,p2,q2,q3) ) by JORDAN6:26; ::_thesis: verum
end;
supposeA11: p3 <> q2 ; ::_thesis: ( x in Segment (P,p1,p2,q1,q2) or x in Segment (P,p1,p2,q2,q3) )
now__::_thesis:_(_(_LE_p3,q2,P,p1,p2_&_not_LE_q2,p3,P,p1,p2_&_(_x_in_Segment_(P,p1,p2,q1,q2)_or_x_in_Segment_(P,p1,p2,q2,q3)_)_)_or_(_LE_q2,p3,P,p1,p2_&_not_LE_p3,q2,P,p1,p2_&_(_x_in_Segment_(P,p1,p2,q1,q2)_or_x_in_Segment_(P,p1,p2,q2,q3)_)_)_)
percases ( ( LE p3,q2,P,p1,p2 & not LE q2,p3,P,p1,p2 ) or ( LE q2,p3,P,p1,p2 & not LE p3,q2,P,p1,p2 ) ) by A1, A4, A9, A11, JORDAN5C:14;
case ( LE p3,q2,P,p1,p2 & not LE q2,p3,P,p1,p2 ) ; ::_thesis: ( x in Segment (P,p1,p2,q1,q2) or x in Segment (P,p1,p2,q2,q3) )
then x in { p31 where p31 is Point of (TOP-REAL 2) : ( LE q1,p31,P,p1,p2 & LE p31,q2,P,p1,p2 ) } by A6, A7;
hence ( x in Segment (P,p1,p2,q1,q2) or x in Segment (P,p1,p2,q2,q3) ) by JORDAN6:26; ::_thesis: verum
end;
case ( LE q2,p3,P,p1,p2 & not LE p3,q2,P,p1,p2 ) ; ::_thesis: ( x in Segment (P,p1,p2,q1,q2) or x in Segment (P,p1,p2,q2,q3) )
then x in { p31 where p31 is Point of (TOP-REAL 2) : ( LE q2,p31,P,p1,p2 & LE p31,q3,P,p1,p2 ) } by A6, A8;
hence ( x in Segment (P,p1,p2,q1,q2) or x in Segment (P,p1,p2,q2,q3) ) by JORDAN6:26; ::_thesis: verum
end;
end;
end;
hence ( x in Segment (P,p1,p2,q1,q2) or x in Segment (P,p1,p2,q2,q3) ) ; ::_thesis: verum
end;
end;
end;
hence x in (Segment (P,p1,p2,q1,q2)) \/ (Segment (P,p1,p2,q2,q3)) by XBOOLE_0:def_3; ::_thesis: verum
end;
(Segment (P,p1,p2,q1,q2)) \/ (Segment (P,p1,p2,q2,q3)) c= Segment (P,p1,p2,q1,q3)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (Segment (P,p1,p2,q1,q2)) \/ (Segment (P,p1,p2,q2,q3)) or x in Segment (P,p1,p2,q1,q3) )
assume A12: x in (Segment (P,p1,p2,q1,q2)) \/ (Segment (P,p1,p2,q2,q3)) ; ::_thesis: x in Segment (P,p1,p2,q1,q3)
percases ( x in Segment (P,p1,p2,q1,q2) or x in Segment (P,p1,p2,q2,q3) ) by A12, XBOOLE_0:def_3;
suppose x in Segment (P,p1,p2,q1,q2) ; ::_thesis: x in Segment (P,p1,p2,q1,q3)
then x in { p where p is Point of (TOP-REAL 2) : ( LE q1,p,P,p1,p2 & LE p,q2,P,p1,p2 ) } by JORDAN6:26;
then consider p being Point of (TOP-REAL 2) such that
A13: ( x = p & LE q1,p,P,p1,p2 ) and
A14: LE p,q2,P,p1,p2 ;
LE p,q3,P,p1,p2 by A3, A14, JORDAN5C:13;
then x in { p3 where p3 is Point of (TOP-REAL 2) : ( LE q1,p3,P,p1,p2 & LE p3,q3,P,p1,p2 ) } by A13;
hence x in Segment (P,p1,p2,q1,q3) by JORDAN6:26; ::_thesis: verum
end;
suppose x in Segment (P,p1,p2,q2,q3) ; ::_thesis: x in Segment (P,p1,p2,q1,q3)
then x in { p where p is Point of (TOP-REAL 2) : ( LE q2,p,P,p1,p2 & LE p,q3,P,p1,p2 ) } by JORDAN6:26;
then consider p being Point of (TOP-REAL 2) such that
A15: x = p and
A16: LE q2,p,P,p1,p2 and
A17: LE p,q3,P,p1,p2 ;
LE q1,p,P,p1,p2 by A2, A16, JORDAN5C:13;
then x in { p3 where p3 is Point of (TOP-REAL 2) : ( LE q1,p3,P,p1,p2 & LE p3,q3,P,p1,p2 ) } by A15, A17;
hence x in Segment (P,p1,p2,q1,q3) by JORDAN6:26; ::_thesis: verum
end;
end;
end;
hence (Segment (P,p1,p2,q1,q2)) \/ (Segment (P,p1,p2,q2,q3)) = Segment (P,p1,p2,q1,q3) by A5, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem :: JORDAN20:23
for P being non empty Subset of (TOP-REAL 2)
for p1, p2, q1, q2, q3 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & LE q1,q2,P,p1,p2 & LE q2,q3,P,p1,p2 holds
(Segment (P,p1,p2,q1,q2)) /\ (Segment (P,p1,p2,q2,q3)) = {q2}
proof
let P be non empty Subset of (TOP-REAL 2); ::_thesis: for p1, p2, q1, q2, q3 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & LE q1,q2,P,p1,p2 & LE q2,q3,P,p1,p2 holds
(Segment (P,p1,p2,q1,q2)) /\ (Segment (P,p1,p2,q2,q3)) = {q2}
let p1, p2, q1, q2, q3 be Point of (TOP-REAL 2); ::_thesis: ( P is_an_arc_of p1,p2 & LE q1,q2,P,p1,p2 & LE q2,q3,P,p1,p2 implies (Segment (P,p1,p2,q1,q2)) /\ (Segment (P,p1,p2,q2,q3)) = {q2} )
assume that
A1: P is_an_arc_of p1,p2 and
A2: LE q1,q2,P,p1,p2 and
A3: LE q2,q3,P,p1,p2 ; ::_thesis: (Segment (P,p1,p2,q1,q2)) /\ (Segment (P,p1,p2,q2,q3)) = {q2}
A4: q2 in P by A2, JORDAN5C:def_3;
A5: {q2} c= (Segment (P,p1,p2,q1,q2)) /\ (Segment (P,p1,p2,q2,q3))
proof
set p3 = q2;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {q2} or x in (Segment (P,p1,p2,q1,q2)) /\ (Segment (P,p1,p2,q2,q3)) )
assume x in {q2} ; ::_thesis: x in (Segment (P,p1,p2,q1,q2)) /\ (Segment (P,p1,p2,q2,q3))
then A6: x = q2 by TARSKI:def_1;
LE q2,q2,P,p1,p2 by A4, JORDAN5C:9;
then x in { p31 where p31 is Point of (TOP-REAL 2) : ( LE q2,p31,P,p1,p2 & LE p31,q3,P,p1,p2 ) } by A3, A6;
then A7: x in Segment (P,p1,p2,q2,q3) by JORDAN6:26;
LE q2,q2,P,p1,p2 by A4, JORDAN5C:9;
then x in { p31 where p31 is Point of (TOP-REAL 2) : ( LE q1,p31,P,p1,p2 & LE p31,q2,P,p1,p2 ) } by A2, A6;
then x in Segment (P,p1,p2,q1,q2) by JORDAN6:26;
hence x in (Segment (P,p1,p2,q1,q2)) /\ (Segment (P,p1,p2,q2,q3)) by A7, XBOOLE_0:def_4; ::_thesis: verum
end;
(Segment (P,p1,p2,q1,q2)) /\ (Segment (P,p1,p2,q2,q3)) c= {q2}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (Segment (P,p1,p2,q1,q2)) /\ (Segment (P,p1,p2,q2,q3)) or x in {q2} )
assume A8: x in (Segment (P,p1,p2,q1,q2)) /\ (Segment (P,p1,p2,q2,q3)) ; ::_thesis: x in {q2}
then x in Segment (P,p1,p2,q2,q3) by XBOOLE_0:def_4;
then x in { p4 where p4 is Point of (TOP-REAL 2) : ( LE q2,p4,P,p1,p2 & LE p4,q3,P,p1,p2 ) } by JORDAN6:26;
then A9: ex p4 being Point of (TOP-REAL 2) st
( x = p4 & LE q2,p4,P,p1,p2 & LE p4,q3,P,p1,p2 ) ;
x in Segment (P,p1,p2,q1,q2) by A8, XBOOLE_0:def_4;
then x in { p where p is Point of (TOP-REAL 2) : ( LE q1,p,P,p1,p2 & LE p,q2,P,p1,p2 ) } by JORDAN6:26;
then ex p being Point of (TOP-REAL 2) st
( x = p & LE q1,p,P,p1,p2 & LE p,q2,P,p1,p2 ) ;
then x = q2 by A1, A9, JORDAN5C:12;
hence x in {q2} by TARSKI:def_1; ::_thesis: verum
end;
hence (Segment (P,p1,p2,q1,q2)) /\ (Segment (P,p1,p2,q2,q3)) = {q2} by A5, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th24: :: JORDAN20:24
for P being non empty Subset of (TOP-REAL 2)
for p1, p2 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 holds
Segment (P,p1,p2,p1,p2) = P
proof
let P be non empty Subset of (TOP-REAL 2); ::_thesis: for p1, p2 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 holds
Segment (P,p1,p2,p1,p2) = P
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( P is_an_arc_of p1,p2 implies Segment (P,p1,p2,p1,p2) = P )
assume P is_an_arc_of p1,p2 ; ::_thesis: Segment (P,p1,p2,p1,p2) = P
then A1: ( R_Segment (P,p1,p2,p1) = P & L_Segment (P,p1,p2,p2) = P ) by JORDAN6:22, JORDAN6:24;
(R_Segment (P,p1,p2,p1)) /\ (L_Segment (P,p1,p2,p2)) = Segment (P,p1,p2,p1,p2) by JORDAN6:def_5;
hence Segment (P,p1,p2,p1,p2) = P by A1; ::_thesis: verum
end;
theorem Th25: :: JORDAN20:25
for P, Q1 being non empty Subset of (TOP-REAL 2)
for p1, p2, q1, q2 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & Q1 is_an_arc_of q1,q2 & LE q1,q2,P,p1,p2 & Q1 c= P holds
Q1 = Segment (P,p1,p2,q1,q2)
proof
let P, Q1 be non empty Subset of (TOP-REAL 2); ::_thesis: for p1, p2, q1, q2 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & Q1 is_an_arc_of q1,q2 & LE q1,q2,P,p1,p2 & Q1 c= P holds
Q1 = Segment (P,p1,p2,q1,q2)
let p1, p2, q1, q2 be Point of (TOP-REAL 2); ::_thesis: ( P is_an_arc_of p1,p2 & Q1 is_an_arc_of q1,q2 & LE q1,q2,P,p1,p2 & Q1 c= P implies Q1 = Segment (P,p1,p2,q1,q2) )
assume that
A1: P is_an_arc_of p1,p2 and
A2: Q1 is_an_arc_of q1,q2 and
A3: LE q1,q2,P,p1,p2 and
A4: Q1 c= P ; ::_thesis: Q1 = Segment (P,p1,p2,q1,q2)
reconsider Q0 = Segment (P,p1,p2,q1,q2) as non empty Subset of (TOP-REAL 2) by A3, JORDAN16:18;
A5: q1 <> q2 by A2, JORDAN6:37;
then A6: Segment (P,p1,p2,q1,q2) is_an_arc_of q1,q2 by A1, A3, JORDAN16:21;
A7: q2 in P by A3, JORDAN5C:def_3;
A8: now__::_thesis:_not_q1_=_p2
assume A9: q1 = p2 ; ::_thesis: contradiction
LE q2,p2,P,p1,p2 by A1, A7, JORDAN5C:10;
hence contradiction by A1, A2, A3, A9, JORDAN5C:12, JORDAN6:37; ::_thesis: verum
end;
A10: q1 in P by A3, JORDAN5C:def_3;
A11: now__::_thesis:_not_q2_=_p1
assume A12: q2 = p1 ; ::_thesis: contradiction
LE p1,q1,P,p1,p2 by A1, A10, JORDAN5C:10;
hence contradiction by A1, A2, A3, A12, JORDAN5C:12, JORDAN6:37; ::_thesis: verum
end;
A13: ( p1 in P & p2 in P ) by A1, TOPREAL1:1;
now__::_thesis:_Q1_c=_Q0
A14: LE p1,q1,P,p1,p2 by A1, A10, JORDAN5C:10;
then A15: (Segment (P,p1,p2,p1,q1)) \/ (Segment (P,p1,p2,q1,q2)) = Segment (P,p1,p2,p1,q2) by A1, A3, Th22;
A16: [#] ((TOP-REAL 2) | P) = P by PRE_TOPC:def_5;
A17: LE q2,p2,P,p1,p2 by A1, A7, JORDAN5C:10;
A18: [#] I[01] = the carrier of I[01] ;
Q0 is_an_arc_of q1,q2 by A1, A3, A5, JORDAN16:21;
then A19: q2 in Q0 by TOPREAL1:1;
assume not Q1 c= Q0 ; ::_thesis: contradiction
then consider x8 being set such that
A20: x8 in Q1 and
A21: not x8 in Q0 by TARSKI:def_3;
reconsider q = x8 as Point of (TOP-REAL 2) by A20;
A22: q <> q1 by A3, A21, JORDAN16:5;
LE p1,q2,P,p1,p2 by A3, A14, JORDAN5C:13;
then (Segment (P,p1,p2,p1,q2)) \/ (Segment (P,p1,p2,q2,p2)) = Segment (P,p1,p2,p1,p2) by A1, A17, Th22
.= P by A1, Th24 ;
then A23: ( q in Segment (P,p1,p2,p1,q2) or q in Segment (P,p1,p2,q2,p2) ) by A4, A20, XBOOLE_0:def_3;
now__::_thesis:_(_(_q_in_Segment_(P,p1,p2,p1,q1)_&_contradiction_)_or_(_q_in_Segment_(P,p1,p2,q2,p2)_&_contradiction_)_)
percases ( q in Segment (P,p1,p2,p1,q1) or q in Segment (P,p1,p2,q2,p2) ) by A21, A15, A23, XBOOLE_0:def_3;
caseA24: q in Segment (P,p1,p2,p1,q1) ; ::_thesis: contradiction
A25: not q in {q1} by A22, TARSKI:def_1;
not q2 in {q1} by A5, TARSKI:def_1;
then reconsider Qa = P \ {q1} as non empty Subset of ((TOP-REAL 2) | P) by A7, A16, XBOOLE_0:def_5, XBOOLE_1:36;
A26: the carrier of (((TOP-REAL 2) | P) | Qa) = Qa by PRE_TOPC:8;
reconsider Qa9 = Qa as Subset of (TOP-REAL 2) ;
A27: the carrier of (((TOP-REAL 2) | P) | Qa) = Qa by PRE_TOPC:8;
A28: Segment (Q1,q1,q2,q,q2) is_an_arc_of q,q2 by A2, A20, A21, A19, Th21;
then consider f2 being Function of I[01],((TOP-REAL 2) | (Segment (Q1,q1,q2,q,q2))) such that
A29: f2 is being_homeomorphism and
A30: ( f2 . 0 = q & f2 . 1 = q2 ) by TOPREAL1:def_1;
A31: rng f2 = [#] ((TOP-REAL 2) | (Segment (Q1,q1,q2,q,q2))) by A29, TOPS_2:def_5
.= Segment (Q1,q1,q2,q,q2) by PRE_TOPC:def_5 ;
A32: ( not p2 in {q1} & not q2 in {q1} ) by A5, A8, TARSKI:def_1;
q in { p3 where p3 is Point of (TOP-REAL 2) : ( LE p1,p3,P,p1,p2 & LE p3,q1,P,p1,p2 ) } by A24, JORDAN6:26;
then A33: ex p3 being Point of (TOP-REAL 2) st
( q = p3 & LE p1,p3,P,p1,p2 & LE p3,q1,P,p1,p2 ) ;
A34: now__::_thesis:_not_p1_=_q1
assume A35: p1 = q1 ; ::_thesis: contradiction
then q = p1 by A1, A33, JORDAN5C:12;
hence contradiction by A6, A21, A35, TOPREAL1:1; ::_thesis: verum
end;
then not p1 in {q1} by TARSKI:def_1;
then reconsider p19 = p1, q9 = q, q29 = q2, p29 = p2 as Point of (((TOP-REAL 2) | P) | Qa) by A4, A7, A13, A20, A26, A32, A25, XBOOLE_0:def_5;
now__::_thesis:_(_(_q_<>_p1_&_ex_G1_being_Path_of_p19,q9_st_
(_G1_is_continuous_&_G1_._0_=_p19_&_G1_._1_=_q9_)_)_or_(_q_=_p1_&_ex_G1_being_Path_of_p19,q9_st_
(_G1_is_continuous_&_G1_._0_=_p19_&_G1_._1_=_q9_)_)_)
percases ( q <> p1 or q = p1 ) ;
case q <> p1 ; ::_thesis: ex G1 being Path of p19,q9 st
( G1 is continuous & G1 . 0 = p19 & G1 . 1 = q9 )
then A36: Segment (P,p1,p2,p1,q) is_an_arc_of p1,q by A1, A4, A20, JORDAN16:24;
then consider f1 being Function of I[01],((TOP-REAL 2) | (Segment (P,p1,p2,p1,q))) such that
A37: f1 is being_homeomorphism and
A38: ( f1 . 0 = p1 & f1 . 1 = q ) by TOPREAL1:def_1;
A39: rng f1 = [#] ((TOP-REAL 2) | (Segment (P,p1,p2,p1,q))) by A37, TOPS_2:def_5
.= Segment (P,p1,p2,p1,q) by PRE_TOPC:def_5 ;
{ p where p is Point of (TOP-REAL 2) : ( LE p1,p,P,p1,p2 & LE p,q,P,p1,p2 ) } c= Qa
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of (TOP-REAL 2) : ( LE p1,p,P,p1,p2 & LE p,q,P,p1,p2 ) } or x in Qa )
assume x in { p where p is Point of (TOP-REAL 2) : ( LE p1,p,P,p1,p2 & LE p,q,P,p1,p2 ) } ; ::_thesis: x in Qa
then A40: ex p being Point of (TOP-REAL 2) st
( x = p & LE p1,p,P,p1,p2 & LE p,q,P,p1,p2 ) ;
then x <> q1 by A1, A22, A33, JORDAN5C:12;
then A41: not x in {q1} by TARSKI:def_1;
x in P by A40, JORDAN5C:def_3;
hence x in Qa by A41, XBOOLE_0:def_5; ::_thesis: verum
end;
then A42: Segment (P,p1,p2,p1,q) c= Qa by JORDAN6:26;
dom f1 = the carrier of I[01] by A18, A37, TOPS_2:def_5;
then reconsider g1 = f1 as Function of I[01],(((TOP-REAL 2) | P) | Qa) by A26, A39, A42, FUNCT_2:2;
A43: f1 is continuous by A37, TOPS_2:def_5;
A44: for G being Subset of (((TOP-REAL 2) | P) | Qa) st G is open holds
g1 " G is open
proof
let G be Subset of (((TOP-REAL 2) | P) | Qa); ::_thesis: ( G is open implies g1 " G is open )
A45: ((TOP-REAL 2) | P) | Qa = (TOP-REAL 2) | Qa9 by PRE_TOPC:7, XBOOLE_1:36;
assume G is open ; ::_thesis: g1 " G is open
then consider G4 being Subset of (TOP-REAL 2) such that
A46: G4 is open and
A47: G = G4 /\ ([#] ((TOP-REAL 2) | Qa9)) by A45, TOPS_2:24;
reconsider G5 = G4 /\ ([#] ((TOP-REAL 2) | (Segment (P,p1,p2,p1,q)))) as Subset of ((TOP-REAL 2) | (Segment (P,p1,p2,p1,q))) ;
A48: G5 is open by A46, TOPS_2:24;
A49: rng g1 = [#] ((TOP-REAL 2) | (Segment (P,p1,p2,p1,q))) by A37, TOPS_2:def_5
.= Segment (P,p1,p2,p1,q) by PRE_TOPC:def_5 ;
A50: p1 in Segment (P,p1,p2,p1,q) by A36, TOPREAL1:1;
A51: f1 " G5 = g1 " (G4 /\ (Segment (P,p1,p2,p1,q))) by PRE_TOPC:def_5
.= (g1 " G4) /\ (g1 " (Segment (P,p1,p2,p1,q))) by FUNCT_1:68 ;
g1 " G = (g1 " G4) /\ (g1 " ([#] ((TOP-REAL 2) | Qa9))) by A47, FUNCT_1:68
.= (g1 " G4) /\ (g1 " Qa9) by PRE_TOPC:def_5
.= (g1 " G4) /\ (g1 " ((rng g1) /\ Qa9)) by RELAT_1:133
.= (g1 " G4) /\ (g1 " (Segment (P,p1,p2,p1,q))) by A42, A49, XBOOLE_1:28 ;
hence g1 " G is open by A43, A50, A48, A51, TOPS_2:43; ::_thesis: verum
end;
[#] (((TOP-REAL 2) | P) | Qa) <> {} ;
then A52: g1 is continuous by A44, TOPS_2:43;
then p19,q9 are_connected by A38, BORSUK_2:def_1;
then g1 is Path of p19,q9 by A38, A52, BORSUK_2:def_2;
hence ex G1 being Path of p19,q9 st
( G1 is continuous & G1 . 0 = p19 & G1 . 1 = q9 ) by A38, A52; ::_thesis: verum
end;
caseA53: q = p1 ; ::_thesis: ex G1 being Path of p19,q9 st
( G1 is continuous & G1 . 0 = p19 & G1 . 1 = q9 )
consider g01 being Function of I[01],(((TOP-REAL 2) | P) | Qa) such that
A54: ( g01 is continuous & g01 . 0 = p19 & g01 . 1 = p19 ) by BORSUK_2:3;
p19,p19 are_connected ;
then g01 is Path of p19,p19 by A54, BORSUK_2:def_2;
hence ex G1 being Path of p19,q9 st
( G1 is continuous & G1 . 0 = p19 & G1 . 1 = q9 ) by A53, A54; ::_thesis: verum
end;
end;
end;
then consider G1 being Path of p19,q9 such that
A55: ( G1 is continuous & G1 . 0 = p19 & G1 . 1 = q9 ) ;
now__::_thesis:_(_(_q2_<>_p2_&_ex_G3_being_Path_of_q29,p29_st_
(_G3_is_continuous_&_G3_._0_=_q29_&_G3_._1_=_p29_)_)_or_(_q2_=_p2_&_ex_G3_being_Path_of_q29,p29_st_
(_G3_is_continuous_&_G3_._0_=_q29_&_G3_._1_=_p29_)_)_)
percases ( q2 <> p2 or q2 = p2 ) ;
case q2 <> p2 ; ::_thesis: ex G3 being Path of q29,p29 st
( G3 is continuous & G3 . 0 = q29 & G3 . 1 = p29 )
then A56: Segment (P,p1,p2,q2,p2) is_an_arc_of q2,p2 by A1, A7, Th21;
then consider f3 being Function of I[01],((TOP-REAL 2) | (Segment (P,p1,p2,q2,p2))) such that
A57: f3 is being_homeomorphism and
A58: ( f3 . 0 = q2 & f3 . 1 = p2 ) by TOPREAL1:def_1;
A59: rng f3 = [#] ((TOP-REAL 2) | (Segment (P,p1,p2,q2,p2))) by A57, TOPS_2:def_5
.= Segment (P,p1,p2,q2,p2) by PRE_TOPC:def_5 ;
{ p where p is Point of (TOP-REAL 2) : ( LE q2,p,P,p1,p2 & LE p,p2,P,p1,p2 ) } c= Qa
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of (TOP-REAL 2) : ( LE q2,p,P,p1,p2 & LE p,p2,P,p1,p2 ) } or x in Qa )
assume x in { p where p is Point of (TOP-REAL 2) : ( LE q2,p,P,p1,p2 & LE p,p2,P,p1,p2 ) } ; ::_thesis: x in Qa
then A60: ex p being Point of (TOP-REAL 2) st
( x = p & LE q2,p,P,p1,p2 & LE p,p2,P,p1,p2 ) ;
then x <> q1 by A1, A2, A3, JORDAN5C:12, JORDAN6:37;
then A61: not x in {q1} by TARSKI:def_1;
x in P by A60, JORDAN5C:def_3;
hence x in Qa by A61, XBOOLE_0:def_5; ::_thesis: verum
end;
then A62: Segment (P,p1,p2,q2,p2) c= Qa by JORDAN6:26;
A63: the carrier of (((TOP-REAL 2) | P) | Qa) = Qa by PRE_TOPC:8;
dom f3 = the carrier of I[01] by A18, A57, TOPS_2:def_5;
then reconsider g3 = f3 as Function of I[01],(((TOP-REAL 2) | P) | Qa) by A59, A63, A62, FUNCT_2:2;
A64: f3 is continuous by A57, TOPS_2:def_5;
A65: for G being Subset of (((TOP-REAL 2) | P) | Qa) st G is open holds
g3 " G is open
proof
let G be Subset of (((TOP-REAL 2) | P) | Qa); ::_thesis: ( G is open implies g3 " G is open )
A66: ((TOP-REAL 2) | P) | Qa = (TOP-REAL 2) | Qa9 by PRE_TOPC:7, XBOOLE_1:36;
assume G is open ; ::_thesis: g3 " G is open
then consider G4 being Subset of (TOP-REAL 2) such that
A67: G4 is open and
A68: G = G4 /\ ([#] ((TOP-REAL 2) | Qa9)) by A66, TOPS_2:24;
reconsider G5 = G4 /\ ([#] ((TOP-REAL 2) | (Segment (P,p1,p2,q2,p2)))) as Subset of ((TOP-REAL 2) | (Segment (P,p1,p2,q2,p2))) ;
A69: G5 is open by A67, TOPS_2:24;
A70: rng g3 = [#] ((TOP-REAL 2) | (Segment (P,p1,p2,q2,p2))) by A57, TOPS_2:def_5
.= Segment (P,p1,p2,q2,p2) by PRE_TOPC:def_5 ;
A71: p2 in Segment (P,p1,p2,q2,p2) by A56, TOPREAL1:1;
A72: f3 " G5 = g3 " (G4 /\ (Segment (P,p1,p2,q2,p2))) by PRE_TOPC:def_5
.= (g3 " G4) /\ (g3 " (Segment (P,p1,p2,q2,p2))) by FUNCT_1:68 ;
g3 " G = (g3 " G4) /\ (g3 " ([#] ((TOP-REAL 2) | Qa9))) by A68, FUNCT_1:68
.= (g3 " G4) /\ (g3 " Qa9) by PRE_TOPC:def_5
.= (g3 " G4) /\ (g3 " ((rng g3) /\ Qa9)) by RELAT_1:133
.= (g3 " G4) /\ (g3 " (Segment (P,p1,p2,q2,p2))) by A62, A70, XBOOLE_1:28 ;
hence g3 " G is open by A64, A71, A69, A72, TOPS_2:43; ::_thesis: verum
end;
[#] (((TOP-REAL 2) | P) | Qa) <> {} ;
then A73: g3 is continuous by A65, TOPS_2:43;
then q29,p29 are_connected by A58, BORSUK_2:def_1;
then g3 is Path of q29,p29 by A58, A73, BORSUK_2:def_2;
hence ex G3 being Path of q29,p29 st
( G3 is continuous & G3 . 0 = q29 & G3 . 1 = p29 ) by A58, A73; ::_thesis: verum
end;
caseA74: q2 = p2 ; ::_thesis: ex G3 being Path of q29,p29 st
( G3 is continuous & G3 . 0 = q29 & G3 . 1 = p29 )
consider g01 being Function of I[01],(((TOP-REAL 2) | P) | Qa) such that
A75: ( g01 is continuous & g01 . 0 = q29 & g01 . 1 = q29 ) by BORSUK_2:3;
q29,q29 are_connected ;
then g01 is Path of q29,q29 by A75, BORSUK_2:def_2;
hence ex G3 being Path of q29,p29 st
( G3 is continuous & G3 . 0 = q29 & G3 . 1 = p29 ) by A74, A75; ::_thesis: verum
end;
end;
end;
then consider G3 being Path of q29,p29 such that
A76: ( G3 is continuous & G3 . 0 = q29 & G3 . 1 = p29 ) ;
{ p where p is Point of (TOP-REAL 2) : ( LE q,p,Q1,q1,q2 & LE p,q2,Q1,q1,q2 ) } c= Qa
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of (TOP-REAL 2) : ( LE q,p,Q1,q1,q2 & LE p,q2,Q1,q1,q2 ) } or x in Qa )
assume x in { p where p is Point of (TOP-REAL 2) : ( LE q,p,Q1,q1,q2 & LE p,q2,Q1,q1,q2 ) } ; ::_thesis: x in Qa
then A77: ex p being Point of (TOP-REAL 2) st
( x = p & LE q,p,Q1,q1,q2 & LE p,q2,Q1,q1,q2 ) ;
now__::_thesis:_not_x_=_q1
assume A78: x = q1 ; ::_thesis: contradiction
LE q1,q,Q1,q1,q2 by A2, A20, JORDAN5C:10;
hence contradiction by A2, A22, A77, A78, JORDAN5C:12; ::_thesis: verum
end;
then A79: not x in {q1} by TARSKI:def_1;
x in Q1 by A77, JORDAN5C:def_3;
hence x in Qa by A4, A79, XBOOLE_0:def_5; ::_thesis: verum
end;
then A80: Segment (Q1,q1,q2,q,q2) c= Qa by JORDAN6:26;
dom f2 = the carrier of I[01] by A18, A29, TOPS_2:def_5;
then reconsider g2 = f2 as Function of I[01],(((TOP-REAL 2) | P) | Qa) by A31, A27, A80, FUNCT_2:2;
A81: f2 is continuous by A29, TOPS_2:def_5;
A82: for G being Subset of (((TOP-REAL 2) | P) | Qa) st G is open holds
g2 " G is open
proof
let G be Subset of (((TOP-REAL 2) | P) | Qa); ::_thesis: ( G is open implies g2 " G is open )
A83: ((TOP-REAL 2) | P) | Qa = (TOP-REAL 2) | Qa9 by PRE_TOPC:7, XBOOLE_1:36;
assume G is open ; ::_thesis: g2 " G is open
then consider G4 being Subset of (TOP-REAL 2) such that
A84: G4 is open and
A85: G = G4 /\ ([#] ((TOP-REAL 2) | Qa9)) by A83, TOPS_2:24;
reconsider G5 = G4 /\ ([#] ((TOP-REAL 2) | (Segment (Q1,q1,q2,q,q2)))) as Subset of ((TOP-REAL 2) | (Segment (Q1,q1,q2,q,q2))) ;
A86: G5 is open by A84, TOPS_2:24;
A87: rng g2 = [#] ((TOP-REAL 2) | (Segment (Q1,q1,q2,q,q2))) by A29, TOPS_2:def_5
.= Segment (Q1,q1,q2,q,q2) by PRE_TOPC:def_5 ;
A88: q2 in Segment (Q1,q1,q2,q,q2) by A28, TOPREAL1:1;
A89: f2 " G5 = g2 " (G4 /\ (Segment (Q1,q1,q2,q,q2))) by PRE_TOPC:def_5
.= (g2 " G4) /\ (g2 " (Segment (Q1,q1,q2,q,q2))) by FUNCT_1:68 ;
g2 " G = (g2 " G4) /\ (g2 " ([#] ((TOP-REAL 2) | Qa9))) by A85, FUNCT_1:68
.= (g2 " G4) /\ (g2 " Qa9) by PRE_TOPC:def_5
.= (g2 " G4) /\ (g2 " ((rng g2) /\ Qa9)) by RELAT_1:133
.= (g2 " G4) /\ (g2 " (Segment (Q1,q1,q2,q,q2))) by A80, A87, XBOOLE_1:28 ;
hence g2 " G is open by A81, A88, A86, A89, TOPS_2:43; ::_thesis: verum
end;
[#] (((TOP-REAL 2) | P) | Qa) <> {} ;
then A90: g2 is continuous by A82, TOPS_2:43;
then q9,q29 are_connected by A30, BORSUK_2:def_1;
then reconsider G2 = g2 as Path of q9,q29 by A30, A90, BORSUK_2:def_2;
A91: (G1 + G2) . 1 = q29 by A55, A30, A90, BORSUK_2:14;
A92: ( G1 + G2 is continuous & (G1 + G2) . 0 = p19 ) by A55, A30, A90, BORSUK_2:14;
then A93: ((G1 + G2) + G3) . 1 = p29 by A91, A76, BORSUK_2:14;
( (G1 + G2) + G3 is continuous & ((G1 + G2) + G3) . 0 = p19 ) by A92, A91, A76, BORSUK_2:14;
hence contradiction by A1, A10, A8, A34, A93, Th18; ::_thesis: verum
end;
caseA94: q in Segment (P,p1,p2,q2,p2) ; ::_thesis: contradiction
A95: ( not p1 in {q2} & not q1 in {q2} ) by A5, A11, TARSKI:def_1;
not q1 in {q2} by A5, TARSKI:def_1;
then reconsider Qa = P \ {q2} as non empty Subset of ((TOP-REAL 2) | P) by A10, A16, XBOOLE_0:def_5, XBOOLE_1:36;
A96: the carrier of (((TOP-REAL 2) | P) | Qa) = Qa by PRE_TOPC:8;
reconsider Qa9 = Qa as Subset of (TOP-REAL 2) ;
A97: the carrier of (((TOP-REAL 2) | P) | Qa) = Qa by PRE_TOPC:8;
A98: Segment (Q1,q1,q2,q1,q) is_an_arc_of q1,q by A2, A20, A22, JORDAN16:24;
then consider f2 being Function of I[01],((TOP-REAL 2) | (Segment (Q1,q1,q2,q1,q))) such that
A99: f2 is being_homeomorphism and
A100: ( f2 . 0 = q1 & f2 . 1 = q ) by TOPREAL1:def_1;
A101: rng f2 = [#] ((TOP-REAL 2) | (Segment (Q1,q1,q2,q1,q))) by A99, TOPS_2:def_5
.= Segment (Q1,q1,q2,q1,q) by PRE_TOPC:def_5 ;
A102: not q in {q2} by A21, A19, TARSKI:def_1;
q in { p3 where p3 is Point of (TOP-REAL 2) : ( LE q2,p3,P,p1,p2 & LE p3,p2,P,p1,p2 ) } by A94, JORDAN6:26;
then A103: ex p3 being Point of (TOP-REAL 2) st
( q = p3 & LE q2,p3,P,p1,p2 & LE p3,p2,P,p1,p2 ) ;
A104: now__::_thesis:_not_p2_=_q2
assume A105: p2 = q2 ; ::_thesis: contradiction
then q = p2 by A1, A103, JORDAN5C:12;
hence contradiction by A6, A21, A105, TOPREAL1:1; ::_thesis: verum
end;
then not p2 in {q2} by TARSKI:def_1;
then reconsider p19 = p1, q9 = q, q19 = q1, p29 = p2 as Point of (((TOP-REAL 2) | P) | Qa) by A4, A10, A13, A20, A96, A95, A102, XBOOLE_0:def_5;
now__::_thesis:_(_(_q_<>_p2_&_ex_G1_being_Path_of_q9,p29_st_
(_G1_is_continuous_&_G1_._0_=_q9_&_G1_._1_=_p29_)_)_or_(_q_=_p2_&_ex_G1_being_Path_of_q9,p29_st_
(_G1_is_continuous_&_G1_._0_=_q9_&_G1_._1_=_p29_)_)_)
percases ( q <> p2 or q = p2 ) ;
case q <> p2 ; ::_thesis: ex G1 being Path of q9,p29 st
( G1 is continuous & G1 . 0 = q9 & G1 . 1 = p29 )
then A106: Segment (P,p1,p2,q,p2) is_an_arc_of q,p2 by A1, A4, A20, Th21;
then consider f1 being Function of I[01],((TOP-REAL 2) | (Segment (P,p1,p2,q,p2))) such that
A107: f1 is being_homeomorphism and
A108: ( f1 . 0 = q & f1 . 1 = p2 ) by TOPREAL1:def_1;
A109: rng f1 = [#] ((TOP-REAL 2) | (Segment (P,p1,p2,q,p2))) by A107, TOPS_2:def_5
.= Segment (P,p1,p2,q,p2) by PRE_TOPC:def_5 ;
{ p where p is Point of (TOP-REAL 2) : ( LE q,p,P,p1,p2 & LE p,p2,P,p1,p2 ) } c= Qa
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of (TOP-REAL 2) : ( LE q,p,P,p1,p2 & LE p,p2,P,p1,p2 ) } or x in Qa )
assume x in { p where p is Point of (TOP-REAL 2) : ( LE q,p,P,p1,p2 & LE p,p2,P,p1,p2 ) } ; ::_thesis: x in Qa
then A110: ex p being Point of (TOP-REAL 2) st
( x = p & LE q,p,P,p1,p2 & LE p,p2,P,p1,p2 ) ;
then x <> q2 by A1, A21, A19, A103, JORDAN5C:12;
then A111: not x in {q2} by TARSKI:def_1;
x in P by A110, JORDAN5C:def_3;
hence x in Qa by A111, XBOOLE_0:def_5; ::_thesis: verum
end;
then A112: Segment (P,p1,p2,q,p2) c= Qa by JORDAN6:26;
dom f1 = the carrier of I[01] by A18, A107, TOPS_2:def_5;
then reconsider g1 = f1 as Function of I[01],(((TOP-REAL 2) | P) | Qa) by A96, A109, A112, FUNCT_2:2;
A113: f1 is continuous by A107, TOPS_2:def_5;
A114: for G being Subset of (((TOP-REAL 2) | P) | Qa) st G is open holds
g1 " G is open
proof
let G be Subset of (((TOP-REAL 2) | P) | Qa); ::_thesis: ( G is open implies g1 " G is open )
A115: ((TOP-REAL 2) | P) | Qa = (TOP-REAL 2) | Qa9 by PRE_TOPC:7, XBOOLE_1:36;
assume G is open ; ::_thesis: g1 " G is open
then consider G4 being Subset of (TOP-REAL 2) such that
A116: G4 is open and
A117: G = G4 /\ ([#] ((TOP-REAL 2) | Qa9)) by A115, TOPS_2:24;
reconsider G5 = G4 /\ ([#] ((TOP-REAL 2) | (Segment (P,p1,p2,q,p2)))) as Subset of ((TOP-REAL 2) | (Segment (P,p1,p2,q,p2))) ;
A118: G5 is open by A116, TOPS_2:24;
A119: rng g1 = [#] ((TOP-REAL 2) | (Segment (P,p1,p2,q,p2))) by A107, TOPS_2:def_5
.= Segment (P,p1,p2,q,p2) by PRE_TOPC:def_5 ;
A120: p2 in Segment (P,p1,p2,q,p2) by A106, TOPREAL1:1;
A121: f1 " G5 = g1 " (G4 /\ (Segment (P,p1,p2,q,p2))) by PRE_TOPC:def_5
.= (g1 " G4) /\ (g1 " (Segment (P,p1,p2,q,p2))) by FUNCT_1:68 ;
g1 " G = (g1 " G4) /\ (g1 " ([#] ((TOP-REAL 2) | Qa9))) by A117, FUNCT_1:68
.= (g1 " G4) /\ (g1 " Qa9) by PRE_TOPC:def_5
.= (g1 " G4) /\ (g1 " ((rng g1) /\ Qa9)) by RELAT_1:133
.= (g1 " G4) /\ (g1 " (Segment (P,p1,p2,q,p2))) by A112, A119, XBOOLE_1:28 ;
hence g1 " G is open by A113, A120, A118, A121, TOPS_2:43; ::_thesis: verum
end;
[#] (((TOP-REAL 2) | P) | Qa) <> {} ;
then A122: g1 is continuous by A114, TOPS_2:43;
then q9,p29 are_connected by A108, BORSUK_2:def_1;
then g1 is Path of q9,p29 by A108, A122, BORSUK_2:def_2;
hence ex G1 being Path of q9,p29 st
( G1 is continuous & G1 . 0 = q9 & G1 . 1 = p29 ) by A108, A122; ::_thesis: verum
end;
caseA123: q = p2 ; ::_thesis: ex G1 being Path of q9,p29 st
( G1 is continuous & G1 . 0 = q9 & G1 . 1 = p29 )
consider g01 being Function of I[01],(((TOP-REAL 2) | P) | Qa) such that
A124: ( g01 is continuous & g01 . 0 = p29 & g01 . 1 = p29 ) by BORSUK_2:3;
p29,p29 are_connected ;
then g01 is Path of p29,p29 by A124, BORSUK_2:def_2;
hence ex G1 being Path of q9,p29 st
( G1 is continuous & G1 . 0 = q9 & G1 . 1 = p29 ) by A123, A124; ::_thesis: verum
end;
end;
end;
then consider G1 being Path of q9,p29 such that
A125: ( G1 is continuous & G1 . 0 = q9 & G1 . 1 = p29 ) ;
now__::_thesis:_(_(_q1_<>_p1_&_ex_G3_being_Path_of_p19,q19_st_
(_G3_is_continuous_&_G3_._0_=_p19_&_G3_._1_=_q19_)_)_or_(_q1_=_p1_&_ex_G3_being_Path_of_p19,q19_st_
(_G3_is_continuous_&_G3_._0_=_p19_&_G3_._1_=_q19_)_)_)
percases ( q1 <> p1 or q1 = p1 ) ;
case q1 <> p1 ; ::_thesis: ex G3 being Path of p19,q19 st
( G3 is continuous & G3 . 0 = p19 & G3 . 1 = q19 )
then A126: Segment (P,p1,p2,p1,q1) is_an_arc_of p1,q1 by A1, A10, JORDAN16:24;
then consider f3 being Function of I[01],((TOP-REAL 2) | (Segment (P,p1,p2,p1,q1))) such that
A127: f3 is being_homeomorphism and
A128: ( f3 . 0 = p1 & f3 . 1 = q1 ) by TOPREAL1:def_1;
A129: rng f3 = [#] ((TOP-REAL 2) | (Segment (P,p1,p2,p1,q1))) by A127, TOPS_2:def_5
.= Segment (P,p1,p2,p1,q1) by PRE_TOPC:def_5 ;
{ p where p is Point of (TOP-REAL 2) : ( LE p1,p,P,p1,p2 & LE p,q1,P,p1,p2 ) } c= Qa
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of (TOP-REAL 2) : ( LE p1,p,P,p1,p2 & LE p,q1,P,p1,p2 ) } or x in Qa )
assume x in { p where p is Point of (TOP-REAL 2) : ( LE p1,p,P,p1,p2 & LE p,q1,P,p1,p2 ) } ; ::_thesis: x in Qa
then A130: ex p being Point of (TOP-REAL 2) st
( x = p & LE p1,p,P,p1,p2 & LE p,q1,P,p1,p2 ) ;
then x <> q2 by A1, A2, A3, JORDAN5C:12, JORDAN6:37;
then A131: not x in {q2} by TARSKI:def_1;
x in P by A130, JORDAN5C:def_3;
hence x in Qa by A131, XBOOLE_0:def_5; ::_thesis: verum
end;
then A132: Segment (P,p1,p2,p1,q1) c= Qa by JORDAN6:26;
A133: the carrier of (((TOP-REAL 2) | P) | Qa) = Qa by PRE_TOPC:8;
dom f3 = the carrier of I[01] by A18, A127, TOPS_2:def_5;
then reconsider g3 = f3 as Function of I[01],(((TOP-REAL 2) | P) | Qa) by A129, A133, A132, FUNCT_2:2;
A134: f3 is continuous by A127, TOPS_2:def_5;
A135: for G being Subset of (((TOP-REAL 2) | P) | Qa) st G is open holds
g3 " G is open
proof
let G be Subset of (((TOP-REAL 2) | P) | Qa); ::_thesis: ( G is open implies g3 " G is open )
A136: ((TOP-REAL 2) | P) | Qa = (TOP-REAL 2) | Qa9 by PRE_TOPC:7, XBOOLE_1:36;
assume G is open ; ::_thesis: g3 " G is open
then consider G4 being Subset of (TOP-REAL 2) such that
A137: G4 is open and
A138: G = G4 /\ ([#] ((TOP-REAL 2) | Qa9)) by A136, TOPS_2:24;
reconsider G5 = G4 /\ ([#] ((TOP-REAL 2) | (Segment (P,p1,p2,p1,q1)))) as Subset of ((TOP-REAL 2) | (Segment (P,p1,p2,p1,q1))) ;
A139: G5 is open by A137, TOPS_2:24;
A140: rng g3 = [#] ((TOP-REAL 2) | (Segment (P,p1,p2,p1,q1))) by A127, TOPS_2:def_5
.= Segment (P,p1,p2,p1,q1) by PRE_TOPC:def_5 ;
A141: p1 in Segment (P,p1,p2,p1,q1) by A126, TOPREAL1:1;
A142: f3 " G5 = g3 " (G4 /\ (Segment (P,p1,p2,p1,q1))) by PRE_TOPC:def_5
.= (g3 " G4) /\ (g3 " (Segment (P,p1,p2,p1,q1))) by FUNCT_1:68 ;
g3 " G = (g3 " G4) /\ (g3 " ([#] ((TOP-REAL 2) | Qa9))) by A138, FUNCT_1:68
.= (g3 " G4) /\ (g3 " Qa9) by PRE_TOPC:def_5
.= (g3 " G4) /\ (g3 " ((rng g3) /\ Qa9)) by RELAT_1:133
.= (g3 " G4) /\ (g3 " (Segment (P,p1,p2,p1,q1))) by A132, A140, XBOOLE_1:28 ;
hence g3 " G is open by A134, A141, A139, A142, TOPS_2:43; ::_thesis: verum
end;
[#] (((TOP-REAL 2) | P) | Qa) <> {} ;
then A143: g3 is continuous by A135, TOPS_2:43;
then p19,q19 are_connected by A128, BORSUK_2:def_1;
then g3 is Path of p19,q19 by A128, A143, BORSUK_2:def_2;
hence ex G3 being Path of p19,q19 st
( G3 is continuous & G3 . 0 = p19 & G3 . 1 = q19 ) by A128, A143; ::_thesis: verum
end;
caseA144: q1 = p1 ; ::_thesis: ex G3 being Path of p19,q19 st
( G3 is continuous & G3 . 0 = p19 & G3 . 1 = q19 )
consider g01 being Function of I[01],(((TOP-REAL 2) | P) | Qa) such that
A145: ( g01 is continuous & g01 . 0 = q19 & g01 . 1 = q19 ) by BORSUK_2:3;
q19,q19 are_connected ;
then g01 is Path of q19,q19 by A145, BORSUK_2:def_2;
hence ex G3 being Path of p19,q19 st
( G3 is continuous & G3 . 0 = p19 & G3 . 1 = q19 ) by A144, A145; ::_thesis: verum
end;
end;
end;
then consider G3 being Path of p19,q19 such that
A146: ( G3 is continuous & G3 . 0 = p19 & G3 . 1 = q19 ) ;
{ p where p is Point of (TOP-REAL 2) : ( LE q1,p,Q1,q1,q2 & LE p,q,Q1,q1,q2 ) } c= Qa
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of (TOP-REAL 2) : ( LE q1,p,Q1,q1,q2 & LE p,q,Q1,q1,q2 ) } or x in Qa )
assume x in { p where p is Point of (TOP-REAL 2) : ( LE q1,p,Q1,q1,q2 & LE p,q,Q1,q1,q2 ) } ; ::_thesis: x in Qa
then A147: ex p being Point of (TOP-REAL 2) st
( x = p & LE q1,p,Q1,q1,q2 & LE p,q,Q1,q1,q2 ) ;
now__::_thesis:_not_x_=_q2
assume A148: x = q2 ; ::_thesis: contradiction
LE q,q2,Q1,q1,q2 by A2, A20, JORDAN5C:10;
hence contradiction by A2, A21, A19, A147, A148, JORDAN5C:12; ::_thesis: verum
end;
then A149: not x in {q2} by TARSKI:def_1;
x in Q1 by A147, JORDAN5C:def_3;
hence x in Qa by A4, A149, XBOOLE_0:def_5; ::_thesis: verum
end;
then A150: Segment (Q1,q1,q2,q1,q) c= Qa by JORDAN6:26;
dom f2 = the carrier of I[01] by A18, A99, TOPS_2:def_5;
then reconsider g2 = f2 as Function of I[01],(((TOP-REAL 2) | P) | Qa) by A101, A97, A150, FUNCT_2:2;
A151: f2 is continuous by A99, TOPS_2:def_5;
A152: for G being Subset of (((TOP-REAL 2) | P) | Qa) st G is open holds
g2 " G is open
proof
let G be Subset of (((TOP-REAL 2) | P) | Qa); ::_thesis: ( G is open implies g2 " G is open )
A153: ((TOP-REAL 2) | P) | Qa = (TOP-REAL 2) | Qa9 by PRE_TOPC:7, XBOOLE_1:36;
assume G is open ; ::_thesis: g2 " G is open
then consider G4 being Subset of (TOP-REAL 2) such that
A154: G4 is open and
A155: G = G4 /\ ([#] ((TOP-REAL 2) | Qa9)) by A153, TOPS_2:24;
reconsider G5 = G4 /\ ([#] ((TOP-REAL 2) | (Segment (Q1,q1,q2,q1,q)))) as Subset of ((TOP-REAL 2) | (Segment (Q1,q1,q2,q1,q))) ;
A156: G5 is open by A154, TOPS_2:24;
A157: rng g2 = [#] ((TOP-REAL 2) | (Segment (Q1,q1,q2,q1,q))) by A99, TOPS_2:def_5
.= Segment (Q1,q1,q2,q1,q) by PRE_TOPC:def_5 ;
A158: q1 in Segment (Q1,q1,q2,q1,q) by A98, TOPREAL1:1;
A159: f2 " G5 = g2 " (G4 /\ (Segment (Q1,q1,q2,q1,q))) by PRE_TOPC:def_5
.= (g2 " G4) /\ (g2 " (Segment (Q1,q1,q2,q1,q))) by FUNCT_1:68 ;
g2 " G = (g2 " G4) /\ (g2 " ([#] ((TOP-REAL 2) | Qa9))) by A155, FUNCT_1:68
.= (g2 " G4) /\ (g2 " Qa9) by PRE_TOPC:def_5
.= (g2 " G4) /\ (g2 " ((rng g2) /\ Qa9)) by RELAT_1:133
.= (g2 " G4) /\ (g2 " (Segment (Q1,q1,q2,q1,q))) by A150, A157, XBOOLE_1:28 ;
hence g2 " G is open by A151, A158, A156, A159, TOPS_2:43; ::_thesis: verum
end;
[#] (((TOP-REAL 2) | P) | Qa) <> {} ;
then A160: g2 is continuous by A152, TOPS_2:43;
then q19,q9 are_connected by A100, BORSUK_2:def_1;
then reconsider G2 = g2 as Path of q19,q9 by A100, A160, BORSUK_2:def_2;
A161: (G2 + G1) . 1 = p29 by A125, A100, A160, BORSUK_2:14;
A162: ( G2 + G1 is continuous & (G2 + G1) . 0 = q19 ) by A125, A100, A160, BORSUK_2:14;
then A163: (G3 + (G2 + G1)) . 1 = p29 by A161, A146, BORSUK_2:14;
( G3 + (G2 + G1) is continuous & (G3 + (G2 + G1)) . 0 = p19 ) by A162, A161, A146, BORSUK_2:14;
hence contradiction by A1, A7, A11, A104, A163, Th18; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
hence Q1 = Segment (P,p1,p2,q1,q2) by A2, A6, Th20; ::_thesis: verum
end;
theorem :: JORDAN20:26
for P being non empty Subset of (TOP-REAL 2)
for p1, p2, q1, q2, p being Point of (TOP-REAL 2)
for e being Real st q1 is_Lin P,p1,p2,e & q2 `1 = e & LSeg (q1,q2) c= P & p in LSeg (q1,q2) holds
p is_Lin P,p1,p2,e
proof
let P be non empty Subset of (TOP-REAL 2); ::_thesis: for p1, p2, q1, q2, p being Point of (TOP-REAL 2)
for e being Real st q1 is_Lin P,p1,p2,e & q2 `1 = e & LSeg (q1,q2) c= P & p in LSeg (q1,q2) holds
p is_Lin P,p1,p2,e
let p1, p2, q1, q2, p be Point of (TOP-REAL 2); ::_thesis: for e being Real st q1 is_Lin P,p1,p2,e & q2 `1 = e & LSeg (q1,q2) c= P & p in LSeg (q1,q2) holds
p is_Lin P,p1,p2,e
let e be Real; ::_thesis: ( q1 is_Lin P,p1,p2,e & q2 `1 = e & LSeg (q1,q2) c= P & p in LSeg (q1,q2) implies p is_Lin P,p1,p2,e )
assume that
A1: q1 is_Lin P,p1,p2,e and
A2: q2 `1 = e and
A3: LSeg (q1,q2) c= P and
A4: p in LSeg (q1,q2) ; ::_thesis: p is_Lin P,p1,p2,e
A5: q1 in P by A1, Def1;
A6: q2 in LSeg (q1,q2) by RLTOPSP1:68;
A7: q1 `1 = e by A1, Def1;
consider p4 being Point of (TOP-REAL 2) such that
A8: p4 `1 < e and
A9: LE p4,q1,P,p1,p2 and
A10: for p5 being Point of (TOP-REAL 2) st LE p4,p5,P,p1,p2 & LE p5,q1,P,p1,p2 holds
p5 `1 <= e by A1, Def1;
A11: P is_an_arc_of p1,p2 by A1, Def1;
A12: p4 in P by A9, JORDAN5C:def_3;
now__::_thesis:_(_(_LE_q1,q2,P,p1,p2_&_p_is_Lin_P,p1,p2,e_)_or_(_LE_q2,q1,P,p1,p2_&_p_is_Lin_P,p1,p2,e_)_)
percases ( LE q1,q2,P,p1,p2 or LE q2,q1,P,p1,p2 ) by A3, A11, A5, A6, Th19;
caseA13: LE q1,q2,P,p1,p2 ; ::_thesis: p is_Lin P,p1,p2,e
A14: now__::_thesis:_(_(_q1_<>_q2_&_Segment_(P,p1,p2,q1,q2)_=_LSeg_(q1,q2)_)_or_(_q1_=_q2_&_Segment_(P,p1,p2,q1,q2)_=_LSeg_(q1,q2)_)_)
percases ( q1 <> q2 or q1 = q2 ) ;
case q1 <> q2 ; ::_thesis: Segment (P,p1,p2,q1,q2) = LSeg (q1,q2)
then LSeg (q1,q2) is_an_arc_of q1,q2 by TOPREAL1:9;
hence Segment (P,p1,p2,q1,q2) = LSeg (q1,q2) by A3, A11, A13, Th25; ::_thesis: verum
end;
caseA15: q1 = q2 ; ::_thesis: Segment (P,p1,p2,q1,q2) = LSeg (q1,q2)
then LSeg (q1,q2) = {q1} by RLTOPSP1:70;
hence Segment (P,p1,p2,q1,q2) = LSeg (q1,q2) by A11, A5, A15, Th1; ::_thesis: verum
end;
end;
end;
Segment (P,p1,p2,q1,q2) = { p3 where p3 is Point of (TOP-REAL 2) : ( LE q1,p3,P,p1,p2 & LE p3,q2,P,p1,p2 ) } by JORDAN6:26;
then A16: ex p3 being Point of (TOP-REAL 2) st
( p = p3 & LE q1,p3,P,p1,p2 & LE p3,q2,P,p1,p2 ) by A4, A14;
then A17: LE p4,p,P,p1,p2 by A9, JORDAN5C:13;
A18: for p6 being Point of (TOP-REAL 2) st LE p4,p6,P,p1,p2 & LE p6,p,P,p1,p2 holds
p6 `1 <= e
proof
let p6 be Point of (TOP-REAL 2); ::_thesis: ( LE p4,p6,P,p1,p2 & LE p6,p,P,p1,p2 implies p6 `1 <= e )
assume that
A19: LE p4,p6,P,p1,p2 and
A20: LE p6,p,P,p1,p2 ; ::_thesis: p6 `1 <= e
A21: p6 in P by A19, JORDAN5C:def_3;
now__::_thesis:_(_(_LE_p6,q1,P,p1,p2_&_p6_`1_<=_e_)_or_(_LE_q1,p6,P,p1,p2_&_p6_`1_<=_e_)_)
percases ( LE p6,q1,P,p1,p2 or LE q1,p6,P,p1,p2 ) by A11, A5, A21, Th19;
case LE p6,q1,P,p1,p2 ; ::_thesis: p6 `1 <= e
hence p6 `1 <= e by A10, A19; ::_thesis: verum
end;
caseA22: LE q1,p6,P,p1,p2 ; ::_thesis: p6 `1 <= e
LE p6,q2,P,p1,p2 by A16, A20, JORDAN5C:13;
then p6 in { qq where qq is Point of (TOP-REAL 2) : ( LE q1,qq,P,p1,p2 & LE qq,q2,P,p1,p2 ) } by A22;
then p6 in LSeg (q1,q2) by A14, JORDAN6:26;
hence p6 `1 <= e by A2, A7, GOBOARD7:5; ::_thesis: verum
end;
end;
end;
hence p6 `1 <= e ; ::_thesis: verum
end;
p `1 = e by A2, A4, A7, GOBOARD7:5;
hence p is_Lin P,p1,p2,e by A3, A4, A11, A8, A17, A18, Def1; ::_thesis: verum
end;
caseA23: LE q2,q1,P,p1,p2 ; ::_thesis: p is_Lin P,p1,p2,e
A24: now__::_thesis:_(_(_q1_<>_q2_&_Segment_(P,p1,p2,q2,q1)_=_LSeg_(q2,q1)_)_or_(_q1_=_q2_&_Segment_(P,p1,p2,q2,q1)_=_LSeg_(q2,q1)_)_)
percases ( q1 <> q2 or q1 = q2 ) ;
case q1 <> q2 ; ::_thesis: Segment (P,p1,p2,q2,q1) = LSeg (q2,q1)
then LSeg (q2,q1) is_an_arc_of q2,q1 by TOPREAL1:9;
hence Segment (P,p1,p2,q2,q1) = LSeg (q2,q1) by A3, A11, A23, Th25; ::_thesis: verum
end;
caseA25: q1 = q2 ; ::_thesis: Segment (P,p1,p2,q2,q1) = LSeg (q2,q1)
then LSeg (q2,q1) = {q1} by RLTOPSP1:70;
hence Segment (P,p1,p2,q2,q1) = LSeg (q2,q1) by A11, A5, A25, Th1; ::_thesis: verum
end;
end;
end;
A26: now__::_thesis:_not_LE_q2,p4,P,p1,p2
assume LE q2,p4,P,p1,p2 ; ::_thesis: contradiction
then p4 in { qq where qq is Point of (TOP-REAL 2) : ( LE q2,qq,P,p1,p2 & LE qq,q1,P,p1,p2 ) } by A9;
then p4 in Segment (P,p1,p2,q2,q1) by JORDAN6:26;
hence contradiction by A2, A7, A8, A24, GOBOARD7:5; ::_thesis: verum
end;
Segment (P,p1,p2,q2,q1) = { p3 where p3 is Point of (TOP-REAL 2) : ( LE q2,p3,P,p1,p2 & LE p3,q1,P,p1,p2 ) } by JORDAN6:26;
then A27: ex p3 being Point of (TOP-REAL 2) st
( p = p3 & LE q2,p3,P,p1,p2 & LE p3,q1,P,p1,p2 ) by A4, A24;
A28: for p6 being Point of (TOP-REAL 2) st LE p4,p6,P,p1,p2 & LE p6,p,P,p1,p2 holds
p6 `1 <= e
proof
let p6 be Point of (TOP-REAL 2); ::_thesis: ( LE p4,p6,P,p1,p2 & LE p6,p,P,p1,p2 implies p6 `1 <= e )
assume that
A29: LE p4,p6,P,p1,p2 and
A30: LE p6,p,P,p1,p2 ; ::_thesis: p6 `1 <= e
LE p6,q1,P,p1,p2 by A27, A30, JORDAN5C:13;
hence p6 `1 <= e by A10, A29; ::_thesis: verum
end;
( LE q2,p4,P,p1,p2 or LE p4,q2,P,p1,p2 ) by A3, A11, A6, A12, Th19;
then A31: LE p4,p,P,p1,p2 by A27, A26, JORDAN5C:13;
p `1 = e by A2, A4, A7, GOBOARD7:5;
hence p is_Lin P,p1,p2,e by A3, A4, A11, A8, A31, A28, Def1; ::_thesis: verum
end;
end;
end;
hence p is_Lin P,p1,p2,e ; ::_thesis: verum
end;
theorem :: JORDAN20:27
for P being non empty Subset of (TOP-REAL 2)
for p1, p2, q1, q2, p being Point of (TOP-REAL 2)
for e being Real st q1 is_Rin P,p1,p2,e & q2 `1 = e & LSeg (q1,q2) c= P & p in LSeg (q1,q2) holds
p is_Rin P,p1,p2,e
proof
let P be non empty Subset of (TOP-REAL 2); ::_thesis: for p1, p2, q1, q2, p being Point of (TOP-REAL 2)
for e being Real st q1 is_Rin P,p1,p2,e & q2 `1 = e & LSeg (q1,q2) c= P & p in LSeg (q1,q2) holds
p is_Rin P,p1,p2,e
let p1, p2, q1, q2, p be Point of (TOP-REAL 2); ::_thesis: for e being Real st q1 is_Rin P,p1,p2,e & q2 `1 = e & LSeg (q1,q2) c= P & p in LSeg (q1,q2) holds
p is_Rin P,p1,p2,e
let e be Real; ::_thesis: ( q1 is_Rin P,p1,p2,e & q2 `1 = e & LSeg (q1,q2) c= P & p in LSeg (q1,q2) implies p is_Rin P,p1,p2,e )
assume that
A1: q1 is_Rin P,p1,p2,e and
A2: q2 `1 = e and
A3: LSeg (q1,q2) c= P and
A4: p in LSeg (q1,q2) ; ::_thesis: p is_Rin P,p1,p2,e
A5: q1 in P by A1, Def2;
A6: q2 in LSeg (q1,q2) by RLTOPSP1:68;
A7: q1 `1 = e by A1, Def2;
consider p4 being Point of (TOP-REAL 2) such that
A8: p4 `1 > e and
A9: LE p4,q1,P,p1,p2 and
A10: for p5 being Point of (TOP-REAL 2) st LE p4,p5,P,p1,p2 & LE p5,q1,P,p1,p2 holds
p5 `1 >= e by A1, Def2;
A11: P is_an_arc_of p1,p2 by A1, Def2;
A12: p4 in P by A9, JORDAN5C:def_3;
now__::_thesis:_(_(_LE_q1,q2,P,p1,p2_&_p_is_Rin_P,p1,p2,e_)_or_(_LE_q2,q1,P,p1,p2_&_p_is_Rin_P,p1,p2,e_)_)
percases ( LE q1,q2,P,p1,p2 or LE q2,q1,P,p1,p2 ) by A3, A11, A5, A6, Th19;
caseA13: LE q1,q2,P,p1,p2 ; ::_thesis: p is_Rin P,p1,p2,e
A14: now__::_thesis:_(_(_q1_<>_q2_&_Segment_(P,p1,p2,q1,q2)_=_LSeg_(q1,q2)_)_or_(_q1_=_q2_&_Segment_(P,p1,p2,q1,q2)_=_LSeg_(q1,q2)_)_)
percases ( q1 <> q2 or q1 = q2 ) ;
case q1 <> q2 ; ::_thesis: Segment (P,p1,p2,q1,q2) = LSeg (q1,q2)
then LSeg (q1,q2) is_an_arc_of q1,q2 by TOPREAL1:9;
hence Segment (P,p1,p2,q1,q2) = LSeg (q1,q2) by A3, A11, A13, Th25; ::_thesis: verum
end;
caseA15: q1 = q2 ; ::_thesis: Segment (P,p1,p2,q1,q2) = LSeg (q1,q2)
then LSeg (q1,q2) = {q1} by RLTOPSP1:70;
hence Segment (P,p1,p2,q1,q2) = LSeg (q1,q2) by A11, A5, A15, Th1; ::_thesis: verum
end;
end;
end;
Segment (P,p1,p2,q1,q2) = { p3 where p3 is Point of (TOP-REAL 2) : ( LE q1,p3,P,p1,p2 & LE p3,q2,P,p1,p2 ) } by JORDAN6:26;
then A16: ex p3 being Point of (TOP-REAL 2) st
( p = p3 & LE q1,p3,P,p1,p2 & LE p3,q2,P,p1,p2 ) by A4, A14;
then A17: LE p4,p,P,p1,p2 by A9, JORDAN5C:13;
A18: for p6 being Point of (TOP-REAL 2) st LE p4,p6,P,p1,p2 & LE p6,p,P,p1,p2 holds
p6 `1 >= e
proof
let p6 be Point of (TOP-REAL 2); ::_thesis: ( LE p4,p6,P,p1,p2 & LE p6,p,P,p1,p2 implies p6 `1 >= e )
assume that
A19: LE p4,p6,P,p1,p2 and
A20: LE p6,p,P,p1,p2 ; ::_thesis: p6 `1 >= e
A21: p6 in P by A19, JORDAN5C:def_3;
now__::_thesis:_(_(_LE_p6,q1,P,p1,p2_&_p6_`1_>=_e_)_or_(_LE_q1,p6,P,p1,p2_&_p6_`1_>=_e_)_)
percases ( LE p6,q1,P,p1,p2 or LE q1,p6,P,p1,p2 ) by A11, A5, A21, Th19;
case LE p6,q1,P,p1,p2 ; ::_thesis: p6 `1 >= e
hence p6 `1 >= e by A10, A19; ::_thesis: verum
end;
caseA22: LE q1,p6,P,p1,p2 ; ::_thesis: p6 `1 >= e
LE p6,q2,P,p1,p2 by A16, A20, JORDAN5C:13;
then p6 in { qq where qq is Point of (TOP-REAL 2) : ( LE q1,qq,P,p1,p2 & LE qq,q2,P,p1,p2 ) } by A22;
then p6 in LSeg (q1,q2) by A14, JORDAN6:26;
hence p6 `1 >= e by A2, A7, GOBOARD7:5; ::_thesis: verum
end;
end;
end;
hence p6 `1 >= e ; ::_thesis: verum
end;
p `1 = e by A2, A4, A7, GOBOARD7:5;
hence p is_Rin P,p1,p2,e by A3, A4, A11, A8, A17, A18, Def2; ::_thesis: verum
end;
caseA23: LE q2,q1,P,p1,p2 ; ::_thesis: p is_Rin P,p1,p2,e
A24: now__::_thesis:_(_(_q1_<>_q2_&_Segment_(P,p1,p2,q2,q1)_=_LSeg_(q2,q1)_)_or_(_q1_=_q2_&_Segment_(P,p1,p2,q2,q1)_=_LSeg_(q2,q1)_)_)
percases ( q1 <> q2 or q1 = q2 ) ;
case q1 <> q2 ; ::_thesis: Segment (P,p1,p2,q2,q1) = LSeg (q2,q1)
then LSeg (q2,q1) is_an_arc_of q2,q1 by TOPREAL1:9;
hence Segment (P,p1,p2,q2,q1) = LSeg (q2,q1) by A3, A11, A23, Th25; ::_thesis: verum
end;
caseA25: q1 = q2 ; ::_thesis: Segment (P,p1,p2,q2,q1) = LSeg (q2,q1)
then LSeg (q2,q1) = {q1} by RLTOPSP1:70;
hence Segment (P,p1,p2,q2,q1) = LSeg (q2,q1) by A11, A5, A25, Th1; ::_thesis: verum
end;
end;
end;
A26: now__::_thesis:_not_LE_q2,p4,P,p1,p2
assume LE q2,p4,P,p1,p2 ; ::_thesis: contradiction
then p4 in { qq where qq is Point of (TOP-REAL 2) : ( LE q2,qq,P,p1,p2 & LE qq,q1,P,p1,p2 ) } by A9;
then p4 in Segment (P,p1,p2,q2,q1) by JORDAN6:26;
hence contradiction by A2, A7, A8, A24, GOBOARD7:5; ::_thesis: verum
end;
Segment (P,p1,p2,q2,q1) = { p3 where p3 is Point of (TOP-REAL 2) : ( LE q2,p3,P,p1,p2 & LE p3,q1,P,p1,p2 ) } by JORDAN6:26;
then A27: ex p3 being Point of (TOP-REAL 2) st
( p = p3 & LE q2,p3,P,p1,p2 & LE p3,q1,P,p1,p2 ) by A4, A24;
A28: for p6 being Point of (TOP-REAL 2) st LE p4,p6,P,p1,p2 & LE p6,p,P,p1,p2 holds
p6 `1 >= e
proof
let p6 be Point of (TOP-REAL 2); ::_thesis: ( LE p4,p6,P,p1,p2 & LE p6,p,P,p1,p2 implies p6 `1 >= e )
assume that
A29: LE p4,p6,P,p1,p2 and
A30: LE p6,p,P,p1,p2 ; ::_thesis: p6 `1 >= e
LE p6,q1,P,p1,p2 by A27, A30, JORDAN5C:13;
hence p6 `1 >= e by A10, A29; ::_thesis: verum
end;
( LE q2,p4,P,p1,p2 or LE p4,q2,P,p1,p2 ) by A3, A11, A6, A12, Th19;
then A31: LE p4,p,P,p1,p2 by A27, A26, JORDAN5C:13;
p `1 = e by A2, A4, A7, GOBOARD7:5;
hence p is_Rin P,p1,p2,e by A3, A4, A11, A8, A31, A28, Def2; ::_thesis: verum
end;
end;
end;
hence p is_Rin P,p1,p2,e ; ::_thesis: verum
end;
theorem :: JORDAN20:28
for P being non empty Subset of (TOP-REAL 2)
for p1, p2, q1, q2, p being Point of (TOP-REAL 2)
for e being Real st q1 is_Lout P,p1,p2,e & q2 `1 = e & LSeg (q1,q2) c= P & p in LSeg (q1,q2) holds
p is_Lout P,p1,p2,e
proof
let P be non empty Subset of (TOP-REAL 2); ::_thesis: for p1, p2, q1, q2, p being Point of (TOP-REAL 2)
for e being Real st q1 is_Lout P,p1,p2,e & q2 `1 = e & LSeg (q1,q2) c= P & p in LSeg (q1,q2) holds
p is_Lout P,p1,p2,e
let p1, p2, q1, q2, p be Point of (TOP-REAL 2); ::_thesis: for e being Real st q1 is_Lout P,p1,p2,e & q2 `1 = e & LSeg (q1,q2) c= P & p in LSeg (q1,q2) holds
p is_Lout P,p1,p2,e
let e be Real; ::_thesis: ( q1 is_Lout P,p1,p2,e & q2 `1 = e & LSeg (q1,q2) c= P & p in LSeg (q1,q2) implies p is_Lout P,p1,p2,e )
assume that
A1: q1 is_Lout P,p1,p2,e and
A2: q2 `1 = e and
A3: LSeg (q1,q2) c= P and
A4: p in LSeg (q1,q2) ; ::_thesis: p is_Lout P,p1,p2,e
A5: q1 in P by A1, Def3;
A6: q2 in LSeg (q1,q2) by RLTOPSP1:68;
A7: q1 `1 = e by A1, Def3;
consider p4 being Point of (TOP-REAL 2) such that
A8: p4 `1 < e and
A9: LE q1,p4,P,p1,p2 and
A10: for p5 being Point of (TOP-REAL 2) st LE p5,p4,P,p1,p2 & LE q1,p5,P,p1,p2 holds
p5 `1 <= e by A1, Def3;
A11: P is_an_arc_of p1,p2 by A1, Def3;
A12: p4 in P by A9, JORDAN5C:def_3;
now__::_thesis:_(_(_LE_q2,q1,P,p1,p2_&_p_is_Lout_P,p1,p2,e_)_or_(_LE_q1,q2,P,p1,p2_&_p_is_Lout_P,p1,p2,e_)_)
percases ( LE q2,q1,P,p1,p2 or LE q1,q2,P,p1,p2 ) by A3, A11, A5, A6, Th19;
caseA13: LE q2,q1,P,p1,p2 ; ::_thesis: p is_Lout P,p1,p2,e
A14: now__::_thesis:_(_(_q1_<>_q2_&_Segment_(P,p1,p2,q2,q1)_=_LSeg_(q2,q1)_)_or_(_q1_=_q2_&_Segment_(P,p1,p2,q2,q1)_=_LSeg_(q2,q1)_)_)
percases ( q1 <> q2 or q1 = q2 ) ;
case q1 <> q2 ; ::_thesis: Segment (P,p1,p2,q2,q1) = LSeg (q2,q1)
then LSeg (q2,q1) is_an_arc_of q2,q1 by TOPREAL1:9;
hence Segment (P,p1,p2,q2,q1) = LSeg (q2,q1) by A3, A11, A13, Th25; ::_thesis: verum
end;
caseA15: q1 = q2 ; ::_thesis: Segment (P,p1,p2,q2,q1) = LSeg (q2,q1)
then LSeg (q1,q2) = {q1} by RLTOPSP1:70;
hence Segment (P,p1,p2,q2,q1) = LSeg (q2,q1) by A11, A5, A15, Th1; ::_thesis: verum
end;
end;
end;
Segment (P,p1,p2,q2,q1) = { p3 where p3 is Point of (TOP-REAL 2) : ( LE q2,p3,P,p1,p2 & LE p3,q1,P,p1,p2 ) } by JORDAN6:26;
then A16: ex p3 being Point of (TOP-REAL 2) st
( p = p3 & LE q2,p3,P,p1,p2 & LE p3,q1,P,p1,p2 ) by A4, A14;
then A17: LE p,p4,P,p1,p2 by A9, JORDAN5C:13;
A18: for p6 being Point of (TOP-REAL 2) st LE p6,p4,P,p1,p2 & LE p,p6,P,p1,p2 holds
p6 `1 <= e
proof
let p6 be Point of (TOP-REAL 2); ::_thesis: ( LE p6,p4,P,p1,p2 & LE p,p6,P,p1,p2 implies p6 `1 <= e )
assume that
A19: LE p6,p4,P,p1,p2 and
A20: LE p,p6,P,p1,p2 ; ::_thesis: p6 `1 <= e
A21: p6 in P by A19, JORDAN5C:def_3;
now__::_thesis:_(_(_LE_q1,p6,P,p1,p2_&_p6_`1_<=_e_)_or_(_LE_p6,q1,P,p1,p2_&_p6_`1_<=_e_)_)
percases ( LE q1,p6,P,p1,p2 or LE p6,q1,P,p1,p2 ) by A11, A5, A21, Th19;
case LE q1,p6,P,p1,p2 ; ::_thesis: p6 `1 <= e
hence p6 `1 <= e by A10, A19; ::_thesis: verum
end;
caseA22: LE p6,q1,P,p1,p2 ; ::_thesis: p6 `1 <= e
LE q2,p6,P,p1,p2 by A16, A20, JORDAN5C:13;
then p6 in { qq where qq is Point of (TOP-REAL 2) : ( LE q2,qq,P,p1,p2 & LE qq,q1,P,p1,p2 ) } by A22;
then p6 in LSeg (q2,q1) by A14, JORDAN6:26;
hence p6 `1 <= e by A2, A7, GOBOARD7:5; ::_thesis: verum
end;
end;
end;
hence p6 `1 <= e ; ::_thesis: verum
end;
p `1 = e by A2, A4, A7, GOBOARD7:5;
hence p is_Lout P,p1,p2,e by A3, A4, A11, A8, A17, A18, Def3; ::_thesis: verum
end;
caseA23: LE q1,q2,P,p1,p2 ; ::_thesis: p is_Lout P,p1,p2,e
A24: now__::_thesis:_(_(_q1_<>_q2_&_Segment_(P,p1,p2,q1,q2)_=_LSeg_(q1,q2)_)_or_(_q1_=_q2_&_Segment_(P,p1,p2,q1,q2)_=_LSeg_(q1,q2)_)_)
percases ( q1 <> q2 or q1 = q2 ) ;
case q1 <> q2 ; ::_thesis: Segment (P,p1,p2,q1,q2) = LSeg (q1,q2)
then LSeg (q1,q2) is_an_arc_of q1,q2 by TOPREAL1:9;
hence Segment (P,p1,p2,q1,q2) = LSeg (q1,q2) by A3, A11, A23, Th25; ::_thesis: verum
end;
caseA25: q1 = q2 ; ::_thesis: Segment (P,p1,p2,q1,q2) = LSeg (q1,q2)
then LSeg (q2,q1) = {q1} by RLTOPSP1:70;
hence Segment (P,p1,p2,q1,q2) = LSeg (q1,q2) by A11, A5, A25, Th1; ::_thesis: verum
end;
end;
end;
A26: now__::_thesis:_not_LE_p4,q2,P,p1,p2
assume LE p4,q2,P,p1,p2 ; ::_thesis: contradiction
then p4 in { qq where qq is Point of (TOP-REAL 2) : ( LE q1,qq,P,p1,p2 & LE qq,q2,P,p1,p2 ) } by A9;
then p4 in Segment (P,p1,p2,q1,q2) by JORDAN6:26;
hence contradiction by A2, A7, A8, A24, GOBOARD7:5; ::_thesis: verum
end;
Segment (P,p1,p2,q1,q2) = { p3 where p3 is Point of (TOP-REAL 2) : ( LE q1,p3,P,p1,p2 & LE p3,q2,P,p1,p2 ) } by JORDAN6:26;
then A27: ex p3 being Point of (TOP-REAL 2) st
( p = p3 & LE q1,p3,P,p1,p2 & LE p3,q2,P,p1,p2 ) by A4, A24;
A28: for p6 being Point of (TOP-REAL 2) st LE p6,p4,P,p1,p2 & LE p,p6,P,p1,p2 holds
p6 `1 <= e
proof
let p6 be Point of (TOP-REAL 2); ::_thesis: ( LE p6,p4,P,p1,p2 & LE p,p6,P,p1,p2 implies p6 `1 <= e )
assume that
A29: LE p6,p4,P,p1,p2 and
A30: LE p,p6,P,p1,p2 ; ::_thesis: p6 `1 <= e
LE q1,p6,P,p1,p2 by A27, A30, JORDAN5C:13;
hence p6 `1 <= e by A10, A29; ::_thesis: verum
end;
( LE q2,p4,P,p1,p2 or LE p4,q2,P,p1,p2 ) by A3, A11, A6, A12, Th19;
then A31: LE p,p4,P,p1,p2 by A27, A26, JORDAN5C:13;
p `1 = e by A2, A4, A7, GOBOARD7:5;
hence p is_Lout P,p1,p2,e by A3, A4, A11, A8, A31, A28, Def3; ::_thesis: verum
end;
end;
end;
hence p is_Lout P,p1,p2,e ; ::_thesis: verum
end;
theorem :: JORDAN20:29
for P being non empty Subset of (TOP-REAL 2)
for p1, p2, q1, q2, p being Point of (TOP-REAL 2)
for e being Real st q1 is_Rout P,p1,p2,e & q2 `1 = e & LSeg (q1,q2) c= P & p in LSeg (q1,q2) holds
p is_Rout P,p1,p2,e
proof
let P be non empty Subset of (TOP-REAL 2); ::_thesis: for p1, p2, q1, q2, p being Point of (TOP-REAL 2)
for e being Real st q1 is_Rout P,p1,p2,e & q2 `1 = e & LSeg (q1,q2) c= P & p in LSeg (q1,q2) holds
p is_Rout P,p1,p2,e
let p1, p2, q1, q2, p be Point of (TOP-REAL 2); ::_thesis: for e being Real st q1 is_Rout P,p1,p2,e & q2 `1 = e & LSeg (q1,q2) c= P & p in LSeg (q1,q2) holds
p is_Rout P,p1,p2,e
let e be Real; ::_thesis: ( q1 is_Rout P,p1,p2,e & q2 `1 = e & LSeg (q1,q2) c= P & p in LSeg (q1,q2) implies p is_Rout P,p1,p2,e )
assume that
A1: q1 is_Rout P,p1,p2,e and
A2: q2 `1 = e and
A3: LSeg (q1,q2) c= P and
A4: p in LSeg (q1,q2) ; ::_thesis: p is_Rout P,p1,p2,e
A5: q1 in P by A1, Def4;
A6: q2 in LSeg (q1,q2) by RLTOPSP1:68;
A7: q1 `1 = e by A1, Def4;
consider p4 being Point of (TOP-REAL 2) such that
A8: p4 `1 > e and
A9: LE q1,p4,P,p1,p2 and
A10: for p5 being Point of (TOP-REAL 2) st LE p5,p4,P,p1,p2 & LE q1,p5,P,p1,p2 holds
p5 `1 >= e by A1, Def4;
A11: P is_an_arc_of p1,p2 by A1, Def4;
A12: p4 in P by A9, JORDAN5C:def_3;
now__::_thesis:_(_(_LE_q2,q1,P,p1,p2_&_p_is_Rout_P,p1,p2,e_)_or_(_LE_q1,q2,P,p1,p2_&_p_is_Rout_P,p1,p2,e_)_)
percases ( LE q2,q1,P,p1,p2 or LE q1,q2,P,p1,p2 ) by A3, A11, A5, A6, Th19;
caseA13: LE q2,q1,P,p1,p2 ; ::_thesis: p is_Rout P,p1,p2,e
A14: now__::_thesis:_(_(_q1_<>_q2_&_Segment_(P,p1,p2,q2,q1)_=_LSeg_(q2,q1)_)_or_(_q1_=_q2_&_Segment_(P,p1,p2,q2,q1)_=_LSeg_(q2,q1)_)_)
percases ( q1 <> q2 or q1 = q2 ) ;
case q1 <> q2 ; ::_thesis: Segment (P,p1,p2,q2,q1) = LSeg (q2,q1)
then LSeg (q2,q1) is_an_arc_of q2,q1 by TOPREAL1:9;
hence Segment (P,p1,p2,q2,q1) = LSeg (q2,q1) by A3, A11, A13, Th25; ::_thesis: verum
end;
caseA15: q1 = q2 ; ::_thesis: Segment (P,p1,p2,q2,q1) = LSeg (q2,q1)
then LSeg (q1,q2) = {q1} by RLTOPSP1:70;
hence Segment (P,p1,p2,q2,q1) = LSeg (q2,q1) by A11, A5, A15, Th1; ::_thesis: verum
end;
end;
end;
Segment (P,p1,p2,q2,q1) = { p3 where p3 is Point of (TOP-REAL 2) : ( LE q2,p3,P,p1,p2 & LE p3,q1,P,p1,p2 ) } by JORDAN6:26;
then A16: ex p3 being Point of (TOP-REAL 2) st
( p = p3 & LE q2,p3,P,p1,p2 & LE p3,q1,P,p1,p2 ) by A4, A14;
then A17: LE p,p4,P,p1,p2 by A9, JORDAN5C:13;
A18: for p6 being Point of (TOP-REAL 2) st LE p6,p4,P,p1,p2 & LE p,p6,P,p1,p2 holds
p6 `1 >= e
proof
let p6 be Point of (TOP-REAL 2); ::_thesis: ( LE p6,p4,P,p1,p2 & LE p,p6,P,p1,p2 implies p6 `1 >= e )
assume that
A19: LE p6,p4,P,p1,p2 and
A20: LE p,p6,P,p1,p2 ; ::_thesis: p6 `1 >= e
A21: p6 in P by A19, JORDAN5C:def_3;
now__::_thesis:_(_(_LE_q1,p6,P,p1,p2_&_p6_`1_>=_e_)_or_(_LE_p6,q1,P,p1,p2_&_p6_`1_>=_e_)_)
percases ( LE q1,p6,P,p1,p2 or LE p6,q1,P,p1,p2 ) by A11, A5, A21, Th19;
case LE q1,p6,P,p1,p2 ; ::_thesis: p6 `1 >= e
hence p6 `1 >= e by A10, A19; ::_thesis: verum
end;
caseA22: LE p6,q1,P,p1,p2 ; ::_thesis: p6 `1 >= e
LE q2,p6,P,p1,p2 by A16, A20, JORDAN5C:13;
then p6 in { qq where qq is Point of (TOP-REAL 2) : ( LE q2,qq,P,p1,p2 & LE qq,q1,P,p1,p2 ) } by A22;
then p6 in LSeg (q2,q1) by A14, JORDAN6:26;
hence p6 `1 >= e by A2, A7, GOBOARD7:5; ::_thesis: verum
end;
end;
end;
hence p6 `1 >= e ; ::_thesis: verum
end;
p `1 = e by A2, A4, A7, GOBOARD7:5;
hence p is_Rout P,p1,p2,e by A3, A4, A11, A8, A17, A18, Def4; ::_thesis: verum
end;
caseA23: LE q1,q2,P,p1,p2 ; ::_thesis: p is_Rout P,p1,p2,e
A24: now__::_thesis:_(_(_q1_<>_q2_&_Segment_(P,p1,p2,q1,q2)_=_LSeg_(q1,q2)_)_or_(_q1_=_q2_&_Segment_(P,p1,p2,q1,q2)_=_LSeg_(q1,q2)_)_)
percases ( q1 <> q2 or q1 = q2 ) ;
case q1 <> q2 ; ::_thesis: Segment (P,p1,p2,q1,q2) = LSeg (q1,q2)
then LSeg (q1,q2) is_an_arc_of q1,q2 by TOPREAL1:9;
hence Segment (P,p1,p2,q1,q2) = LSeg (q1,q2) by A3, A11, A23, Th25; ::_thesis: verum
end;
caseA25: q1 = q2 ; ::_thesis: Segment (P,p1,p2,q1,q2) = LSeg (q1,q2)
then LSeg (q2,q1) = {q1} by RLTOPSP1:70;
hence Segment (P,p1,p2,q1,q2) = LSeg (q1,q2) by A11, A5, A25, Th1; ::_thesis: verum
end;
end;
end;
A26: now__::_thesis:_not_LE_p4,q2,P,p1,p2
assume LE p4,q2,P,p1,p2 ; ::_thesis: contradiction
then p4 in { qq where qq is Point of (TOP-REAL 2) : ( LE q1,qq,P,p1,p2 & LE qq,q2,P,p1,p2 ) } by A9;
then p4 in Segment (P,p1,p2,q1,q2) by JORDAN6:26;
hence contradiction by A2, A7, A8, A24, GOBOARD7:5; ::_thesis: verum
end;
Segment (P,p1,p2,q1,q2) = { p3 where p3 is Point of (TOP-REAL 2) : ( LE q1,p3,P,p1,p2 & LE p3,q2,P,p1,p2 ) } by JORDAN6:26;
then A27: ex p3 being Point of (TOP-REAL 2) st
( p = p3 & LE q1,p3,P,p1,p2 & LE p3,q2,P,p1,p2 ) by A4, A24;
A28: for p6 being Point of (TOP-REAL 2) st LE p6,p4,P,p1,p2 & LE p,p6,P,p1,p2 holds
p6 `1 >= e
proof
let p6 be Point of (TOP-REAL 2); ::_thesis: ( LE p6,p4,P,p1,p2 & LE p,p6,P,p1,p2 implies p6 `1 >= e )
assume that
A29: LE p6,p4,P,p1,p2 and
A30: LE p,p6,P,p1,p2 ; ::_thesis: p6 `1 >= e
LE q1,p6,P,p1,p2 by A27, A30, JORDAN5C:13;
hence p6 `1 >= e by A10, A29; ::_thesis: verum
end;
( LE q2,p4,P,p1,p2 or LE p4,q2,P,p1,p2 ) by A3, A11, A6, A12, Th19;
then A31: LE p,p4,P,p1,p2 by A27, A26, JORDAN5C:13;
p `1 = e by A2, A4, A7, GOBOARD7:5;
hence p is_Rout P,p1,p2,e by A3, A4, A11, A8, A31, A28, Def4; ::_thesis: verum
end;
end;
end;
hence p is_Rout P,p1,p2,e ; ::_thesis: verum
end;
theorem :: JORDAN20:30
for P being non empty Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2)
for e being Real st P is_S-P_arc_joining p1,p2 & p1 `1 < e & p in P & p `1 = e & not p is_Lin P,p1,p2,e holds
p is_Rin P,p1,p2,e
proof
let P be non empty Subset of (TOP-REAL 2); ::_thesis: for p1, p2, p being Point of (TOP-REAL 2)
for e being Real st P is_S-P_arc_joining p1,p2 & p1 `1 < e & p in P & p `1 = e & not p is_Lin P,p1,p2,e holds
p is_Rin P,p1,p2,e
let p1, p2, p be Point of (TOP-REAL 2); ::_thesis: for e being Real st P is_S-P_arc_joining p1,p2 & p1 `1 < e & p in P & p `1 = e & not p is_Lin P,p1,p2,e holds
p is_Rin P,p1,p2,e
let e be Real; ::_thesis: ( P is_S-P_arc_joining p1,p2 & p1 `1 < e & p in P & p `1 = e & not p is_Lin P,p1,p2,e implies p is_Rin P,p1,p2,e )
assume that
A1: P is_S-P_arc_joining p1,p2 and
A2: p1 `1 < e and
A3: p in P and
A4: p `1 = e ; ::_thesis: ( p is_Lin P,p1,p2,e or p is_Rin P,p1,p2,e )
consider f being FinSequence of (TOP-REAL 2) such that
A5: f is being_S-Seq and
A6: P = L~ f and
A7: p1 = f /. 1 and
A8: p2 = f /. (len f) by A1, TOPREAL4:def_1;
A9: P is_an_arc_of p1,p2 by A1, TOPREAL4:2;
len f >= 2 by A5, TOPREAL1:def_8;
then A10: len f > 1 by XXREAL_0:2;
A11: L~ f = union { (LSeg (f,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len f ) } by TOPREAL1:def_4;
then consider Y being set such that
A12: p in Y and
A13: Y in { (LSeg (f,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len f ) } by A3, A6, TARSKI:def_4;
consider i being Element of NAT such that
A14: Y = LSeg (f,i) and
A15: 1 <= i and
A16: i + 1 <= len f by A13;
A17: i - 1 >= 0 by A15, XREAL_1:48;
A18: 1 < i + 1 by A15, NAT_1:13;
A19: Y c= L~ f
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Y or x in L~ f )
assume x in Y ; ::_thesis: x in L~ f
hence x in L~ f by A11, A13, TARSKI:def_4; ::_thesis: verum
end;
defpred S1[ Nat] means for p being Point of (TOP-REAL 2) st p = f . (i -' $1) holds
p `1 <> e;
A20: i < len f by A16, NAT_1:13;
then A21: i in dom f by A15, FINSEQ_3:25;
A22: 1 < len f by A15, A20, XXREAL_0:2;
then 1 in dom f by FINSEQ_3:25;
then f /. 1 = f . 1 by PARTFUN1:def_6;
then A23: S1[i -' 1] by A2, A7, A15, NAT_D:58;
then A24: ex k being Nat st S1[k] ;
ex k being Nat st
( S1[k] & ( for n being Nat st S1[n] holds
k <= n ) ) from NAT_1:sch_5(A24);
then consider k being Nat such that
A25: S1[k] and
A26: for n being Nat st S1[n] holds
k <= n ;
k <= i -' 1 by A23, A26;
then k <= i - 1 by A17, XREAL_0:def_2;
then k + 1 <= (i - 1) + 1 by XREAL_1:7;
then A27: (1 + k) - k <= i - k by XREAL_1:9;
then A28: i -' k >= 1 by XREAL_0:def_2;
i -' k <= i by NAT_D:35;
then A29: i -' k < len f by A20, XXREAL_0:2;
then A30: i -' k in dom f by A28, FINSEQ_3:25;
then A31: f /. (i -' k) = f . (i -' k) by PARTFUN1:def_6;
then reconsider pk = f . (i -' k) as Point of (TOP-REAL 2) ;
A32: i -' k = i - k by A27, XREAL_0:def_2;
now__::_thesis:_(_(_pk_`1_<_e_&_(_p_is_Lin_P,p1,p2,e_or_p_is_Rin_P,p1,p2,e_)_)_or_(_pk_`1_>_e_&_(_p_is_Lin_P,p1,p2,e_or_p_is_Rin_P,p1,p2,e_)_)_)
percases ( pk `1 < e or pk `1 > e ) by A25, XXREAL_0:1;
caseA33: pk `1 < e ; ::_thesis: ( p is_Lin P,p1,p2,e or p is_Rin P,p1,p2,e )
now__::_thesis:_(_(_k_=_0_&_(_p_is_Lin_P,p1,p2,e_or_p_is_Rin_P,p1,p2,e_)_)_or_(_k_<>_0_&_(_p_is_Lin_P,p1,p2,e_or_p_is_Rin_P,p1,p2,e_)_)_)
percases ( k = 0 or k <> 0 ) ;
caseA34: k = 0 ; ::_thesis: ( p is_Lin P,p1,p2,e or p is_Rin P,p1,p2,e )
set p44 = f /. i;
A35: pk = f . i by A34, NAT_D:40
.= f /. i by A21, PARTFUN1:def_6 ;
reconsider ia = i + 1 as Element of NAT ;
reconsider g = mid (f,i,(len f)) as FinSequence of (TOP-REAL 2) ;
A36: i <= len f by A16, NAT_1:13;
ia in Seg (len f) by A16, A18, FINSEQ_1:1;
then A37: i + 1 in dom f by FINSEQ_1:def_3;
1 + (1 + i) <= 1 + (len f) by A16, XREAL_1:7;
then A38: ((1 + 1) + i) - i <= ((len f) + 1) - i by XREAL_1:9;
then A39: 1 <= ((len f) + 1) - i by XXREAL_0:2;
A40: (len f) - i > 0 by A20, XREAL_1:50;
then (len f) -' i = (len f) - i by XREAL_0:def_2;
then A41: ((len f) -' i) + 1 > 0 + 1 by A40, XREAL_1:8;
A42: len g = ((len f) -' i) + 1 by A10, A15, A20, FINSEQ_6:118;
then A43: 1 + 1 <= len g by A41, NAT_1:13;
then 1 + 1 in Seg (len g) by FINSEQ_1:1;
then 1 + 1 in dom g by FINSEQ_1:def_3;
then A44: g /. (1 + 1) = g . (1 + 1) by PARTFUN1:def_6
.= f . (((1 + 1) - 1) + i) by A15, A20, A38, FINSEQ_6:122
.= f /. (i + 1) by A37, PARTFUN1:def_6 ;
1 in dom g by A42, A41, FINSEQ_3:25;
then A45: g /. 1 = g . 1 by PARTFUN1:def_6
.= f . ((1 - 1) + i) by A15, A36, A39, FINSEQ_6:122
.= f /. i by A21, PARTFUN1:def_6 ;
LSeg (f,i) = LSeg ((f /. i),(f /. (i + 1))) by A15, A16, TOPREAL1:def_3
.= LSeg (g,1) by A43, A45, A44, TOPREAL1:def_3 ;
then Y in { (LSeg (g,i2)) where i2 is Element of NAT : ( 1 <= i2 & i2 + 1 <= len g ) } by A14, A43;
then p in union { (LSeg (g,i2)) where i2 is Element of NAT : ( 1 <= i2 & i2 + 1 <= len g ) } by A12, TARSKI:def_4;
then A46: p in L~ (mid (f,i,(len f))) by TOPREAL1:def_4;
A47: LSeg (f,i) = LSeg ((f /. i),(f /. (i + 1))) by A15, A16, TOPREAL1:def_3;
A48: for p5 being Point of (TOP-REAL 2) st LE f /. i,p5,P,p1,p2 & LE p5,p,P,p1,p2 holds
p5 `1 <= e
proof
f /. i in LSeg ((f /. i),(f /. (i + 1))) by RLTOPSP1:68;
then LSeg ((f /. i),p) c= LSeg (f,i) by A12, A14, A47, TOPREAL1:6;
then A49: LSeg ((f /. i),p) c= P by A6, A19, A14, XBOOLE_1:1;
let p5 be Point of (TOP-REAL 2); ::_thesis: ( LE f /. i,p5,P,p1,p2 & LE p5,p,P,p1,p2 implies p5 `1 <= e )
A50: Segment (P,p1,p2,(f /. i),p) = { p8 where p8 is Point of (TOP-REAL 2) : ( LE f /. i,p8,P,p1,p2 & LE p8,p,P,p1,p2 ) } by JORDAN6:26;
assume ( LE f /. i,p5,P,p1,p2 & LE p5,p,P,p1,p2 ) ; ::_thesis: p5 `1 <= e
then A51: p5 in Segment (P,p1,p2,(f /. i),p) by A50;
now__::_thesis:_(_(_f_/._i_<>_p_&_p5_`1_<=_e_)_or_(_f_/._i_=_p_&_p5_`1_<=_e_)_)
percases ( f /. i <> p or f /. i = p ) ;
case f /. i <> p ; ::_thesis: p5 `1 <= e
then LSeg ((f /. i),p) is_an_arc_of f /. i,p by TOPREAL1:9;
then Segment (P,p1,p2,(f /. i),p) = LSeg ((f /. i),p) by A9, A5, A6, A7, A8, A15, A20, A46, A49, Th25, SPRECT_4:3;
hence p5 `1 <= e by A4, A33, A35, A51, TOPREAL1:3; ::_thesis: verum
end;
case f /. i = p ; ::_thesis: p5 `1 <= e
hence p5 `1 <= e by A4, A33, A35; ::_thesis: verum
end;
end;
end;
hence p5 `1 <= e ; ::_thesis: verum
end;
LE f /. i,p,P,p1,p2 by A5, A6, A7, A8, A15, A20, A46, SPRECT_4:3;
hence ( p is_Lin P,p1,p2,e or p is_Rin P,p1,p2,e ) by A3, A4, A9, A33, A35, A48, Def1; ::_thesis: verum
end;
caseA52: k <> 0 ; ::_thesis: ( p is_Lin P,p1,p2,e or p is_Rin P,p1,p2,e )
reconsider ia = i + 1 as Element of NAT ;
reconsider g = mid (f,i,(len f)) as FinSequence of (TOP-REAL 2) ;
A53: i <= len f by A16, NAT_1:13;
ia in Seg (len f) by A16, A18, FINSEQ_1:1;
then A54: i + 1 in dom f by FINSEQ_1:def_3;
1 + (1 + i) <= 1 + (len f) by A16, XREAL_1:7;
then A55: ((1 + 1) + i) - i <= ((len f) + 1) - i by XREAL_1:9;
then A56: 1 <= ((len f) + 1) - i by XXREAL_0:2;
A57: (len f) - i > 0 by A20, XREAL_1:50;
then (len f) -' i = (len f) - i by XREAL_0:def_2;
then A58: ((len f) -' i) + 1 > 0 + 1 by A57, XREAL_1:8;
A59: len g = ((len f) -' i) + 1 by A10, A15, A20, FINSEQ_6:118;
then A60: 1 + 1 <= len g by A58, NAT_1:13;
then 1 + 1 in Seg (len g) by FINSEQ_1:1;
then 1 + 1 in dom g by FINSEQ_1:def_3;
then A61: g /. (1 + 1) = g . (1 + 1) by PARTFUN1:def_6
.= f . (((1 + 1) - 1) + i) by A15, A20, A55, FINSEQ_6:122
.= f /. (i + 1) by A54, PARTFUN1:def_6 ;
1 in dom g by A59, A58, FINSEQ_3:25;
then A62: g /. 1 = g . 1 by PARTFUN1:def_6
.= f . ((1 - 1) + i) by A15, A53, A56, FINSEQ_6:122
.= f /. i by A21, PARTFUN1:def_6 ;
LSeg (f,i) = LSeg ((f /. i),(f /. (i + 1))) by A15, A16, TOPREAL1:def_3
.= LSeg (g,1) by A60, A62, A61, TOPREAL1:def_3 ;
then Y in { (LSeg (g,i2)) where i2 is Element of NAT : ( 1 <= i2 & i2 + 1 <= len g ) } by A14, A60;
then p in union { (LSeg (g,i2)) where i2 is Element of NAT : ( 1 <= i2 & i2 + 1 <= len g ) } by A12, TARSKI:def_4;
then A63: p in L~ (mid (f,i,(len f))) by TOPREAL1:def_4;
reconsider g = mid (f,1,i) as FinSequence of (TOP-REAL 2) ;
set p44 = f /. i;
A64: ( i <= len f & 1 <= i -' k ) by A16, A27, NAT_1:13, XREAL_0:def_2;
A65: k >= 0 + 1 by A52, NAT_1:13;
then A66: i -' k <= (i + 1) - 1 by A28, NAT_D:51;
A67: i > i -' k by A28, A65, NAT_D:51;
then A68: i > 1 by A28, XXREAL_0:2;
then i - 1 > 0 by XREAL_1:50;
then A69: i -' 1 = i - 1 by XREAL_0:def_2;
A70: now__::_thesis:_not_(f_/._i)_`1_<>_e
assume A71: (f /. i) `1 <> e ; ::_thesis: contradiction
f . i = f /. i by A21, PARTFUN1:def_6;
then for p9 being Point of (TOP-REAL 2) st p9 = f . (i -' 0) holds
p9 `1 <> e by A71, NAT_D:40;
hence contradiction by A26, A52; ::_thesis: verum
end;
A72: now__::_thesis:_for_p51_being_Point_of_(TOP-REAL_2)_st_LE_pk,p51,P,p1,p2_&_LE_p51,f_/._i,P,p1,p2_holds_
p51_`1_<=_e
assume ex p51 being Point of (TOP-REAL 2) st
( LE pk,p51,P,p1,p2 & LE p51,f /. i,P,p1,p2 & not p51 `1 <= e ) ; ::_thesis: contradiction
then consider p51 being Point of (TOP-REAL 2) such that
A73: LE pk,p51,P,p1,p2 and
A74: LE p51,f /. i,P,p1,p2 and
A75: p51 `1 > e ;
p51 in P by A73, JORDAN5C:def_3;
then consider Y3 being set such that
A76: p51 in Y3 and
A77: Y3 in { (LSeg (f,i5)) where i5 is Element of NAT : ( 1 <= i5 & i5 + 1 <= len f ) } by A6, A11, TARSKI:def_4;
consider kk being Element of NAT such that
A78: Y3 = LSeg (f,kk) and
A79: 1 <= kk and
A80: kk + 1 <= len f by A77;
A81: LSeg (f,kk) = LSeg ((f /. kk),(f /. (kk + 1))) by A79, A80, TOPREAL1:def_3;
1 < kk + 1 by A79, NAT_1:13;
then kk + 1 in Seg (len f) by A80, FINSEQ_1:1;
then A82: kk + 1 in dom f by FINSEQ_1:def_3;
A83: kk < len f by A80, NAT_1:13;
then kk in Seg (len f) by A79, FINSEQ_1:1;
then A84: kk in dom f by FINSEQ_1:def_3;
A85: LE p51,f /. (kk + 1), L~ f,f /. 1,f /. (len f) by A5, A76, A78, A79, A80, JORDAN5C:26;
now__::_thesis:_(_(_(f_/._kk)_`1_>_e_&_contradiction_)_or_(_(f_/._(kk_+_1))_`1_>_e_&_(f_/._kk)_`1_<=_e_&_contradiction_)_)
percases ( (f /. kk) `1 > e or ( (f /. (kk + 1)) `1 > e & (f /. kk) `1 <= e ) ) by A75, A76, A78, A81, Th2;
caseA86: (f /. kk) `1 > e ; ::_thesis: contradiction
A87: LSeg ((f /. kk),(f /. (kk + 1))) c= L~ f
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in LSeg ((f /. kk),(f /. (kk + 1))) or z in L~ f )
assume A88: z in LSeg ((f /. kk),(f /. (kk + 1))) ; ::_thesis: z in L~ f
LSeg ((f /. kk),(f /. (kk + 1))) in { (LSeg (f,i7)) where i7 is Element of NAT : ( 1 <= i7 & i7 + 1 <= len f ) } by A79, A80, A81;
hence z in L~ f by A11, A88, TARSKI:def_4; ::_thesis: verum
end;
f is special by A5, TOPREAL1:def_8;
then A89: ( (f /. kk) `1 = (f /. (kk + 1)) `1 or (f /. kk) `2 = (f /. (kk + 1)) `2 ) by A79, A80, TOPREAL1:def_5;
( f is one-to-one & kk < kk + 1 ) by A5, NAT_1:13, TOPREAL1:def_8;
then A90: f . kk <> f . (kk + 1) by A84, A82, FUNCT_1:def_4;
A91: LE f /. (i -' k),p51, L~ f,f /. 1,f /. (len f) by A6, A7, A8, A30, A73, PARTFUN1:def_6;
A92: LE f /. (i -' k),f /. (kk + 1), L~ f,f /. 1,f /. (len f) by A6, A7, A8, A31, A73, A85, JORDAN5C:13;
set k2 = i -' kk;
LE f /. kk,p51, L~ f,f /. 1,f /. (len f) by A5, A76, A78, A79, A80, JORDAN5C:25;
then A93: LE f /. kk,f /. i, L~ f,f /. 1,f /. (len f) by A6, A7, A8, A74, JORDAN5C:13;
now__::_thesis:_not_i_-_kk_<=_0
assume i - kk <= 0 ; ::_thesis: contradiction
then (i - kk) + kk <= 0 + kk by XREAL_1:7;
then LE f /. i,f /. kk, L~ f,f /. 1,f /. (len f) by A5, A68, A83, JORDAN5C:24;
hence contradiction by A1, A6, A7, A8, A70, A86, A93, JORDAN5C:12, TOPREAL4:2; ::_thesis: verum
end;
then A94: i -' kk = i - kk by XREAL_0:def_2;
then A95: i - (i -' kk) = i -' (i -' kk) by XREAL_0:def_2;
i - (i -' kk) > 0 by A79, A94;
then i -' (i -' kk) > 0 by XREAL_0:def_2;
then i -' (i -' kk) >= 0 + 1 by NAT_1:13;
then S1[i -' kk] by A20, A86, A94, A95, FINSEQ_4:15, NAT_D:50;
then i -' kk >= k by A26;
then i - (i -' kk) <= i - k by XREAL_1:10;
then A96: LE f /. (i -' (i -' kk)),f /. (i -' k), L~ f,f /. 1,f /. (len f) by A5, A29, A32, A79, A94, A95, JORDAN5C:24;
( f /. kk = f . kk & f /. (kk + 1) = f . (kk + 1) ) by A84, A82, PARTFUN1:def_6;
then LSeg ((f /. kk),(f /. (kk + 1))) is_an_arc_of f /. kk,f /. (kk + 1) by A90, TOPREAL1:9;
then A97: Segment ((L~ f),(f /. 1),(f /. (len f)),(f /. kk),(f /. (kk + 1))) = LSeg ((f /. kk),(f /. (kk + 1))) by A9, A6, A7, A8, A94, A95, A96, A92, A87, Th25, JORDAN5C:13;
Segment ((L~ f),(f /. 1),(f /. (len f)),(f /. kk),(f /. (kk + 1))) = { p8 where p8 is Point of (TOP-REAL 2) : ( LE f /. kk,p8, L~ f,f /. 1,f /. (len f) & LE p8,f /. (kk + 1), L~ f,f /. 1,f /. (len f) ) } by JORDAN6:26;
then A98: f /. (i -' k) in Segment ((L~ f),(f /. 1),(f /. (len f)),(f /. kk),(f /. (kk + 1))) by A94, A95, A96, A92;
then (f /. (kk + 1)) `1 < e by A31, A33, A86, A97, Th3;
then (f /. kk) `1 > (f /. (kk + 1)) `1 by A86, XXREAL_0:2;
then (f /. (i -' k)) `1 >= p51 `1 by A5, A76, A78, A79, A83, A81, A91, A98, A97, A89, Th6;
hence contradiction by A31, A33, A75, XXREAL_0:2; ::_thesis: verum
end;
caseA99: ( (f /. (kk + 1)) `1 > e & (f /. kk) `1 <= e ) ; ::_thesis: contradiction
set k2 = (i -' kk) -' 1;
A100: LSeg ((f /. kk),(f /. (kk + 1))) c= L~ f
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in LSeg ((f /. kk),(f /. (kk + 1))) or z in L~ f )
assume A101: z in LSeg ((f /. kk),(f /. (kk + 1))) ; ::_thesis: z in L~ f
LSeg ((f /. kk),(f /. (kk + 1))) in { (LSeg (f,i7)) where i7 is Element of NAT : ( 1 <= i7 & i7 + 1 <= len f ) } by A79, A80, A81;
hence z in L~ f by A11, A101, TARSKI:def_4; ::_thesis: verum
end;
( f is one-to-one & kk < kk + 1 ) by A5, NAT_1:13, TOPREAL1:def_8;
then A102: f . kk <> f . (kk + 1) by A84, A82, FUNCT_1:def_4;
A103: (f /. kk) `1 < (f /. (kk + 1)) `1 by A99, XXREAL_0:2;
LE f /. kk,p51, L~ f,f /. 1,f /. (len f) by A5, A76, A78, A79, A80, JORDAN5C:25;
then A104: LE f /. kk,f /. i, L~ f,f /. 1,f /. (len f) by A6, A7, A8, A74, JORDAN5C:13;
( f /. kk = f . kk & f /. (kk + 1) = f . (kk + 1) ) by A84, A82, PARTFUN1:def_6;
then LSeg ((f /. kk),(f /. (kk + 1))) is_an_arc_of f /. kk,f /. (kk + 1) by A102, TOPREAL1:9;
then A105: ( Segment ((L~ f),(f /. 1),(f /. (len f)),(f /. kk),(f /. (kk + 1))) = { p8 where p8 is Point of (TOP-REAL 2) : ( LE f /. kk,p8, L~ f,f /. 1,f /. (len f) & LE p8,f /. (kk + 1), L~ f,f /. 1,f /. (len f) ) } & Segment ((L~ f),(f /. 1),(f /. (len f)),(f /. kk),(f /. (kk + 1))) = LSeg ((f /. kk),(f /. (kk + 1))) ) by A9, A5, A6, A7, A8, A79, A80, A100, Th25, JORDAN5C:23, JORDAN6:26;
A106: now__::_thesis:_not_(i_-_kk)_-_1_<=_0
assume (i - kk) - 1 <= 0 ; ::_thesis: contradiction
then (i - (kk + 1)) + (kk + 1) <= 0 + (kk + 1) by XREAL_1:7;
then LE f /. i,f /. (kk + 1), L~ f,f /. 1,f /. (len f) by A5, A68, A80, JORDAN5C:24;
then A107: f /. i in LSeg ((f /. kk),(f /. (kk + 1))) by A105, A104;
f is special by A5, TOPREAL1:def_8;
then A108: ( (f /. kk) `1 = (f /. (kk + 1)) `1 or (f /. kk) `2 = (f /. (kk + 1)) `2 ) by A79, A80, TOPREAL1:def_5;
LSeg (f,kk) = LSeg ((f /. kk),(f /. (kk + 1))) by A79, A80, TOPREAL1:def_3;
hence contradiction by A5, A6, A7, A8, A70, A74, A75, A76, A78, A79, A83, A103, A107, A108, Th7; ::_thesis: verum
end;
then ((i - kk) - 1) + 1 >= 0 + 1 by XREAL_1:7;
then i -' kk = i - kk by XREAL_0:def_2;
then A109: i - ((i -' kk) -' 1) = i - ((i - kk) - 1) by A106, XREAL_0:def_2
.= kk + 1 ;
then i -' ((i -' kk) -' 1) > 0 by XREAL_0:def_2;
then A110: i -' ((i -' kk) -' 1) >= 0 + 1 by NAT_1:13;
A111: i - ((i -' kk) -' 1) = i -' ((i -' kk) -' 1) by A109, XREAL_0:def_2;
then S1[(i -' kk) -' 1] by A20, A99, A109, A110, FINSEQ_4:15, NAT_D:50;
then (i -' kk) -' 1 >= k by A26;
then i - ((i -' kk) -' 1) <= i - k by XREAL_1:10;
then A112: LE f /. (kk + 1),f /. (i -' k), L~ f,f /. 1,f /. (len f) by A5, A29, A32, A109, A111, A110, JORDAN5C:24;
LE f /. (i -' k),f /. (kk + 1), L~ f,f /. 1,f /. (len f) by A6, A7, A8, A31, A73, A85, JORDAN5C:13;
hence contradiction by A1, A6, A7, A8, A31, A33, A99, A112, JORDAN5C:12, TOPREAL4:2; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
A113: LSeg (f,i) = LSeg ((f /. i),(f /. (i + 1))) by A15, A16, TOPREAL1:def_3;
A114: for p5 being Point of (TOP-REAL 2) st LE f /. i,p5,P,p1,p2 & LE p5,p,P,p1,p2 holds
p5 `1 <= e
proof
let p5 be Point of (TOP-REAL 2); ::_thesis: ( LE f /. i,p5,P,p1,p2 & LE p5,p,P,p1,p2 implies p5 `1 <= e )
A115: Segment (P,p1,p2,(f /. i),p) = { p8 where p8 is Point of (TOP-REAL 2) : ( LE f /. i,p8,P,p1,p2 & LE p8,p,P,p1,p2 ) } by JORDAN6:26;
assume ( LE f /. i,p5,P,p1,p2 & LE p5,p,P,p1,p2 ) ; ::_thesis: p5 `1 <= e
then A116: p5 in Segment (P,p1,p2,(f /. i),p) by A115;
f /. i in LSeg ((f /. i),(f /. (i + 1))) by RLTOPSP1:68;
then LSeg ((f /. i),p) c= LSeg (f,i) by A12, A14, A113, TOPREAL1:6;
then A117: LSeg ((f /. i),p) c= P by A6, A19, A14, XBOOLE_1:1;
now__::_thesis:_(_(_f_/._i_<>_p_&_p5_`1_<=_e_)_or_(_f_/._i_=_p_&_p5_`1_<=_e_)_)
percases ( f /. i <> p or f /. i = p ) ;
case f /. i <> p ; ::_thesis: p5 `1 <= e
then LSeg ((f /. i),p) is_an_arc_of f /. i,p by TOPREAL1:9;
then Segment (P,p1,p2,(f /. i),p) = LSeg ((f /. i),p) by A9, A5, A6, A7, A8, A15, A20, A63, A117, Th25, SPRECT_4:3;
hence p5 `1 <= e by A4, A70, A116, TOPREAL1:3; ::_thesis: verum
end;
case f /. i = p ; ::_thesis: p5 `1 <= e
then Segment (P,p1,p2,(f /. i),p) = {(f /. i)} by A1, A3, Th1, TOPREAL4:2;
hence p5 `1 <= e by A70, A116, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
hence p5 `1 <= e ; ::_thesis: verum
end;
A118: len g = (i -' 1) + 1 by A15, A20, A22, FINSEQ_6:118;
then (i -' k) + 1 <= len g by A67, A69, NAT_1:13;
then A119: LSeg (g,(i -' k)) = LSeg ((g /. (i -' k)),(g /. ((i -' k) + 1))) by A28, TOPREAL1:def_3;
i -' k < i by A28, A65, NAT_D:51;
then i -' k in Seg (len g) by A28, A118, A69, FINSEQ_1:1;
then i -' k in dom g by FINSEQ_1:def_3;
then g /. (i -' k) = g . (i -' k) by PARTFUN1:def_6
.= f . (((i -' k) - 1) + 1) by A15, A64, A66, FINSEQ_6:122
.= f /. (i -' k) by A30, PARTFUN1:def_6 ;
then A120: pk in LSeg (g,(i -' k)) by A31, A119, RLTOPSP1:68;
A121: (i -' k) + 1 <= i by A67, NAT_1:13;
1 <= i -' k by A27, XREAL_0:def_2;
then LSeg (g,(i -' k)) in { (LSeg (g,i2)) where i2 is Element of NAT : ( 1 <= i2 & i2 + 1 <= len g ) } by A118, A69, A121;
then pk in union { (LSeg (g,i2)) where i2 is Element of NAT : ( 1 <= i2 & i2 + 1 <= len g ) } by A120, TARSKI:def_4;
then pk in L~ (mid (f,1,i)) by TOPREAL1:def_4;
then A122: LE pk,f /. i,P,p1,p2 by A5, A6, A7, A8, A20, A68, SPRECT_3:17;
then A123: f /. i in P by JORDAN5C:def_3;
A124: for p5 being Point of (TOP-REAL 2) st LE pk,p5,P,p1,p2 & LE p5,p,P,p1,p2 holds
p5 `1 <= e
proof
let p5 be Point of (TOP-REAL 2); ::_thesis: ( LE pk,p5,P,p1,p2 & LE p5,p,P,p1,p2 implies p5 `1 <= e )
assume that
A125: LE pk,p5,P,p1,p2 and
A126: LE p5,p,P,p1,p2 ; ::_thesis: p5 `1 <= e
A127: p5 in P by A125, JORDAN5C:def_3;
now__::_thesis:_(_(_LE_p5,f_/._i,P,p1,p2_&_p5_`1_<=_e_)_or_(_LE_f_/._i,p5,P,p1,p2_&_p5_`1_<=_e_)_)
percases ( LE p5,f /. i,P,p1,p2 or LE f /. i,p5,P,p1,p2 ) by A1, A123, A127, Th19, TOPREAL4:2;
case LE p5,f /. i,P,p1,p2 ; ::_thesis: p5 `1 <= e
hence p5 `1 <= e by A72, A125; ::_thesis: verum
end;
case LE f /. i,p5,P,p1,p2 ; ::_thesis: p5 `1 <= e
hence p5 `1 <= e by A114, A126; ::_thesis: verum
end;
end;
end;
hence p5 `1 <= e ; ::_thesis: verum
end;
LE f /. i,p,P,p1,p2 by A5, A6, A7, A8, A15, A20, A63, SPRECT_4:3;
then LE pk,p,P,p1,p2 by A122, JORDAN5C:13;
hence ( p is_Lin P,p1,p2,e or p is_Rin P,p1,p2,e ) by A3, A4, A9, A33, A124, Def1; ::_thesis: verum
end;
end;
end;
hence ( p is_Lin P,p1,p2,e or p is_Rin P,p1,p2,e ) ; ::_thesis: verum
end;
caseA128: pk `1 > e ; ::_thesis: ( p is_Lin P,p1,p2,e or p is_Rin P,p1,p2,e )
now__::_thesis:_(_(_k_=_0_&_(_p_is_Lin_P,p1,p2,e_or_p_is_Rin_P,p1,p2,e_)_)_or_(_k_<>_0_&_(_p_is_Lin_P,p1,p2,e_or_p_is_Rin_P,p1,p2,e_)_)_)
percases ( k = 0 or k <> 0 ) ;
caseA129: k = 0 ; ::_thesis: ( p is_Lin P,p1,p2,e or p is_Rin P,p1,p2,e )
set p44 = f /. i;
A130: pk = f . i by A129, NAT_D:40
.= f /. i by A21, PARTFUN1:def_6 ;
reconsider ia = i + 1 as Element of NAT ;
reconsider g = mid (f,i,(len f)) as FinSequence of (TOP-REAL 2) ;
A131: i <= len f by A16, NAT_1:13;
ia in Seg (len f) by A16, A18, FINSEQ_1:1;
then A132: i + 1 in dom f by FINSEQ_1:def_3;
1 + (1 + i) <= 1 + (len f) by A16, XREAL_1:7;
then A133: ((1 + 1) + i) - i <= ((len f) + 1) - i by XREAL_1:9;
then A134: 1 <= ((len f) + 1) - i by XXREAL_0:2;
A135: (len f) - i > 0 by A20, XREAL_1:50;
then (len f) -' i = (len f) - i by XREAL_0:def_2;
then A136: ((len f) -' i) + 1 > 0 + 1 by A135, XREAL_1:8;
A137: len g = ((len f) -' i) + 1 by A10, A15, A20, FINSEQ_6:118;
then A138: 1 + 1 <= len g by A136, NAT_1:13;
then 1 + 1 in Seg (len g) by FINSEQ_1:1;
then 1 + 1 in dom g by FINSEQ_1:def_3;
then A139: g /. (1 + 1) = g . (1 + 1) by PARTFUN1:def_6
.= f . (((1 + 1) - 1) + i) by A15, A20, A133, FINSEQ_6:122
.= f /. (i + 1) by A132, PARTFUN1:def_6 ;
1 in Seg (len g) by A137, A136, FINSEQ_1:1;
then 1 in dom g by FINSEQ_1:def_3;
then A140: g /. 1 = g . 1 by PARTFUN1:def_6
.= f . ((1 - 1) + i) by A15, A131, A134, FINSEQ_6:122
.= f /. i by A21, PARTFUN1:def_6 ;
LSeg (f,i) = LSeg ((f /. i),(f /. (i + 1))) by A15, A16, TOPREAL1:def_3
.= LSeg (g,1) by A138, A140, A139, TOPREAL1:def_3 ;
then Y in { (LSeg (g,i2)) where i2 is Element of NAT : ( 1 <= i2 & i2 + 1 <= len g ) } by A14, A138;
then p in union { (LSeg (g,i2)) where i2 is Element of NAT : ( 1 <= i2 & i2 + 1 <= len g ) } by A12, TARSKI:def_4;
then A141: p in L~ (mid (f,i,(len f))) by TOPREAL1:def_4;
A142: LSeg (f,i) = LSeg ((f /. i),(f /. (i + 1))) by A15, A16, TOPREAL1:def_3;
A143: for p5 being Point of (TOP-REAL 2) st LE f /. i,p5,P,p1,p2 & LE p5,p,P,p1,p2 holds
p5 `1 >= e
proof
f /. i in LSeg ((f /. i),(f /. (i + 1))) by RLTOPSP1:68;
then LSeg ((f /. i),p) c= LSeg (f,i) by A12, A14, A142, TOPREAL1:6;
then A144: LSeg ((f /. i),p) c= P by A6, A19, A14, XBOOLE_1:1;
let p5 be Point of (TOP-REAL 2); ::_thesis: ( LE f /. i,p5,P,p1,p2 & LE p5,p,P,p1,p2 implies p5 `1 >= e )
A145: Segment (P,p1,p2,(f /. i),p) = { p8 where p8 is Point of (TOP-REAL 2) : ( LE f /. i,p8,P,p1,p2 & LE p8,p,P,p1,p2 ) } by JORDAN6:26;
assume ( LE f /. i,p5,P,p1,p2 & LE p5,p,P,p1,p2 ) ; ::_thesis: p5 `1 >= e
then A146: p5 in Segment (P,p1,p2,(f /. i),p) by A145;
now__::_thesis:_(_(_f_/._i_<>_p_&_p5_`1_>=_e_)_or_(_f_/._i_=_p_&_p5_`1_>=_e_)_)
percases ( f /. i <> p or f /. i = p ) ;
case f /. i <> p ; ::_thesis: p5 `1 >= e
then LSeg ((f /. i),p) is_an_arc_of f /. i,p by TOPREAL1:9;
then Segment (P,p1,p2,(f /. i),p) = LSeg ((f /. i),p) by A9, A5, A6, A7, A8, A15, A20, A141, A144, Th25, SPRECT_4:3;
hence p5 `1 >= e by A4, A128, A130, A146, TOPREAL1:3; ::_thesis: verum
end;
case f /. i = p ; ::_thesis: p5 `1 >= e
hence p5 `1 >= e by A4, A128, A130; ::_thesis: verum
end;
end;
end;
hence p5 `1 >= e ; ::_thesis: verum
end;
LE f /. i,p,P,p1,p2 by A5, A6, A7, A8, A15, A20, A141, SPRECT_4:3;
hence ( p is_Lin P,p1,p2,e or p is_Rin P,p1,p2,e ) by A3, A4, A9, A128, A130, A143, Def2; ::_thesis: verum
end;
caseA147: k <> 0 ; ::_thesis: ( p is_Lin P,p1,p2,e or p is_Rin P,p1,p2,e )
reconsider ia = i + 1 as Element of NAT ;
reconsider g = mid (f,i,(len f)) as FinSequence of (TOP-REAL 2) ;
A148: i <= len f by A16, NAT_1:13;
ia in Seg (len f) by A16, A18, FINSEQ_1:1;
then A149: i + 1 in dom f by FINSEQ_1:def_3;
1 + (1 + i) <= 1 + (len f) by A16, XREAL_1:7;
then A150: ((1 + 1) + i) - i <= ((len f) + 1) - i by XREAL_1:9;
then A151: 1 <= ((len f) + 1) - i by XXREAL_0:2;
A152: (len f) - i > 0 by A20, XREAL_1:50;
then (len f) -' i = (len f) - i by XREAL_0:def_2;
then A153: ((len f) -' i) + 1 > 0 + 1 by A152, XREAL_1:8;
A154: len g = ((len f) -' i) + 1 by A10, A15, A20, FINSEQ_6:118;
then A155: 1 + 1 <= len g by A153, NAT_1:13;
then 1 + 1 in Seg (len g) by FINSEQ_1:1;
then 1 + 1 in dom g by FINSEQ_1:def_3;
then A156: g /. (1 + 1) = g . (1 + 1) by PARTFUN1:def_6
.= f . (((1 + 1) - 1) + i) by A15, A20, A150, FINSEQ_6:122
.= f /. (i + 1) by A149, PARTFUN1:def_6 ;
1 in Seg (len g) by A154, A153, FINSEQ_1:1;
then 1 in dom g by FINSEQ_1:def_3;
then A157: g /. 1 = g . 1 by PARTFUN1:def_6
.= f . ((1 - 1) + i) by A15, A148, A151, FINSEQ_6:122
.= f /. i by A21, PARTFUN1:def_6 ;
LSeg (f,i) = LSeg ((f /. i),(f /. (i + 1))) by A15, A16, TOPREAL1:def_3
.= LSeg (g,1) by A155, A157, A156, TOPREAL1:def_3 ;
then Y in { (LSeg (g,i2)) where i2 is Element of NAT : ( 1 <= i2 & i2 + 1 <= len g ) } by A14, A155;
then p in union { (LSeg (g,i2)) where i2 is Element of NAT : ( 1 <= i2 & i2 + 1 <= len g ) } by A12, TARSKI:def_4;
then A158: p in L~ (mid (f,i,(len f))) by TOPREAL1:def_4;
reconsider g = mid (f,1,i) as FinSequence of (TOP-REAL 2) ;
set p44 = f /. i;
A159: ( i <= len f & 1 <= i -' k ) by A16, A27, NAT_1:13, XREAL_0:def_2;
A160: k >= 0 + 1 by A147, NAT_1:13;
then A161: i -' k <= (i + 1) - 1 by A28, NAT_D:51;
A162: i > i -' k by A28, A160, NAT_D:51;
then A163: i > 1 by A28, XXREAL_0:2;
then i - 1 > 0 by XREAL_1:50;
then A164: i -' 1 = i - 1 by XREAL_0:def_2;
A165: now__::_thesis:_not_(f_/._i)_`1_<>_e
assume A166: (f /. i) `1 <> e ; ::_thesis: contradiction
f . i = f /. i by A21, PARTFUN1:def_6;
then for p9 being Point of (TOP-REAL 2) st p9 = f . (i -' 0) holds
p9 `1 <> e by A166, NAT_D:40;
hence contradiction by A26, A147; ::_thesis: verum
end;
A167: now__::_thesis:_for_p51_being_Point_of_(TOP-REAL_2)_st_LE_pk,p51,P,p1,p2_&_LE_p51,f_/._i,P,p1,p2_holds_
p51_`1_>=_e
assume ex p51 being Point of (TOP-REAL 2) st
( LE pk,p51,P,p1,p2 & LE p51,f /. i,P,p1,p2 & not p51 `1 >= e ) ; ::_thesis: contradiction
then consider p51 being Point of (TOP-REAL 2) such that
A168: LE pk,p51,P,p1,p2 and
A169: LE p51,f /. i,P,p1,p2 and
A170: p51 `1 < e ;
p51 in P by A168, JORDAN5C:def_3;
then consider Y3 being set such that
A171: p51 in Y3 and
A172: Y3 in { (LSeg (f,i5)) where i5 is Element of NAT : ( 1 <= i5 & i5 + 1 <= len f ) } by A6, A11, TARSKI:def_4;
consider kk being Element of NAT such that
A173: Y3 = LSeg (f,kk) and
A174: 1 <= kk and
A175: kk + 1 <= len f by A172;
A176: LSeg (f,kk) = LSeg ((f /. kk),(f /. (kk + 1))) by A174, A175, TOPREAL1:def_3;
1 < kk + 1 by A174, NAT_1:13;
then kk + 1 in Seg (len f) by A175, FINSEQ_1:1;
then A177: kk + 1 in dom f by FINSEQ_1:def_3;
A178: kk < len f by A175, NAT_1:13;
then kk in Seg (len f) by A174, FINSEQ_1:1;
then A179: kk in dom f by FINSEQ_1:def_3;
A180: LE p51,f /. (kk + 1), L~ f,f /. 1,f /. (len f) by A5, A171, A173, A174, A175, JORDAN5C:26;
now__::_thesis:_(_(_(f_/._kk)_`1_<_e_&_contradiction_)_or_(_(f_/._(kk_+_1))_`1_<_e_&_(f_/._kk)_`1_>=_e_&_contradiction_)_)
percases ( (f /. kk) `1 < e or ( (f /. (kk + 1)) `1 < e & (f /. kk) `1 >= e ) ) by A170, A171, A173, A176, Th3;
caseA181: (f /. kk) `1 < e ; ::_thesis: contradiction
set k2 = i -' kk;
LE f /. kk,p51, L~ f,f /. 1,f /. (len f) by A5, A171, A173, A174, A175, JORDAN5C:25;
then A182: LE f /. kk,f /. i, L~ f,f /. 1,f /. (len f) by A6, A7, A8, A169, JORDAN5C:13;
now__::_thesis:_not_i_-_kk_<=_0
assume i - kk <= 0 ; ::_thesis: contradiction
then (i - kk) + kk <= 0 + kk by XREAL_1:7;
then LE f /. i,f /. kk, L~ f,f /. 1,f /. (len f) by A5, A163, A178, JORDAN5C:24;
hence contradiction by A1, A6, A7, A8, A165, A181, A182, JORDAN5C:12, TOPREAL4:2; ::_thesis: verum
end;
then A183: i - (i -' kk) = i - (i - kk) by XREAL_0:def_2
.= kk ;
then A184: i - (i -' kk) = i -' (i -' kk) by XREAL_0:def_2;
then S1[i -' kk] by A20, A174, A181, A183, FINSEQ_4:15, NAT_D:50;
then i -' kk >= k by A26;
then i - (i -' kk) <= i - k by XREAL_1:10;
then A185: LE f /. (i -' (i -' kk)),f /. (i -' k), L~ f,f /. 1,f /. (len f) by A5, A29, A32, A174, A183, A184, JORDAN5C:24;
A186: LSeg ((f /. kk),(f /. (kk + 1))) c= L~ f
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in LSeg ((f /. kk),(f /. (kk + 1))) or z in L~ f )
assume A187: z in LSeg ((f /. kk),(f /. (kk + 1))) ; ::_thesis: z in L~ f
LSeg ((f /. kk),(f /. (kk + 1))) in { (LSeg (f,i7)) where i7 is Element of NAT : ( 1 <= i7 & i7 + 1 <= len f ) } by A174, A175, A176;
hence z in L~ f by A11, A187, TARSKI:def_4; ::_thesis: verum
end;
f is special by A5, TOPREAL1:def_8;
then A188: ( (f /. kk) `1 = (f /. (kk + 1)) `1 or (f /. kk) `2 = (f /. (kk + 1)) `2 ) by A174, A175, TOPREAL1:def_5;
( f is one-to-one & kk < kk + 1 ) by A5, NAT_1:13, TOPREAL1:def_8;
then A189: f . kk <> f . (kk + 1) by A179, A177, FUNCT_1:def_4;
A190: LE f /. (i -' k),p51, L~ f,f /. 1,f /. (len f) by A6, A7, A8, A30, A168, PARTFUN1:def_6;
A191: LE f /. (i -' k),f /. (kk + 1), L~ f,f /. 1,f /. (len f) by A6, A7, A8, A31, A168, A180, JORDAN5C:13;
( f /. kk = f . kk & f /. (kk + 1) = f . (kk + 1) ) by A179, A177, PARTFUN1:def_6;
then LSeg ((f /. kk),(f /. (kk + 1))) is_an_arc_of f /. kk,f /. (kk + 1) by A189, TOPREAL1:9;
then A192: Segment ((L~ f),(f /. 1),(f /. (len f)),(f /. kk),(f /. (kk + 1))) = LSeg ((f /. kk),(f /. (kk + 1))) by A9, A6, A7, A8, A183, A184, A185, A191, A186, Th25, JORDAN5C:13;
Segment ((L~ f),(f /. 1),(f /. (len f)),(f /. kk),(f /. (kk + 1))) = { p8 where p8 is Point of (TOP-REAL 2) : ( LE f /. kk,p8, L~ f,f /. 1,f /. (len f) & LE p8,f /. (kk + 1), L~ f,f /. 1,f /. (len f) ) } by JORDAN6:26;
then A193: f /. (i -' k) in Segment ((L~ f),(f /. 1),(f /. (len f)),(f /. kk),(f /. (kk + 1))) by A183, A184, A185, A191;
then (f /. (kk + 1)) `1 > e by A31, A128, A181, A192, Th2;
then (f /. kk) `1 < (f /. (kk + 1)) `1 by A181, XXREAL_0:2;
then (f /. (i -' k)) `1 <= p51 `1 by A5, A171, A173, A174, A178, A176, A190, A193, A192, A188, Th7;
hence contradiction by A31, A128, A170, XXREAL_0:2; ::_thesis: verum
end;
caseA194: ( (f /. (kk + 1)) `1 < e & (f /. kk) `1 >= e ) ; ::_thesis: contradiction
set k2 = (i -' kk) -' 1;
A195: LSeg ((f /. kk),(f /. (kk + 1))) c= L~ f
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in LSeg ((f /. kk),(f /. (kk + 1))) or z in L~ f )
assume A196: z in LSeg ((f /. kk),(f /. (kk + 1))) ; ::_thesis: z in L~ f
LSeg ((f /. kk),(f /. (kk + 1))) in { (LSeg (f,i7)) where i7 is Element of NAT : ( 1 <= i7 & i7 + 1 <= len f ) } by A174, A175, A176;
hence z in L~ f by A11, A196, TARSKI:def_4; ::_thesis: verum
end;
( f is one-to-one & kk < kk + 1 ) by A5, NAT_1:13, TOPREAL1:def_8;
then A197: f . kk <> f . (kk + 1) by A179, A177, FUNCT_1:def_4;
A198: (f /. kk) `1 > (f /. (kk + 1)) `1 by A194, XXREAL_0:2;
LE f /. kk,p51, L~ f,f /. 1,f /. (len f) by A5, A171, A173, A174, A175, JORDAN5C:25;
then A199: LE f /. kk,f /. i, L~ f,f /. 1,f /. (len f) by A6, A7, A8, A169, JORDAN5C:13;
( f /. kk = f . kk & f /. (kk + 1) = f . (kk + 1) ) by A179, A177, PARTFUN1:def_6;
then LSeg ((f /. kk),(f /. (kk + 1))) is_an_arc_of f /. kk,f /. (kk + 1) by A197, TOPREAL1:9;
then A200: ( Segment ((L~ f),(f /. 1),(f /. (len f)),(f /. kk),(f /. (kk + 1))) = { p8 where p8 is Point of (TOP-REAL 2) : ( LE f /. kk,p8, L~ f,f /. 1,f /. (len f) & LE p8,f /. (kk + 1), L~ f,f /. 1,f /. (len f) ) } & Segment ((L~ f),(f /. 1),(f /. (len f)),(f /. kk),(f /. (kk + 1))) = LSeg ((f /. kk),(f /. (kk + 1))) ) by A9, A5, A6, A7, A8, A174, A175, A195, Th25, JORDAN5C:23, JORDAN6:26;
A201: now__::_thesis:_not_(i_-_kk)_-_1_<=_0
assume (i - kk) - 1 <= 0 ; ::_thesis: contradiction
then (i - (kk + 1)) + (kk + 1) <= 0 + (kk + 1) by XREAL_1:7;
then LE f /. i,f /. (kk + 1), L~ f,f /. 1,f /. (len f) by A5, A163, A175, JORDAN5C:24;
then A202: f /. i in LSeg ((f /. kk),(f /. (kk + 1))) by A200, A199;
f is special by A5, TOPREAL1:def_8;
then A203: ( (f /. kk) `1 = (f /. (kk + 1)) `1 or (f /. kk) `2 = (f /. (kk + 1)) `2 ) by A174, A175, TOPREAL1:def_5;
LSeg (f,kk) = LSeg ((f /. kk),(f /. (kk + 1))) by A174, A175, TOPREAL1:def_3;
hence contradiction by A5, A6, A7, A8, A165, A169, A170, A171, A173, A174, A178, A198, A202, A203, Th6; ::_thesis: verum
end;
then ((i - kk) - 1) + 1 >= 0 + 1 by XREAL_1:7;
then i -' kk = i - kk by XREAL_0:def_2;
then A204: i - ((i -' kk) -' 1) = i - ((i - kk) - 1) by A201, XREAL_0:def_2
.= kk + 1 ;
then i -' ((i -' kk) -' 1) > 0 by XREAL_0:def_2;
then A205: i -' ((i -' kk) -' 1) >= 0 + 1 by NAT_1:13;
A206: i - ((i -' kk) -' 1) = i -' ((i -' kk) -' 1) by A204, XREAL_0:def_2;
then S1[(i -' kk) -' 1] by A20, A194, A204, A205, FINSEQ_4:15, NAT_D:50;
then (i -' kk) -' 1 >= k by A26;
then i - ((i -' kk) -' 1) <= i - k by XREAL_1:10;
then A207: LE f /. (kk + 1),f /. (i -' k), L~ f,f /. 1,f /. (len f) by A5, A29, A32, A204, A206, A205, JORDAN5C:24;
LE f /. (i -' k),f /. (kk + 1), L~ f,f /. 1,f /. (len f) by A6, A7, A8, A31, A168, A180, JORDAN5C:13;
hence contradiction by A1, A6, A7, A8, A31, A128, A194, A207, JORDAN5C:12, TOPREAL4:2; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
A208: LSeg (f,i) = LSeg ((f /. i),(f /. (i + 1))) by A15, A16, TOPREAL1:def_3;
A209: for p5 being Point of (TOP-REAL 2) st LE f /. i,p5,P,p1,p2 & LE p5,p,P,p1,p2 holds
p5 `1 >= e
proof
let p5 be Point of (TOP-REAL 2); ::_thesis: ( LE f /. i,p5,P,p1,p2 & LE p5,p,P,p1,p2 implies p5 `1 >= e )
A210: Segment (P,p1,p2,(f /. i),p) = { p8 where p8 is Point of (TOP-REAL 2) : ( LE f /. i,p8,P,p1,p2 & LE p8,p,P,p1,p2 ) } by JORDAN6:26;
assume ( LE f /. i,p5,P,p1,p2 & LE p5,p,P,p1,p2 ) ; ::_thesis: p5 `1 >= e
then A211: p5 in Segment (P,p1,p2,(f /. i),p) by A210;
f /. i in LSeg ((f /. i),(f /. (i + 1))) by RLTOPSP1:68;
then LSeg ((f /. i),p) c= LSeg (f,i) by A12, A14, A208, TOPREAL1:6;
then A212: LSeg ((f /. i),p) c= P by A6, A19, A14, XBOOLE_1:1;
now__::_thesis:_(_(_f_/._i_<>_p_&_p5_`1_>=_e_)_or_(_f_/._i_=_p_&_p5_`1_>=_e_)_)
percases ( f /. i <> p or f /. i = p ) ;
case f /. i <> p ; ::_thesis: p5 `1 >= e
then LSeg ((f /. i),p) is_an_arc_of f /. i,p by TOPREAL1:9;
then Segment (P,p1,p2,(f /. i),p) = LSeg ((f /. i),p) by A9, A5, A6, A7, A8, A15, A20, A158, A212, Th25, SPRECT_4:3;
hence p5 `1 >= e by A4, A165, A211, TOPREAL1:3; ::_thesis: verum
end;
case f /. i = p ; ::_thesis: p5 `1 >= e
then Segment (P,p1,p2,(f /. i),p) = {(f /. i)} by A1, A3, Th1, TOPREAL4:2;
hence p5 `1 >= e by A165, A211, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
hence p5 `1 >= e ; ::_thesis: verum
end;
A213: len g = (i -' 1) + 1 by A15, A20, A22, FINSEQ_6:118;
then (i -' k) + 1 <= len g by A162, A164, NAT_1:13;
then A214: LSeg (g,(i -' k)) = LSeg ((g /. (i -' k)),(g /. ((i -' k) + 1))) by A28, TOPREAL1:def_3;
i -' k < i by A28, A160, NAT_D:51;
then i -' k in Seg (len g) by A28, A213, A164, FINSEQ_1:1;
then i -' k in dom g by FINSEQ_1:def_3;
then g /. (i -' k) = g . (i -' k) by PARTFUN1:def_6
.= f . (((i -' k) - 1) + 1) by A15, A159, A161, FINSEQ_6:122
.= f /. (i -' k) by A30, PARTFUN1:def_6 ;
then A215: pk in LSeg (g,(i -' k)) by A31, A214, RLTOPSP1:68;
A216: (i -' k) + 1 <= i by A162, NAT_1:13;
1 <= i -' k by A27, XREAL_0:def_2;
then LSeg (g,(i -' k)) in { (LSeg (g,i2)) where i2 is Element of NAT : ( 1 <= i2 & i2 + 1 <= len g ) } by A213, A164, A216;
then pk in union { (LSeg (g,i2)) where i2 is Element of NAT : ( 1 <= i2 & i2 + 1 <= len g ) } by A215, TARSKI:def_4;
then pk in L~ (mid (f,1,i)) by TOPREAL1:def_4;
then A217: LE pk,f /. i,P,p1,p2 by A5, A6, A7, A8, A20, A163, SPRECT_3:17;
then A218: f /. i in P by JORDAN5C:def_3;
A219: for p5 being Point of (TOP-REAL 2) st LE pk,p5,P,p1,p2 & LE p5,p,P,p1,p2 holds
p5 `1 >= e
proof
let p5 be Point of (TOP-REAL 2); ::_thesis: ( LE pk,p5,P,p1,p2 & LE p5,p,P,p1,p2 implies p5 `1 >= e )
assume that
A220: LE pk,p5,P,p1,p2 and
A221: LE p5,p,P,p1,p2 ; ::_thesis: p5 `1 >= e
A222: p5 in P by A220, JORDAN5C:def_3;
now__::_thesis:_(_(_LE_p5,f_/._i,P,p1,p2_&_p5_`1_>=_e_)_or_(_LE_f_/._i,p5,P,p1,p2_&_p5_`1_>=_e_)_)
percases ( LE p5,f /. i,P,p1,p2 or LE f /. i,p5,P,p1,p2 ) by A1, A218, A222, Th19, TOPREAL4:2;
case LE p5,f /. i,P,p1,p2 ; ::_thesis: p5 `1 >= e
hence p5 `1 >= e by A167, A220; ::_thesis: verum
end;
case LE f /. i,p5,P,p1,p2 ; ::_thesis: p5 `1 >= e
hence p5 `1 >= e by A209, A221; ::_thesis: verum
end;
end;
end;
hence p5 `1 >= e ; ::_thesis: verum
end;
LE f /. i,p,P,p1,p2 by A5, A6, A7, A8, A15, A20, A158, SPRECT_4:3;
then LE pk,p,P,p1,p2 by A217, JORDAN5C:13;
hence ( p is_Lin P,p1,p2,e or p is_Rin P,p1,p2,e ) by A3, A4, A9, A128, A219, Def2; ::_thesis: verum
end;
end;
end;
hence ( p is_Lin P,p1,p2,e or p is_Rin P,p1,p2,e ) ; ::_thesis: verum
end;
end;
end;
hence ( p is_Lin P,p1,p2,e or p is_Rin P,p1,p2,e ) ; ::_thesis: verum
end;
theorem :: JORDAN20:31
for P being non empty Subset of (TOP-REAL 2)
for p1, p2, p being Point of (TOP-REAL 2)
for e being Real st P is_S-P_arc_joining p1,p2 & p2 `1 > e & p in P & p `1 = e & not p is_Lout P,p1,p2,e holds
p is_Rout P,p1,p2,e
proof
let P be non empty Subset of (TOP-REAL 2); ::_thesis: for p1, p2, p being Point of (TOP-REAL 2)
for e being Real st P is_S-P_arc_joining p1,p2 & p2 `1 > e & p in P & p `1 = e & not p is_Lout P,p1,p2,e holds
p is_Rout P,p1,p2,e
let p1, p2, p be Point of (TOP-REAL 2); ::_thesis: for e being Real st P is_S-P_arc_joining p1,p2 & p2 `1 > e & p in P & p `1 = e & not p is_Lout P,p1,p2,e holds
p is_Rout P,p1,p2,e
let e be Real; ::_thesis: ( P is_S-P_arc_joining p1,p2 & p2 `1 > e & p in P & p `1 = e & not p is_Lout P,p1,p2,e implies p is_Rout P,p1,p2,e )
assume that
A1: P is_S-P_arc_joining p1,p2 and
A2: p2 `1 > e and
A3: p in P and
A4: p `1 = e ; ::_thesis: ( p is_Lout P,p1,p2,e or p is_Rout P,p1,p2,e )
consider f being FinSequence of (TOP-REAL 2) such that
A5: f is being_S-Seq and
A6: P = L~ f and
A7: p1 = f /. 1 and
A8: p2 = f /. (len f) by A1, TOPREAL4:def_1;
A9: P is_an_arc_of p1,p2 by A1, TOPREAL4:2;
A10: L~ f = union { (LSeg (f,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len f ) } by TOPREAL1:def_4;
then consider Y being set such that
A11: p in Y and
A12: Y in { (LSeg (f,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len f ) } by A3, A6, TARSKI:def_4;
consider i being Element of NAT such that
A13: Y = LSeg (f,i) and
A14: 1 <= i and
A15: i + 1 <= len f by A12;
A16: 1 < i + 1 by A14, NAT_1:13;
A17: 1 < i + 1 by A14, NAT_1:13;
then i + 1 in Seg (len f) by A15, FINSEQ_1:1;
then A18: i + 1 in dom f by FINSEQ_1:def_3;
A19: Y c= L~ f
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Y or x in L~ f )
assume x in Y ; ::_thesis: x in L~ f
hence x in L~ f by A10, A12, TARSKI:def_4; ::_thesis: verum
end;
defpred S1[ Nat] means for p being Point of (TOP-REAL 2) st p = f . ((i + 1) + $1) holds
p `1 <> e;
A20: (len f) - (i + 1) >= 0 by A15, XREAL_1:48;
then A21: (i + 1) + ((len f) -' (i + 1)) = (i + 1) + ((len f) - (i + 1)) by XREAL_0:def_2
.= len f ;
A22: (len f) -' (i + 1) = (len f) - (i + 1) by A20, XREAL_0:def_2;
A23: i < len f by A15, NAT_1:13;
then 1 < len f by A14, XXREAL_0:2;
then len f in Seg (len f) by FINSEQ_1:1;
then len f in dom f by FINSEQ_1:def_3;
then A24: S1[(len f) -' (i + 1)] by A2, A8, A21, PARTFUN1:def_6;
then A25: ex k being Nat st S1[k] ;
ex k being Nat st
( S1[k] & ( for n being Nat st S1[n] holds
k <= n ) ) from NAT_1:sch_5(A25);
then consider k being Nat such that
A26: S1[k] and
A27: for n being Nat st S1[n] holds
k <= n ;
k <= (len f) -' (i + 1) by A24, A27;
then A28: k + (i + 1) <= ((len f) - (i + 1)) + (i + 1) by A22, XREAL_1:7;
i + k >= i by NAT_1:11;
then A29: (i + k) + 1 >= i + 1 by XREAL_1:7;
then A30: (i + 1) + k > 1 by A16, XXREAL_0:2;
1 <= (i + 1) + k by A17, NAT_1:12;
then (i + 1) + k in Seg (len f) by A28, FINSEQ_1:1;
then A31: (i + 1) + k in dom f by FINSEQ_1:def_3;
then A32: f /. ((i + 1) + k) = f . ((i + 1) + k) by PARTFUN1:def_6;
then reconsider pk = f . ((i + 1) + k) as Point of (TOP-REAL 2) ;
A33: (k + i) + 1 > 1 by A16, A29, XXREAL_0:2;
now__::_thesis:_(_(_pk_`1_<_e_&_(_p_is_Lout_P,p1,p2,e_or_p_is_Rout_P,p1,p2,e_)_)_or_(_pk_`1_>_e_&_(_p_is_Lout_P,p1,p2,e_or_p_is_Rout_P,p1,p2,e_)_)_)
percases ( pk `1 < e or pk `1 > e ) by A26, XXREAL_0:1;
caseA34: pk `1 < e ; ::_thesis: ( p is_Lout P,p1,p2,e or p is_Rout P,p1,p2,e )
now__::_thesis:_(_(_k_=_0_&_(_p_is_Lout_P,p1,p2,e_or_p_is_Rout_P,p1,p2,e_)_)_or_(_k_<>_0_&_(_p_is_Lout_P,p1,p2,e_or_p_is_Rout_P,p1,p2,e_)_)_)
percases ( k = 0 or k <> 0 ) ;
caseA35: k = 0 ; ::_thesis: ( p is_Lout P,p1,p2,e or p is_Rout P,p1,p2,e )
set p44 = f /. (i + 1);
A36: pk = f /. (i + 1) by A18, A35, PARTFUN1:def_6;
A37: f /. (i + 1) in LSeg (p,(f /. (i + 1))) by RLTOPSP1:68;
A38: LSeg (f,i) = LSeg ((f /. i),(f /. (i + 1))) by A14, A15, TOPREAL1:def_3;
A39: for p5 being Point of (TOP-REAL 2) st LE p5,f /. (i + 1),P,p1,p2 & LE p,p5,P,p1,p2 holds
p5 `1 <= e
proof
f /. (i + 1) in LSeg ((f /. i),(f /. (i + 1))) by RLTOPSP1:68;
then LSeg (p,(f /. (i + 1))) c= LSeg (f,i) by A11, A13, A38, TOPREAL1:6;
then A40: LSeg (p,(f /. (i + 1))) c= P by A6, A19, A13, XBOOLE_1:1;
let p5 be Point of (TOP-REAL 2); ::_thesis: ( LE p5,f /. (i + 1),P,p1,p2 & LE p,p5,P,p1,p2 implies p5 `1 <= e )
A41: Segment (P,p1,p2,p,(f /. (i + 1))) = { p8 where p8 is Point of (TOP-REAL 2) : ( LE p,p8,P,p1,p2 & LE p8,f /. (i + 1),P,p1,p2 ) } by JORDAN6:26;
assume ( LE p5,f /. (i + 1),P,p1,p2 & LE p,p5,P,p1,p2 ) ; ::_thesis: p5 `1 <= e
then A42: p5 in Segment (P,p1,p2,p,(f /. (i + 1))) by A41;
now__::_thesis:_(_(_f_/._(i_+_1)_<>_p_&_p5_`1_<=_e_)_or_(_f_/._(i_+_1)_=_p_&_p5_`1_<=_e_)_)
percases ( f /. (i + 1) <> p or f /. (i + 1) = p ) ;
case f /. (i + 1) <> p ; ::_thesis: p5 `1 <= e
then LSeg (p,(f /. (i + 1))) is_an_arc_of p,f /. (i + 1) by TOPREAL1:9;
then Segment (P,p1,p2,p,(f /. (i + 1))) = LSeg (p,(f /. (i + 1))) by A9, A5, A6, A7, A8, A11, A13, A14, A23, A37, A40, Th25, SPRECT_4:4;
hence p5 `1 <= e by A4, A34, A36, A42, TOPREAL1:3; ::_thesis: verum
end;
case f /. (i + 1) = p ; ::_thesis: p5 `1 <= e
hence p5 `1 <= e by A4, A18, A34, A35, PARTFUN1:def_6; ::_thesis: verum
end;
end;
end;
hence p5 `1 <= e ; ::_thesis: verum
end;
LE p,f /. (i + 1),P,p1,p2 by A5, A6, A7, A8, A11, A13, A14, A23, A37, SPRECT_4:4;
hence ( p is_Lout P,p1,p2,e or p is_Rout P,p1,p2,e ) by A3, A4, A9, A34, A36, A39, Def3; ::_thesis: verum
end;
caseA43: k <> 0 ; ::_thesis: ( p is_Lout P,p1,p2,e or p is_Rout P,p1,p2,e )
set p44 = f /. (i + 1);
A44: now__::_thesis:_not_(f_/._(i_+_1))_`1_<>_e
assume (f /. (i + 1)) `1 <> e ; ::_thesis: contradiction
then for p9 being Point of (TOP-REAL 2) st p9 = f . ((i + 1) + 0) holds
p9 `1 <> e by A18, PARTFUN1:def_6;
hence contradiction by A27, A43; ::_thesis: verum
end;
A45: LSeg (f,i) = LSeg ((f /. i),(f /. (i + 1))) by A14, A15, TOPREAL1:def_3;
A46: now__::_thesis:_for_p51_being_Point_of_(TOP-REAL_2)_st_LE_f_/._(i_+_1),p51,P,p1,p2_&_LE_p51,pk,P,p1,p2_holds_
p51_`1_<=_e
assume ex p51 being Point of (TOP-REAL 2) st
( LE f /. (i + 1),p51,P,p1,p2 & LE p51,pk,P,p1,p2 & not p51 `1 <= e ) ; ::_thesis: contradiction
then consider p51 being Point of (TOP-REAL 2) such that
A47: LE f /. (i + 1),p51,P,p1,p2 and
A48: LE p51,pk,P,p1,p2 and
A49: p51 `1 > e ;
p51 in P by A47, JORDAN5C:def_3;
then consider Y3 being set such that
A50: p51 in Y3 and
A51: Y3 in { (LSeg (f,i5)) where i5 is Element of NAT : ( 1 <= i5 & i5 + 1 <= len f ) } by A6, A10, TARSKI:def_4;
consider kk being Element of NAT such that
A52: Y3 = LSeg (f,kk) and
A53: 1 <= kk and
A54: kk + 1 <= len f by A51;
A55: LE p51,f /. (kk + 1), L~ f,f /. 1,f /. (len f) by A5, A50, A52, A53, A54, JORDAN5C:26;
A56: LE f /. kk,p51, L~ f,f /. 1,f /. (len f) by A5, A50, A52, A53, A54, JORDAN5C:25;
A57: kk - 1 >= 0 by A53, XREAL_1:48;
A58: LSeg (f,kk) = LSeg ((f /. kk),(f /. (kk + 1))) by A53, A54, TOPREAL1:def_3;
A59: kk < len f by A54, NAT_1:13;
then A60: kk in dom f by A53, FINSEQ_3:25;
then A61: f /. kk = f . kk by PARTFUN1:def_6;
A62: 1 < kk + 1 by A53, NAT_1:13;
then A63: kk + 1 in dom f by A54, FINSEQ_3:25;
( f is one-to-one & kk < kk + 1 ) by A5, NAT_1:13, TOPREAL1:def_8;
then A64: f . kk <> f . (kk + 1) by A60, A63, FUNCT_1:def_4;
now__::_thesis:_(_(_(f_/._(kk_+_1))_`1_>_e_&_contradiction_)_or_(_(f_/._kk)_`1_>_e_&_(f_/._(kk_+_1))_`1_<=_e_&_contradiction_)_)
percases ( (f /. (kk + 1)) `1 > e or ( (f /. kk) `1 > e & (f /. (kk + 1)) `1 <= e ) ) by A49, A50, A52, A58, Th2;
caseA65: (f /. (kk + 1)) `1 > e ; ::_thesis: contradiction
set k2 = kk -' i;
A66: LE f /. (i + 1),f /. (kk + 1), L~ f,f /. 1,f /. (len f) by A6, A7, A8, A47, A55, JORDAN5C:13;
now__::_thesis:_not_kk_-_i_<_0
assume kk - i < 0 ; ::_thesis: contradiction
then (kk - i) + i < 0 + i by XREAL_1:6;
then LE f /. (kk + 1),f /. (i + 1), L~ f,f /. 1,f /. (len f) by A5, A15, A62, JORDAN5C:24, XREAL_1:7;
hence contradiction by A1, A6, A7, A8, A44, A65, A66, JORDAN5C:12, TOPREAL4:2; ::_thesis: verum
end;
then A67: (i + 1) + (kk -' i) = (1 + i) + (kk - i) by XREAL_0:def_2
.= kk + 1 ;
A68: LSeg ((f /. kk),(f /. (kk + 1))) c= L~ f
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in LSeg ((f /. kk),(f /. (kk + 1))) or z in L~ f )
assume A69: z in LSeg ((f /. kk),(f /. (kk + 1))) ; ::_thesis: z in L~ f
LSeg ((f /. kk),(f /. (kk + 1))) in { (LSeg (f,i7)) where i7 is Element of NAT : ( 1 <= i7 & i7 + 1 <= len f ) } by A53, A54, A58;
hence z in L~ f by A10, A69, TARSKI:def_4; ::_thesis: verum
end;
f is special by A5, TOPREAL1:def_8;
then A70: ( (f /. kk) `1 = (f /. (kk + 1)) `1 or (f /. kk) `2 = (f /. (kk + 1)) `2 ) by A53, A54, TOPREAL1:def_5;
( f is one-to-one & kk < kk + 1 ) by A5, NAT_1:13, TOPREAL1:def_8;
then A71: f . kk <> f . (kk + 1) by A60, A63, FUNCT_1:def_4;
A72: LE p51,f /. ((i + 1) + k), L~ f,f /. 1,f /. (len f) by A6, A7, A8, A31, A48, PARTFUN1:def_6;
A73: LE f /. kk,f /. ((i + 1) + k), L~ f,f /. 1,f /. (len f) by A6, A7, A8, A32, A48, A56, JORDAN5C:13;
1 < kk + 1 by A53, NAT_1:13;
then S1[kk -' i] by A54, A65, A67, FINSEQ_4:15;
then kk -' i >= k by A27;
then A74: LE f /. ((i + 1) + k),f /. ((i + 1) + (kk -' i)), L~ f,f /. 1,f /. (len f) by A5, A33, A54, A67, JORDAN5C:24, XREAL_1:7;
( f /. kk = f . kk & f /. (kk + 1) = f . (kk + 1) ) by A60, A63, PARTFUN1:def_6;
then LSeg ((f /. kk),(f /. (kk + 1))) is_an_arc_of f /. kk,f /. (kk + 1) by A71, TOPREAL1:9;
then A75: Segment ((L~ f),(f /. 1),(f /. (len f)),(f /. kk),(f /. (kk + 1))) = LSeg ((f /. kk),(f /. (kk + 1))) by A9, A6, A7, A8, A67, A74, A73, A68, Th25, JORDAN5C:13;
Segment ((L~ f),(f /. 1),(f /. (len f)),(f /. kk),(f /. (kk + 1))) = { p8 where p8 is Point of (TOP-REAL 2) : ( LE f /. kk,p8, L~ f,f /. 1,f /. (len f) & LE p8,f /. (kk + 1), L~ f,f /. 1,f /. (len f) ) } by JORDAN6:26;
then A76: f /. ((i + 1) + k) in Segment ((L~ f),(f /. 1),(f /. (len f)),(f /. kk),(f /. (kk + 1))) by A67, A74, A73;
then (f /. kk) `1 < e by A32, A34, A65, A75, Th3;
then (f /. kk) `1 < (f /. (kk + 1)) `1 by A65, XXREAL_0:2;
then (f /. ((i + 1) + k)) `1 >= p51 `1 by A5, A50, A52, A53, A59, A58, A72, A76, A75, A70, Th7;
hence contradiction by A32, A34, A49, XXREAL_0:2; ::_thesis: verum
end;
caseA77: ( (f /. kk) `1 > e & (f /. (kk + 1)) `1 <= e ) ; ::_thesis: contradiction
set k2 = kk -' (i + 1);
A78: LSeg ((f /. kk),(f /. (kk + 1))) c= L~ f
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in LSeg ((f /. kk),(f /. (kk + 1))) or z in L~ f )
assume A79: z in LSeg ((f /. kk),(f /. (kk + 1))) ; ::_thesis: z in L~ f
LSeg ((f /. kk),(f /. (kk + 1))) in { (LSeg (f,i7)) where i7 is Element of NAT : ( 1 <= i7 & i7 + 1 <= len f ) } by A53, A54, A58;
hence z in L~ f by A10, A79, TARSKI:def_4; ::_thesis: verum
end;
LE p51,f /. (kk + 1), L~ f,f /. 1,f /. (len f) by A5, A50, A52, A53, A54, JORDAN5C:26;
then A80: LE f /. (i + 1),f /. (kk + 1), L~ f,f /. 1,f /. (len f) by A6, A7, A8, A47, JORDAN5C:13;
f /. (kk + 1) = f . (kk + 1) by A63, PARTFUN1:def_6;
then LSeg ((f /. kk),(f /. (kk + 1))) is_an_arc_of f /. kk,f /. (kk + 1) by A64, A61, TOPREAL1:9;
then A81: ( Segment ((L~ f),(f /. 1),(f /. (len f)),(f /. kk),(f /. (kk + 1))) = { p8 where p8 is Point of (TOP-REAL 2) : ( LE f /. kk,p8, L~ f,f /. 1,f /. (len f) & LE p8,f /. (kk + 1), L~ f,f /. 1,f /. (len f) ) } & Segment ((L~ f),(f /. 1),(f /. (len f)),(f /. kk),(f /. (kk + 1))) = LSeg ((f /. kk),(f /. (kk + 1))) ) by A9, A5, A6, A7, A8, A53, A54, A78, Th25, JORDAN5C:23, JORDAN6:26;
A82: now__::_thesis:_not_kk_-_(i_+_1)_<_0
assume kk - (i + 1) < 0 ; ::_thesis: contradiction
then (kk - (i + 1)) + (i + 1) < 0 + (i + 1) by XREAL_1:6;
then kk <= i by NAT_1:13;
then A83: LE f /. kk,f /. i, L~ f,f /. 1,f /. (len f) by A5, A23, A53, JORDAN5C:24;
A84: f /. i in LSeg ((f /. i),(f /. (i + 1))) by RLTOPSP1:68;
LE f /. i,f /. (i + 1), L~ f,f /. 1,f /. (len f) by A5, A14, A15, JORDAN5C:23;
then LE f /. i,f /. (kk + 1), L~ f,f /. 1,f /. (len f) by A80, JORDAN5C:13;
then f /. i in LSeg ((f /. kk),(f /. (kk + 1))) by A81, A83;
then (LSeg ((f /. kk),(f /. (kk + 1)))) /\ (LSeg ((f /. i),(f /. (i + 1)))) <> {} by A84, XBOOLE_0:def_4;
then A85: not LSeg ((f /. kk),(f /. (kk + 1))) misses LSeg ((f /. i),(f /. (i + 1))) by XBOOLE_0:def_7;
A86: kk - 1 = kk -' 1 by A57, XREAL_0:def_2;
A87: now__::_thesis:_not_i_=_(kk_-'_1)_+_2
assume A88: i = (kk -' 1) + 2 ; ::_thesis: contradiction
then kk + 1 < i + 1 by A86, NAT_1:13;
then LE f /. (kk + 1),f /. (i + 1), L~ f,f /. 1,f /. (len f) by A5, A15, A62, JORDAN5C:24;
then f /. (i + 1) = f /. (kk + 1) by A1, A6, A7, A8, A80, JORDAN5C:12, TOPREAL4:2;
then f . (i + 1) = f /. (kk + 1) by A18, PARTFUN1:def_6;
then A89: f . (i + 1) = f . (kk + 1) by A63, PARTFUN1:def_6;
f is one-to-one by A5, TOPREAL1:def_8;
then i + 1 = kk + 1 by A18, A63, A89, FUNCT_1:def_4;
hence contradiction by A86, A88; ::_thesis: verum
end;
A90: f is s.n.c. by A5, TOPREAL1:def_8;
A91: ( LSeg (f,kk) = LSeg ((f /. kk),(f /. (kk + 1))) & LSeg (f,i) = LSeg ((f /. i),(f /. (i + 1))) ) by A14, A15, A53, A54, TOPREAL1:def_3;
then i + 1 >= kk by A85, A90, TOPREAL1:def_7;
then A92: (i + 1) - 1 >= kk - 1 by XREAL_1:9;
kk + 1 >= i by A85, A91, A90, TOPREAL1:def_7;
then A93: ( i = kk -' 1 or i = (kk -' 1) + 1 or i = (kk -' 1) + 2 ) by A86, A92, NAT_1:56;
A94: now__::_thesis:_(_(_i_=_kk_&_f_/._(i_+_1)_in_LSeg_(f,kk)_)_or_(_i_=_kk_-_1_&_f_/._(i_+_1)_in_LSeg_(f,kk)_)_)
percases ( i = kk or i = kk - 1 ) by A86, A93, A87;
case i = kk ; ::_thesis: f /. (i + 1) in LSeg (f,kk)
hence f /. (i + 1) in LSeg (f,kk) by A45, RLTOPSP1:68; ::_thesis: verum
end;
case i = kk - 1 ; ::_thesis: f /. (i + 1) in LSeg (f,kk)
hence f /. (i + 1) in LSeg (f,kk) by A58, RLTOPSP1:68; ::_thesis: verum
end;
end;
end;
f is special by A5, TOPREAL1:def_8;
then ( (f /. kk) `1 = (f /. (kk + 1)) `1 or (f /. kk) `2 = (f /. (kk + 1)) `2 ) by A53, A54, TOPREAL1:def_5;
hence contradiction by A5, A6, A7, A8, A44, A47, A49, A50, A52, A53, A59, A77, A94, Th6; ::_thesis: verum
end;
then (i + 1) + (kk -' (i + 1)) = (i + 1) + (kk - (i + 1)) by XREAL_0:def_2
.= kk ;
then S1[kk -' (i + 1)] by A53, A59, A77, FINSEQ_4:15;
then A95: kk -' (i + 1) >= k by A27;
kk -' (i + 1) = kk - (i + 1) by A82, XREAL_0:def_2;
then (kk - (i + 1)) + (i + 1) >= k + (i + 1) by A95, XREAL_1:7;
then A96: LE f /. ((i + 1) + k),f /. kk, L~ f,f /. 1,f /. (len f) by A5, A30, A59, JORDAN5C:24;
LE f /. kk,f /. ((i + 1) + k), L~ f,f /. 1,f /. (len f) by A6, A7, A8, A32, A48, A56, JORDAN5C:13;
hence contradiction by A1, A6, A7, A8, A32, A34, A77, A96, JORDAN5C:12, TOPREAL4:2; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
A97: f /. (i + 1) in LSeg (p,(f /. (i + 1))) by RLTOPSP1:68;
A98: for p5 being Point of (TOP-REAL 2) st LE p,p5,P,p1,p2 & LE p5,f /. (i + 1),P,p1,p2 holds
p5 `1 <= e
proof
let p5 be Point of (TOP-REAL 2); ::_thesis: ( LE p,p5,P,p1,p2 & LE p5,f /. (i + 1),P,p1,p2 implies p5 `1 <= e )
A99: Segment (P,p1,p2,p,(f /. (i + 1))) = { p8 where p8 is Point of (TOP-REAL 2) : ( LE p,p8,P,p1,p2 & LE p8,f /. (i + 1),P,p1,p2 ) } by JORDAN6:26;
assume ( LE p,p5,P,p1,p2 & LE p5,f /. (i + 1),P,p1,p2 ) ; ::_thesis: p5 `1 <= e
then A100: p5 in Segment (P,p1,p2,p,(f /. (i + 1))) by A99;
f /. (i + 1) in LSeg ((f /. i),(f /. (i + 1))) by RLTOPSP1:68;
then LSeg (p,(f /. (i + 1))) c= LSeg (f,i) by A11, A13, A45, TOPREAL1:6;
then A101: LSeg (p,(f /. (i + 1))) c= P by A6, A19, A13, XBOOLE_1:1;
now__::_thesis:_(_(_f_/._(i_+_1)_<>_p_&_p5_`1_<=_e_)_or_(_f_/._(i_+_1)_=_p_&_p5_`1_<=_e_)_)
percases ( f /. (i + 1) <> p or f /. (i + 1) = p ) ;
case f /. (i + 1) <> p ; ::_thesis: p5 `1 <= e
then LSeg (p,(f /. (i + 1))) is_an_arc_of p,f /. (i + 1) by TOPREAL1:9;
then Segment (P,p1,p2,p,(f /. (i + 1))) = LSeg (p,(f /. (i + 1))) by A9, A5, A6, A7, A8, A11, A13, A14, A23, A97, A101, Th25, SPRECT_4:4;
hence p5 `1 <= e by A4, A44, A100, TOPREAL1:3; ::_thesis: verum
end;
case f /. (i + 1) = p ; ::_thesis: p5 `1 <= e
then Segment (P,p1,p2,p,(f /. (i + 1))) = {(f /. (i + 1))} by A1, A3, Th1, TOPREAL4:2;
hence p5 `1 <= e by A44, A100, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
hence p5 `1 <= e ; ::_thesis: verum
end;
i + 1 <= (i + 1) + k by NAT_1:11;
then A102: LE f /. (i + 1),pk,P,p1,p2 by A5, A6, A7, A8, A17, A28, A32, JORDAN5C:24;
then A103: f /. (i + 1) in P by JORDAN5C:def_3;
A104: for p5 being Point of (TOP-REAL 2) st LE p5,pk,P,p1,p2 & LE p,p5,P,p1,p2 holds
p5 `1 <= e
proof
let p5 be Point of (TOP-REAL 2); ::_thesis: ( LE p5,pk,P,p1,p2 & LE p,p5,P,p1,p2 implies p5 `1 <= e )
assume that
A105: LE p5,pk,P,p1,p2 and
A106: LE p,p5,P,p1,p2 ; ::_thesis: p5 `1 <= e
A107: p5 in P by A105, JORDAN5C:def_3;
now__::_thesis:_(_(_LE_f_/._(i_+_1),p5,P,p1,p2_&_p5_`1_<=_e_)_or_(_LE_p5,f_/._(i_+_1),P,p1,p2_&_p5_`1_<=_e_)_)
percases ( LE f /. (i + 1),p5,P,p1,p2 or LE p5,f /. (i + 1),P,p1,p2 ) by A1, A103, A107, Th19, TOPREAL4:2;
case LE f /. (i + 1),p5,P,p1,p2 ; ::_thesis: p5 `1 <= e
hence p5 `1 <= e by A46, A105; ::_thesis: verum
end;
case LE p5,f /. (i + 1),P,p1,p2 ; ::_thesis: p5 `1 <= e
hence p5 `1 <= e by A98, A106; ::_thesis: verum
end;
end;
end;
hence p5 `1 <= e ; ::_thesis: verum
end;
LE p,f /. (i + 1),P,p1,p2 by A5, A6, A7, A8, A11, A13, A14, A23, A97, SPRECT_4:4;
then LE p,pk,P,p1,p2 by A102, JORDAN5C:13;
hence ( p is_Lout P,p1,p2,e or p is_Rout P,p1,p2,e ) by A3, A4, A9, A34, A104, Def3; ::_thesis: verum
end;
end;
end;
hence ( p is_Lout P,p1,p2,e or p is_Rout P,p1,p2,e ) ; ::_thesis: verum
end;
caseA108: pk `1 > e ; ::_thesis: ( p is_Lout P,p1,p2,e or p is_Rout P,p1,p2,e )
now__::_thesis:_(_(_k_=_0_&_(_p_is_Lout_P,p1,p2,e_or_p_is_Rout_P,p1,p2,e_)_)_or_(_k_<>_0_&_(_p_is_Lout_P,p1,p2,e_or_p_is_Rout_P,p1,p2,e_)_)_)
percases ( k = 0 or k <> 0 ) ;
caseA109: k = 0 ; ::_thesis: ( p is_Lout P,p1,p2,e or p is_Rout P,p1,p2,e )
set p44 = f /. (i + 1);
A110: pk = f /. (i + 1) by A18, A109, PARTFUN1:def_6;
A111: f /. (i + 1) in LSeg (p,(f /. (i + 1))) by RLTOPSP1:68;
A112: LSeg (f,i) = LSeg ((f /. i),(f /. (i + 1))) by A14, A15, TOPREAL1:def_3;
A113: for p5 being Point of (TOP-REAL 2) st LE p5,f /. (i + 1),P,p1,p2 & LE p,p5,P,p1,p2 holds
p5 `1 >= e
proof
f /. (i + 1) in LSeg ((f /. i),(f /. (i + 1))) by RLTOPSP1:68;
then LSeg (p,(f /. (i + 1))) c= LSeg (f,i) by A11, A13, A112, TOPREAL1:6;
then A114: LSeg (p,(f /. (i + 1))) c= P by A6, A19, A13, XBOOLE_1:1;
let p5 be Point of (TOP-REAL 2); ::_thesis: ( LE p5,f /. (i + 1),P,p1,p2 & LE p,p5,P,p1,p2 implies p5 `1 >= e )
A115: Segment (P,p1,p2,p,(f /. (i + 1))) = { p8 where p8 is Point of (TOP-REAL 2) : ( LE p,p8,P,p1,p2 & LE p8,f /. (i + 1),P,p1,p2 ) } by JORDAN6:26;
assume ( LE p5,f /. (i + 1),P,p1,p2 & LE p,p5,P,p1,p2 ) ; ::_thesis: p5 `1 >= e
then A116: p5 in Segment (P,p1,p2,p,(f /. (i + 1))) by A115;
now__::_thesis:_(_(_f_/._(i_+_1)_<>_p_&_p5_`1_>=_e_)_or_(_f_/._(i_+_1)_=_p_&_p5_`1_>=_e_)_)
percases ( f /. (i + 1) <> p or f /. (i + 1) = p ) ;
case f /. (i + 1) <> p ; ::_thesis: p5 `1 >= e
then LSeg (p,(f /. (i + 1))) is_an_arc_of p,f /. (i + 1) by TOPREAL1:9;
then Segment (P,p1,p2,p,(f /. (i + 1))) = LSeg (p,(f /. (i + 1))) by A9, A5, A6, A7, A8, A11, A13, A14, A23, A111, A114, Th25, SPRECT_4:4;
hence p5 `1 >= e by A4, A108, A110, A116, TOPREAL1:3; ::_thesis: verum
end;
case f /. (i + 1) = p ; ::_thesis: p5 `1 >= e
hence p5 `1 >= e by A4, A18, A108, A109, PARTFUN1:def_6; ::_thesis: verum
end;
end;
end;
hence p5 `1 >= e ; ::_thesis: verum
end;
LE p,f /. (i + 1),P,p1,p2 by A5, A6, A7, A8, A11, A13, A14, A23, A111, SPRECT_4:4;
hence ( p is_Lout P,p1,p2,e or p is_Rout P,p1,p2,e ) by A3, A4, A9, A108, A110, A113, Def4; ::_thesis: verum
end;
caseA117: k <> 0 ; ::_thesis: ( p is_Lout P,p1,p2,e or p is_Rout P,p1,p2,e )
set p44 = f /. (i + 1);
A118: now__::_thesis:_not_(f_/._(i_+_1))_`1_<>_e
assume (f /. (i + 1)) `1 <> e ; ::_thesis: contradiction
then for p9 being Point of (TOP-REAL 2) st p9 = f . ((i + 1) + 0) holds
p9 `1 <> e by A18, PARTFUN1:def_6;
hence contradiction by A27, A117; ::_thesis: verum
end;
A119: LSeg (f,i) = LSeg ((f /. i),(f /. (i + 1))) by A14, A15, TOPREAL1:def_3;
A120: now__::_thesis:_for_p51_being_Point_of_(TOP-REAL_2)_st_LE_f_/._(i_+_1),p51,P,p1,p2_&_LE_p51,pk,P,p1,p2_holds_
p51_`1_>=_e
assume ex p51 being Point of (TOP-REAL 2) st
( LE f /. (i + 1),p51,P,p1,p2 & LE p51,pk,P,p1,p2 & not p51 `1 >= e ) ; ::_thesis: contradiction
then consider p51 being Point of (TOP-REAL 2) such that
A121: LE f /. (i + 1),p51,P,p1,p2 and
A122: LE p51,pk,P,p1,p2 and
A123: p51 `1 < e ;
p51 in P by A121, JORDAN5C:def_3;
then consider Y3 being set such that
A124: p51 in Y3 and
A125: Y3 in { (LSeg (f,i5)) where i5 is Element of NAT : ( 1 <= i5 & i5 + 1 <= len f ) } by A6, A10, TARSKI:def_4;
consider kk being Element of NAT such that
A126: Y3 = LSeg (f,kk) and
A127: 1 <= kk and
A128: kk + 1 <= len f by A125;
A129: LE p51,f /. (kk + 1), L~ f,f /. 1,f /. (len f) by A5, A124, A126, A127, A128, JORDAN5C:26;
A130: LE f /. kk,p51, L~ f,f /. 1,f /. (len f) by A5, A124, A126, A127, A128, JORDAN5C:25;
A131: kk - 1 >= 0 by A127, XREAL_1:48;
A132: LSeg (f,kk) = LSeg ((f /. kk),(f /. (kk + 1))) by A127, A128, TOPREAL1:def_3;
A133: kk < len f by A128, NAT_1:13;
then kk in Seg (len f) by A127, FINSEQ_1:1;
then A134: kk in dom f by FINSEQ_1:def_3;
then A135: f /. kk = f . kk by PARTFUN1:def_6;
A136: 1 < kk + 1 by A127, NAT_1:13;
then A137: kk + 1 in dom f by A128, FINSEQ_3:25;
( f is one-to-one & kk < kk + 1 ) by A5, NAT_1:13, TOPREAL1:def_8;
then A138: f . kk <> f . (kk + 1) by A134, A137, FUNCT_1:def_4;
now__::_thesis:_(_(_(f_/._(kk_+_1))_`1_<_e_&_contradiction_)_or_(_(f_/._kk)_`1_<_e_&_(f_/._(kk_+_1))_`1_>=_e_&_contradiction_)_)
percases ( (f /. (kk + 1)) `1 < e or ( (f /. kk) `1 < e & (f /. (kk + 1)) `1 >= e ) ) by A123, A124, A126, A132, Th3;
caseA139: (f /. (kk + 1)) `1 < e ; ::_thesis: contradiction
set k2 = kk -' i;
A140: LE f /. (i + 1),f /. (kk + 1), L~ f,f /. 1,f /. (len f) by A6, A7, A8, A121, A129, JORDAN5C:13;
now__::_thesis:_not_kk_-_i_<_0
assume kk - i < 0 ; ::_thesis: contradiction
then (kk - i) + i < 0 + i by XREAL_1:6;
then LE f /. (kk + 1),f /. (i + 1), L~ f,f /. 1,f /. (len f) by A5, A15, A136, JORDAN5C:24, XREAL_1:7;
hence contradiction by A1, A6, A7, A8, A118, A139, A140, JORDAN5C:12, TOPREAL4:2; ::_thesis: verum
end;
then A141: (i + 1) + (kk -' i) = (1 + i) + (kk - i) by XREAL_0:def_2
.= kk + 1 ;
A142: LSeg ((f /. kk),(f /. (kk + 1))) c= L~ f
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in LSeg ((f /. kk),(f /. (kk + 1))) or z in L~ f )
assume A143: z in LSeg ((f /. kk),(f /. (kk + 1))) ; ::_thesis: z in L~ f
LSeg ((f /. kk),(f /. (kk + 1))) in { (LSeg (f,i7)) where i7 is Element of NAT : ( 1 <= i7 & i7 + 1 <= len f ) } by A127, A128, A132;
hence z in L~ f by A10, A143, TARSKI:def_4; ::_thesis: verum
end;
f is special by A5, TOPREAL1:def_8;
then A144: ( (f /. kk) `1 = (f /. (kk + 1)) `1 or (f /. kk) `2 = (f /. (kk + 1)) `2 ) by A127, A128, TOPREAL1:def_5;
( f is one-to-one & kk < kk + 1 ) by A5, NAT_1:13, TOPREAL1:def_8;
then A145: f . kk <> f . (kk + 1) by A134, A137, FUNCT_1:def_4;
A146: LE p51,f /. ((i + 1) + k), L~ f,f /. 1,f /. (len f) by A6, A7, A8, A31, A122, PARTFUN1:def_6;
A147: LE f /. kk,f /. ((i + 1) + k), L~ f,f /. 1,f /. (len f) by A6, A7, A8, A32, A122, A130, JORDAN5C:13;
1 < kk + 1 by A127, NAT_1:13;
then S1[kk -' i] by A128, A139, A141, FINSEQ_4:15;
then kk -' i >= k by A27;
then A148: LE f /. ((i + 1) + k),f /. ((i + 1) + (kk -' i)), L~ f,f /. 1,f /. (len f) by A5, A33, A128, A141, JORDAN5C:24, XREAL_1:7;
( f /. kk = f . kk & f /. (kk + 1) = f . (kk + 1) ) by A134, A137, PARTFUN1:def_6;
then LSeg ((f /. kk),(f /. (kk + 1))) is_an_arc_of f /. kk,f /. (kk + 1) by A145, TOPREAL1:9;
then A149: Segment ((L~ f),(f /. 1),(f /. (len f)),(f /. kk),(f /. (kk + 1))) = LSeg ((f /. kk),(f /. (kk + 1))) by A9, A6, A7, A8, A141, A148, A147, A142, Th25, JORDAN5C:13;
Segment ((L~ f),(f /. 1),(f /. (len f)),(f /. kk),(f /. (kk + 1))) = { p8 where p8 is Point of (TOP-REAL 2) : ( LE f /. kk,p8, L~ f,f /. 1,f /. (len f) & LE p8,f /. (kk + 1), L~ f,f /. 1,f /. (len f) ) } by JORDAN6:26;
then A150: f /. ((i + 1) + k) in Segment ((L~ f),(f /. 1),(f /. (len f)),(f /. kk),(f /. (kk + 1))) by A141, A148, A147;
then (f /. kk) `1 > e by A32, A108, A139, A149, Th2;
then (f /. kk) `1 > (f /. (kk + 1)) `1 by A139, XXREAL_0:2;
then (f /. ((i + 1) + k)) `1 <= p51 `1 by A5, A124, A126, A127, A133, A132, A146, A150, A149, A144, Th6;
hence contradiction by A32, A108, A123, XXREAL_0:2; ::_thesis: verum
end;
caseA151: ( (f /. kk) `1 < e & (f /. (kk + 1)) `1 >= e ) ; ::_thesis: contradiction
set k2 = kk -' (i + 1);
A152: LSeg ((f /. kk),(f /. (kk + 1))) c= L~ f
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in LSeg ((f /. kk),(f /. (kk + 1))) or z in L~ f )
assume A153: z in LSeg ((f /. kk),(f /. (kk + 1))) ; ::_thesis: z in L~ f
LSeg ((f /. kk),(f /. (kk + 1))) in { (LSeg (f,i7)) where i7 is Element of NAT : ( 1 <= i7 & i7 + 1 <= len f ) } by A127, A128, A132;
hence z in L~ f by A10, A153, TARSKI:def_4; ::_thesis: verum
end;
LE p51,f /. (kk + 1), L~ f,f /. 1,f /. (len f) by A5, A124, A126, A127, A128, JORDAN5C:26;
then A154: LE f /. (i + 1),f /. (kk + 1), L~ f,f /. 1,f /. (len f) by A6, A7, A8, A121, JORDAN5C:13;
f /. (kk + 1) = f . (kk + 1) by A137, PARTFUN1:def_6;
then LSeg ((f /. kk),(f /. (kk + 1))) is_an_arc_of f /. kk,f /. (kk + 1) by A138, A135, TOPREAL1:9;
then A155: ( Segment ((L~ f),(f /. 1),(f /. (len f)),(f /. kk),(f /. (kk + 1))) = { p8 where p8 is Point of (TOP-REAL 2) : ( LE f /. kk,p8, L~ f,f /. 1,f /. (len f) & LE p8,f /. (kk + 1), L~ f,f /. 1,f /. (len f) ) } & Segment ((L~ f),(f /. 1),(f /. (len f)),(f /. kk),(f /. (kk + 1))) = LSeg ((f /. kk),(f /. (kk + 1))) ) by A9, A5, A6, A7, A8, A127, A128, A152, Th25, JORDAN5C:23, JORDAN6:26;
A156: now__::_thesis:_not_kk_-_(i_+_1)_<_0
assume kk - (i + 1) < 0 ; ::_thesis: contradiction
then (kk - (i + 1)) + (i + 1) < 0 + (i + 1) by XREAL_1:6;
then kk <= i by NAT_1:13;
then A157: LE f /. kk,f /. i, L~ f,f /. 1,f /. (len f) by A5, A23, A127, JORDAN5C:24;
A158: f /. i in LSeg ((f /. i),(f /. (i + 1))) by RLTOPSP1:68;
LE f /. i,f /. (i + 1), L~ f,f /. 1,f /. (len f) by A5, A14, A15, JORDAN5C:23;
then LE f /. i,f /. (kk + 1), L~ f,f /. 1,f /. (len f) by A154, JORDAN5C:13;
then f /. i in LSeg ((f /. kk),(f /. (kk + 1))) by A155, A157;
then (LSeg ((f /. kk),(f /. (kk + 1)))) /\ (LSeg ((f /. i),(f /. (i + 1)))) <> {} by A158, XBOOLE_0:def_4;
then A159: not LSeg ((f /. kk),(f /. (kk + 1))) misses LSeg ((f /. i),(f /. (i + 1))) by XBOOLE_0:def_7;
A160: kk - 1 = kk -' 1 by A131, XREAL_0:def_2;
A161: now__::_thesis:_not_i_=_(kk_-'_1)_+_2
assume A162: i = (kk -' 1) + 2 ; ::_thesis: contradiction
then kk + 1 < i + 1 by A160, NAT_1:13;
then LE f /. (kk + 1),f /. (i + 1), L~ f,f /. 1,f /. (len f) by A5, A15, A136, JORDAN5C:24;
then f /. (i + 1) = f /. (kk + 1) by A1, A6, A7, A8, A154, JORDAN5C:12, TOPREAL4:2;
then f . (i + 1) = f /. (kk + 1) by A18, PARTFUN1:def_6;
then A163: f . (i + 1) = f . (kk + 1) by A137, PARTFUN1:def_6;
f is one-to-one by A5, TOPREAL1:def_8;
then i + 1 = kk + 1 by A18, A137, A163, FUNCT_1:def_4;
hence contradiction by A160, A162; ::_thesis: verum
end;
A164: f is s.n.c. by A5, TOPREAL1:def_8;
A165: ( LSeg (f,kk) = LSeg ((f /. kk),(f /. (kk + 1))) & LSeg (f,i) = LSeg ((f /. i),(f /. (i + 1))) ) by A14, A15, A127, A128, TOPREAL1:def_3;
then i + 1 >= kk by A159, A164, TOPREAL1:def_7;
then A166: (i + 1) - 1 >= kk - 1 by XREAL_1:9;
kk + 1 >= i by A159, A165, A164, TOPREAL1:def_7;
then A167: ( i = kk -' 1 or i = (kk -' 1) + 1 or i = (kk -' 1) + 2 ) by A160, A166, NAT_1:56;
A168: now__::_thesis:_(_(_i_=_kk_&_f_/._(i_+_1)_in_LSeg_(f,kk)_)_or_(_i_=_kk_-_1_&_f_/._(i_+_1)_in_LSeg_(f,kk)_)_)
percases ( i = kk or i = kk - 1 ) by A160, A167, A161;
case i = kk ; ::_thesis: f /. (i + 1) in LSeg (f,kk)
hence f /. (i + 1) in LSeg (f,kk) by A119, RLTOPSP1:68; ::_thesis: verum
end;
case i = kk - 1 ; ::_thesis: f /. (i + 1) in LSeg (f,kk)
hence f /. (i + 1) in LSeg (f,kk) by A132, RLTOPSP1:68; ::_thesis: verum
end;
end;
end;
f is special by A5, TOPREAL1:def_8;
then ( (f /. kk) `1 = (f /. (kk + 1)) `1 or (f /. kk) `2 = (f /. (kk + 1)) `2 ) by A127, A128, TOPREAL1:def_5;
hence contradiction by A5, A6, A7, A8, A118, A121, A123, A124, A126, A127, A133, A151, A168, Th7; ::_thesis: verum
end;
then (i + 1) + (kk -' (i + 1)) = (i + 1) + (kk - (i + 1)) by XREAL_0:def_2
.= kk ;
then S1[kk -' (i + 1)] by A127, A133, A151, FINSEQ_4:15;
then A169: kk -' (i + 1) >= k by A27;
kk -' (i + 1) = kk - (i + 1) by A156, XREAL_0:def_2;
then (kk - (i + 1)) + (i + 1) >= k + (i + 1) by A169, XREAL_1:7;
then A170: LE f /. ((i + 1) + k),f /. kk, L~ f,f /. 1,f /. (len f) by A5, A30, A133, JORDAN5C:24;
LE f /. kk,f /. ((i + 1) + k), L~ f,f /. 1,f /. (len f) by A6, A7, A8, A32, A122, A130, JORDAN5C:13;
hence contradiction by A1, A6, A7, A8, A32, A108, A151, A170, JORDAN5C:12, TOPREAL4:2; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
A171: f /. (i + 1) in LSeg (p,(f /. (i + 1))) by RLTOPSP1:68;
A172: for p5 being Point of (TOP-REAL 2) st LE p,p5,P,p1,p2 & LE p5,f /. (i + 1),P,p1,p2 holds
p5 `1 >= e
proof
let p5 be Point of (TOP-REAL 2); ::_thesis: ( LE p,p5,P,p1,p2 & LE p5,f /. (i + 1),P,p1,p2 implies p5 `1 >= e )
A173: Segment (P,p1,p2,p,(f /. (i + 1))) = { p8 where p8 is Point of (TOP-REAL 2) : ( LE p,p8,P,p1,p2 & LE p8,f /. (i + 1),P,p1,p2 ) } by JORDAN6:26;
assume ( LE p,p5,P,p1,p2 & LE p5,f /. (i + 1),P,p1,p2 ) ; ::_thesis: p5 `1 >= e
then A174: p5 in Segment (P,p1,p2,p,(f /. (i + 1))) by A173;
f /. (i + 1) in LSeg ((f /. i),(f /. (i + 1))) by RLTOPSP1:68;
then LSeg (p,(f /. (i + 1))) c= LSeg (f,i) by A11, A13, A119, TOPREAL1:6;
then A175: LSeg (p,(f /. (i + 1))) c= P by A6, A19, A13, XBOOLE_1:1;
now__::_thesis:_(_(_f_/._(i_+_1)_<>_p_&_p5_`1_>=_e_)_or_(_f_/._(i_+_1)_=_p_&_p5_`1_>=_e_)_)
percases ( f /. (i + 1) <> p or f /. (i + 1) = p ) ;
case f /. (i + 1) <> p ; ::_thesis: p5 `1 >= e
then LSeg (p,(f /. (i + 1))) is_an_arc_of p,f /. (i + 1) by TOPREAL1:9;
then Segment (P,p1,p2,p,(f /. (i + 1))) = LSeg (p,(f /. (i + 1))) by A9, A5, A6, A7, A8, A11, A13, A14, A23, A171, A175, Th25, SPRECT_4:4;
hence p5 `1 >= e by A4, A118, A174, TOPREAL1:3; ::_thesis: verum
end;
case f /. (i + 1) = p ; ::_thesis: p5 `1 >= e
then Segment (P,p1,p2,p,(f /. (i + 1))) = {(f /. (i + 1))} by A1, A3, Th1, TOPREAL4:2;
hence p5 `1 >= e by A118, A174, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
hence p5 `1 >= e ; ::_thesis: verum
end;
i + 1 <= (i + 1) + k by NAT_1:11;
then A176: LE f /. (i + 1),pk,P,p1,p2 by A5, A6, A7, A8, A17, A28, A32, JORDAN5C:24;
then A177: f /. (i + 1) in P by JORDAN5C:def_3;
A178: for p5 being Point of (TOP-REAL 2) st LE p5,pk,P,p1,p2 & LE p,p5,P,p1,p2 holds
p5 `1 >= e
proof
let p5 be Point of (TOP-REAL 2); ::_thesis: ( LE p5,pk,P,p1,p2 & LE p,p5,P,p1,p2 implies p5 `1 >= e )
assume that
A179: LE p5,pk,P,p1,p2 and
A180: LE p,p5,P,p1,p2 ; ::_thesis: p5 `1 >= e
A181: p5 in P by A179, JORDAN5C:def_3;
now__::_thesis:_(_(_LE_f_/._(i_+_1),p5,P,p1,p2_&_p5_`1_>=_e_)_or_(_LE_p5,f_/._(i_+_1),P,p1,p2_&_p5_`1_>=_e_)_)
percases ( LE f /. (i + 1),p5,P,p1,p2 or LE p5,f /. (i + 1),P,p1,p2 ) by A1, A177, A181, Th19, TOPREAL4:2;
case LE f /. (i + 1),p5,P,p1,p2 ; ::_thesis: p5 `1 >= e
hence p5 `1 >= e by A120, A179; ::_thesis: verum
end;
case LE p5,f /. (i + 1),P,p1,p2 ; ::_thesis: p5 `1 >= e
hence p5 `1 >= e by A172, A180; ::_thesis: verum
end;
end;
end;
hence p5 `1 >= e ; ::_thesis: verum
end;
LE p,f /. (i + 1),P,p1,p2 by A5, A6, A7, A8, A11, A13, A14, A23, A171, SPRECT_4:4;
then LE p,pk,P,p1,p2 by A176, JORDAN5C:13;
hence ( p is_Lout P,p1,p2,e or p is_Rout P,p1,p2,e ) by A3, A4, A9, A108, A178, Def4; ::_thesis: verum
end;
end;
end;
hence ( p is_Lout P,p1,p2,e or p is_Rout P,p1,p2,e ) ; ::_thesis: verum
end;
end;
end;
hence ( p is_Lout P,p1,p2,e or p is_Rout P,p1,p2,e ) ; ::_thesis: verum
end;