:: TOPREAL2 semantic presentation

begin


Lm1: now__::_thesis:_for_x,_X_being_set_st_not_x_in_X_holds_
{x}_/\_X_=_{}
let x, X be set ; ::_thesis: ( not x in X implies {x} /\ X = {} )
assume  not x in X ; ::_thesis: {x} /\ X = {}
then  {x} misses X by ZFMISC_1:50;
hence  {x} /\ X = {} by XBOOLE_0:def_7; ::_thesis: verum
end;

Lm2:  (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {}
by TOPREAL1:19, XBOOLE_0:def_7;

Lm3:  (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {}
by TOPREAL1:20, XBOOLE_0:def_7;

set p00 = |[0,0]|;

set p01 = |[0,1]|;

set p10 = |[1,0]|;

set p11 = |[1,1]|;

set L1 = LSeg (|[0,0]|,|[0,1]|);

set L2 = LSeg (|[0,1]|,|[1,1]|);

set L3 = LSeg (|[0,0]|,|[1,0]|);

set L4 = LSeg (|[1,0]|,|[1,1]|);

Lm4:  |[0,0]| `1 = 0
by EUCLID:52;

Lm5:  |[0,0]| `2 = 0
by EUCLID:52;

Lm6:  |[0,1]| `1 = 0
by EUCLID:52;

Lm7:  |[0,1]| `2 = 1
by EUCLID:52;

Lm8:  |[1,0]| `1 = 1
by EUCLID:52;

Lm9:  |[1,0]| `2 = 0
by EUCLID:52;

Lm10:  |[1,1]| `1 = 1
by EUCLID:52;

Lm11:  |[1,1]| `2 = 1
by EUCLID:52;

Lm12:  not |[0,0]| in LSeg (|[1,0]|,|[1,1]|)
by Lm4, Lm8, Lm10, TOPREAL1:3;

Lm13:  not |[0,0]| in LSeg (|[0,1]|,|[1,1]|)
by Lm5, Lm7, Lm11, TOPREAL1:4;

Lm14:  not |[0,1]| in LSeg (|[0,0]|,|[1,0]|)
by Lm5, Lm7, Lm9, TOPREAL1:4;

Lm15:  not |[0,1]| in LSeg (|[1,0]|,|[1,1]|)
by Lm6, Lm8, Lm10, TOPREAL1:3;

Lm16:  not |[1,0]| in LSeg (|[0,0]|,|[0,1]|)
by Lm4, Lm6, Lm8, TOPREAL1:3;

Lm17:  not |[1,0]| in LSeg (|[0,1]|,|[1,1]|)
by Lm7, Lm9, Lm11, TOPREAL1:4;

Lm18:  not |[1,1]| in LSeg (|[0,0]|,|[0,1]|)
by Lm4, Lm6, Lm10, TOPREAL1:3;

Lm19:  not |[1,1]| in LSeg (|[0,0]|,|[1,0]|)
by Lm5, Lm9, Lm11, TOPREAL1:4;

Lm20:  |[0,0]| in LSeg (|[0,0]|,|[0,1]|)
by RLTOPSP1:68;

Lm21:  |[0,0]| in LSeg (|[0,0]|,|[1,0]|)
by RLTOPSP1:68;

Lm22:  |[0,1]| in LSeg (|[0,0]|,|[0,1]|)
by RLTOPSP1:68;

Lm23:  |[0,1]| in LSeg (|[0,1]|,|[1,1]|)
by RLTOPSP1:68;

Lm24:  |[1,0]| in LSeg (|[0,0]|,|[1,0]|)
by RLTOPSP1:68;

Lm25:  |[1,0]| in LSeg (|[1,0]|,|[1,1]|)
by RLTOPSP1:68;

Lm26:  |[1,1]| in LSeg (|[0,1]|,|[1,1]|)
by RLTOPSP1:68;

Lm27:  |[1,1]| in LSeg (|[1,0]|,|[1,1]|)
by RLTOPSP1:68;

set L =  {  p where p is Point of (TOP-REAL 2) : ( ( p `1 = 0 & p `2 <= 1 & p `2 >= 0 ) or ( p `1 <= 1 & p `1 >= 0 & p `2 = 1 ) or ( p `1 <= 1 & p `1 >= 0 & p `2 = 0 ) or ( p `1 = 1 & p `2 <= 1 & p `2 >= 0 ) )  }  ;

Lm28:  |[0,0]| in  {  p where p is Point of (TOP-REAL 2) : ( ( p `1 = 0 & p `2 <= 1 & p `2 >= 0 ) or ( p `1 <= 1 & p `1 >= 0 & p `2 = 1 ) or ( p `1 <= 1 & p `1 >= 0 & p `2 = 0 ) or ( p `1 = 1 & p `2 <= 1 & p `2 >= 0 ) )  }
by Lm4, Lm5;

Lm29:  |[1,1]| in  {  p where p is Point of (TOP-REAL 2) : ( ( p `1 = 0 & p `2 <= 1 & p `2 >= 0 ) or ( p `1 <= 1 & p `1 >= 0 & p `2 = 1 ) or ( p `1 <= 1 & p `1 >= 0 & p `2 = 0 ) or ( p `1 = 1 & p `2 <= 1 & p `2 >= 0 ) )  }
by Lm10, Lm11;

Lm30:  for p1, p2 being Point of (TOP-REAL 2) st p1 <> p2 & p2 in R^2-unit_square & p1 in LSeg (|[0,0]|,|[0,1]|) holds
ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 <> p2 & p2 in R^2-unit_square & p1 in LSeg (|[0,0]|,|[0,1]|) implies ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) )
assume that
A1: p1 <> p2 and
A2: p2 in R^2-unit_square and
A3: p1 in LSeg (|[0,0]|,|[0,1]|) ; ::_thesis: ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A4: LSeg (|[0,0]|,p1) c= LSeg (|[0,0]|,|[0,1]|) by A3, Lm20, TOPREAL1:6;
 |[0,0]| in LSeg (p1,|[0,0]|) by RLTOPSP1:68;
then  |[0,0]| in (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by Lm21, XBOOLE_0:def_4;
then A5: {|[0,0]|} c= (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by ZFMISC_1:31;
A6: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) c= (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by A3, Lm20, TOPREAL1:6, XBOOLE_1:26;
then A7: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {|[0,0]|} by A5, TOPREAL1:17, XBOOLE_0:def_10;
A8: (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by TOPREAL1:20, XBOOLE_0:def_7;
then A9: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by A4, XBOOLE_1:3, XBOOLE_1:26;
 |[0,1]| in LSeg (|[0,1]|,p1) by RLTOPSP1:68;
then  |[0,1]| in (LSeg (|[0,1]|,p1)) /\ (LSeg (|[0,1]|,|[1,1]|)) by Lm23, XBOOLE_0:def_4;
then A10: {|[0,1]|} c= (LSeg (|[0,1]|,p1)) /\ (LSeg (|[0,1]|,|[1,1]|)) by ZFMISC_1:31;
A11: ( p2 in (LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|)) or p2 in (LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)) ) by A2, TOPREAL1:def_2, XBOOLE_0:def_3;
A12: (LSeg (|[0,1]|,p1)) /\ (LSeg (|[0,1]|,|[1,1]|)) c= {|[0,1]|} by A3, Lm22, TOPREAL1:6, TOPREAL1:15, XBOOLE_1:26;
A13: LSeg (p1,|[0,1]|) c= LSeg (|[0,0]|,|[0,1]|) by A3, Lm22, TOPREAL1:6;
then A14: (LSeg (|[0,1]|,p1)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by A8, XBOOLE_1:3, XBOOLE_1:26;
consider p being Point of (TOP-REAL 2) such that
A15: p = p1 and
A16: p `1 = 0 and
A17: p `2 <= 1 and
A18: p `2 >= 0 by A3, TOPREAL1:13;
percases ( p2 in LSeg (|[0,0]|,|[0,1]|) or p2 in LSeg (|[0,1]|,|[1,1]|) or p2 in LSeg (|[0,0]|,|[1,0]|) or p2 in LSeg (|[1,0]|,|[1,1]|) ) by A11, XBOOLE_0:def_3;
supposeA19: p2 in LSeg (|[0,0]|,|[0,1]|) ; ::_thesis: ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
then A20: LSeg (p2,p1) c= LSeg (|[0,0]|,|[0,1]|) by A3, TOPREAL1:6;
A21: p = |[(p `1),(p `2)]| by EUCLID:53;
consider q being Point of (TOP-REAL 2) such that
A22: q = p2 and
A23: q `1 = 0 and
A24: q `2 <= 1 and
A25: q `2 >= 0 by A19, TOPREAL1:13;
A26: q = |[(q `1),(q `2)]| by EUCLID:53;
now__::_thesis:_(_(_p_`2_<_q_`2_&_LSeg_(p1,p2)_is_an_arc_of_p1,p2_&_(LSeg_(p1,|[0,0]|))_\/_((((LSeg_(|[0,0]|,|[1,0]|))_\/_(LSeg_(|[1,0]|,|[1,1]|)))_\/_(LSeg_(|[1,1]|,|[0,1]|)))_\/_(LSeg_(|[0,1]|,p2)))_is_an_arc_of_p1,p2_&_R^2-unit_square_=_(LSeg_(p1,p2))_\/_((LSeg_(p1,|[0,0]|))_\/_((((LSeg_(|[0,0]|,|[1,0]|))_\/_(LSeg_(|[1,0]|,|[1,1]|)))_\/_(LSeg_(|[1,1]|,|[0,1]|)))_\/_(LSeg_(|[0,1]|,p2))))_&_(LSeg_(p1,p2))_/\_((LSeg_(p1,|[0,0]|))_\/_((((LSeg_(|[0,0]|,|[1,0]|))_\/_(LSeg_(|[1,0]|,|[1,1]|)))_\/_(LSeg_(|[1,1]|,|[0,1]|)))_\/_(LSeg_(|[0,1]|,p2))))_=_{p1,p2}_)_or_(_p_`2_>_q_`2_&_ex_P1_being_Element_of_K19(_the_carrier_of_(TOP-REAL_2))_ex_P2_being_Element_of_K19(_the_carrier_of_(TOP-REAL_2))_st_
(_P1_is_an_arc_of_p1,p2_&_P2_is_an_arc_of_p1,p2_&_R^2-unit_square_=_P1_\/_P2_&_P1_/\_P2_=_{p1,p2}_)_)_)
percases ( p `2 < q `2 or p `2 > q `2 ) by A1, A15, A16, A22, A23, A21, A26, XXREAL_0:1;
caseA27: p `2 < q `2 ; ::_thesis: ( LSeg (p1,p2) is_an_arc_of p1,p2 & (LSeg (p1,|[0,0]|)) \/ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2))) is_an_arc_of p1,p2 & R^2-unit_square = (LSeg (p1,p2)) \/ ((LSeg (p1,|[0,0]|)) \/ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2)))) & (LSeg (p1,p2)) /\ ((LSeg (p1,|[0,0]|)) \/ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2)))) = {p1,p2} )
A28: (LSeg (p1,p2)) /\ (LSeg (p1,|[0,0]|)) c= {p1}
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in (LSeg (p1,p2)) /\ (LSeg (p1,|[0,0]|)) or a in {p1} )
assume A29: a in (LSeg (p1,p2)) /\ (LSeg (p1,|[0,0]|)) ; ::_thesis: a in {p1}
then reconsider p = a as Point of (TOP-REAL 2) ;
A30: p in LSeg (|[0,0]|,p1) by A29, XBOOLE_0:def_4;
 |[0,0]| `2 <= p1 `2 by A15, A18, EUCLID:52;
then A31: p `2 <= p1 `2 by A30, TOPREAL1:4;
A32: p in LSeg (p1,p2) by A29, XBOOLE_0:def_4;
then  p1 `2 <= p `2 by A15, A22, A27, TOPREAL1:4;
then A33: p1 `2 = p `2 by A31, XXREAL_0:1;
 p1 `1 <= p `1 by A15, A16, A22, A23, A32, TOPREAL1:3;
then  p `1 = 0 by A15, A16, A22, A23, A32, TOPREAL1:3;
then p = |[0,(p1 `2)]| by A33, EUCLID:53
.= p1 by A15, A16, EUCLID:53 ;
hence  a in {p1} by TARSKI:def_1; ::_thesis: verum
end;
A34: (LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) c= (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) by A19, Lm22, TOPREAL1:6, XBOOLE_1:26;
A35: now__::_thesis:_not_(LSeg_(p1,|[0,0]|))_/\_(LSeg_(|[0,1]|,p2))_<>_{}
set a = the Element of (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2));
assume A36: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2)) <> {} ; ::_thesis: contradiction
then reconsider p = the Element of (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2)) as Point of (TOP-REAL 2) by TARSKI:def_3;
A37: p in LSeg (|[0,0]|,p1) by A36, XBOOLE_0:def_4;
A38: p in LSeg (p2,|[0,1]|) by A36, XBOOLE_0:def_4;
 p2 `2 <= |[0,1]| `2 by A22, A24, EUCLID:52;
then A39: p2 `2 <= p `2 by A38, TOPREAL1:4;
 |[0,0]| `2 <= p1 `2 by A15, A18, EUCLID:52;
then  p `2 <= p1 `2 by A37, TOPREAL1:4;
hence  contradiction by A15, A22, A27, A39, XXREAL_0:2; ::_thesis: verum
end;
 |[0,1]| in LSeg (|[0,1]|,p2) by RLTOPSP1:68;
then  |[0,1]| in (LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) by Lm23, XBOOLE_0:def_4;
then A40: {|[0,1]|} c= (LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) by ZFMISC_1:31;
now__::_thesis:_not_|[0,0]|_in_(LSeg_(|[0,1]|,p2))_/\_(LSeg_(|[0,0]|,|[1,0]|))
assume  |[0,0]| in (LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) ; ::_thesis: contradiction
then A41: |[0,0]| in LSeg (p2,|[0,1]|) by XBOOLE_0:def_4;
 p2 `2 <= |[0,1]| `2 by A22, A24, EUCLID:52;
hence  contradiction by A18, A22, A27, A41, Lm5, TOPREAL1:4; ::_thesis: verum
end;
then A42: {|[0,0]|} <> (LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) by ZFMISC_1:31;
 (LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) c= {|[0,0]|} by A19, Lm22, TOPREAL1:6, TOPREAL1:17, XBOOLE_1:26;
then A43: (LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {} by A42, ZFMISC_1:33;
A44: (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) c= {|[0,0]|} by A3, A19, TOPREAL1:6, TOPREAL1:17, XBOOLE_1:26;
A45: (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,p2)) c= {p2}
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,p2)) or a in {p2} )
assume A46: a in (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,p2)) ; ::_thesis: a in {p2}
then reconsider p = a as Point of (TOP-REAL 2) ;
A47: p in LSeg (p2,|[0,1]|) by A46, XBOOLE_0:def_4;
 p2 `2 <= |[0,1]| `2 by A22, A24, EUCLID:52;
then A48: p2 `2 <= p `2 by A47, TOPREAL1:4;
A49: p in LSeg (p1,p2) by A46, XBOOLE_0:def_4;
then  p `2 <= p2 `2 by A15, A22, A27, TOPREAL1:4;
then A50: p2 `2 = p `2 by A48, XXREAL_0:1;
 p1 `1 <= p `1 by A15, A16, A22, A23, A49, TOPREAL1:3;
then  p `1 = 0 by A15, A16, A22, A23, A49, TOPREAL1:3;
then p = |[0,(p2 `2)]| by A50, EUCLID:53
.= p2 by A22, A23, EUCLID:53 ;
hence  a in {p2} by TARSKI:def_1; ::_thesis: verum
end;
A51: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) c= {|[0,1]|} by A3, Lm20, TOPREAL1:6, TOPREAL1:15, XBOOLE_1:26;
now__::_thesis:_not_|[0,1]|_in_(LSeg_(p1,|[0,0]|))_/\_(LSeg_(|[0,1]|,|[1,1]|))
assume  |[0,1]| in (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) ; ::_thesis: contradiction
then A52: |[0,1]| in LSeg (|[0,0]|,p1) by XBOOLE_0:def_4;
 |[0,0]| `2 <= p1 `2 by A15, A18, EUCLID:52;
then  |[0,1]| `2 <= p1 `2 by A52, TOPREAL1:4;
hence  contradiction by A15, A17, A24, A27, Lm7, XXREAL_0:1; ::_thesis: verum
end;
then A53: {|[0,1]|} <> (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) by ZFMISC_1:31;
set P1 = LSeg (p1,p2);
set P2 = (LSeg (p1,|[0,0]|)) \/ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2)));
A54: p1 in LSeg (p1,|[0,0]|) by RLTOPSP1:68;
A55: LSeg (|[0,1]|,p2) c= LSeg (|[0,0]|,|[0,1]|) by A19, Lm22, TOPREAL1:6;
 p1 in LSeg (p1,p2) by RLTOPSP1:68;
then  p1 in (LSeg (p1,p2)) /\ (LSeg (p1,|[0,0]|)) by A54, XBOOLE_0:def_4;
then  {p1} c= (LSeg (p1,p2)) /\ (LSeg (p1,|[0,0]|)) by ZFMISC_1:31;
then A56: (LSeg (p1,p2)) /\ (LSeg (p1,|[0,0]|)) = {p1} by A28, XBOOLE_0:def_10;
thus  LSeg (p1,p2) is_an_arc_of p1,p2 by A1, TOPREAL1:9; ::_thesis: ( (LSeg (p1,|[0,0]|)) \/ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2))) is_an_arc_of p1,p2 & R^2-unit_square = (LSeg (p1,p2)) \/ ((LSeg (p1,|[0,0]|)) \/ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2)))) & (LSeg (p1,p2)) /\ ((LSeg (p1,|[0,0]|)) \/ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2)))) = {p1,p2} )
A57: ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) /\ (LSeg (|[1,1]|,|[0,1]|)) = {} \/ {|[1,1]|} by Lm2, TOPREAL1:18, XBOOLE_1:23
.= {|[1,1]|} ;
 (LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)) is_an_arc_of |[0,0]|,|[1,1]| by Lm4, Lm8, TOPREAL1:12, TOPREAL1:16;
then A58: ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|)) is_an_arc_of |[0,0]|,|[0,1]| by A57, TOPREAL1:10;
(((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) /\ (LSeg (|[0,1]|,p2)) = ((LSeg (|[0,1]|,p2)) /\ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,1]|,|[0,1]|))) by XBOOLE_1:23
.= ({} \/ ((LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,1]|,|[0,1]|))) by A43, XBOOLE_1:23
.= {} \/ ((LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,1]|,|[0,1]|))) by A55, Lm3, XBOOLE_1:3, XBOOLE_1:26
.= {|[0,1]|} by A40, A34, TOPREAL1:15, XBOOLE_0:def_10 ;
then A59: (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2)) is_an_arc_of |[0,0]|,p2 by A58, TOPREAL1:10;
(LSeg (p1,|[0,0]|)) /\ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2))) = ((LSeg (p1,|[0,0]|)) /\ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|)))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2))) by XBOOLE_1:23
.= (((LSeg (p1,|[0,0]|)) /\ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,1]|,|[0,1]|)))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2))) by XBOOLE_1:23
.= ((((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,1]|,|[0,1]|)))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2))) by XBOOLE_1:23
.= {|[0,0]|} by A9, A7, A35, A51, A53, ZFMISC_1:33 ;
hence  (LSeg (p1,|[0,0]|)) \/ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2))) is_an_arc_of p1,p2 by A59, TOPREAL1:11; ::_thesis: ( R^2-unit_square = (LSeg (p1,p2)) \/ ((LSeg (p1,|[0,0]|)) \/ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2)))) & (LSeg (p1,p2)) /\ ((LSeg (p1,|[0,0]|)) \/ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2)))) = {p1,p2} )
 ((LSeg (|[0,1]|,p2)) \/ (LSeg (p2,p1))) \/ (LSeg (p1,|[0,0]|)) = LSeg (|[0,0]|,|[0,1]|) by A3, A19, TOPREAL1:7;
hence  R^2-unit_square = (((LSeg (p1,p2)) \/ (LSeg (|[0,1]|,p2))) \/ (LSeg (p1,|[0,0]|))) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) by TOPREAL1:def_2, XBOOLE_1:4
.= ((LSeg (p1,p2)) \/ ((LSeg (p1,|[0,0]|)) \/ (LSeg (|[0,1]|,p2)))) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) by XBOOLE_1:4
.= (LSeg (p1,p2)) \/ (((LSeg (p1,|[0,0]|)) \/ (LSeg (|[0,1]|,p2))) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|)))) by XBOOLE_1:4
.= (LSeg (p1,p2)) \/ ((LSeg (p1,|[0,0]|)) \/ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2)))) by XBOOLE_1:4 ;
::_thesis: (LSeg (p1,p2)) /\ ((LSeg (p1,|[0,0]|)) \/ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2)))) = {p1,p2}
A60: p2 in LSeg (|[0,1]|,p2) by RLTOPSP1:68;
 p2 in LSeg (p1,p2) by RLTOPSP1:68;
then  p2 in (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,p2)) by A60, XBOOLE_0:def_4;
then  {p2} c= (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,p2)) by ZFMISC_1:31;
then A61: (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,p2)) = {p2} by A45, XBOOLE_0:def_10;
A62: LSeg (p1,p2) c= LSeg (|[0,0]|,|[0,1]|) by A3, A19, TOPREAL1:6;
A63: (LSeg (p1,p2)) /\ ((LSeg (p1,|[0,0]|)) \/ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2)))) = ((LSeg (p1,p2)) /\ (LSeg (p1,|[0,0]|))) \/ ((LSeg (p1,p2)) /\ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2)))) by XBOOLE_1:23
.= ((LSeg (p1,p2)) /\ (LSeg (p1,|[0,0]|))) \/ (((LSeg (p1,p2)) /\ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|)))) \/ ((LSeg (p1,p2)) /\ (LSeg (|[0,1]|,p2)))) by XBOOLE_1:23
.= ((LSeg (p1,p2)) /\ (LSeg (p1,|[0,0]|))) \/ ((((LSeg (p1,p2)) /\ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (p1,p2)) /\ (LSeg (|[1,1]|,|[0,1]|)))) \/ ((LSeg (p1,p2)) /\ (LSeg (|[0,1]|,p2)))) by XBOOLE_1:23
.= {p1} \/ (((((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (p1,p2)) /\ (LSeg (|[1,1]|,|[0,1]|)))) \/ ((LSeg (p1,p2)) /\ (LSeg (|[0,1]|,p2)))) by A56, XBOOLE_1:23
.= {p1} \/ (((((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {}) \/ ((LSeg (p1,p2)) /\ (LSeg (|[1,1]|,|[0,1]|)))) \/ ((LSeg (p1,p2)) /\ (LSeg (|[0,1]|,p2)))) by A62, Lm3, XBOOLE_1:3, XBOOLE_1:26
.= {p1} \/ (((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ (((LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2})) by A61, XBOOLE_1:4
.= ({p1} \/ ((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)))) \/ (((LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2}) by XBOOLE_1:4 ;
A64: now__::_thesis:_(LSeg_(p1,p2))_/\_((LSeg_(p1,|[0,0]|))_\/_((((LSeg_(|[0,0]|,|[1,0]|))_\/_(LSeg_(|[1,0]|,|[1,1]|)))_\/_(LSeg_(|[1,1]|,|[0,1]|)))_\/_(LSeg_(|[0,1]|,p2))))_=_{p1}_\/_(((LSeg_(p1,p2))_/\_(LSeg_(|[0,1]|,|[1,1]|)))_\/_{p2})
percases ( p1 = |[0,0]| or p1 <> |[0,0]| ) ;
supposeA65: p1 = |[0,0]| ; ::_thesis: (LSeg (p1,p2)) /\ ((LSeg (p1,|[0,0]|)) \/ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2)))) = {p1} \/ (((LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2})
then  |[0,0]| in LSeg (p1,p2) by RLTOPSP1:68;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) <> {} by Lm21, XBOOLE_0:def_4;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {p1} by A44, A65, ZFMISC_1:33;
hence  (LSeg (p1,p2)) /\ ((LSeg (p1,|[0,0]|)) \/ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2)))) = {p1} \/ (((LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2}) by A63; ::_thesis: verum
end;
supposeA66: p1 <> |[0,0]| ; ::_thesis: (LSeg (p1,p2)) /\ ((LSeg (p1,|[0,0]|)) \/ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2)))) = {p1} \/ (((LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2})
now__::_thesis:_not_|[0,0]|_in_(LSeg_(p1,p2))_/\_(LSeg_(|[0,0]|,|[1,0]|))
assume  |[0,0]| in (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) ; ::_thesis: contradiction
then  |[0,0]| in LSeg (p1,p2) by XBOOLE_0:def_4;
then  p1 `2 <= |[0,0]| `2 by A15, A22, A27, TOPREAL1:4;
then  |[0,0]| `2 = p1 `2 by A3, Lm5, Lm7, TOPREAL1:4;
hence  contradiction by A15, A16, A66, Lm5, EUCLID:53; ::_thesis: verum
end;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) <> {|[0,0]|} by ZFMISC_1:31;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {} by A44, ZFMISC_1:33;
hence  (LSeg (p1,p2)) /\ ((LSeg (p1,|[0,0]|)) \/ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2)))) = {p1} \/ (((LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2}) by A63; ::_thesis: verum
end;
end;
end;
A67: (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) c= {|[0,1]|} by A3, A19, TOPREAL1:6, TOPREAL1:15, XBOOLE_1:26;
now__::_thesis:_(LSeg_(p1,p2))_/\_((LSeg_(p1,|[0,0]|))_\/_((((LSeg_(|[0,0]|,|[1,0]|))_\/_(LSeg_(|[1,0]|,|[1,1]|)))_\/_(LSeg_(|[1,1]|,|[0,1]|)))_\/_(LSeg_(|[0,1]|,p2))))_=_{p1,p2}
percases ( p2 <> |[0,1]| or p2 = |[0,1]| ) ;
supposeA68: p2 <> |[0,1]| ; ::_thesis: (LSeg (p1,p2)) /\ ((LSeg (p1,|[0,0]|)) \/ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2)))) = {p1,p2}
now__::_thesis:_not_|[0,1]|_in_(LSeg_(p1,p2))_/\_(LSeg_(|[0,1]|,|[1,1]|))
assume  |[0,1]| in (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) ; ::_thesis: contradiction
then  |[0,1]| in LSeg (p1,p2) by XBOOLE_0:def_4;
then A69: |[0,1]| `2 <= p2 `2 by A15, A22, A27, TOPREAL1:4;
 p2 `2 <= |[0,1]| `2 by A19, Lm5, Lm7, TOPREAL1:4;
then A70: |[0,1]| `2 = p2 `2 by A69, XXREAL_0:1;
p2 = |[(p2 `1),(p2 `2)]| by EUCLID:53
.= |[0,1]| by A22, A23, A70, EUCLID:52 ;
hence  contradiction by A68; ::_thesis: verum
end;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) <> {|[0,1]|} by ZFMISC_1:31;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by A67, ZFMISC_1:33;
hence  (LSeg (p1,p2)) /\ ((LSeg (p1,|[0,0]|)) \/ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2)))) = {p1,p2} by A64, ENUMSET1:1; ::_thesis: verum
end;
supposeA71: p2 = |[0,1]| ; ::_thesis: (LSeg (p1,p2)) /\ ((LSeg (p1,|[0,0]|)) \/ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2)))) = {p1,p2}
then  |[0,1]| in LSeg (p1,p2) by RLTOPSP1:68;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) <> {} by Lm23, XBOOLE_0:def_4;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {p2} by A67, A71, ZFMISC_1:33;
hence  (LSeg (p1,p2)) /\ ((LSeg (p1,|[0,0]|)) \/ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2)))) = {p1,p2} by A64, ENUMSET1:1; ::_thesis: verum
end;
end;
end;
hence  (LSeg (p1,p2)) /\ ((LSeg (p1,|[0,0]|)) \/ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2)))) = {p1,p2} ; ::_thesis: verum
end;
caseA72: p `2 > q `2 ; ::_thesis: ex P1 being Element of K19( the carrier of (TOP-REAL 2)) ex P2 being Element of K19( the carrier of (TOP-REAL 2)) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A73: (LSeg (p2,p1)) /\ (LSeg (|[0,1]|,p1)) c= {p1}
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in (LSeg (p2,p1)) /\ (LSeg (|[0,1]|,p1)) or a in {p1} )
assume A74: a in (LSeg (p2,p1)) /\ (LSeg (|[0,1]|,p1)) ; ::_thesis: a in {p1}
then reconsider p = a as Point of (TOP-REAL 2) ;
A75: p in LSeg (p1,|[0,1]|) by A74, XBOOLE_0:def_4;
 p1 `2 <= |[0,1]| `2 by A15, A17, EUCLID:52;
then A76: p1 `2 <= p `2 by A75, TOPREAL1:4;
A77: p in LSeg (p2,p1) by A74, XBOOLE_0:def_4;
then  p `2 <= p1 `2 by A15, A22, A72, TOPREAL1:4;
then A78: p1 `2 = p `2 by A76, XXREAL_0:1;
 p2 `1 <= p `1 by A15, A16, A22, A23, A77, TOPREAL1:3;
then  p `1 = 0 by A15, A16, A22, A23, A77, TOPREAL1:3;
then p = |[0,(p1 `2)]| by A78, EUCLID:53
.= p1 by A15, A16, EUCLID:53 ;
hence  a in {p1} by TARSKI:def_1; ::_thesis: verum
end;
A79: LSeg (p2,|[0,0]|) c= LSeg (|[0,0]|,|[0,1]|) by A19, Lm20, TOPREAL1:6;
A80: now__::_thesis:_not_(LSeg_(p2,|[0,0]|))_/\_(LSeg_(|[0,1]|,p1))_<>_{}
set a = the Element of (LSeg (p2,|[0,0]|)) /\ (LSeg (|[0,1]|,p1));
assume A81: (LSeg (p2,|[0,0]|)) /\ (LSeg (|[0,1]|,p1)) <> {} ; ::_thesis: contradiction
then reconsider p = the Element of (LSeg (p2,|[0,0]|)) /\ (LSeg (|[0,1]|,p1)) as Point of (TOP-REAL 2) by TARSKI:def_3;
A82: p in LSeg (|[0,0]|,p2) by A81, XBOOLE_0:def_4;
A83: p in LSeg (p1,|[0,1]|) by A81, XBOOLE_0:def_4;
 p1 `2 <= |[0,1]| `2 by A15, A17, EUCLID:52;
then A84: p1 `2 <= p `2 by A83, TOPREAL1:4;
 |[0,0]| `2 <= p2 `2 by A22, A25, EUCLID:52;
then  p `2 <= p2 `2 by A82, TOPREAL1:4;
hence  contradiction by A15, A22, A72, A84, XXREAL_0:2; ::_thesis: verum
end;
A85: (LSeg (p2,p1)) /\ (LSeg (|[0,0]|,|[1,0]|)) c= {|[0,0]|} by A3, A19, TOPREAL1:6, TOPREAL1:17, XBOOLE_1:26;
now__::_thesis:_not_|[0,1]|_in_(LSeg_(p2,|[0,0]|))_/\_(LSeg_(|[0,1]|,|[1,1]|))
assume  |[0,1]| in (LSeg (p2,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) ; ::_thesis: contradiction
then A86: |[0,1]| in LSeg (|[0,0]|,p2) by XBOOLE_0:def_4;
 |[0,0]| `2 <= p2 `2 by A22, A25, EUCLID:52;
then  |[0,1]| `2 <= p2 `2 by A86, TOPREAL1:4;
hence  contradiction by A17, A22, A24, A72, Lm7, XXREAL_0:1; ::_thesis: verum
end;
then A87: {|[0,1]|} <> (LSeg (p2,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) by ZFMISC_1:31;
A88: (LSeg (p2,|[0,0]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) c= {|[0,0]|} by A19, Lm20, TOPREAL1:6, TOPREAL1:17, XBOOLE_1:26;
now__::_thesis:_not_|[0,0]|_in_(LSeg_(|[0,1]|,p1))_/\_(LSeg_(|[0,0]|,|[1,0]|))
assume  |[0,0]| in (LSeg (|[0,1]|,p1)) /\ (LSeg (|[0,0]|,|[1,0]|)) ; ::_thesis: contradiction
then A89: |[0,0]| in LSeg (p1,|[0,1]|) by XBOOLE_0:def_4;
 p1 `2 <= |[0,1]| `2 by A15, A17, EUCLID:52;
hence  contradiction by A15, A25, A72, A89, Lm5, TOPREAL1:4; ::_thesis: verum
end;
then A90: {|[0,0]|} <> (LSeg (|[0,1]|,p1)) /\ (LSeg (|[0,0]|,|[1,0]|)) by ZFMISC_1:31;
A91: (LSeg (p2,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) c= {|[0,1]|} by A19, Lm20, TOPREAL1:6, TOPREAL1:15, XBOOLE_1:26;
A92: (LSeg (p2,p1)) /\ (LSeg (p2,|[0,0]|)) c= {p2}
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in (LSeg (p2,p1)) /\ (LSeg (p2,|[0,0]|)) or a in {p2} )
assume A93: a in (LSeg (p2,p1)) /\ (LSeg (p2,|[0,0]|)) ; ::_thesis: a in {p2}
then reconsider p = a as Point of (TOP-REAL 2) ;
A94: p in LSeg (|[0,0]|,p2) by A93, XBOOLE_0:def_4;
 |[0,0]| `2 <= p2 `2 by A22, A25, EUCLID:52;
then A95: p `2 <= p2 `2 by A94, TOPREAL1:4;
A96: p in LSeg (p2,p1) by A93, XBOOLE_0:def_4;
then  p2 `2 <= p `2 by A15, A22, A72, TOPREAL1:4;
then A97: p2 `2 = p `2 by A95, XXREAL_0:1;
 p2 `1 <= p `1 by A15, A16, A22, A23, A96, TOPREAL1:3;
then  p `1 = 0 by A15, A16, A22, A23, A96, TOPREAL1:3;
then p = |[0,(p2 `2)]| by A97, EUCLID:53
.= p2 by A22, A23, EUCLID:53 ;
hence  a in {p2} by TARSKI:def_1; ::_thesis: verum
end;
A98: (LSeg (|[0,1]|,p1)) /\ (LSeg (|[0,0]|,|[1,0]|)) c= {|[0,0]|} by A3, Lm22, TOPREAL1:6, TOPREAL1:17, XBOOLE_1:26;
take P1 = LSeg (p2,p1); ::_thesis: ex P2 being Element of K19( the carrier of (TOP-REAL 2)) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
take P2 = (LSeg (p2,|[0,0]|)) \/ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p1))); ::_thesis: ( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A99: p2 in LSeg (p2,|[0,0]|) by RLTOPSP1:68;
 p2 in LSeg (p2,p1) by RLTOPSP1:68;
then  p2 in (LSeg (p2,p1)) /\ (LSeg (p2,|[0,0]|)) by A99, XBOOLE_0:def_4;
then A100: {p2} c= (LSeg (p2,p1)) /\ (LSeg (p2,|[0,0]|)) by ZFMISC_1:31;
thus  P1 is_an_arc_of p1,p2 by A1, TOPREAL1:9; ::_thesis: ( P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A101: (LSeg (|[0,1]|,|[1,1]|)) /\ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) = {} \/ {|[1,1]|} by Lm2, TOPREAL1:18, XBOOLE_1:23
.= {|[1,1]|} ;
 (LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)) is_an_arc_of |[1,1]|,|[0,0]| by Lm4, Lm8, TOPREAL1:12, TOPREAL1:16;
then A102: ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|)) is_an_arc_of |[0,1]|,|[0,0]| by A101, TOPREAL1:11;
 |[0,0]| in LSeg (p2,|[0,0]|) by RLTOPSP1:68;
then  |[0,0]| in (LSeg (p2,|[0,0]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by Lm21, XBOOLE_0:def_4;
then A103: {|[0,0]|} c= (LSeg (p2,|[0,0]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by ZFMISC_1:31;
(LSeg (p1,|[0,1]|)) /\ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) = ((LSeg (|[0,1]|,p1)) /\ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (|[0,1]|,p1)) /\ (LSeg (|[1,1]|,|[0,1]|))) by XBOOLE_1:23
.= (((LSeg (|[0,1]|,p1)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (|[0,1]|,p1)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (|[0,1]|,p1)) /\ (LSeg (|[1,1]|,|[0,1]|))) by XBOOLE_1:23
.= ({} \/ ((LSeg (|[0,1]|,p1)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (|[0,1]|,p1)) /\ (LSeg (|[1,1]|,|[0,1]|))) by A98, A90, ZFMISC_1:33
.= {|[0,1]|} by A14, A10, A12, XBOOLE_0:def_10 ;
then A104: (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p1)) is_an_arc_of p1,|[0,0]| by A102, TOPREAL1:11;
((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p1))) /\ (LSeg (|[0,0]|,p2)) = ((LSeg (p2,|[0,0]|)) /\ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|)))) \/ ((LSeg (p2,|[0,0]|)) /\ (LSeg (|[0,1]|,p1))) by XBOOLE_1:23
.= (((LSeg (p2,|[0,0]|)) /\ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (p2,|[0,0]|)) /\ (LSeg (|[1,1]|,|[0,1]|)))) \/ ((LSeg (p2,|[0,0]|)) /\ (LSeg (|[0,1]|,p1))) by XBOOLE_1:23
.= ((((LSeg (p2,|[0,0]|)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (p2,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (p2,|[0,0]|)) /\ (LSeg (|[1,1]|,|[0,1]|)))) \/ ((LSeg (p2,|[0,0]|)) /\ (LSeg (|[0,1]|,p1))) by XBOOLE_1:23
.= ((((LSeg (p2,|[0,0]|)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {}) \/ ((LSeg (p2,|[0,0]|)) /\ (LSeg (|[1,1]|,|[0,1]|)))) \/ ((LSeg (p2,|[0,0]|)) /\ (LSeg (|[0,1]|,p1))) by A79, Lm3, XBOOLE_1:3, XBOOLE_1:26
.= ((LSeg (p2,|[0,0]|)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {} by A80, A91, A87, ZFMISC_1:33
.= {|[0,0]|} by A103, A88, XBOOLE_0:def_10 ;
hence  P2 is_an_arc_of p1,p2 by A104, TOPREAL1:10; ::_thesis: ( R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
 ((LSeg (|[0,1]|,p1)) \/ (LSeg (p1,p2))) \/ (LSeg (p2,|[0,0]|)) = LSeg (|[0,0]|,|[0,1]|) by A3, A19, TOPREAL1:7;
hence  R^2-unit_square = (((LSeg (p2,p1)) \/ (LSeg (|[0,1]|,p1))) \/ (LSeg (p2,|[0,0]|))) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) by TOPREAL1:def_2, XBOOLE_1:4
.= ((LSeg (p2,p1)) \/ ((LSeg (p2,|[0,0]|)) \/ (LSeg (|[0,1]|,p1)))) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) by XBOOLE_1:4
.= (LSeg (p2,p1)) \/ (((LSeg (p2,|[0,0]|)) \/ (LSeg (|[0,1]|,p1))) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|)))) by XBOOLE_1:4
.= P1 \/ P2 by XBOOLE_1:4 ;
::_thesis: P1 /\ P2 = {p1,p2}
A105: p1 in LSeg (|[0,1]|,p1) by RLTOPSP1:68;
 p1 in LSeg (p2,p1) by RLTOPSP1:68;
then  p1 in (LSeg (p2,p1)) /\ (LSeg (|[0,1]|,p1)) by A105, XBOOLE_0:def_4;
then A106: {p1} c= (LSeg (p2,p1)) /\ (LSeg (|[0,1]|,p1)) by ZFMISC_1:31;
A107: P1 /\ P2 = ((LSeg (p2,p1)) /\ (LSeg (p2,|[0,0]|))) \/ ((LSeg (p2,p1)) /\ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p1)))) by XBOOLE_1:23
.= ((LSeg (p2,p1)) /\ (LSeg (p2,|[0,0]|))) \/ (((LSeg (p2,p1)) /\ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,|[0,1]|)))) \/ ((LSeg (p2,p1)) /\ (LSeg (|[0,1]|,p1)))) by XBOOLE_1:23
.= ((LSeg (p2,p1)) /\ (LSeg (p2,|[0,0]|))) \/ ((((LSeg (p2,p1)) /\ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (p2,p1)) /\ (LSeg (|[1,1]|,|[0,1]|)))) \/ ((LSeg (p2,p1)) /\ (LSeg (|[0,1]|,p1)))) by XBOOLE_1:23
.= ((LSeg (p2,p1)) /\ (LSeg (p2,|[0,0]|))) \/ (((((LSeg (p2,p1)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (p2,p1)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (p2,p1)) /\ (LSeg (|[1,1]|,|[0,1]|)))) \/ ((LSeg (p2,p1)) /\ (LSeg (|[0,1]|,p1)))) by XBOOLE_1:23
.= {p2} \/ (((((LSeg (p2,p1)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (p2,p1)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (p2,p1)) /\ (LSeg (|[1,1]|,|[0,1]|)))) \/ ((LSeg (p2,p1)) /\ (LSeg (|[0,1]|,p1)))) by A100, A92, XBOOLE_0:def_10
.= {p2} \/ (((((LSeg (p2,p1)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {}) \/ ((LSeg (p2,p1)) /\ (LSeg (|[1,1]|,|[0,1]|)))) \/ ((LSeg (p2,p1)) /\ (LSeg (|[0,1]|,p1)))) by A20, Lm3, XBOOLE_1:3, XBOOLE_1:26
.= {p2} \/ ((((LSeg (p2,p1)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (p2,p1)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ {p1}) by A106, A73, XBOOLE_0:def_10
.= {p2} \/ (((LSeg (p2,p1)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ (((LSeg (p2,p1)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p1})) by XBOOLE_1:4
.= ({p2} \/ ((LSeg (p2,p1)) /\ (LSeg (|[0,0]|,|[1,0]|)))) \/ (((LSeg (p2,p1)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p1}) by XBOOLE_1:4 ;
A108: now__::_thesis:_P1_/\_P2_=_{p2}_\/_(((LSeg_(p2,p1))_/\_(LSeg_(|[0,1]|,|[1,1]|)))_\/_{p1})
percases ( p2 = |[0,0]| or p2 <> |[0,0]| ) ;
supposeA109: p2 = |[0,0]| ; ::_thesis: P1 /\ P2 = {p2} \/ (((LSeg (p2,p1)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p1})
 p2 in LSeg (p2,p1) by RLTOPSP1:68;
then  (LSeg (p2,p1)) /\ (LSeg (|[0,0]|,|[1,0]|)) <> {} by A109, Lm21, XBOOLE_0:def_4;
then  (LSeg (p2,p1)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {p2} by A85, A109, ZFMISC_1:33;
hence  P1 /\ P2 = {p2} \/ (((LSeg (p2,p1)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p1}) by A107; ::_thesis: verum
end;
supposeA110: p2 <> |[0,0]| ; ::_thesis: P1 /\ P2 = {p2} \/ (((LSeg (p2,p1)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p1})
now__::_thesis:_not_|[0,0]|_in_(LSeg_(p2,p1))_/\_(LSeg_(|[0,0]|,|[1,0]|))
assume  |[0,0]| in (LSeg (p2,p1)) /\ (LSeg (|[0,0]|,|[1,0]|)) ; ::_thesis: contradiction
then  |[0,0]| in LSeg (p2,p1) by XBOOLE_0:def_4;
then  p2 `2 <= |[0,0]| `2 by A15, A22, A72, TOPREAL1:4;
then  |[0,0]| `2 = p2 `2 by A19, Lm5, Lm7, TOPREAL1:4;
hence  contradiction by A22, A23, A110, Lm5, EUCLID:53; ::_thesis: verum
end;
then  (LSeg (p2,p1)) /\ (LSeg (|[0,0]|,|[1,0]|)) <> {|[0,0]|} by ZFMISC_1:31;
then  (LSeg (p2,p1)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {} by A85, ZFMISC_1:33;
hence  P1 /\ P2 = {p2} \/ (((LSeg (p2,p1)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p1}) by A107; ::_thesis: verum
end;
end;
end;
A111: (LSeg (p2,p1)) /\ (LSeg (|[0,1]|,|[1,1]|)) c= {|[0,1]|} by A3, A19, TOPREAL1:6, TOPREAL1:15, XBOOLE_1:26;
now__::_thesis:_P1_/\_P2_=_{p1,p2}
percases ( p1 <> |[0,1]| or p1 = |[0,1]| ) ;
supposeA112: p1 <> |[0,1]| ; ::_thesis: P1 /\ P2 = {p1,p2}
now__::_thesis:_not_|[0,1]|_in_(LSeg_(p2,p1))_/\_(LSeg_(|[0,1]|,|[1,1]|))
assume  |[0,1]| in (LSeg (p2,p1)) /\ (LSeg (|[0,1]|,|[1,1]|)) ; ::_thesis: contradiction
then  |[0,1]| in LSeg (p2,p1) by XBOOLE_0:def_4;
then A113: |[0,1]| `2 <= p1 `2 by A15, A22, A72, TOPREAL1:4;
 p1 `2 <= |[0,1]| `2 by A3, Lm5, Lm7, TOPREAL1:4;
then A114: |[0,1]| `2 = p1 `2 by A113, XXREAL_0:1;
p1 = |[(p1 `1),(p1 `2)]| by EUCLID:53
.= |[0,1]| by A15, A16, A114, EUCLID:52 ;
hence  contradiction by A112; ::_thesis: verum
end;
then  (LSeg (p2,p1)) /\ (LSeg (|[0,1]|,|[1,1]|)) <> {|[0,1]|} by ZFMISC_1:31;
then  (LSeg (p2,p1)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by A111, ZFMISC_1:33;
hence  P1 /\ P2 = {p1,p2} by A108, ENUMSET1:1; ::_thesis: verum
end;
supposeA115: p1 = |[0,1]| ; ::_thesis: P1 /\ P2 = {p1,p2}
then  |[0,1]| in LSeg (p2,p1) by RLTOPSP1:68;
then  (LSeg (p2,p1)) /\ (LSeg (|[0,1]|,|[1,1]|)) <> {} by Lm23, XBOOLE_0:def_4;
then  (LSeg (p2,p1)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {p1} by A111, A115, ZFMISC_1:33;
hence  P1 /\ P2 = {p1,p2} by A108, ENUMSET1:1; ::_thesis: verum
end;
end;
end;
hence  P1 /\ P2 = {p1,p2} ; ::_thesis: verum
end;
end;
end;
hence  ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) ; ::_thesis: verum
end;
supposeA116: p2 in LSeg (|[0,1]|,|[1,1]|) ; ::_thesis: ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
then A117: LSeg (|[0,1]|,p2) c= LSeg (|[0,1]|,|[1,1]|) by Lm23, TOPREAL1:6;
 LSeg (p1,|[0,1]|) c= LSeg (|[0,0]|,|[0,1]|) by A3, Lm22, TOPREAL1:6;
then A118: (LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,1]|,p2)) c= (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) by A117, XBOOLE_1:27;
take P1 = (LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,1]|,p2)); ::_thesis: ex P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
take P2 = (LSeg (p1,|[0,0]|)) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))); ::_thesis: ( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A119: |[0,1]| in LSeg (|[0,1]|,p2) by RLTOPSP1:68;
 |[1,1]| in LSeg (|[1,1]|,p2) by RLTOPSP1:68;
then A120: |[1,1]| in (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[1,1]|,p2)) by Lm27, XBOOLE_0:def_4;
 |[0,1]| in LSeg (p1,|[0,1]|) by RLTOPSP1:68;
then  (LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,1]|,p2)) <> {} by A119, XBOOLE_0:def_4;
then A121: (LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,1]|,p2)) = {|[0,1]|} by A118, TOPREAL1:15, ZFMISC_1:33;
 ( p1 <> |[0,1]| or p2 <> |[0,1]| ) by A1;
hence  P1 is_an_arc_of p1,p2 by A121, TOPREAL1:12; ::_thesis: ( P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A122: LSeg (|[0,0]|,|[0,1]|) = (LSeg (p1,|[0,1]|)) \/ (LSeg (p1,|[0,0]|)) by A3, TOPREAL1:5;
A123: LSeg (|[1,0]|,|[1,1]|) is_an_arc_of |[1,0]|,|[1,1]| by Lm9, Lm11, TOPREAL1:9;
 LSeg (|[0,0]|,|[1,0]|) is_an_arc_of |[0,0]|,|[1,0]| by Lm4, Lm8, TOPREAL1:9;
then A124: (LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)) is_an_arc_of |[0,0]|,|[1,1]| by A123, TOPREAL1:2, TOPREAL1:16;
A125: LSeg (|[1,1]|,p2) c= LSeg (|[0,1]|,|[1,1]|) by A116, Lm26, TOPREAL1:6;
then A126: (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[1,1]|,p2)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) by XBOOLE_1:27;
A127: (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)) = {} by A125, Lm2, XBOOLE_1:3, XBOOLE_1:26;
((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) /\ (LSeg (|[1,1]|,p2)) = ((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,1]|,p2))) \/ ((LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[1,1]|,p2))) by XBOOLE_1:23
.= {|[1,1]|} by A127, A126, A120, TOPREAL1:18, ZFMISC_1:33 ;
then A128: ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2)) is_an_arc_of |[0,0]|,p2 by A124, TOPREAL1:10;
A129: (LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,1]|,p2)) = {p2} by A116, TOPREAL1:8;
A130: LSeg (|[0,1]|,|[1,1]|) = (LSeg (|[1,1]|,p2)) \/ (LSeg (|[0,1]|,p2)) by A116, TOPREAL1:5;
 (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,1]|,p2)) c= {|[0,1]|} by A4, A125, TOPREAL1:15, XBOOLE_1:27;
then A131: ( (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,1]|,p2)) = {|[0,1]|} or (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,1]|,p2)) = {} ) by ZFMISC_1:33;
A132: LSeg (|[0,1]|,p2) c= LSeg (|[0,1]|,|[1,1]|) by A116, Lm23, TOPREAL1:6;
then A133: (LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {} by Lm2, XBOOLE_1:3, XBOOLE_1:27;
A134: ex q being Point of (TOP-REAL 2) st
( q = p2 & q `1 <= 1 & q `1 >= 0 & q `2 = 1 ) by A116, TOPREAL1:13;
A135: now__::_thesis:_not_|[0,1]|_in_(LSeg_(p1,|[0,0]|))_/\_(LSeg_(|[1,1]|,p2))
A136: p2 `1 <= |[1,1]| `1 by A134, EUCLID:52;
assume A137: |[0,1]| in (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,1]|,p2)) ; ::_thesis: contradiction
then A138: |[0,1]| in LSeg (|[0,0]|,p1) by XBOOLE_0:def_4;
 |[0,1]| in LSeg (p2,|[1,1]|) by A137, XBOOLE_0:def_4;
then A139: |[0,1]| `1 = p2 `1 by A134, A136, Lm6, TOPREAL1:3;
 |[0,0]| `2 <= p1 `2 by A15, A18, EUCLID:52;
then  |[0,1]| `2 <= p1 `2 by A138, TOPREAL1:4;
then  |[0,1]| `2 = p1 `2 by A15, A17, Lm7, XXREAL_0:1;
then p1 = |[(|[0,1]| `1),(|[0,1]| `2)]| by A15, A16, Lm6, EUCLID:53
.= p2 by A134, A139, Lm7, EUCLID:53 ;
hence  contradiction by A1; ::_thesis: verum
end;
(LSeg (p1,|[0,0]|)) /\ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))) = ((LSeg (p1,|[0,0]|)) /\ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,1]|,p2))) by XBOOLE_1:23
.= ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|))) by A131, A135, XBOOLE_1:23, ZFMISC_1:31
.= {|[0,0]|} by A9, A5, A6, TOPREAL1:17, XBOOLE_0:def_10 ;
hence  P2 is_an_arc_of p1,p2 by A128, TOPREAL1:11; ::_thesis: ( R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
thus P1 \/ P2 = (LSeg (|[0,1]|,p2)) \/ ((LSeg (p1,|[0,1]|)) \/ ((LSeg (p1,|[0,0]|)) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))))) by XBOOLE_1:4
.= ((LSeg (|[0,0]|,|[0,1]|)) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2)))) \/ (LSeg (|[0,1]|,p2)) by A122, XBOOLE_1:4
.= (LSeg (|[0,0]|,|[0,1]|)) \/ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))) \/ (LSeg (|[0,1]|,p2))) by XBOOLE_1:4
.= (LSeg (|[0,0]|,|[0,1]|)) \/ ((LSeg (|[0,1]|,|[1,1]|)) \/ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)))) by A130, XBOOLE_1:4
.= R^2-unit_square by TOPREAL1:def_2, XBOOLE_1:4 ; ::_thesis: P1 /\ P2 = {p1,p2}
A140: {p1} = (LSeg (p1,|[0,1]|)) /\ (LSeg (p1,|[0,0]|)) by A3, TOPREAL1:8;
A141: P1 /\ P2 = ((LSeg (p1,|[0,1]|)) /\ ((LSeg (p1,|[0,0]|)) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))))) \/ ((LSeg (|[0,1]|,p2)) /\ ((LSeg (p1,|[0,0]|)) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))))) by XBOOLE_1:23
.= (((LSeg (p1,|[0,1]|)) /\ (LSeg (p1,|[0,0]|))) \/ ((LSeg (p1,|[0,1]|)) /\ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))))) \/ ((LSeg (|[0,1]|,p2)) /\ ((LSeg (p1,|[0,0]|)) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))))) by XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[0,1]|)) /\ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,1]|,p2))))) \/ ((LSeg (|[0,1]|,p2)) /\ ((LSeg (p1,|[0,0]|)) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))))) by A140, XBOOLE_1:23
.= ({p1} \/ ((((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,1]|,p2))))) \/ ((LSeg (|[0,1]|,p2)) /\ ((LSeg (p1,|[0,0]|)) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))))) by XBOOLE_1:23
.= ({p1} \/ ((((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,1]|,p2))))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[0,0]|))) \/ ((LSeg (|[0,1]|,p2)) /\ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))))) by XBOOLE_1:23
.= ({p1} \/ ((((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,1]|,p2))))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[0,0]|))) \/ (((LSeg (|[0,1]|,p2)) /\ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)))) \/ {p2})) by A129, XBOOLE_1:23
.= ({p1} \/ ((((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,1]|,p2))))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[0,0]|))) \/ ((((LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ {p2})) by XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,1]|,p2))))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[0,0]|))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ {p2})) by A14, A133 ;
A142: now__::_thesis:_P1_/\_P2_=_({p1}_\/_((LSeg_(p1,|[0,1]|))_/\_(LSeg_(|[0,0]|,|[1,0]|))))_\/_(((LSeg_(|[0,1]|,p2))_/\_(LSeg_(p1,|[0,0]|)))_\/_{p2})
percases ( p2 = |[1,1]| or ( p2 <> |[1,1]| & p2 <> |[0,1]| ) or p2 = |[0,1]| ) ;
supposeA143: p2 = |[1,1]| ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[0,0]|))) \/ {p2})
then A144: not p2 in LSeg (p1,|[0,1]|) by A13, Lm4, Lm6, Lm10, TOPREAL1:3;
(LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)) = (LSeg (p1,|[0,1]|)) /\ {p2} by A143, RLTOPSP1:70
.= {} by A144, Lm1 ;
hence  P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[0,0]|))) \/ {p2}) by A141, A143, TOPREAL1:18; ::_thesis: verum
end;
supposeA145: ( p2 <> |[1,1]| & p2 <> |[0,1]| ) ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[0,0]|))) \/ {p2})
now__::_thesis:_not_|[0,1]|_in_(LSeg_(p1,|[0,1]|))_/\_(LSeg_(|[1,1]|,p2))
assume  |[0,1]| in (LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)) ; ::_thesis: contradiction
then A146: |[0,1]| in LSeg (p2,|[1,1]|) by XBOOLE_0:def_4;
 p2 `1 <= |[1,1]| `1 by A134, EUCLID:52;
then  p2 `1 = 0 by A134, A146, Lm6, TOPREAL1:3;
hence  contradiction by A134, A145, EUCLID:53; ::_thesis: verum
end;
then A147: {|[0,1]|} <> (LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)) by ZFMISC_1:31;
 (LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)) c= {|[0,1]|} by A13, A125, TOPREAL1:15, XBOOLE_1:27;
then A148: (LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)) = {} by A147, ZFMISC_1:33;
now__::_thesis:_not_|[1,1]|_in_(LSeg_(|[0,1]|,p2))_/\_(LSeg_(|[1,0]|,|[1,1]|))
assume  |[1,1]| in (LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) ; ::_thesis: contradiction
then A149: |[1,1]| in LSeg (|[0,1]|,p2) by XBOOLE_0:def_4;
 |[0,1]| `1 <= p2 `1 by A134, EUCLID:52;
then  |[1,1]| `1 <= p2 `1 by A149, TOPREAL1:3;
then  p2 `1 = |[1,1]| `1 by A134, Lm10, XXREAL_0:1;
hence  contradiction by A134, A145, Lm10, EUCLID:53; ::_thesis: verum
end;
then A150: {|[1,1]|} <> (LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) by ZFMISC_1:31;
 (LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) c= {|[1,1]|} by A132, TOPREAL1:18, XBOOLE_1:27;
then  (LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by A150, ZFMISC_1:33;
hence  P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[0,0]|))) \/ {p2}) by A141, A148; ::_thesis: verum
end;
supposeA151: p2 = |[0,1]| ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[0,0]|))) \/ {p2})
then  p2 in LSeg (p1,|[0,1]|) by RLTOPSP1:68;
then A152: (LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)) <> {} by A151, Lm23, XBOOLE_0:def_4;
 (LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)) c= {p2} by A13, A151, TOPREAL1:15, XBOOLE_1:27;
then A153: (LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)) = {p2} by A152, ZFMISC_1:33;
(LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {|[0,1]|} /\ (LSeg (|[1,0]|,|[1,1]|)) by A151, RLTOPSP1:70
.= {} by Lm1, Lm15 ;
hence P1 /\ P2 = (({p1} \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)))) \/ {p2}) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[0,0]|))) \/ {p2}) by A141, A153, XBOOLE_1:4
.= ({p1} \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)))) \/ ((((LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[0,0]|))) \/ {p2}) \/ {p2}) by XBOOLE_1:4
.= ({p1} \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[0,0]|))) \/ ({p2} \/ {p2})) by XBOOLE_1:4
.= ({p1} \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[0,0]|))) \/ {p2}) ;
::_thesis: verum
end;
end;
end;
now__::_thesis:_P1_/\_P2_=_{p1,p2}
percases ( p1 = |[0,1]| or p1 = |[0,0]| or ( p1 <> |[0,0]| & p1 <> |[0,1]| ) ) ;
supposeA154: p1 = |[0,1]| ; ::_thesis: P1 /\ P2 = {p1,p2}
then  p1 in LSeg (|[0,1]|,p2) by RLTOPSP1:68;
then A155: (LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[0,0]|)) <> {} by A154, Lm22, XBOOLE_0:def_4;
 (LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[0,0]|)) c= {p1} by A132, A154, TOPREAL1:15, XBOOLE_1:27;
then A156: (LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[0,0]|)) = {p1} by A155, ZFMISC_1:33;
(LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {p1} /\ (LSeg (|[0,0]|,|[1,0]|)) by A154, RLTOPSP1:70
.= {} by A154, Lm1, Lm14 ;
hence P1 /\ P2 = ({p1} \/ {p1}) \/ {p2} by A142, A156, XBOOLE_1:4
.= {p1,p2} by ENUMSET1:1 ;
::_thesis: verum
end;
supposeA157: p1 = |[0,0]| ; ::_thesis: P1 /\ P2 = {p1,p2}
A158: not |[0,0]| in LSeg (|[0,1]|,p2) by A132, Lm5, Lm7, Lm11, TOPREAL1:4;
(LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[0,0]|)) = (LSeg (|[0,1]|,p2)) /\ {|[0,0]|} by A157, RLTOPSP1:70
.= {} by A158, Lm1 ;
hence  P1 /\ P2 = {p1,p2} by A142, A157, ENUMSET1:1, TOPREAL1:17; ::_thesis: verum
end;
supposeA159: ( p1 <> |[0,0]| & p1 <> |[0,1]| ) ; ::_thesis: P1 /\ P2 = {p1,p2}
now__::_thesis:_not_|[0,1]|_in_(LSeg_(|[0,1]|,p2))_/\_(LSeg_(p1,|[0,0]|))
assume  |[0,1]| in (LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[0,0]|)) ; ::_thesis: contradiction
then A160: |[0,1]| in LSeg (|[0,0]|,p1) by XBOOLE_0:def_4;
 |[0,0]| `2 <= p1 `2 by A15, A18, EUCLID:52;
then  |[0,1]| `2 <= p1 `2 by A160, TOPREAL1:4;
then  p1 `2 = 1 by A15, A17, Lm7, XXREAL_0:1;
hence  contradiction by A15, A16, A159, EUCLID:53; ::_thesis: verum
end;
then A161: {|[0,1]|} <> (LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[0,0]|)) by ZFMISC_1:31;
 (LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[0,0]|)) c= {|[0,1]|} by A4, A132, TOPREAL1:15, XBOOLE_1:27;
then A162: (LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[0,0]|)) = {} by A161, ZFMISC_1:33;
now__::_thesis:_not_|[0,0]|_in_(LSeg_(p1,|[0,1]|))_/\_(LSeg_(|[0,0]|,|[1,0]|))
assume  |[0,0]| in (LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) ; ::_thesis: contradiction
then A163: |[0,0]| in LSeg (p1,|[0,1]|) by XBOOLE_0:def_4;
 p1 `2 <= |[0,1]| `2 by A15, A17, EUCLID:52;
then  p1 `2 = 0 by A15, A18, A163, Lm5, TOPREAL1:4;
hence  contradiction by A15, A16, A159, EUCLID:53; ::_thesis: verum
end;
then A164: {|[0,0]|} <> (LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by ZFMISC_1:31;
 (LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) c= {|[0,0]|} by A13, TOPREAL1:17, XBOOLE_1:27;
then  (LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {} by A164, ZFMISC_1:33;
hence  P1 /\ P2 = {p1,p2} by A142, A162, ENUMSET1:1; ::_thesis: verum
end;
end;
end;
hence  P1 /\ P2 = {p1,p2} ; ::_thesis: verum
end;
supposeA165: p2 in LSeg (|[0,0]|,|[1,0]|) ; ::_thesis: ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
then A166: LSeg (|[0,0]|,p2) c= LSeg (|[0,0]|,|[1,0]|) by Lm21, TOPREAL1:6;
 LSeg (p1,|[0,0]|) c= LSeg (|[0,0]|,|[0,1]|) by A3, Lm20, TOPREAL1:6;
then A167: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,0]|,p2)) c= (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by A166, XBOOLE_1:27;
take P1 = (LSeg (p1,|[0,0]|)) \/ (LSeg (|[0,0]|,p2)); ::_thesis: ex P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
take P2 = (LSeg (p1,|[0,1]|)) \/ (((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,0]|,p2))); ::_thesis: ( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A168: |[0,0]| in LSeg (|[0,0]|,p2) by RLTOPSP1:68;
 |[1,0]| in LSeg (|[1,0]|,p2) by RLTOPSP1:68;
then A169: |[1,0]| in (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[1,0]|,p2)) by Lm25, XBOOLE_0:def_4;
 |[0,0]| in LSeg (p1,|[0,0]|) by RLTOPSP1:68;
then  (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,0]|,p2)) <> {} by A168, XBOOLE_0:def_4;
then A170: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,0]|,p2)) = {|[0,0]|} by A167, TOPREAL1:17, ZFMISC_1:33;
 ( p1 <> |[0,0]| or |[0,0]| <> p2 ) by A1;
hence  P1 is_an_arc_of p1,p2 by A170, TOPREAL1:12; ::_thesis: ( P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A171: LSeg (|[0,0]|,|[0,1]|) = (LSeg (p1,|[0,0]|)) \/ (LSeg (p1,|[0,1]|)) by A3, TOPREAL1:5;
A172: LSeg (|[1,0]|,|[1,1]|) is_an_arc_of |[1,1]|,|[1,0]| by Lm9, Lm11, TOPREAL1:9;
 LSeg (|[0,1]|,|[1,1]|) is_an_arc_of |[0,1]|,|[1,1]| by Lm6, Lm10, TOPREAL1:9;
then A173: (LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[1,0]|,|[1,1]|)) is_an_arc_of |[0,1]|,|[1,0]| by A172, TOPREAL1:2, TOPREAL1:18;
A174: LSeg (|[1,0]|,p2) c= LSeg (|[0,0]|,|[1,0]|) by A165, Lm24, TOPREAL1:6;
then A175: (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[1,0]|,p2)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by XBOOLE_1:27;
A176: (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[1,0]|,p2)) = {} by A174, Lm2, XBOOLE_1:3, XBOOLE_1:26;
((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) /\ (LSeg (|[1,0]|,p2)) = ((LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[1,0]|,p2))) \/ ((LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[1,0]|,p2))) by XBOOLE_1:23
.= {|[1,0]|} by A176, A175, A169, TOPREAL1:16, ZFMISC_1:33 ;
then A177: ((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,0]|,p2)) is_an_arc_of |[0,1]|,p2 by A173, TOPREAL1:10;
A178: (LSeg (|[0,0]|,p2)) /\ (LSeg (|[1,0]|,p2)) = {p2} by A165, TOPREAL1:8;
A179: LSeg (|[0,0]|,|[1,0]|) = (LSeg (|[1,0]|,p2)) \/ (LSeg (|[0,0]|,p2)) by A165, TOPREAL1:5;
 (LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,p2)) c= {|[0,0]|} by A13, A174, TOPREAL1:17, XBOOLE_1:27;
then A180: ( (LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,p2)) = {|[0,0]|} or (LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,p2)) = {} ) by ZFMISC_1:33;
A181: LSeg (|[0,0]|,p2) c= LSeg (|[0,0]|,|[1,0]|) by A165, Lm21, TOPREAL1:6;
then A182: (LSeg (|[0,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by Lm2, XBOOLE_1:3, XBOOLE_1:27;
A183: ex q being Point of (TOP-REAL 2) st
( q = p2 & q `1 <= 1 & q `1 >= 0 & q `2 = 0 ) by A165, TOPREAL1:13;
A184: now__::_thesis:_not_|[0,0]|_in_(LSeg_(p1,|[0,1]|))_/\_(LSeg_(|[1,0]|,p2))
A185: p2 `1 <= |[1,0]| `1 by A183, EUCLID:52;
assume A186: |[0,0]| in (LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,p2)) ; ::_thesis: contradiction
then A187: |[0,0]| in LSeg (p1,|[0,1]|) by XBOOLE_0:def_4;
 |[0,0]| in LSeg (p2,|[1,0]|) by A186, XBOOLE_0:def_4;
then A188: |[0,0]| `1 = p2 `1 by A183, A185, Lm4, TOPREAL1:3;
 p1 `2 <= |[0,1]| `2 by A15, A17, EUCLID:52;
then  |[0,0]| `2 = p1 `2 by A15, A18, A187, Lm5, TOPREAL1:4;
then p1 = |[(|[0,0]| `1),(|[0,0]| `2)]| by A15, A16, Lm4, EUCLID:53
.= p2 by A183, A188, Lm5, EUCLID:53 ;
hence  contradiction by A1; ::_thesis: verum
end;
(LSeg (p1,|[0,1]|)) /\ (((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,0]|,p2))) = ((LSeg (p1,|[0,1]|)) /\ ((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,p2))) by XBOOLE_1:23
.= ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ ((LSeg (|[0,1]|,p1)) /\ (LSeg (|[1,0]|,|[1,1]|))) by A180, A184, XBOOLE_1:23, ZFMISC_1:31
.= {|[0,1]|} by A14, A10, A12, XBOOLE_0:def_10 ;
hence  P2 is_an_arc_of p1,p2 by A177, TOPREAL1:11; ::_thesis: ( R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
thus P1 \/ P2 = (LSeg (|[0,0]|,p2)) \/ ((LSeg (p1,|[0,0]|)) \/ ((LSeg (p1,|[0,1]|)) \/ (((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,0]|,p2))))) by XBOOLE_1:4
.= ((LSeg (|[0,0]|,|[0,1]|)) \/ (((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,0]|,p2)))) \/ (LSeg (|[0,0]|,p2)) by A171, XBOOLE_1:4
.= (LSeg (|[0,0]|,|[0,1]|)) \/ ((((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,0]|,p2))) \/ (LSeg (|[0,0]|,p2))) by XBOOLE_1:4
.= (LSeg (|[0,0]|,|[0,1]|)) \/ (((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ ((LSeg (|[1,0]|,p2)) \/ (LSeg (|[0,0]|,p2)))) by XBOOLE_1:4
.= (LSeg (|[0,0]|,|[0,1]|)) \/ ((LSeg (|[0,1]|,|[1,1]|)) \/ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)))) by A179, XBOOLE_1:4
.= R^2-unit_square by TOPREAL1:def_2, XBOOLE_1:4 ; ::_thesis: P1 /\ P2 = {p1,p2}
A189: {p1} = (LSeg (p1,|[0,0]|)) /\ (LSeg (p1,|[0,1]|)) by A3, TOPREAL1:8;
A190: P1 /\ P2 = ((LSeg (p1,|[0,0]|)) /\ ((LSeg (p1,|[0,1]|)) \/ (((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,0]|,p2))))) \/ ((LSeg (|[0,0]|,p2)) /\ ((LSeg (p1,|[0,1]|)) \/ (((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,0]|,p2))))) by XBOOLE_1:23
.= (((LSeg (p1,|[0,0]|)) /\ (LSeg (p1,|[0,1]|))) \/ ((LSeg (p1,|[0,0]|)) /\ (((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,0]|,p2))))) \/ ((LSeg (|[0,0]|,p2)) /\ ((LSeg (p1,|[0,1]|)) \/ (((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,0]|,p2))))) by XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[0,0]|)) /\ ((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,p2))))) \/ ((LSeg (|[0,0]|,p2)) /\ ((LSeg (p1,|[0,1]|)) \/ (((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,0]|,p2))))) by A189, XBOOLE_1:23
.= ({p1} \/ ((((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,p2))))) \/ ((LSeg (|[0,0]|,p2)) /\ ((LSeg (p1,|[0,1]|)) \/ (((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,0]|,p2))))) by XBOOLE_1:23
.= ({p1} \/ ((((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,p2))))) \/ (((LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[0,1]|))) \/ ((LSeg (|[0,0]|,p2)) /\ (((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,0]|,p2))))) by XBOOLE_1:23
.= ({p1} \/ ((((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,p2))))) \/ (((LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[0,1]|))) \/ (((LSeg (|[0,0]|,p2)) /\ ((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[1,0]|,|[1,1]|)))) \/ {p2})) by A178, XBOOLE_1:23
.= ({p1} \/ ((((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,p2))))) \/ (((LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[0,1]|))) \/ ((((LSeg (|[0,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ ((LSeg (|[0,0]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ {p2})) by XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,p2))))) \/ (((LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[0,1]|))) \/ (((LSeg (|[0,0]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ {p2})) by A9, A182 ;
A191: now__::_thesis:_P1_/\_P2_=_({p1}_\/_((LSeg_(p1,|[0,0]|))_/\_(LSeg_(|[0,1]|,|[1,1]|))))_\/_(((LSeg_(|[0,0]|,p2))_/\_(LSeg_(p1,|[0,1]|)))_\/_{p2})
percases ( p2 = |[1,0]| or ( p2 <> |[1,0]| & p2 <> |[0,0]| ) or p2 = |[0,0]| ) ;
supposeA192: p2 = |[1,0]| ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ (((LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[0,1]|))) \/ {p2})
then  not p2 in LSeg (p1,|[0,0]|) by A4, Lm4, Lm6, Lm8, TOPREAL1:3;
then A193: LSeg (p1,|[0,0]|) misses {p2} by ZFMISC_1:50;
(LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,p2)) = (LSeg (p1,|[0,0]|)) /\ {p2} by A192, RLTOPSP1:70
.= {} by A193, XBOOLE_0:def_7 ;
hence  P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ (((LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[0,1]|))) \/ {p2}) by A190, A192, TOPREAL1:16; ::_thesis: verum
end;
supposeA194: ( p2 <> |[1,0]| & p2 <> |[0,0]| ) ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ (((LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[0,1]|))) \/ {p2})
now__::_thesis:_not_|[0,0]|_in_(LSeg_(p1,|[0,0]|))_/\_(LSeg_(|[1,0]|,p2))
assume  |[0,0]| in (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,p2)) ; ::_thesis: contradiction
then A195: |[0,0]| in LSeg (p2,|[1,0]|) by XBOOLE_0:def_4;
 p2 `1 <= |[1,0]| `1 by A183, EUCLID:52;
then  p2 `1 = 0 by A183, A195, Lm4, TOPREAL1:3;
hence  contradiction by A183, A194, EUCLID:53; ::_thesis: verum
end;
then A196: {|[0,0]|} <> (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,p2)) by ZFMISC_1:31;
 (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,p2)) c= {|[0,0]|} by A4, A174, TOPREAL1:17, XBOOLE_1:27;
then A197: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,p2)) = {} by A196, ZFMISC_1:33;
now__::_thesis:_not_|[1,0]|_in_(LSeg_(|[0,0]|,p2))_/\_(LSeg_(|[1,0]|,|[1,1]|))
assume  |[1,0]| in (LSeg (|[0,0]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) ; ::_thesis: contradiction
then A198: |[1,0]| in LSeg (|[0,0]|,p2) by XBOOLE_0:def_4;
 |[0,0]| `1 <= p2 `1 by A183, EUCLID:52;
then  |[1,0]| `1 <= p2 `1 by A198, TOPREAL1:3;
then  p2 `1 = |[1,0]| `1 by A183, Lm8, XXREAL_0:1;
hence  contradiction by A183, A194, Lm8, EUCLID:53; ::_thesis: verum
end;
then A199: {|[1,0]|} <> (LSeg (|[0,0]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) by ZFMISC_1:31;
 (LSeg (|[0,0]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) c= {|[1,0]|} by A181, TOPREAL1:16, XBOOLE_1:27;
then  (LSeg (|[0,0]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by A199, ZFMISC_1:33;
hence  P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ (((LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[0,1]|))) \/ {p2}) by A190, A197; ::_thesis: verum
end;
supposeA200: p2 = |[0,0]| ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ (((LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[0,1]|))) \/ {p2})
then  p2 in LSeg (p1,|[0,0]|) by RLTOPSP1:68;
then A201: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,p2)) <> {} by A200, Lm21, XBOOLE_0:def_4;
 (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,p2)) c= {p2} by A4, A200, TOPREAL1:17, XBOOLE_1:27;
then A202: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,p2)) = {p2} by A201, ZFMISC_1:33;
(LSeg (|[0,0]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {|[0,0]|} /\ (LSeg (|[1,0]|,|[1,1]|)) by A200, RLTOPSP1:70
.= {} by Lm1, Lm12 ;
hence P1 /\ P2 = (({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ {p2}) \/ (((LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[0,1]|))) \/ {p2}) by A190, A202, XBOOLE_1:4
.= ({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((((LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[0,1]|))) \/ {p2}) \/ {p2}) by XBOOLE_1:4
.= ({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ (((LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[0,1]|))) \/ ({p2} \/ {p2})) by XBOOLE_1:4
.= ({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ (((LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[0,1]|))) \/ {p2}) ;
::_thesis: verum
end;
end;
end;
now__::_thesis:_P1_/\_P2_=_{p1,p2}
percases ( p1 = |[0,1]| or p1 = |[0,0]| or ( p1 <> |[0,0]| & p1 <> |[0,1]| ) ) ;
supposeA203: p1 = |[0,1]| ; ::_thesis: P1 /\ P2 = {p1,p2}
then A204: (LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[0,1]|)) = (LSeg (|[0,0]|,p2)) /\ {p1} by RLTOPSP1:70;
 not p1 in LSeg (|[0,0]|,p2) by A181, A203, Lm5, Lm7, Lm9, TOPREAL1:4;
then  (LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[0,1]|)) = {} by A204, Lm1;
hence  P1 /\ P2 = {p1,p2} by A191, A203, ENUMSET1:1, TOPREAL1:15; ::_thesis: verum
end;
supposeA205: p1 = |[0,0]| ; ::_thesis: P1 /\ P2 = {p1,p2}
 |[0,0]| in LSeg (|[0,0]|,p2) by RLTOPSP1:68;
then A206: (LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[0,1]|)) <> {} by A205, Lm20, XBOOLE_0:def_4;
 (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {p1} /\ (LSeg (|[0,1]|,|[1,1]|)) by A205, RLTOPSP1:70;
then A207: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by A205, Lm1, Lm13;
 (LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[0,1]|)) c= (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A165, A205, Lm21, TOPREAL1:6, XBOOLE_1:26;
then  (LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[0,1]|)) = {p1} by A205, A206, TOPREAL1:17, ZFMISC_1:33;
hence P1 /\ P2 = ({p1} \/ {p1}) \/ {p2} by A191, A207, XBOOLE_1:4
.= {p1,p2} by ENUMSET1:1 ;
::_thesis: verum
end;
supposeA208: ( p1 <> |[0,0]| & p1 <> |[0,1]| ) ; ::_thesis: P1 /\ P2 = {p1,p2}
now__::_thesis:_not_|[0,0]|_in_(LSeg_(|[0,0]|,p2))_/\_(LSeg_(p1,|[0,1]|))
assume  |[0,0]| in (LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[0,1]|)) ; ::_thesis: contradiction
then A209: |[0,0]| in LSeg (p1,|[0,1]|) by XBOOLE_0:def_4;
 p1 `2 <= |[0,1]| `2 by A15, A17, EUCLID:52;
then  p1 `2 = 0 by A15, A18, A209, Lm5, TOPREAL1:4;
hence  contradiction by A15, A16, A208, EUCLID:53; ::_thesis: verum
end;
then A210: {|[0,0]|} <> (LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[0,1]|)) by ZFMISC_1:31;
 (LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[0,1]|)) c= (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A13, A181, XBOOLE_1:27;
then A211: (LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[0,1]|)) = {} by A210, TOPREAL1:17, ZFMISC_1:33;
now__::_thesis:_not_|[0,1]|_in_(LSeg_(p1,|[0,0]|))_/\_(LSeg_(|[0,1]|,|[1,1]|))
assume  |[0,1]| in (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) ; ::_thesis: contradiction
then A212: |[0,1]| in LSeg (|[0,0]|,p1) by XBOOLE_0:def_4;
 |[0,0]| `2 <= p1 `2 by A15, A18, EUCLID:52;
then  |[0,1]| `2 <= p1 `2 by A212, TOPREAL1:4;
then  p1 `2 = 1 by A15, A17, Lm7, XXREAL_0:1;
hence  contradiction by A15, A16, A208, EUCLID:53; ::_thesis: verum
end;
then A213: {|[0,1]|} <> (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) by ZFMISC_1:31;
 (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) c= {|[0,1]|} by A4, TOPREAL1:15, XBOOLE_1:27;
then  (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by A213, ZFMISC_1:33;
hence  P1 /\ P2 = {p1,p2} by A191, A211, ENUMSET1:1; ::_thesis: verum
end;
end;
end;
hence  P1 /\ P2 = {p1,p2} ; ::_thesis: verum
end;
supposeA214: p2 in LSeg (|[1,0]|,|[1,1]|) ; ::_thesis: ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
now__::_thesis:_for_a_being_set_st_a_in_(LSeg_(p1,|[0,0]|))_/\_(LSeg_(p1,|[0,1]|))_holds_
a_in_{p1}
let a be set ; ::_thesis: ( a in (LSeg (p1,|[0,0]|)) /\ (LSeg (p1,|[0,1]|)) implies a in {p1} )
assume A215: a in (LSeg (p1,|[0,0]|)) /\ (LSeg (p1,|[0,1]|)) ; ::_thesis: a in {p1}
then reconsider p = a as Point of (TOP-REAL 2) ;
 a in LSeg (p1,|[0,1]|) by A215, XBOOLE_0:def_4;
then A216: p1 `2 <= p `2 by A15, A17, Lm7, TOPREAL1:4;
A217: a in LSeg (|[0,0]|,p1) by A215, XBOOLE_0:def_4;
then  p `2 <= p1 `2 by A15, A18, Lm5, TOPREAL1:4;
then A218: p `2 = p1 `2 by A216, XXREAL_0:1;
 p `1 <= p1 `1 by A15, A16, A217, Lm4, TOPREAL1:3;
then  p `1 = p1 `1 by A15, A16, A217, Lm4, TOPREAL1:3;
then a = |[(p1 `1),(p1 `2)]| by A218, EUCLID:53
.= p1 by EUCLID:53 ;
hence  a in {p1} by TARSKI:def_1; ::_thesis: verum
end;
then A219: (LSeg (p1,|[0,0]|)) /\ (LSeg (p1,|[0,1]|)) c= {p1} by TARSKI:def_3;
A220: p2 in LSeg (|[1,1]|,p2) by RLTOPSP1:68;
 p2 in LSeg (|[1,0]|,p2) by RLTOPSP1:68;
then  p2 in (LSeg (|[1,0]|,p2)) /\ (LSeg (|[1,1]|,p2)) by A220, XBOOLE_0:def_4;
then A221: {p2} c= (LSeg (|[1,0]|,p2)) /\ (LSeg (|[1,1]|,p2)) by ZFMISC_1:31;
A222: ex q being Point of (TOP-REAL 2) st
( q = p2 & q `1 = 1 & q `2 <= 1 & q `2 >= 0 ) by A214, TOPREAL1:13;
now__::_thesis:_for_a_being_set_st_a_in_(LSeg_(|[1,0]|,p2))_/\_(LSeg_(|[1,1]|,p2))_holds_
a_in_{p2}
let a be set ; ::_thesis: ( a in (LSeg (|[1,0]|,p2)) /\ (LSeg (|[1,1]|,p2)) implies a in {p2} )
assume A223: a in (LSeg (|[1,0]|,p2)) /\ (LSeg (|[1,1]|,p2)) ; ::_thesis: a in {p2}
then reconsider p = a as Point of (TOP-REAL 2) ;
A224: a in LSeg (|[1,0]|,p2) by A223, XBOOLE_0:def_4;
then A225: p2 `1 <= p `1 by A222, Lm8, TOPREAL1:3;
 a in LSeg (p2,|[1,1]|) by A223, XBOOLE_0:def_4;
then A226: p2 `2 <= p `2 by A222, Lm11, TOPREAL1:4;
 p `1 <= p2 `1 by A222, A224, Lm8, TOPREAL1:3;
then A227: p `1 = p2 `1 by A225, XXREAL_0:1;
 p `2 <= p2 `2 by A222, A224, Lm9, TOPREAL1:4;
then  p `2 = p2 `2 by A226, XXREAL_0:1;
then a = |[(p2 `1),(p2 `2)]| by A227, EUCLID:53
.= p2 by EUCLID:53 ;
hence  a in {p2} by TARSKI:def_1; ::_thesis: verum
end;
then A228: (LSeg (|[1,0]|,p2)) /\ (LSeg (|[1,1]|,p2)) c= {p2} by TARSKI:def_3;
 LSeg (|[1,0]|,p2) c= LSeg (|[1,0]|,|[1,1]|) by A214, Lm25, TOPREAL1:6;
then A229: (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,0]|,p2)) c= {|[1,0]|} by TOPREAL1:16, XBOOLE_1:27;
take P1 = ((LSeg (p1,|[0,0]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,p2)); ::_thesis: ex P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
take P2 = ((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2)); ::_thesis: ( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A230: |[1,0]| in LSeg (|[1,0]|,p2) by RLTOPSP1:68;
 |[1,0]| in LSeg (|[0,0]|,|[1,0]|) by RLTOPSP1:68;
then  (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,0]|,p2)) <> {} by A230, XBOOLE_0:def_4;
then  (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,0]|,p2)) = {|[1,0]|} by A229, ZFMISC_1:33;
then A231: (LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,p2)) is_an_arc_of |[0,0]|,p2 by Lm4, Lm8, TOPREAL1:12;
 LSeg (|[1,0]|,p2) c= LSeg (|[1,0]|,|[1,1]|) by A214, Lm25, TOPREAL1:6;
then A232: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,p2)) = {} by A4, Lm3, XBOOLE_1:3, XBOOLE_1:27;
(LSeg (p1,|[0,0]|)) /\ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,p2))) = ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,p2))) by XBOOLE_1:23
.= {|[0,0]|} by A5, A6, A232, TOPREAL1:17, XBOOLE_0:def_10 ;
then  (LSeg (p1,|[0,0]|)) \/ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,p2))) is_an_arc_of p1,p2 by A231, TOPREAL1:11;
hence  P1 is_an_arc_of p1,p2 by XBOOLE_1:4; ::_thesis: ( P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
 |[1,1]| in LSeg (|[1,1]|,p2) by RLTOPSP1:68;
then A233: (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[1,1]|,p2)) <> {} by Lm26, XBOOLE_0:def_4;
A234: LSeg (|[1,1]|,p2) c= LSeg (|[1,0]|,|[1,1]|) by A214, Lm27, TOPREAL1:6;
then A235: (LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)) = {} by A13, Lm3, XBOOLE_1:3, XBOOLE_1:27;
 (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[1,1]|,p2)) c= {|[1,1]|} by A234, TOPREAL1:18, XBOOLE_1:27;
then  (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[1,1]|,p2)) = {|[1,1]|} by A233, ZFMISC_1:33;
then A236: (LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[1,1]|,p2)) is_an_arc_of |[0,1]|,p2 by Lm6, Lm10, TOPREAL1:12;
(LSeg (p1,|[0,1]|)) /\ ((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[1,1]|,p2))) = ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,1]|,p2))) by XBOOLE_1:23
.= {|[0,1]|} by A10, A12, A235, XBOOLE_0:def_10 ;
then  (LSeg (p1,|[0,1]|)) \/ ((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[1,1]|,p2))) is_an_arc_of p1,p2 by A236, TOPREAL1:11;
hence  P2 is_an_arc_of p1,p2 by XBOOLE_1:4; ::_thesis: ( R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
thus  R^2-unit_square = (((LSeg (p1,|[0,0]|)) \/ (LSeg (p1,|[0,1]|))) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) by A3, TOPREAL1:5, TOPREAL1:def_2
.= ((LSeg (p1,|[0,0]|)) \/ ((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) by XBOOLE_1:4
.= (LSeg (p1,|[0,0]|)) \/ (((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)))) by XBOOLE_1:4
.= (LSeg (p1,|[0,0]|)) \/ ((LSeg (|[0,0]|,|[1,0]|)) \/ (((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,0]|,|[1,1]|)))) by XBOOLE_1:4
.= ((LSeg (p1,|[0,0]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,0]|,|[1,1]|))) by XBOOLE_1:4
.= ((LSeg (p1,|[0,0]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ ((LSeg (|[1,1]|,p2)) \/ (LSeg (|[1,0]|,p2)))) by A214, TOPREAL1:5
.= ((LSeg (p1,|[0,0]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (|[1,0]|,p2)) \/ (((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2)))) by XBOOLE_1:4
.= P1 \/ P2 by XBOOLE_1:4 ; ::_thesis: P1 /\ P2 = {p1,p2}
A237: p1 in LSeg (p1,|[0,1]|) by RLTOPSP1:68;
 p1 in LSeg (p1,|[0,0]|) by RLTOPSP1:68;
then  p1 in (LSeg (p1,|[0,0]|)) /\ (LSeg (p1,|[0,1]|)) by A237, XBOOLE_0:def_4;
then  {p1} c= (LSeg (p1,|[0,0]|)) /\ (LSeg (p1,|[0,1]|)) by ZFMISC_1:31;
then A238: (LSeg (p1,|[0,0]|)) /\ (LSeg (p1,|[0,1]|)) = {p1} by A219, XBOOLE_0:def_10;
A239: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,1]|,p2)) = {} by A4, A234, Lm3, XBOOLE_1:3, XBOOLE_1:27;
A240: LSeg (|[1,0]|,p2) c= LSeg (|[1,0]|,|[1,1]|) by A214, Lm25, TOPREAL1:6;
then A241: (LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[0,1]|)) = {} by A13, Lm3, XBOOLE_1:3, XBOOLE_1:27;
A242: P1 /\ P2 = (((LSeg (p1,|[0,0]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) /\ (((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2)))) \/ ((LSeg (|[1,0]|,p2)) /\ (((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2)))) by XBOOLE_1:23
.= (((LSeg (p1,|[0,0]|)) /\ (((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2)))) \/ ((LSeg (|[0,0]|,|[1,0]|)) /\ (((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))))) \/ ((LSeg (|[1,0]|,p2)) /\ (((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2)))) by XBOOLE_1:23
.= ((((LSeg (p1,|[0,0]|)) /\ ((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,1]|,p2)))) \/ ((LSeg (|[0,0]|,|[1,0]|)) /\ (((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))))) \/ ((LSeg (|[1,0]|,p2)) /\ (((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2)))) by XBOOLE_1:23
.= ((((LSeg (p1,|[0,0]|)) /\ (LSeg (p1,|[0,1]|))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((LSeg (|[0,0]|,|[1,0]|)) /\ (((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))))) \/ ((LSeg (|[1,0]|,p2)) /\ (((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2)))) by A239, XBOOLE_1:23
.= (({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ (((LSeg (|[0,0]|,|[1,0]|)) /\ ((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,1]|,p2))))) \/ ((LSeg (|[1,0]|,p2)) /\ (((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2)))) by A238, XBOOLE_1:23
.= (({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (p1,|[0,1]|))) \/ ((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,1]|,p2))))) \/ ((LSeg (|[1,0]|,p2)) /\ (((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2)))) by XBOOLE_1:23
.= (({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ (((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (p1,|[0,1]|))) \/ ((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,1]|,p2))))) \/ (((LSeg (|[1,0]|,p2)) /\ ((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((LSeg (|[1,0]|,p2)) /\ (LSeg (|[1,1]|,p2)))) by Lm2, XBOOLE_1:23
.= (({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ (((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (p1,|[0,1]|))) \/ ((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,1]|,p2))))) \/ (((LSeg (|[1,0]|,p2)) /\ ((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|)))) \/ {p2}) by A221, A228, XBOOLE_0:def_10
.= (({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ (((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (p1,|[0,1]|))) \/ ((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,1]|,p2))))) \/ ((((LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[0,1]|))) \/ ((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ {p2}) by XBOOLE_1:23
.= (({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ (((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (p1,|[0,1]|))) \/ ((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,1]|,p2))))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2}) by A241 ;
A243: now__::_thesis:_P1_/\_P2_=_(({p1}_\/_((LSeg_(p1,|[0,0]|))_/\_(LSeg_(|[0,1]|,|[1,1]|))))_\/_((LSeg_(|[0,0]|,|[1,0]|))_/\_(LSeg_(p1,|[0,1]|))))_\/_{p2}
percases ( p2 = |[1,1]| or p2 = |[1,0]| or ( p2 <> |[1,0]| & p2 <> |[1,1]| ) ) ;
supposeA244: p2 = |[1,1]| ; ::_thesis: P1 /\ P2 = (({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (p1,|[0,1]|)))) \/ {p2}
then (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)) = (LSeg (|[0,0]|,|[1,0]|)) /\ {|[1,1]|} by RLTOPSP1:70
.= {} by Lm1, Lm19 ;
hence  P1 /\ P2 = (({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (p1,|[0,1]|)))) \/ {p2} by A242, A244, TOPREAL1:18; ::_thesis: verum
end;
supposeA245: p2 = |[1,0]| ; ::_thesis: P1 /\ P2 = (({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (p1,|[0,1]|)))) \/ {p2}
then (LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {|[1,0]|} /\ (LSeg (|[0,1]|,|[1,1]|)) by RLTOPSP1:70
.= {} by Lm1, Lm17 ;
hence P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (p1,|[0,1]|))) \/ {p2}) \/ {p2}) by A242, A245, TOPREAL1:16, XBOOLE_1:4
.= ({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ (((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (p1,|[0,1]|))) \/ ({p2} \/ {p2})) by XBOOLE_1:4
.= (({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (p1,|[0,1]|)))) \/ {p2} by XBOOLE_1:4 ;
::_thesis: verum
end;
supposeA246: ( p2 <> |[1,0]| & p2 <> |[1,1]| ) ; ::_thesis: P1 /\ P2 = (({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (p1,|[0,1]|)))) \/ {p2}
now__::_thesis:_not_|[1,1]|_in_(LSeg_(|[1,0]|,p2))_/\_(LSeg_(|[0,1]|,|[1,1]|))
assume  |[1,1]| in (LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) ; ::_thesis: contradiction
then A247: |[1,1]| in LSeg (|[1,0]|,p2) by XBOOLE_0:def_4;
 |[1,0]| `2 <= p2 `2 by A222, EUCLID:52;
then  |[1,1]| `2 <= p2 `2 by A247, TOPREAL1:4;
then  |[1,1]| `2 = p2 `2 by A222, Lm11, XXREAL_0:1;
then p2 = |[(|[1,1]| `1),(|[1,1]| `2)]| by A222, Lm10, EUCLID:53
.= |[1,1]| by EUCLID:53 ;
hence  contradiction by A246; ::_thesis: verum
end;
then A248: {|[1,1]|} <> (LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) by ZFMISC_1:31;
 (LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) by A240, XBOOLE_1:27;
then A249: (LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by A248, TOPREAL1:18, ZFMISC_1:33;
now__::_thesis:_not_|[1,0]|_in_(LSeg_(|[0,0]|,|[1,0]|))_/\_(LSeg_(|[1,1]|,p2))
assume  |[1,0]| in (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)) ; ::_thesis: contradiction
then A250: |[1,0]| in LSeg (p2,|[1,1]|) by XBOOLE_0:def_4;
 p2 `2 <= |[1,1]| `2 by A222, EUCLID:52;
then  p2 `2 = |[1,0]| `2 by A222, A250, Lm9, TOPREAL1:4;
then p2 = |[(|[1,0]| `1),(|[1,0]| `2)]| by A222, Lm8, EUCLID:53
.= |[1,0]| by EUCLID:53 ;
hence  contradiction by A246; ::_thesis: verum
end;
then A251: (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)) <> {|[1,0]|} by ZFMISC_1:31;
 (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)) c= {|[1,0]|} by A234, TOPREAL1:16, XBOOLE_1:27;
then  (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)) = {} by A251, ZFMISC_1:33;
hence  P1 /\ P2 = (({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (p1,|[0,1]|)))) \/ {p2} by A242, A249; ::_thesis: verum
end;
end;
end;
now__::_thesis:_P1_/\_P2_=_{p1,p2}
percases ( p1 = |[0,1]| or ( p1 <> |[0,1]| & p1 <> |[0,0]| ) or p1 = |[0,0]| ) ;
supposeA252: p1 = |[0,1]| ; ::_thesis: P1 /\ P2 = {p1,p2}
then (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (p1,|[0,1]|)) = (LSeg (|[0,0]|,|[1,0]|)) /\ {|[0,1]|} by RLTOPSP1:70
.= {} by Lm1, Lm14 ;
hence  P1 /\ P2 = {p1,p2} by A243, A252, ENUMSET1:1, TOPREAL1:15; ::_thesis: verum
end;
supposeA253: ( p1 <> |[0,1]| & p1 <> |[0,0]| ) ; ::_thesis: P1 /\ P2 = {p1,p2}
now__::_thesis:_not_|[0,1]|_in_(LSeg_(p1,|[0,0]|))_/\_(LSeg_(|[0,1]|,|[1,1]|))
assume  |[0,1]| in (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) ; ::_thesis: contradiction
then A254: |[0,1]| in LSeg (|[0,0]|,p1) by XBOOLE_0:def_4;
 |[0,0]| `2 <= p1 `2 by A15, A18, EUCLID:52;
then  |[0,1]| `2 <= p1 `2 by A254, TOPREAL1:4;
then  p1 `2 = |[0,1]| `2 by A15, A17, Lm7, XXREAL_0:1;
then p1 = |[(|[0,1]| `1),(|[0,1]| `2)]| by A15, A16, Lm6, EUCLID:53
.= |[0,1]| by EUCLID:53 ;
hence  contradiction by A253; ::_thesis: verum
end;
then A255: {|[0,1]|} <> (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) by ZFMISC_1:31;
 (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) c= {|[0,1]|} by A4, TOPREAL1:15, XBOOLE_1:27;
then A256: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by A255, ZFMISC_1:33;
now__::_thesis:_not_|[0,0]|_in_(LSeg_(|[0,0]|,|[1,0]|))_/\_(LSeg_(p1,|[0,1]|))
assume  |[0,0]| in (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (p1,|[0,1]|)) ; ::_thesis: contradiction
then  |[0,0]| in LSeg (p1,|[0,1]|) by XBOOLE_0:def_4;
then  p1 `2 = |[0,0]| `2 by A15, A17, A18, Lm5, Lm7, TOPREAL1:4;
then p1 = |[(|[0,0]| `1),(|[0,0]| `2)]| by A15, A16, Lm4, EUCLID:53
.= |[0,0]| by EUCLID:53 ;
hence  contradiction by A253; ::_thesis: verum
end;
then A257: {|[0,0]|} <> (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (p1,|[0,1]|)) by ZFMISC_1:31;
 (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (p1,|[0,1]|)) c= (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A13, XBOOLE_1:27;
then  (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (p1,|[0,1]|)) = {} by A257, TOPREAL1:17, ZFMISC_1:33;
hence  P1 /\ P2 = {p1,p2} by A243, A256, ENUMSET1:1; ::_thesis: verum
end;
supposeA258: p1 = |[0,0]| ; ::_thesis: P1 /\ P2 = {p1,p2}
then (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {|[0,0]|} /\ (LSeg (|[0,1]|,|[1,1]|)) by RLTOPSP1:70
.= {} by Lm1, Lm13 ;
hence  P1 /\ P2 = {p1,p2} by A243, A258, ENUMSET1:1, TOPREAL1:17; ::_thesis: verum
end;
end;
end;
hence  P1 /\ P2 = {p1,p2} ; ::_thesis: verum
end;
end;
end;

Lm31:  for p1, p2 being Point of (TOP-REAL 2) st p1 <> p2 & p2 in R^2-unit_square & p1 in LSeg (|[0,1]|,|[1,1]|) holds
ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 <> p2 & p2 in R^2-unit_square & p1 in LSeg (|[0,1]|,|[1,1]|) implies ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) )
assume that
A1: p1 <> p2 and
A2: p2 in R^2-unit_square and
A3: p1 in LSeg (|[0,1]|,|[1,1]|) ; ::_thesis: ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A4: ( p2 in (LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|)) or p2 in (LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)) ) by A2, TOPREAL1:def_2, XBOOLE_0:def_3;
A5: (LSeg (|[0,1]|,p1)) /\ (LSeg (|[0,0]|,|[0,1]|)) c= (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A3, Lm23, TOPREAL1:6, XBOOLE_1:26;
 |[1,1]| in LSeg (p1,|[1,1]|) by RLTOPSP1:68;
then A6: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) <> {} by Lm27, XBOOLE_0:def_4;
 |[0,1]| in LSeg (|[0,1]|,p1) by RLTOPSP1:68;
then A7: (LSeg (|[0,1]|,p1)) /\ (LSeg (|[0,0]|,|[0,1]|)) <> {} by Lm22, XBOOLE_0:def_4;
A8: LSeg (p1,|[1,1]|) c= LSeg (|[0,1]|,|[1,1]|) by A3, Lm26, TOPREAL1:6;
then A9: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {} by Lm2, XBOOLE_1:3, XBOOLE_1:26;
A10: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) c= {|[1,1]|} by A3, Lm26, TOPREAL1:6, TOPREAL1:18, XBOOLE_1:26;
A11: LSeg (|[0,1]|,p1) c= LSeg (|[0,1]|,|[1,1]|) by A3, Lm23, TOPREAL1:6;
then A12: (LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {} by Lm2, XBOOLE_1:3, XBOOLE_1:26;
consider q1 being Point of (TOP-REAL 2) such that
A13: q1 = p1 and
A14: q1 `1 <= 1 and
A15: q1 `1 >= 0 and
A16: q1 `2 = 1 by A3, TOPREAL1:13;
percases ( p2 in LSeg (|[0,0]|,|[0,1]|) or p2 in LSeg (|[0,1]|,|[1,1]|) or p2 in LSeg (|[0,0]|,|[1,0]|) or p2 in LSeg (|[1,0]|,|[1,1]|) ) by A4, XBOOLE_0:def_3;
supposeA17: p2 in LSeg (|[0,0]|,|[0,1]|) ; ::_thesis: ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
then A18: LSeg (|[0,1]|,p2) c= LSeg (|[0,0]|,|[0,1]|) by Lm22, TOPREAL1:6;
 LSeg (p1,|[0,1]|) c= LSeg (|[0,1]|,|[1,1]|) by A3, Lm23, TOPREAL1:6;
then A19: (LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,1]|,p2)) c= (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A18, XBOOLE_1:27;
take P1 = (LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,1]|,p2)); ::_thesis: ex P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
take P2 = (LSeg (p1,|[1,1]|)) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[0,0]|,p2))); ::_thesis: ( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A20: |[0,1]| in LSeg (|[0,1]|,p2) by RLTOPSP1:68;
 |[0,0]| in LSeg (|[0,0]|,p2) by RLTOPSP1:68;
then A21: {} <> (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,0]|,p2)) by Lm21, XBOOLE_0:def_4;
 |[0,1]| in LSeg (p1,|[0,1]|) by RLTOPSP1:68;
then  (LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,1]|,p2)) <> {} by A20, XBOOLE_0:def_4;
then A22: (LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,1]|,p2)) = {|[0,1]|} by A19, TOPREAL1:15, ZFMISC_1:33;
 ( p1 <> |[0,1]| or p2 <> |[0,1]| ) by A1;
hence  P1 is_an_arc_of p1,p2 by A22, TOPREAL1:12; ::_thesis: ( P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A23: (LSeg (p1,|[0,1]|)) \/ (LSeg (p1,|[1,1]|)) = LSeg (|[0,1]|,|[1,1]|) by A3, TOPREAL1:5;
A24: LSeg (|[1,0]|,|[1,1]|) is_an_arc_of |[1,1]|,|[1,0]| by Lm9, Lm11, TOPREAL1:9;
 LSeg (|[0,0]|,|[1,0]|) is_an_arc_of |[1,0]|,|[0,0]| by Lm4, Lm8, TOPREAL1:9;
then A25: (LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)) is_an_arc_of |[1,1]|,|[0,0]| by A24, TOPREAL1:2, TOPREAL1:16;
A26: (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,0]|,p2)) c= {|[0,0]|} by A17, Lm20, TOPREAL1:6, TOPREAL1:17, XBOOLE_1:26;
A27: (LSeg (|[0,0]|,p2)) \/ (LSeg (|[0,1]|,p2)) = LSeg (|[0,0]|,|[0,1]|) by A17, TOPREAL1:5;
A28: (LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,p2)) = {p2} by A17, TOPREAL1:8;
A29: LSeg (|[0,0]|,p2) c= LSeg (|[0,0]|,|[0,1]|) by A17, Lm20, TOPREAL1:6;
then A30: (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) = {} by Lm3, XBOOLE_1:3, XBOOLE_1:27;
A31: ex q2 being Point of (TOP-REAL 2) st
( q2 = p2 & q2 `1 = 0 & q2 `2 <= 1 & q2 `2 >= 0 ) by A17, TOPREAL1:13;
A32: now__::_thesis:_not_|[0,1]|_in_(LSeg_(p1,|[1,1]|))_/\_(LSeg_(|[0,0]|,p2))
A33: |[0,0]| `2 <= p2 `2 by A31, EUCLID:52;
assume A34: |[0,1]| in (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) ; ::_thesis: contradiction
then A35: |[0,1]| in LSeg (p1,|[1,1]|) by XBOOLE_0:def_4;
 |[0,1]| in LSeg (|[0,0]|,p2) by A34, XBOOLE_0:def_4;
then  |[0,1]| `2 <= p2 `2 by A33, TOPREAL1:4;
then A36: |[0,1]| `2 = p2 `2 by A31, Lm7, XXREAL_0:1;
 p1 `1 <= |[1,1]| `1 by A13, A14, EUCLID:52;
then  |[0,1]| `1 = p1 `1 by A13, A15, A35, Lm6, TOPREAL1:3;
then p1 = |[(|[0,1]| `1),(|[0,1]| `2)]| by A13, A16, Lm7, EUCLID:53
.= p2 by A31, A36, Lm6, EUCLID:53 ;
hence  contradiction by A1; ::_thesis: verum
end;
((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) /\ (LSeg (|[0,0]|,p2)) = ((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,0]|,p2))) \/ ((LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2))) by XBOOLE_1:23
.= {|[0,0]|} by A26, A21, A30, ZFMISC_1:33 ;
then A37: ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[0,0]|,p2)) is_an_arc_of |[1,1]|,p2 by A25, TOPREAL1:10;
 (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) c= (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A8, A29, XBOOLE_1:27;
then A38: ( (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) = {|[0,1]|} or (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) = {} ) by TOPREAL1:15, ZFMISC_1:33;
A39: LSeg (p2,|[0,1]|) c= LSeg (|[0,0]|,|[0,1]|) by A17, Lm22, TOPREAL1:6;
then A40: (LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by Lm3, XBOOLE_1:3, XBOOLE_1:27;
(LSeg (p1,|[1,1]|)) /\ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[0,0]|,p2))) = ((LSeg (p1,|[1,1]|)) /\ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,p2))) by XBOOLE_1:23
.= ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|))) by A38, A32, XBOOLE_1:23, ZFMISC_1:31
.= {|[1,1]|} by A9, A6, A10, ZFMISC_1:33 ;
hence  P2 is_an_arc_of p1,p2 by A37, TOPREAL1:11; ::_thesis: ( R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
thus P1 \/ P2 = (LSeg (|[0,1]|,p2)) \/ ((LSeg (p1,|[0,1]|)) \/ ((LSeg (p1,|[1,1]|)) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[0,0]|,p2))))) by XBOOLE_1:4
.= (LSeg (|[0,1]|,p2)) \/ ((LSeg (|[0,1]|,|[1,1]|)) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[0,0]|,p2)))) by A23, XBOOLE_1:4
.= (LSeg (|[0,1]|,p2)) \/ (((LSeg (|[0,1]|,|[1,1]|)) \/ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)))) \/ (LSeg (|[0,0]|,p2))) by XBOOLE_1:4
.= ((LSeg (|[0,0]|,p2)) \/ (LSeg (|[0,1]|,p2))) \/ ((LSeg (|[0,1]|,|[1,1]|)) \/ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)))) by XBOOLE_1:4
.= R^2-unit_square by A27, TOPREAL1:def_2, XBOOLE_1:4 ; ::_thesis: P1 /\ P2 = {p1,p2}
A41: {p1} = (LSeg (p1,|[0,1]|)) /\ (LSeg (p1,|[1,1]|)) by A3, TOPREAL1:8;
A42: P1 /\ P2 = ((LSeg (p1,|[0,1]|)) /\ ((LSeg (p1,|[1,1]|)) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[0,0]|,p2))))) \/ ((LSeg (|[0,1]|,p2)) /\ ((LSeg (p1,|[1,1]|)) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[0,0]|,p2))))) by XBOOLE_1:23
.= (((LSeg (p1,|[0,1]|)) /\ (LSeg (p1,|[1,1]|))) \/ ((LSeg (p1,|[0,1]|)) /\ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[0,0]|,p2))))) \/ ((LSeg (|[0,1]|,p2)) /\ ((LSeg (p1,|[1,1]|)) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[0,0]|,p2))))) by XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[0,1]|)) /\ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,p2))))) \/ ((LSeg (|[0,1]|,p2)) /\ ((LSeg (p1,|[1,1]|)) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[0,0]|,p2))))) by A41, XBOOLE_1:23
.= ({p1} \/ ((((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,p2))))) \/ ((LSeg (|[0,1]|,p2)) /\ ((LSeg (p1,|[1,1]|)) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[0,0]|,p2))))) by XBOOLE_1:23
.= ({p1} \/ ((((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,p2))))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[1,1]|))) \/ ((LSeg (|[0,1]|,p2)) /\ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[0,0]|,p2))))) by XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,p2))))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[1,1]|))) \/ (((LSeg (|[0,1]|,p2)) /\ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)))) \/ {p2})) by A12, A28, XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,p2))))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[1,1]|))) \/ ((((LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ {p2})) by XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,p2))))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[1,1]|))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {p2})) by A40 ;
A43: now__::_thesis:_P1_/\_P2_=_({p1}_\/_((LSeg_(p1,|[0,1]|))_/\_(LSeg_(|[0,0]|,p2))))_\/_(((LSeg_(|[0,1]|,p2))_/\_(LSeg_(|[0,0]|,|[1,0]|)))_\/_{p2})
percases ( p1 = |[0,1]| or p1 = |[1,1]| or ( p1 <> |[1,1]| & p1 <> |[0,1]| ) ) ;
supposeA44: p1 = |[0,1]| ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {p2})
then  p1 in LSeg (|[0,1]|,p2) by RLTOPSP1:68;
then A45: (LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[1,1]|)) <> {} by A44, Lm23, XBOOLE_0:def_4;
 (LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[1,1]|)) c= {p1} by A39, A44, TOPREAL1:15, XBOOLE_1:27;
then A46: (LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[1,1]|)) = {p1} by A45, ZFMISC_1:33;
 (LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {p1} /\ (LSeg (|[1,0]|,|[1,1]|)) by A44, RLTOPSP1:70;
then  (LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by A44, Lm1, Lm15;
hence P1 /\ P2 = ({p1} \/ ({p1} \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,p2))))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {p2}) by A42, A46, XBOOLE_1:4
.= (({p1} \/ {p1}) \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {p2}) by XBOOLE_1:4
.= ({p1} \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {p2}) ;
::_thesis: verum
end;
supposeA47: p1 = |[1,1]| ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {p2})
then A48: (LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[1,1]|)) = (LSeg (|[0,1]|,p2)) /\ {p1} by RLTOPSP1:70;
 not p1 in LSeg (|[0,1]|,p2) by A31, A47, Lm6, Lm10, TOPREAL1:3;
then  (LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[1,1]|)) = {} by A48, Lm1;
hence P1 /\ P2 = (({p1} \/ {p1}) \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {p2}) by A42, A47, TOPREAL1:18, XBOOLE_1:4
.= ({p1} \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {p2}) ;
::_thesis: verum
end;
supposeA49: ( p1 <> |[1,1]| & p1 <> |[0,1]| ) ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {p2})
now__::_thesis:_not_|[0,1]|_in_(LSeg_(|[0,1]|,p2))_/\_(LSeg_(p1,|[1,1]|))
assume  |[0,1]| in (LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[1,1]|)) ; ::_thesis: contradiction
then A50: |[0,1]| in LSeg (p1,|[1,1]|) by XBOOLE_0:def_4;
 p1 `1 <= |[1,1]| `1 by A13, A14, EUCLID:52;
then  p1 `1 = 0 by A13, A15, A50, Lm6, TOPREAL1:3;
hence  contradiction by A13, A16, A49, EUCLID:53; ::_thesis: verum
end;
then A51: {|[0,1]|} <> (LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[1,1]|)) by ZFMISC_1:31;
 (LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[1,1]|)) c= {|[0,1]|} by A8, A39, TOPREAL1:15, XBOOLE_1:27;
then A52: (LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[1,1]|)) = {} by A51, ZFMISC_1:33;
now__::_thesis:_not_|[1,1]|_in_(LSeg_(p1,|[0,1]|))_/\_(LSeg_(|[1,0]|,|[1,1]|))
assume  |[1,1]| in (LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) ; ::_thesis: contradiction
then A53: |[1,1]| in LSeg (|[0,1]|,p1) by XBOOLE_0:def_4;
 |[0,1]| `1 <= p1 `1 by A13, A15, EUCLID:52;
then  |[1,1]| `1 <= p1 `1 by A53, TOPREAL1:3;
then  p1 `1 = 1 by A13, A14, Lm10, XXREAL_0:1;
hence  contradiction by A13, A16, A49, EUCLID:53; ::_thesis: verum
end;
then A54: {|[1,1]|} <> (LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) by ZFMISC_1:31;
 (LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) c= {|[1,1]|} by A11, TOPREAL1:18, XBOOLE_1:27;
then  (LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by A54, ZFMISC_1:33;
hence  P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {p2}) by A42, A52; ::_thesis: verum
end;
end;
end;
now__::_thesis:_P1_/\_P2_=_{p1,p2}
percases ( ( p2 <> |[0,0]| & p2 <> |[0,1]| ) or p2 = |[0,0]| or p2 = |[0,1]| ) ;
supposeA55: ( p2 <> |[0,0]| & p2 <> |[0,1]| ) ; ::_thesis: P1 /\ P2 = {p1,p2}
now__::_thesis:_not_|[0,1]|_in_(LSeg_(p1,|[0,1]|))_/\_(LSeg_(|[0,0]|,p2))
assume  |[0,1]| in (LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,p2)) ; ::_thesis: contradiction
then A56: |[0,1]| in LSeg (|[0,0]|,p2) by XBOOLE_0:def_4;
 |[0,0]| `2 <= p2 `2 by A31, EUCLID:52;
then  |[0,1]| `2 <= p2 `2 by A56, TOPREAL1:4;
then  p2 `2 = 1 by A31, Lm7, XXREAL_0:1;
hence  contradiction by A31, A55, EUCLID:53; ::_thesis: verum
end;
then A57: {|[0,1]|} <> (LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,p2)) by ZFMISC_1:31;
 (LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,p2)) c= (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A11, A29, XBOOLE_1:27;
then A58: (LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,p2)) = {} by A57, TOPREAL1:15, ZFMISC_1:33;
now__::_thesis:_not_|[0,0]|_in_(LSeg_(|[0,1]|,p2))_/\_(LSeg_(|[0,0]|,|[1,0]|))
assume  |[0,0]| in (LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) ; ::_thesis: contradiction
then A59: |[0,0]| in LSeg (p2,|[0,1]|) by XBOOLE_0:def_4;
 p2 `2 <= |[0,1]| `2 by A31, EUCLID:52;
then  0 = p2 `2 by A31, A59, Lm5, TOPREAL1:4;
hence  contradiction by A31, A55, EUCLID:53; ::_thesis: verum
end;
then A60: {|[0,0]|} <> (LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) by ZFMISC_1:31;
 (LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) c= {|[0,0]|} by A39, TOPREAL1:17, XBOOLE_1:27;
then  (LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {} by A60, ZFMISC_1:33;
hence  P1 /\ P2 = {p1,p2} by A43, A58, ENUMSET1:1; ::_thesis: verum
end;
supposeA61: p2 = |[0,0]| ; ::_thesis: P1 /\ P2 = {p1,p2}
then A62: (LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,p2)) = (LSeg (p1,|[0,1]|)) /\ {|[0,0]|} by RLTOPSP1:70;
 not |[0,0]| in LSeg (p1,|[0,1]|) by A11, Lm5, Lm7, Lm11, TOPREAL1:4;
then  (LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,p2)) = {} by A62, Lm1;
hence  P1 /\ P2 = {p1,p2} by A43, A61, ENUMSET1:1, TOPREAL1:17; ::_thesis: verum
end;
supposeA63: p2 = |[0,1]| ; ::_thesis: P1 /\ P2 = {p1,p2}
then  p2 in LSeg (p1,|[0,1]|) by RLTOPSP1:68;
then A64: {} <> (LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,p2)) by A63, Lm22, XBOOLE_0:def_4;
 (LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {|[0,1]|} /\ (LSeg (|[0,0]|,|[1,0]|)) by A63, RLTOPSP1:70;
then A65: (LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {} by Lm1, Lm14;
 (LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,p2)) c= (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A11, A29, XBOOLE_1:27;
then  (LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,p2)) = {p2} by A63, A64, TOPREAL1:15, ZFMISC_1:33;
hence P1 /\ P2 = {p1} \/ ({p2} \/ {p2}) by A43, A65, XBOOLE_1:4
.= {p1,p2} by ENUMSET1:1 ;
::_thesis: verum
end;
end;
end;
hence  P1 /\ P2 = {p1,p2} ; ::_thesis: verum
end;
supposeA66: p2 in LSeg (|[0,1]|,|[1,1]|) ; ::_thesis: ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A67: q1 = |[(q1 `1),(q1 `2)]| by EUCLID:53;
A68: LSeg (p1,p2) c= LSeg (|[0,1]|,|[1,1]|) by A3, A66, TOPREAL1:6;
consider q being Point of (TOP-REAL 2) such that
A69: q = p2 and
A70: q `1 <= 1 and
A71: q `1 >= 0 and
A72: q `2 = 1 by A66, TOPREAL1:13;
A73: q = |[(q `1),(q `2)]| by EUCLID:53;
now__::_thesis:_ex_P1_being_Element_of_K19(_the_carrier_of_(TOP-REAL_2))_ex_P2_being_Element_of_K19(_the_carrier_of_(TOP-REAL_2))_st_
(_P1_is_an_arc_of_p1,p2_&_P2_is_an_arc_of_p1,p2_&_P1_\/_P2_=_R^2-unit_square_&_P1_/\_P2_=_{p1,p2}_)
percases ( q1 `1 < q `1 or q `1 < q1 `1 ) by A1, A13, A16, A69, A72, A67, A73, XXREAL_0:1;
supposeA74: q1 `1 < q `1 ; ::_thesis: ex P1 being Element of K19( the carrier of (TOP-REAL 2)) ex P2 being Element of K19( the carrier of (TOP-REAL 2)) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
A75: (LSeg (p1,p2)) /\ (LSeg (|[1,1]|,p2)) c= {p2}
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in (LSeg (p1,p2)) /\ (LSeg (|[1,1]|,p2)) or a in {p2} )
assume A76: a in (LSeg (p1,p2)) /\ (LSeg (|[1,1]|,p2)) ; ::_thesis: a in {p2}
then reconsider p = a as Point of (TOP-REAL 2) ;
A77: p in LSeg (p2,|[1,1]|) by A76, XBOOLE_0:def_4;
 p2 `1 <= |[1,1]| `1 by A69, A70, EUCLID:52;
then A78: p2 `1 <= p `1 by A77, TOPREAL1:3;
A79: p in LSeg (p1,p2) by A76, XBOOLE_0:def_4;
then A80: p `2 <= p2 `2 by A13, A16, A69, A72, TOPREAL1:4;
 p `1 <= p2 `1 by A13, A69, A74, A79, TOPREAL1:3;
then A81: p2 `1 = p `1 by A78, XXREAL_0:1;
 p1 `2 <= p `2 by A13, A16, A69, A72, A79, TOPREAL1:4;
then  p `2 = 1 by A13, A16, A69, A72, A80, XXREAL_0:1;
then p = |[(p2 `1),1]| by A81, EUCLID:53
.= p2 by A69, A72, EUCLID:53 ;
hence  a in {p2} by TARSKI:def_1; ::_thesis: verum
end;
 (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by TOPREAL1:19, XBOOLE_0:def_7;
then A82: (LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {} by A11, XBOOLE_1:3, XBOOLE_1:26;
A83: now__::_thesis:_not_(LSeg_(p1,|[0,1]|))_/\_(LSeg_(|[1,1]|,p2))_<>_{}
set a = the Element of (LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,1]|,p2));
assume A84: (LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)) <> {} ; ::_thesis: contradiction
then reconsider p = the Element of (LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)) as Point of (TOP-REAL 2) by TARSKI:def_3;
A85: p in LSeg (|[0,1]|,p1) by A84, XBOOLE_0:def_4;
A86: p in LSeg (p2,|[1,1]|) by A84, XBOOLE_0:def_4;
 p2 `1 <= |[1,1]| `1 by A69, A70, EUCLID:52;
then A87: p2 `1 <= p `1 by A86, TOPREAL1:3;
 |[0,1]| `1 <= p1 `1 by A13, A15, EUCLID:52;
then  p `1 <= p1 `1 by A85, TOPREAL1:3;
hence  contradiction by A13, A69, A74, A87, XXREAL_0:2; ::_thesis: verum
end;
A88: ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) /\ (LSeg (|[1,0]|,|[1,1]|)) = ((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ ((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|))) by XBOOLE_1:23
.= {|[1,0]|} by Lm3, TOPREAL1:16 ;
 (LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|)) is_an_arc_of |[0,1]|,|[1,0]| by Lm5, Lm7, TOPREAL1:9, TOPREAL1:10, TOPREAL1:17;
then A89: ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,|[1,1]|)) is_an_arc_of |[0,1]|,|[1,1]| by A88, TOPREAL1:10;
A90: (LSeg (p1,p2)) /\ (LSeg (p1,|[0,1]|)) c= {p1}
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in (LSeg (p1,p2)) /\ (LSeg (p1,|[0,1]|)) or a in {p1} )
assume A91: a in (LSeg (p1,p2)) /\ (LSeg (p1,|[0,1]|)) ; ::_thesis: a in {p1}
then reconsider p = a as Point of (TOP-REAL 2) ;
A92: p in LSeg (|[0,1]|,p1) by A91, XBOOLE_0:def_4;
 |[0,1]| `1 <= p1 `1 by A13, A15, EUCLID:52;
then A93: p `1 <= p1 `1 by A92, TOPREAL1:3;
A94: p in LSeg (p1,p2) by A91, XBOOLE_0:def_4;
then A95: p `2 <= p2 `2 by A13, A16, A69, A72, TOPREAL1:4;
 p1 `1 <= p `1 by A13, A69, A74, A94, TOPREAL1:3;
then A96: p1 `1 = p `1 by A93, XXREAL_0:1;
 p1 `2 <= p `2 by A13, A16, A69, A72, A94, TOPREAL1:4;
then  p `2 = 1 by A13, A16, A69, A72, A95, XXREAL_0:1;
then p = |[(p1 `1),1]| by A96, EUCLID:53
.= p1 by A13, A16, EUCLID:53 ;
hence  a in {p1} by TARSKI:def_1; ::_thesis: verum
end;
A97: (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) c= (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A3, A66, TOPREAL1:6, XBOOLE_1:26;
now__::_thesis:_not_|[1,1]|_in_(LSeg_(p1,|[0,1]|))_/\_(LSeg_(|[1,0]|,|[1,1]|))
assume  |[1,1]| in (LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) ; ::_thesis: contradiction
then A98: |[1,1]| in LSeg (|[0,1]|,p1) by XBOOLE_0:def_4;
 |[0,1]| `1 <= p1 `1 by A13, A15, EUCLID:52;
then  |[1,1]| `1 <= p1 `1 by A98, TOPREAL1:3;
hence  contradiction by A13, A14, A70, A74, Lm10, XXREAL_0:1; ::_thesis: verum
end;
then A99: {|[1,1]|} <> (LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) by ZFMISC_1:31;
 (LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) c= {|[1,1]|} by A3, Lm23, TOPREAL1:6, TOPREAL1:18, XBOOLE_1:26;
then A100: (LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by A99, ZFMISC_1:33;
 |[0,1]| in LSeg (p1,|[0,1]|) by RLTOPSP1:68;
then A101: (LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) <> {} by Lm22, XBOOLE_0:def_4;
now__::_thesis:_not_|[0,1]|_in_(LSeg_(|[0,0]|,|[0,1]|))_/\_(LSeg_(|[1,1]|,p2))
assume  |[0,1]| in (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)) ; ::_thesis: contradiction
then A102: |[0,1]| in LSeg (p2,|[1,1]|) by XBOOLE_0:def_4;
 p2 `1 <= |[1,1]| `1 by A69, A70, EUCLID:52;
hence  contradiction by A15, A69, A74, A102, Lm6, TOPREAL1:3; ::_thesis: verum
end;
then A103: {|[0,1]|} <> (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)) by ZFMISC_1:31;
 (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)) c= {|[0,1]|} by A66, Lm26, TOPREAL1:6, TOPREAL1:15, XBOOLE_1:26;
then A104: (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)) = {} by A103, ZFMISC_1:33;
take P1 = LSeg (p1,p2); ::_thesis: ex P2 being Element of K19( the carrier of (TOP-REAL 2)) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
take P2 = (LSeg (p1,|[0,1]|)) \/ ((((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))); ::_thesis: ( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
A105: p1 in LSeg (p1,|[0,1]|) by RLTOPSP1:68;
A106: (LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) c= (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A3, Lm23, TOPREAL1:6, XBOOLE_1:26;
 |[1,1]| in LSeg (|[1,1]|,p2) by RLTOPSP1:68;
then A107: (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[1,1]|,p2)) <> {} by Lm27, XBOOLE_0:def_4;
 (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[1,1]|,p2)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) by A66, Lm26, TOPREAL1:6, XBOOLE_1:26;
then A108: (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[1,1]|,p2)) = {|[1,1]|} by A107, TOPREAL1:18, ZFMISC_1:33;
thus  P1 is_an_arc_of p1,p2 by A1, TOPREAL1:9; ::_thesis: ( P2 is_an_arc_of p1,p2 & P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
A109: (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by TOPREAL1:19, XBOOLE_0:def_7;
 (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)) c= (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) by A66, Lm26, TOPREAL1:6, XBOOLE_1:26;
then A110: (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)) = {} by A109, XBOOLE_1:3;
(((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,|[1,1]|))) /\ (LSeg (|[1,1]|,p2)) = (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) /\ (LSeg (|[1,1]|,p2))) \/ ((LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[1,1]|,p2))) by XBOOLE_1:23
.= (((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,1]|,p2))) \/ ((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)))) \/ {|[1,1]|} by A108, XBOOLE_1:23
.= {|[1,1]|} by A104, A110 ;
then A111: (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2)) is_an_arc_of |[0,1]|,p2 by A89, TOPREAL1:10;
(LSeg (p1,|[0,1]|)) /\ ((((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))) = ((LSeg (p1,|[0,1]|)) /\ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,1]|,p2))) by XBOOLE_1:23
.= ((LSeg (p1,|[0,1]|)) /\ ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|)))) \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|))) by A83, XBOOLE_1:23
.= ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|))) by A100, XBOOLE_1:23
.= {|[0,1]|} by A82, A106, A101, TOPREAL1:15, ZFMISC_1:33 ;
hence  P2 is_an_arc_of p1,p2 by A111, TOPREAL1:11; ::_thesis: ( P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
thus P1 \/ P2 = ((LSeg (|[0,1]|,p1)) \/ (LSeg (p1,p2))) \/ ((((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))) by XBOOLE_1:4
.= (((LSeg (|[0,1]|,p1)) \/ (LSeg (p1,p2))) \/ (LSeg (p2,|[1,1]|))) \/ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,|[1,1]|))) by XBOOLE_1:4
.= (LSeg (|[0,1]|,|[1,1]|)) \/ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,|[1,1]|))) by A3, A66, TOPREAL1:7
.= (LSeg (|[0,1]|,|[1,1]|)) \/ ((LSeg (|[0,0]|,|[0,1]|)) \/ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)))) by XBOOLE_1:4
.= R^2-unit_square by TOPREAL1:def_2, XBOOLE_1:4 ; ::_thesis: P1 /\ P2 = {p1,p2}
A112: p2 in LSeg (|[1,1]|,p2) by RLTOPSP1:68;
 p2 in LSeg (p1,p2) by RLTOPSP1:68;
then  p2 in (LSeg (p1,p2)) /\ (LSeg (|[1,1]|,p2)) by A112, XBOOLE_0:def_4;
then  {p2} c= (LSeg (p1,p2)) /\ (LSeg (|[1,1]|,p2)) by ZFMISC_1:31;
then A113: (LSeg (p1,p2)) /\ (LSeg (|[1,1]|,p2)) = {p2} by A75, XBOOLE_0:def_10;
 (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by TOPREAL1:19, XBOOLE_0:def_7;
then A114: (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {} by A68, XBOOLE_1:3, XBOOLE_1:26;
 p1 in LSeg (p1,p2) by RLTOPSP1:68;
then  p1 in (LSeg (p1,p2)) /\ (LSeg (p1,|[0,1]|)) by A105, XBOOLE_0:def_4;
then  {p1} c= (LSeg (p1,p2)) /\ (LSeg (p1,|[0,1]|)) by ZFMISC_1:31;
then A115: (LSeg (p1,p2)) /\ (LSeg (p1,|[0,1]|)) = {p1} by A90, XBOOLE_0:def_10;
A116: P1 /\ P2 = ((LSeg (p1,p2)) /\ (LSeg (p1,|[0,1]|))) \/ ((LSeg (p1,p2)) /\ ((((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2)))) by XBOOLE_1:23
.= {p1} \/ (((LSeg (p1,p2)) /\ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,|[1,1]|)))) \/ {p2}) by A115, A113, XBOOLE_1:23
.= {p1} \/ ((((LSeg (p1,p2)) /\ ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|)))) \/ ((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ {p2}) by XBOOLE_1:23
.= {p1} \/ (((((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ ((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)))) \/ ((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ {p2}) by XBOOLE_1:23
.= {p1} \/ (((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ (((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ {p2})) by A114, XBOOLE_1:4
.= ({p1} \/ ((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)))) \/ (((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ {p2}) by XBOOLE_1:4 ;
A117: now__::_thesis:_P1_/\_P2_=_{p1}_\/_(((LSeg_(p1,p2))_/\_(LSeg_(|[1,0]|,|[1,1]|)))_\/_{p2})
percases ( p1 = |[0,1]| or p1 <> |[0,1]| ) ;
supposeA118: p1 = |[0,1]| ; ::_thesis: P1 /\ P2 = {p1} \/ (((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ {p2})
 p1 in LSeg (p1,p2) by RLTOPSP1:68;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) <> {} by A118, Lm22, XBOOLE_0:def_4;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {p1} by A97, A118, TOPREAL1:15, ZFMISC_1:33;
hence  P1 /\ P2 = {p1} \/ (((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ {p2}) by A116; ::_thesis: verum
end;
supposeA119: p1 <> |[0,1]| ; ::_thesis: P1 /\ P2 = {p1} \/ (((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ {p2})
now__::_thesis:_not_|[0,1]|_in_(LSeg_(p1,p2))_/\_(LSeg_(|[0,0]|,|[0,1]|))
assume  |[0,1]| in (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) ; ::_thesis: contradiction
then  |[0,1]| in LSeg (p1,p2) by XBOOLE_0:def_4;
then  p1 `1 = 0 by A13, A15, A69, A74, Lm6, TOPREAL1:3;
hence  contradiction by A13, A16, A119, EUCLID:53; ::_thesis: verum
end;
then  {|[0,1]|} <> (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) by ZFMISC_1:31;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {} by A97, TOPREAL1:15, ZFMISC_1:33;
hence  P1 /\ P2 = {p1} \/ (((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ {p2}) by A116; ::_thesis: verum
end;
end;
end;
A120: (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) c= {|[1,1]|} by A3, A66, TOPREAL1:6, TOPREAL1:18, XBOOLE_1:26;
now__::_thesis:_P1_/\_P2_=_{p1,p2}
percases ( p2 = |[1,1]| or p2 <> |[1,1]| ) ;
supposeA121: p2 = |[1,1]| ; ::_thesis: P1 /\ P2 = {p1,p2}
 p2 in LSeg (p1,p2) by RLTOPSP1:68;
then  (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) <> {} by A121, Lm27, XBOOLE_0:def_4;
then  (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {p2} by A120, A121, ZFMISC_1:33;
hence  P1 /\ P2 = {p1,p2} by A117, ENUMSET1:1; ::_thesis: verum
end;
supposeA122: p2 <> |[1,1]| ; ::_thesis: P1 /\ P2 = {p1,p2}
now__::_thesis:_not_|[1,1]|_in_(LSeg_(p1,p2))_/\_(LSeg_(|[1,0]|,|[1,1]|))
assume  |[1,1]| in (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) ; ::_thesis: contradiction
then  |[1,1]| in LSeg (p1,p2) by XBOOLE_0:def_4;
then  |[1,1]| `1 <= p2 `1 by A13, A69, A74, TOPREAL1:3;
then  p2 `1 = 1 by A69, A70, Lm10, XXREAL_0:1;
hence  contradiction by A69, A72, A122, EUCLID:53; ::_thesis: verum
end;
then  {|[1,1]|} <> (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) by ZFMISC_1:31;
then  (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by A120, ZFMISC_1:33;
hence  P1 /\ P2 = {p1,p2} by A117, ENUMSET1:1; ::_thesis: verum
end;
end;
end;
hence  P1 /\ P2 = {p1,p2} ; ::_thesis: verum
end;
supposeA123: q `1 < q1 `1 ; ::_thesis: ex P1 being Element of K19( the carrier of (TOP-REAL 2)) ex P2 being Element of K19( the carrier of (TOP-REAL 2)) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
A124: (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,p2)) c= {p2}
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,p2)) or a in {p2} )
assume A125: a in (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,p2)) ; ::_thesis: a in {p2}
then reconsider p = a as Point of (TOP-REAL 2) ;
A126: p in LSeg (|[0,1]|,p2) by A125, XBOOLE_0:def_4;
 |[0,1]| `1 <= p2 `1 by A69, A71, EUCLID:52;
then A127: p `1 <= p2 `1 by A126, TOPREAL1:3;
A128: p in LSeg (p2,p1) by A125, XBOOLE_0:def_4;
then A129: p `2 <= p1 `2 by A13, A16, A69, A72, TOPREAL1:4;
 p2 `1 <= p `1 by A13, A69, A123, A128, TOPREAL1:3;
then A130: p2 `1 = p `1 by A127, XXREAL_0:1;
 p2 `2 <= p `2 by A13, A16, A69, A72, A128, TOPREAL1:4;
then  p `2 = 1 by A13, A16, A69, A72, A129, XXREAL_0:1;
then p = |[(p2 `1),1]| by A130, EUCLID:53
.= p2 by A69, A72, EUCLID:53 ;
hence  a in {p2} by TARSKI:def_1; ::_thesis: verum
end;
 (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by TOPREAL1:19, XBOOLE_0:def_7;
then A131: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {} by A8, XBOOLE_1:3, XBOOLE_1:26;
A132: now__::_thesis:_not_(LSeg_(p1,|[1,1]|))_/\_(LSeg_(|[0,1]|,p2))_<>_{}
set a = the Element of (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2));
assume A133: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)) <> {} ; ::_thesis: contradiction
then reconsider p = the Element of (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)) as Point of (TOP-REAL 2) by TARSKI:def_3;
A134: p in LSeg (p1,|[1,1]|) by A133, XBOOLE_0:def_4;
A135: p in LSeg (|[0,1]|,p2) by A133, XBOOLE_0:def_4;
 |[0,1]| `1 <= p2 `1 by A69, A71, EUCLID:52;
then A136: p `1 <= p2 `1 by A135, TOPREAL1:3;
 p1 `1 <= |[1,1]| `1 by A13, A14, EUCLID:52;
then  p1 `1 <= p `1 by A134, TOPREAL1:3;
hence  contradiction by A13, A69, A123, A136, XXREAL_0:2; ::_thesis: verum
end;
A137: ((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) /\ (LSeg (|[0,0]|,|[0,1]|)) = ((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ ((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|))) by XBOOLE_1:23
.= {|[0,0]|} by Lm3, TOPREAL1:17 ;
 (LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|)) is_an_arc_of |[1,1]|,|[0,0]| by Lm9, Lm11, TOPREAL1:9, TOPREAL1:10, TOPREAL1:16;
then A138: ((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[0,0]|,|[0,1]|)) is_an_arc_of |[1,1]|,|[0,1]| by A137, TOPREAL1:10;
now__::_thesis:_not_|[1,1]|_in_(LSeg_(|[1,0]|,|[1,1]|))_/\_(LSeg_(|[0,1]|,p2))
assume  |[1,1]| in (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)) ; ::_thesis: contradiction
then A139: |[1,1]| in LSeg (|[0,1]|,p2) by XBOOLE_0:def_4;
 |[0,1]| `1 <= p2 `1 by A69, A71, EUCLID:52;
then  |[1,1]| `1 <= p2 `1 by A139, TOPREAL1:3;
hence  contradiction by A14, A69, A70, A123, Lm10, XXREAL_0:1; ::_thesis: verum
end;
then A140: {|[1,1]|} <> (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)) by ZFMISC_1:31;
 (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) by A66, Lm23, TOPREAL1:6, XBOOLE_1:26;
then A141: (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)) = {} by A140, TOPREAL1:18, ZFMISC_1:33;
 |[1,1]| in LSeg (p1,|[1,1]|) by RLTOPSP1:68;
then A142: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) <> {} by Lm27, XBOOLE_0:def_4;
now__::_thesis:_not_|[0,1]|_in_(LSeg_(p1,|[1,1]|))_/\_(LSeg_(|[0,0]|,|[0,1]|))
assume  |[0,1]| in (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) ; ::_thesis: contradiction
then A143: |[0,1]| in LSeg (p1,|[1,1]|) by XBOOLE_0:def_4;
 p1 `1 <= |[1,1]| `1 by A13, A14, EUCLID:52;
hence  contradiction by A13, A71, A123, A143, Lm6, TOPREAL1:3; ::_thesis: verum
end;
then A144: {|[0,1]|} <> (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by ZFMISC_1:31;
 (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) c= (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A3, Lm26, TOPREAL1:6, XBOOLE_1:26;
then A145: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {} by A144, TOPREAL1:15, ZFMISC_1:33;
 (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by TOPREAL1:19, XBOOLE_0:def_7;
then A146: (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {} by A68, XBOOLE_1:3, XBOOLE_1:26;
A147: (LSeg (p1,p2)) /\ (LSeg (p1,|[1,1]|)) c= {p1}
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in (LSeg (p1,p2)) /\ (LSeg (p1,|[1,1]|)) or a in {p1} )
assume A148: a in (LSeg (p1,p2)) /\ (LSeg (p1,|[1,1]|)) ; ::_thesis: a in {p1}
then reconsider p = a as Point of (TOP-REAL 2) ;
A149: p in LSeg (p1,|[1,1]|) by A148, XBOOLE_0:def_4;
 p1 `1 <= |[1,1]| `1 by A13, A14, EUCLID:52;
then A150: p1 `1 <= p `1 by A149, TOPREAL1:3;
A151: p in LSeg (p2,p1) by A148, XBOOLE_0:def_4;
then A152: p `2 <= p1 `2 by A13, A16, A69, A72, TOPREAL1:4;
 p `1 <= p1 `1 by A13, A69, A123, A151, TOPREAL1:3;
then A153: p1 `1 = p `1 by A150, XXREAL_0:1;
 p2 `2 <= p `2 by A13, A16, A69, A72, A151, TOPREAL1:4;
then  p `2 = 1 by A13, A16, A69, A72, A152, XXREAL_0:1;
then p = |[(p1 `1),1]| by A153, EUCLID:53
.= p1 by A13, A16, EUCLID:53 ;
hence  a in {p1} by TARSKI:def_1; ::_thesis: verum
end;
A154: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) c= {|[1,1]|} by A3, Lm26, TOPREAL1:6, TOPREAL1:18, XBOOLE_1:26;
 |[0,1]| in LSeg (|[0,1]|,p2) by RLTOPSP1:68;
then A155: (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[0,1]|,p2)) <> {} by Lm22, XBOOLE_0:def_4;
 (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[0,1]|,p2)) c= {|[0,1]|} by A66, Lm23, TOPREAL1:6, TOPREAL1:15, XBOOLE_1:26;
then A156: (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[0,1]|,p2)) = {|[0,1]|} by A155, ZFMISC_1:33;
take P1 = LSeg (p1,p2); ::_thesis: ex P2 being Element of K19( the carrier of (TOP-REAL 2)) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
take P2 = (LSeg (p1,|[1,1]|)) \/ ((((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2))); ::_thesis: ( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
A157: p1 in LSeg (p1,|[1,1]|) by RLTOPSP1:68;
thus  P1 is_an_arc_of p1,p2 by A1, TOPREAL1:9; ::_thesis: ( P2 is_an_arc_of p1,p2 & P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
A158: (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by TOPREAL1:19, XBOOLE_0:def_7;
 (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,p2)) c= (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) by A66, Lm23, TOPREAL1:6, XBOOLE_1:26;
then A159: (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,p2)) = {} by A158, XBOOLE_1:3;
(((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[0,0]|,|[0,1]|))) /\ (LSeg (|[0,1]|,p2)) = (((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) /\ (LSeg (|[0,1]|,p2))) \/ ((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[0,1]|,p2))) by XBOOLE_1:23
.= (((LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,1]|,p2))) \/ ((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,p2)))) \/ {|[0,1]|} by A156, XBOOLE_1:23
.= {|[0,1]|} by A141, A159 ;
then A160: (((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2)) is_an_arc_of |[1,1]|,p2 by A138, TOPREAL1:10;
(LSeg (p1,|[1,1]|)) /\ ((((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2))) = ((LSeg (p1,|[1,1]|)) /\ (((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[0,0]|,|[0,1]|)))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2))) by XBOOLE_1:23
.= ((LSeg (p1,|[1,1]|)) /\ ((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|)))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|))) by A132, XBOOLE_1:23
.= ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|))) by A145, XBOOLE_1:23
.= {|[1,1]|} by A131, A154, A142, ZFMISC_1:33 ;
hence  P2 is_an_arc_of p1,p2 by A160, TOPREAL1:11; ::_thesis: ( P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
thus P1 \/ P2 = ((LSeg (p2,p1)) \/ (LSeg (p1,|[1,1]|))) \/ ((((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2))) by XBOOLE_1:4
.= ((LSeg (|[0,1]|,p2)) \/ ((LSeg (p2,p1)) \/ (LSeg (p1,|[1,1]|)))) \/ (((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[0,0]|,|[0,1]|))) by XBOOLE_1:4
.= (LSeg (|[0,1]|,|[1,1]|)) \/ (((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[0,0]|,|[0,1]|))) by A3, A66, TOPREAL1:7
.= R^2-unit_square by TOPREAL1:def_2, XBOOLE_1:4 ; ::_thesis: P1 /\ P2 = {p1,p2}
A161: p2 in LSeg (|[0,1]|,p2) by RLTOPSP1:68;
 p2 in LSeg (p1,p2) by RLTOPSP1:68;
then  p2 in (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,p2)) by A161, XBOOLE_0:def_4;
then  {p2} c= (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,p2)) by ZFMISC_1:31;
then A162: (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,p2)) = {p2} by A124, XBOOLE_0:def_10;
 p1 in LSeg (p1,p2) by RLTOPSP1:68;
then  p1 in (LSeg (p1,p2)) /\ (LSeg (p1,|[1,1]|)) by A157, XBOOLE_0:def_4;
then  {p1} c= (LSeg (p1,p2)) /\ (LSeg (p1,|[1,1]|)) by ZFMISC_1:31;
then  (LSeg (p1,p2)) /\ (LSeg (p1,|[1,1]|)) = {p1} by A147, XBOOLE_0:def_10;
then A163: P1 /\ P2 = {p1} \/ ((LSeg (p1,p2)) /\ ((((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2)))) by XBOOLE_1:23
.= {p1} \/ (((LSeg (p1,p2)) /\ (((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[0,0]|,|[0,1]|)))) \/ {p2}) by A162, XBOOLE_1:23
.= {p1} \/ ((((LSeg (p1,p2)) /\ ((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|)))) \/ ((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)))) \/ {p2}) by XBOOLE_1:23
.= {p1} \/ (((((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ ((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)))) \/ ((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)))) \/ {p2}) by XBOOLE_1:23
.= {p1} \/ (((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ (((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2})) by A146, XBOOLE_1:4
.= ({p1} \/ ((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ (((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2}) by XBOOLE_1:4 ;
A164: (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) c= (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A3, A66, TOPREAL1:6, XBOOLE_1:26;
A165: now__::_thesis:_P1_/\_P2_=_({p1}_\/_((LSeg_(p1,p2))_/\_(LSeg_(|[1,0]|,|[1,1]|))))_\/_{p2}
percases ( p2 = |[0,1]| or p2 <> |[0,1]| ) ;
supposeA166: p2 = |[0,1]| ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ {p2}
 p2 in LSeg (p1,p2) by RLTOPSP1:68;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) <> {} by A166, Lm22, XBOOLE_0:def_4;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {p2} by A164, A166, TOPREAL1:15, ZFMISC_1:33;
hence  P1 /\ P2 = ({p1} \/ ((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ {p2} by A163; ::_thesis: verum
end;
supposeA167: p2 <> |[0,1]| ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ {p2}
now__::_thesis:_not_|[0,1]|_in_(LSeg_(p1,p2))_/\_(LSeg_(|[0,0]|,|[0,1]|))
assume  |[0,1]| in (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) ; ::_thesis: contradiction
then  |[0,1]| in LSeg (p2,p1) by XBOOLE_0:def_4;
then  p2 `1 = 0 by A13, A69, A71, A123, Lm6, TOPREAL1:3;
hence  contradiction by A69, A72, A167, EUCLID:53; ::_thesis: verum
end;
then  {|[0,1]|} <> (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) by ZFMISC_1:31;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {} by A164, TOPREAL1:15, ZFMISC_1:33;
hence  P1 /\ P2 = ({p1} \/ ((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ {p2} by A163; ::_thesis: verum
end;
end;
end;
A168: (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) c= {|[1,1]|} by A3, A66, TOPREAL1:6, TOPREAL1:18, XBOOLE_1:26;
now__::_thesis:_P1_/\_P2_=_{p1,p2}
percases ( p1 = |[1,1]| or p1 <> |[1,1]| ) ;
supposeA169: p1 = |[1,1]| ; ::_thesis: P1 /\ P2 = {p1,p2}
 p1 in LSeg (p1,p2) by RLTOPSP1:68;
then  (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) <> {} by A169, Lm27, XBOOLE_0:def_4;
then  (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {p1} by A168, A169, ZFMISC_1:33;
hence  P1 /\ P2 = {p1,p2} by A165, ENUMSET1:1; ::_thesis: verum
end;
supposeA170: p1 <> |[1,1]| ; ::_thesis: P1 /\ P2 = {p1,p2}
now__::_thesis:_not_|[1,1]|_in_(LSeg_(p1,p2))_/\_(LSeg_(|[1,0]|,|[1,1]|))
assume  |[1,1]| in (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) ; ::_thesis: contradiction
then  |[1,1]| in LSeg (p2,p1) by XBOOLE_0:def_4;
then  |[1,1]| `1 <= p1 `1 by A13, A69, A123, TOPREAL1:3;
then  p1 `1 = 1 by A13, A14, Lm10, XXREAL_0:1;
hence  contradiction by A13, A16, A170, EUCLID:53; ::_thesis: verum
end;
then  {|[1,1]|} <> (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) by ZFMISC_1:31;
then  (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by A168, ZFMISC_1:33;
hence  P1 /\ P2 = {p1,p2} by A165, ENUMSET1:1; ::_thesis: verum
end;
end;
end;
hence  P1 /\ P2 = {p1,p2} ; ::_thesis: verum
end;
end;
end;
hence  ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) ; ::_thesis: verum
end;
supposeA171: p2 in LSeg (|[0,0]|,|[1,0]|) ; ::_thesis: ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
 |[0,0]| in LSeg (|[0,0]|,p2) by RLTOPSP1:68;
then A172: (LSeg (|[0,1]|,|[0,0]|)) /\ (LSeg (|[0,0]|,p2)) <> {} by Lm20, XBOOLE_0:def_4;
 LSeg (|[0,0]|,p2) c= LSeg (|[0,0]|,|[1,0]|) by A171, Lm21, TOPREAL1:6;
then  (LSeg (|[0,1]|,|[0,0]|)) /\ (LSeg (|[0,0]|,p2)) c= {|[0,0]|} by TOPREAL1:17, XBOOLE_1:27;
then  (LSeg (|[0,1]|,|[0,0]|)) /\ (LSeg (|[0,0]|,p2)) = {|[0,0]|} by A172, ZFMISC_1:33;
then A173: (LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,p2)) is_an_arc_of |[0,1]|,p2 by Lm5, Lm7, TOPREAL1:12;
 LSeg (p2,|[0,0]|) c= LSeg (|[0,0]|,|[1,0]|) by A171, Lm21, TOPREAL1:6;
then A174: (LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,p2)) = {} by A11, Lm2, XBOOLE_1:3, XBOOLE_1:27;
 |[1,0]| in LSeg (|[1,0]|,p2) by RLTOPSP1:68;
then A175: |[1,0]| in (LSeg (|[1,1]|,|[1,0]|)) /\ (LSeg (|[1,0]|,p2)) by Lm25, XBOOLE_0:def_4;
 LSeg (|[1,0]|,p2) c= LSeg (|[0,0]|,|[1,0]|) by A171, Lm24, TOPREAL1:6;
then  (LSeg (|[1,1]|,|[1,0]|)) /\ (LSeg (|[1,0]|,p2)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by XBOOLE_1:27;
then  (LSeg (|[1,1]|,|[1,0]|)) /\ (LSeg (|[1,0]|,p2)) = {|[1,0]|} by A175, TOPREAL1:16, ZFMISC_1:33;
then A176: (LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[1,0]|,p2)) is_an_arc_of |[1,1]|,p2 by Lm9, Lm11, TOPREAL1:12;
take P1 = ((LSeg (p1,|[1,1]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,0]|,p2)); ::_thesis: ex P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
take P2 = ((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,p2)); ::_thesis: ( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A177: (LSeg (p1,|[1,1]|)) \/ (LSeg (p1,|[0,1]|)) = LSeg (|[0,1]|,|[1,1]|) by A3, TOPREAL1:5;
A178: LSeg (p2,|[1,0]|) c= LSeg (|[0,0]|,|[1,0]|) by A171, Lm24, TOPREAL1:6;
then A179: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,0]|,p2)) = {} by A8, Lm2, XBOOLE_1:3, XBOOLE_1:27;
A180: (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) c= (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by A171, Lm21, TOPREAL1:6, XBOOLE_1:26;
 (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) c= (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) by A3, Lm26, TOPREAL1:6, XBOOLE_1:26;
then A181: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) = {} by A180, Lm2, XBOOLE_1:1, XBOOLE_1:3;
A182: (LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[0,1]|)) = {} by A11, A178, Lm2, XBOOLE_1:3, XBOOLE_1:27;
A183: (LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,p2)) = {p2} by A171, TOPREAL1:8;
(LSeg (p1,|[1,1]|)) /\ ((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[1,0]|,p2))) = ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,0]|,p2))) by XBOOLE_1:23
.= {|[1,1]|} by A6, A10, A179, ZFMISC_1:33 ;
then  (LSeg (p1,|[1,1]|)) \/ ((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[1,0]|,p2))) is_an_arc_of p1,p2 by A176, TOPREAL1:11;
hence  P1 is_an_arc_of p1,p2 by XBOOLE_1:4; ::_thesis: ( P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A184: ex q2 being Point of (TOP-REAL 2) st
( q2 = p2 & q2 `1 <= 1 & q2 `1 >= 0 & q2 `2 = 0 ) by A171, TOPREAL1:13;
(LSeg (p1,|[0,1]|)) /\ ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,p2))) = ((LSeg (|[0,1]|,p1)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,p2))) by XBOOLE_1:23
.= {|[0,1]|} by A7, A5, A174, TOPREAL1:15, ZFMISC_1:33 ;
then  (LSeg (p1,|[0,1]|)) \/ ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,p2))) is_an_arc_of p1,p2 by A173, TOPREAL1:11;
hence  P2 is_an_arc_of p1,p2 by XBOOLE_1:4; ::_thesis: ( R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
 (LSeg (|[1,0]|,p2)) \/ (LSeg (|[0,0]|,p2)) = LSeg (|[0,0]|,|[1,0]|) by A171, TOPREAL1:5;
hence  R^2-unit_square = (LSeg (|[0,1]|,|[1,1]|)) \/ (((LSeg (|[1,0]|,|[1,1]|)) \/ ((LSeg (|[1,0]|,p2)) \/ (LSeg (|[0,0]|,p2)))) \/ (LSeg (|[0,0]|,|[0,1]|))) by TOPREAL1:def_2, XBOOLE_1:4
.= (LSeg (|[0,1]|,|[1,1]|)) \/ ((((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[1,0]|,p2))) \/ (LSeg (|[0,0]|,p2))) \/ (LSeg (|[0,0]|,|[0,1]|))) by XBOOLE_1:4
.= (LSeg (|[0,1]|,|[1,1]|)) \/ (((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[1,0]|,p2))) \/ ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,p2)))) by XBOOLE_1:4
.= (LSeg (p1,|[1,1]|)) \/ ((((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[1,0]|,p2))) \/ ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,p2)))) \/ (LSeg (p1,|[0,1]|))) by A177, XBOOLE_1:4
.= (LSeg (p1,|[1,1]|)) \/ (((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[1,0]|,p2))) \/ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,p2))) \/ (LSeg (p1,|[0,1]|)))) by XBOOLE_1:4
.= ((LSeg (p1,|[1,1]|)) \/ ((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[1,0]|,p2)))) \/ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,p2))) \/ (LSeg (p1,|[0,1]|))) by XBOOLE_1:4
.= (((LSeg (p1,|[1,1]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,0]|,p2))) \/ ((LSeg (p1,|[0,1]|)) \/ ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,p2)))) by XBOOLE_1:4
.= P1 \/ P2 by XBOOLE_1:4 ;
::_thesis: P1 /\ P2 = {p1,p2}
A185: (LSeg (p1,|[1,1]|)) /\ (LSeg (p1,|[0,1]|)) = {p1} by A3, TOPREAL1:8;
A186: P1 /\ P2 = (((LSeg (p1,|[1,1]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) /\ (((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,p2)))) \/ ((LSeg (|[1,0]|,p2)) /\ (((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,p2)))) by XBOOLE_1:23
.= (((LSeg (p1,|[1,1]|)) /\ (((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,p2)))) \/ ((LSeg (|[1,0]|,|[1,1]|)) /\ (((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,p2))))) \/ ((LSeg (|[1,0]|,p2)) /\ (((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,p2)))) by XBOOLE_1:23
.= ((((LSeg (p1,|[1,1]|)) /\ ((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|)))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)))) \/ ((LSeg (|[1,0]|,|[1,1]|)) /\ (((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,p2))))) \/ ((LSeg (|[1,0]|,p2)) /\ (((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,p2)))) by XBOOLE_1:23
.= ((((LSeg (p1,|[1,1]|)) /\ (LSeg (p1,|[0,1]|))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)))) \/ ((LSeg (|[1,0]|,|[1,1]|)) /\ (((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,p2))))) \/ ((LSeg (|[1,0]|,p2)) /\ (((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,p2)))) by A181, XBOOLE_1:23
.= (({p1} \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)))) \/ (((LSeg (|[1,0]|,|[1,1]|)) /\ ((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|)))) \/ ((LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2))))) \/ ((LSeg (|[1,0]|,p2)) /\ (((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,p2)))) by A185, XBOOLE_1:23
.= (({p1} \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)))) \/ ((((LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (p1,|[0,1]|))) \/ ((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2))))) \/ ((LSeg (|[1,0]|,p2)) /\ (((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,p2)))) by XBOOLE_1:23
.= (({p1} \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)))) \/ (((LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (p1,|[0,1]|))) \/ ((LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2))))) \/ (((LSeg (|[1,0]|,p2)) /\ ((LSeg (p1,|[0,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|)))) \/ {p2}) by A183, Lm3, XBOOLE_1:23
.= (({p1} \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)))) \/ (((LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (p1,|[0,1]|))) \/ ((LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2))))) \/ ((((LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[0,1]|))) \/ ((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)))) \/ {p2}) by XBOOLE_1:23
.= (({p1} \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)))) \/ (((LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (p1,|[0,1]|))) \/ ((LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2))))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2}) by A182 ;
A187: now__::_thesis:_P1_/\_P2_=_({p1}_\/_((LSeg_(|[1,0]|,|[1,1]|))_/\_(LSeg_(|[0,0]|,p2))))_\/_(((LSeg_(|[1,0]|,p2))_/\_(LSeg_(|[0,0]|,|[0,1]|)))_\/_{p2})
percases ( p1 = |[0,1]| or p1 = |[1,1]| or ( p1 <> |[1,1]| & p1 <> |[0,1]| ) ) ;
supposeA188: p1 = |[0,1]| ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2})
then (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (p1,|[0,1]|)) = (LSeg (|[1,0]|,|[1,1]|)) /\ {|[0,1]|} by RLTOPSP1:70
.= {} by Lm1, Lm15 ;
hence  P1 /\ P2 = ({p1} \/ ((LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2}) by A186, A188, TOPREAL1:15; ::_thesis: verum
end;
supposeA189: p1 = |[1,1]| ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2})
then (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {|[1,1]|} /\ (LSeg (|[0,0]|,|[0,1]|)) by RLTOPSP1:70
.= {} by Lm1, Lm18 ;
hence P1 /\ P2 = (({p1} \/ {p1}) \/ ((LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2}) by A186, A189, TOPREAL1:18, XBOOLE_1:4
.= ({p1} \/ ((LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2}) ;
::_thesis: verum
end;
supposeA190: ( p1 <> |[1,1]| & p1 <> |[0,1]| ) ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2})
now__::_thesis:_not_|[1,1]|_in_(LSeg_(|[1,0]|,|[1,1]|))_/\_(LSeg_(p1,|[0,1]|))
assume  |[1,1]| in (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (p1,|[0,1]|)) ; ::_thesis: contradiction
then A191: |[1,1]| in LSeg (|[0,1]|,p1) by XBOOLE_0:def_4;
 |[0,1]| `1 <= p1 `1 by A13, A15, EUCLID:52;
then  1 <= p1 `1 by A191, Lm10, TOPREAL1:3;
then  p1 `1 = 1 by A13, A14, XXREAL_0:1;
hence  contradiction by A13, A16, A190, EUCLID:53; ::_thesis: verum
end;
then A192: {|[1,1]|} <> (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (p1,|[0,1]|)) by ZFMISC_1:31;
 (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (p1,|[0,1]|)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) by A3, Lm23, TOPREAL1:6, XBOOLE_1:26;
then A193: (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (p1,|[0,1]|)) = {} by A192, TOPREAL1:18, ZFMISC_1:33;
now__::_thesis:_not_|[0,1]|_in_(LSeg_(p1,|[1,1]|))_/\_(LSeg_(|[0,0]|,|[0,1]|))
assume  |[0,1]| in (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) ; ::_thesis: contradiction
then A194: |[0,1]| in LSeg (p1,|[1,1]|) by XBOOLE_0:def_4;
 p1 `1 <= |[1,1]| `1 by A13, A14, EUCLID:52;
then  p1 `1 = 0 by A13, A15, A194, Lm6, TOPREAL1:3;
hence  contradiction by A13, A16, A190, EUCLID:53; ::_thesis: verum
end;
then A195: {|[0,1]|} <> (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by ZFMISC_1:31;
 (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) c= (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A3, Lm26, TOPREAL1:6, XBOOLE_1:26;
then  (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {} by A195, TOPREAL1:15, ZFMISC_1:33;
hence  P1 /\ P2 = ({p1} \/ ((LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2}) by A186, A193; ::_thesis: verum
end;
end;
end;
now__::_thesis:_P1_/\_P2_=_{p1,p2}
percases ( p2 = |[0,0]| or p2 = |[1,0]| or ( p2 <> |[1,0]| & p2 <> |[0,0]| ) ) ;
supposeA196: p2 = |[0,0]| ; ::_thesis: P1 /\ P2 = {p1,p2}
then (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) = (LSeg (|[1,0]|,|[1,1]|)) /\ {|[0,0]|} by RLTOPSP1:70
.= {} by Lm1, Lm12 ;
hence  P1 /\ P2 = {p1,p2} by A187, A196, ENUMSET1:1, TOPREAL1:17; ::_thesis: verum
end;
supposeA197: p2 = |[1,0]| ; ::_thesis: P1 /\ P2 = {p1,p2}
then (LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {|[1,0]|} /\ (LSeg (|[0,0]|,|[0,1]|)) by RLTOPSP1:70
.= {} by Lm1, Lm16 ;
hence P1 /\ P2 = {p1} \/ ({p2} \/ {p2}) by A187, A197, TOPREAL1:16, XBOOLE_1:4
.= {p1,p2} by ENUMSET1:1 ;
::_thesis: verum
end;
supposeA198: ( p2 <> |[1,0]| & p2 <> |[0,0]| ) ; ::_thesis: P1 /\ P2 = {p1,p2}
now__::_thesis:_not_|[0,0]|_in_(LSeg_(|[1,0]|,p2))_/\_(LSeg_(|[0,0]|,|[0,1]|))
assume  |[0,0]| in (LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) ; ::_thesis: contradiction
then A199: |[0,0]| in LSeg (p2,|[1,0]|) by XBOOLE_0:def_4;
 p2 `1 <= |[1,0]| `1 by A184, EUCLID:52;
then  p2 `1 = 0 by A184, A199, Lm4, TOPREAL1:3;
hence  contradiction by A184, A198, EUCLID:53; ::_thesis: verum
end;
then A200: {|[0,0]|} <> (LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) by ZFMISC_1:31;
 (LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) c= (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A171, Lm24, TOPREAL1:6, XBOOLE_1:26;
then A201: (LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {} by A200, TOPREAL1:17, ZFMISC_1:33;
now__::_thesis:_not_|[1,0]|_in_(LSeg_(|[1,0]|,|[1,1]|))_/\_(LSeg_(|[0,0]|,p2))
assume  |[1,0]| in (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) ; ::_thesis: contradiction
then A202: |[1,0]| in LSeg (|[0,0]|,p2) by XBOOLE_0:def_4;
 |[0,0]| `1 <= p2 `1 by A184, EUCLID:52;
then  1 <= p2 `1 by A202, Lm8, TOPREAL1:3;
then  p2 `1 = 1 by A184, XXREAL_0:1;
hence  contradiction by A184, A198, EUCLID:53; ::_thesis: verum
end;
then A203: {|[1,0]|} <> (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) by ZFMISC_1:31;
 (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by A171, Lm21, TOPREAL1:6, XBOOLE_1:26;
then  (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) = {} by A203, TOPREAL1:16, ZFMISC_1:33;
hence  P1 /\ P2 = {p1,p2} by A187, A201, ENUMSET1:1; ::_thesis: verum
end;
end;
end;
hence  P1 /\ P2 = {p1,p2} ; ::_thesis: verum
end;
supposeA204: p2 in LSeg (|[1,0]|,|[1,1]|) ; ::_thesis: ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
then A205: LSeg (|[1,1]|,p2) c= LSeg (|[1,0]|,|[1,1]|) by Lm27, TOPREAL1:6;
 LSeg (p1,|[1,1]|) c= LSeg (|[0,1]|,|[1,1]|) by A3, Lm26, TOPREAL1:6;
then A206: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,1]|,p2)) c= (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) by A205, XBOOLE_1:27;
take P1 = (LSeg (p1,|[1,1]|)) \/ (LSeg (|[1,1]|,p2)); ::_thesis: ex P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
take P2 = (LSeg (p1,|[0,1]|)) \/ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,p2))); ::_thesis: ( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A207: |[1,1]| in LSeg (|[1,1]|,p2) by RLTOPSP1:68;
 |[1,0]| in LSeg (|[1,0]|,p2) by RLTOPSP1:68;
then A208: (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,0]|,p2)) <> {} by Lm24, XBOOLE_0:def_4;
 |[1,1]| in LSeg (p1,|[1,1]|) by RLTOPSP1:68;
then  (LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,1]|,p2)) <> {} by A207, XBOOLE_0:def_4;
then A209: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,1]|,p2)) = {|[1,1]|} by A206, TOPREAL1:18, ZFMISC_1:33;
 ( p1 <> |[1,1]| or |[1,1]| <> p2 ) by A1;
hence  P1 is_an_arc_of p1,p2 by A209, TOPREAL1:12; ::_thesis: ( P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A210: LSeg (|[0,1]|,|[1,1]|) = (LSeg (p1,|[1,1]|)) \/ (LSeg (p1,|[0,1]|)) by A3, TOPREAL1:5;
A211: LSeg (|[0,0]|,|[1,0]|) is_an_arc_of |[0,0]|,|[1,0]| by Lm4, Lm8, TOPREAL1:9;
 LSeg (|[0,0]|,|[0,1]|) is_an_arc_of |[0,1]|,|[0,0]| by Lm5, Lm7, TOPREAL1:9;
then A212: (LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|)) is_an_arc_of |[0,1]|,|[1,0]| by A211, TOPREAL1:2, TOPREAL1:17;
A213: (LSeg (|[1,1]|,p2)) /\ (LSeg (|[1,0]|,p2)) = {p2} by A204, TOPREAL1:8;
A214: LSeg (|[1,0]|,|[1,1]|) = (LSeg (|[1,0]|,p2)) \/ (LSeg (|[1,1]|,p2)) by A204, TOPREAL1:5;
A215: LSeg (|[1,0]|,p2) c= LSeg (|[1,0]|,|[1,1]|) by A204, Lm25, TOPREAL1:6;
then A216: (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,0]|,p2)) c= {|[1,0]|} by TOPREAL1:16, XBOOLE_1:27;
A217: ex q being Point of (TOP-REAL 2) st
( q = p2 & q `1 = 1 & q `2 <= 1 & q `2 >= 0 ) by A204, TOPREAL1:13;
now__::_thesis:_not_|[1,1]|_in_(LSeg_(p1,|[0,1]|))_/\_(LSeg_(|[1,0]|,p2))
A218: |[1,0]| `2 <= p2 `2 by A217, EUCLID:52;
assume A219: |[1,1]| in (LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,p2)) ; ::_thesis: contradiction
then A220: |[1,1]| in LSeg (|[0,1]|,p1) by XBOOLE_0:def_4;
 |[1,1]| in LSeg (|[1,0]|,p2) by A219, XBOOLE_0:def_4;
then  |[1,1]| `2 <= p2 `2 by A218, TOPREAL1:4;
then A221: |[1,1]| `2 = p2 `2 by A217, Lm11, XXREAL_0:1;
 |[0,1]| `1 <= p1 `1 by A13, A15, EUCLID:52;
then  |[1,1]| `1 <= p1 `1 by A220, TOPREAL1:3;
then  |[1,1]| `1 = p1 `1 by A13, A14, Lm10, XXREAL_0:1;
then p1 = |[(|[1,1]| `1),(|[1,1]| `2)]| by A13, A16, Lm11, EUCLID:53
.= p2 by A217, A221, Lm10, EUCLID:53 ;
hence  contradiction by A1; ::_thesis: verum
end;
then A222: {|[1,1]|} <> (LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,p2)) by ZFMISC_1:31;
A223: (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,p2)) = {} by A215, Lm3, XBOOLE_1:3, XBOOLE_1:26;
((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) /\ (LSeg (|[1,0]|,p2)) = ((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,p2))) \/ ((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,0]|,p2))) by XBOOLE_1:23
.= {|[1,0]|} by A223, A216, A208, ZFMISC_1:33 ;
then A224: ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,p2)) is_an_arc_of |[0,1]|,p2 by A212, TOPREAL1:10;
A225: LSeg (p2,|[1,1]|) c= LSeg (|[1,0]|,|[1,1]|) by A204, Lm27, TOPREAL1:6;
then A226: (LSeg (|[1,1]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {} by Lm3, XBOOLE_1:3, XBOOLE_1:27;
 (LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,p2)) c= {|[1,1]|} by A11, A215, TOPREAL1:18, XBOOLE_1:27;
then A227: (LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,p2)) = {} by A222, ZFMISC_1:33;
(LSeg (p1,|[0,1]|)) /\ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,p2))) = ((LSeg (p1,|[0,1]|)) /\ ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|)))) \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[1,0]|,p2))) by XBOOLE_1:23
.= ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ ((LSeg (p1,|[0,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|))) by A227, XBOOLE_1:23
.= {|[0,1]|} by A12, A7, A5, TOPREAL1:15, ZFMISC_1:33 ;
hence  P2 is_an_arc_of p1,p2 by A224, TOPREAL1:11; ::_thesis: ( R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
thus P1 \/ P2 = (LSeg (|[1,1]|,p2)) \/ ((LSeg (p1,|[1,1]|)) \/ ((LSeg (p1,|[0,1]|)) \/ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,p2))))) by XBOOLE_1:4
.= ((LSeg (|[0,1]|,|[1,1]|)) \/ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,p2)))) \/ (LSeg (|[1,1]|,p2)) by A210, XBOOLE_1:4
.= (LSeg (|[0,1]|,|[1,1]|)) \/ ((((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,p2))) \/ (LSeg (|[1,1]|,p2))) by XBOOLE_1:4
.= (LSeg (|[0,1]|,|[1,1]|)) \/ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (|[1,0]|,p2)) \/ (LSeg (|[1,1]|,p2)))) by XBOOLE_1:4
.= (LSeg (|[0,1]|,|[1,1]|)) \/ ((LSeg (|[0,0]|,|[0,1]|)) \/ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)))) by A214, XBOOLE_1:4
.= R^2-unit_square by TOPREAL1:def_2, XBOOLE_1:4 ; ::_thesis: P1 /\ P2 = {p1,p2}
A228: {p1} = (LSeg (p1,|[1,1]|)) /\ (LSeg (p1,|[0,1]|)) by A3, TOPREAL1:8;
A229: P1 /\ P2 = ((LSeg (p1,|[1,1]|)) /\ ((LSeg (p1,|[0,1]|)) \/ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,p2))))) \/ ((LSeg (|[1,1]|,p2)) /\ ((LSeg (p1,|[0,1]|)) \/ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,p2))))) by XBOOLE_1:23
.= (((LSeg (p1,|[1,1]|)) /\ (LSeg (p1,|[0,1]|))) \/ ((LSeg (p1,|[1,1]|)) /\ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,p2))))) \/ ((LSeg (|[1,1]|,p2)) /\ ((LSeg (p1,|[0,1]|)) \/ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,p2))))) by XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[1,1]|)) /\ ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|)))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,0]|,p2))))) \/ ((LSeg (|[1,1]|,p2)) /\ ((LSeg (p1,|[0,1]|)) \/ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,p2))))) by A228, XBOOLE_1:23
.= ({p1} \/ ((((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,0]|,p2))))) \/ ((LSeg (|[1,1]|,p2)) /\ ((LSeg (p1,|[0,1]|)) \/ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,p2))))) by XBOOLE_1:23
.= ({p1} \/ ((((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,0]|,p2))))) \/ (((LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[0,1]|))) \/ ((LSeg (|[1,1]|,p2)) /\ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,p2))))) by XBOOLE_1:23
.= ({p1} \/ ((((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,0]|,p2))))) \/ (((LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[0,1]|))) \/ (((LSeg (|[1,1]|,p2)) /\ ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|)))) \/ {p2})) by A213, XBOOLE_1:23
.= ({p1} \/ ((((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,0]|,p2))))) \/ (((LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[0,1]|))) \/ ((((LSeg (|[1,1]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ ((LSeg (|[1,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)))) \/ {p2})) by XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,0]|,p2))))) \/ (((LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[0,1]|))) \/ (((LSeg (|[1,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {p2})) by A9, A226 ;
A230: now__::_thesis:_P1_/\_P2_=_({p1}_\/_((LSeg_(p1,|[1,1]|))_/\_(LSeg_(|[0,0]|,|[0,1]|))))_\/_(((LSeg_(|[1,1]|,p2))_/\_(LSeg_(p1,|[0,1]|)))_\/_{p2})
percases ( p2 = |[1,0]| or p2 = |[1,1]| or ( p2 <> |[1,1]| & p2 <> |[1,0]| ) ) ;
supposeA231: p2 = |[1,0]| ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)))) \/ (((LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[0,1]|))) \/ {p2})
then A232: not p2 in LSeg (p1,|[1,1]|) by A8, Lm7, Lm9, Lm11, TOPREAL1:4;
(LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,0]|,p2)) = (LSeg (p1,|[1,1]|)) /\ {p2} by A231, RLTOPSP1:70
.= {} by A232, Lm1 ;
hence  P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)))) \/ (((LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[0,1]|))) \/ {p2}) by A229, A231, TOPREAL1:16; ::_thesis: verum
end;
supposeA233: p2 = |[1,1]| ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)))) \/ (((LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[0,1]|))) \/ {p2})
then  p2 in LSeg (p1,|[1,1]|) by RLTOPSP1:68;
then A234: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,0]|,p2)) <> {} by A233, Lm27, XBOOLE_0:def_4;
 (LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,0]|,p2)) c= {p2} by A8, A233, TOPREAL1:18, XBOOLE_1:27;
then A235: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,0]|,p2)) = {p2} by A234, ZFMISC_1:33;
(LSeg (|[1,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {|[1,1]|} /\ (LSeg (|[0,0]|,|[1,0]|)) by A233, RLTOPSP1:70
.= {} by Lm1, Lm19 ;
hence P1 /\ P2 = (({p1} \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)))) \/ {p2}) \/ (((LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[0,1]|))) \/ {p2}) by A229, A235, XBOOLE_1:4
.= ({p1} \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)))) \/ ((((LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[0,1]|))) \/ {p2}) \/ {p2}) by XBOOLE_1:4
.= ({p1} \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)))) \/ (((LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[0,1]|))) \/ ({p2} \/ {p2})) by XBOOLE_1:4
.= ({p1} \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)))) \/ (((LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[0,1]|))) \/ {p2}) ;
::_thesis: verum
end;
supposeA236: ( p2 <> |[1,1]| & p2 <> |[1,0]| ) ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)))) \/ (((LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[0,1]|))) \/ {p2})
now__::_thesis:_not_|[1,1]|_in_(LSeg_(p1,|[1,1]|))_/\_(LSeg_(|[1,0]|,p2))
assume  |[1,1]| in (LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,0]|,p2)) ; ::_thesis: contradiction
then A237: |[1,1]| in LSeg (|[1,0]|,p2) by XBOOLE_0:def_4;
 |[1,0]| `2 <= p2 `2 by A217, EUCLID:52;
then  |[1,1]| `2 <= p2 `2 by A237, TOPREAL1:4;
then  p2 `2 = 1 by A217, Lm11, XXREAL_0:1;
hence  contradiction by A217, A236, EUCLID:53; ::_thesis: verum
end;
then A238: {|[1,1]|} <> (LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,0]|,p2)) by ZFMISC_1:31;
 (LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,0]|,p2)) c= {|[1,1]|} by A8, A215, TOPREAL1:18, XBOOLE_1:27;
then A239: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,0]|,p2)) = {} by A238, ZFMISC_1:33;
now__::_thesis:_not_|[1,0]|_in_(LSeg_(|[1,1]|,p2))_/\_(LSeg_(|[0,0]|,|[1,0]|))
assume  |[1,0]| in (LSeg (|[1,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) ; ::_thesis: contradiction
then A240: |[1,0]| in LSeg (p2,|[1,1]|) by XBOOLE_0:def_4;
 p2 `2 <= |[1,1]| `2 by A217, EUCLID:52;
then  p2 `2 = 0 by A217, A240, Lm9, TOPREAL1:4;
hence  contradiction by A217, A236, EUCLID:53; ::_thesis: verum
end;
then A241: {|[1,0]|} <> (LSeg (|[1,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) by ZFMISC_1:31;
 (LSeg (|[1,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by A225, XBOOLE_1:27;
then  (LSeg (|[1,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {} by A241, TOPREAL1:16, ZFMISC_1:33;
hence  P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)))) \/ (((LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[0,1]|))) \/ {p2}) by A229, A239; ::_thesis: verum
end;
end;
end;
now__::_thesis:_P1_/\_P2_=_{p1,p2}
percases ( p1 = |[0,1]| or p1 = |[1,1]| or ( p1 <> |[1,1]| & p1 <> |[0,1]| ) ) ;
supposeA242: p1 = |[0,1]| ; ::_thesis: P1 /\ P2 = {p1,p2}
then A243: (LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[0,1]|)) = (LSeg (|[1,1]|,p2)) /\ {p1} by RLTOPSP1:70;
 not p1 in LSeg (|[1,1]|,p2) by A225, A242, Lm6, Lm8, Lm10, TOPREAL1:3;
then  (LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[0,1]|)) = {} by A243, Lm1;
hence  P1 /\ P2 = {p1,p2} by A230, A242, ENUMSET1:1, TOPREAL1:15; ::_thesis: verum
end;
supposeA244: p1 = |[1,1]| ; ::_thesis: P1 /\ P2 = {p1,p2}
 |[1,1]| in LSeg (|[1,1]|,p2) by RLTOPSP1:68;
then A245: (LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[0,1]|)) <> {} by A244, Lm26, XBOOLE_0:def_4;
 (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {p1} /\ (LSeg (|[0,0]|,|[0,1]|)) by A244, RLTOPSP1:70;
then A246: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {} by A244, Lm1, Lm18;
 (LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[0,1]|)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) by A11, A225, XBOOLE_1:27;
then  (LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[0,1]|)) = {p1} by A244, A245, TOPREAL1:18, ZFMISC_1:33;
hence P1 /\ P2 = ({p1} \/ {p1}) \/ {p2} by A230, A246, XBOOLE_1:4
.= {p1,p2} by ENUMSET1:1 ;
::_thesis: verum
end;
supposeA247: ( p1 <> |[1,1]| & p1 <> |[0,1]| ) ; ::_thesis: P1 /\ P2 = {p1,p2}
now__::_thesis:_not_|[1,1]|_in_(LSeg_(|[1,1]|,p2))_/\_(LSeg_(p1,|[0,1]|))
assume  |[1,1]| in (LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[0,1]|)) ; ::_thesis: contradiction
then A248: |[1,1]| in LSeg (|[0,1]|,p1) by XBOOLE_0:def_4;
 |[0,1]| `1 <= p1 `1 by A13, A15, EUCLID:52;
then  |[1,1]| `1 <= p1 `1 by A248, TOPREAL1:3;
then  p1 `1 = 1 by A13, A14, Lm10, XXREAL_0:1;
hence  contradiction by A13, A16, A247, EUCLID:53; ::_thesis: verum
end;
then A249: {|[1,1]|} <> (LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[0,1]|)) by ZFMISC_1:31;
 (LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[0,1]|)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) by A11, A225, XBOOLE_1:27;
then A250: (LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[0,1]|)) = {} by A249, TOPREAL1:18, ZFMISC_1:33;
now__::_thesis:_not_|[0,1]|_in_(LSeg_(p1,|[1,1]|))_/\_(LSeg_(|[0,0]|,|[0,1]|))
assume  |[0,1]| in (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) ; ::_thesis: contradiction
then A251: |[0,1]| in LSeg (p1,|[1,1]|) by XBOOLE_0:def_4;
 p1 `1 <= |[1,1]| `1 by A13, A14, EUCLID:52;
then  p1 `1 = 0 by A13, A15, A251, Lm6, TOPREAL1:3;
hence  contradiction by A13, A16, A247, EUCLID:53; ::_thesis: verum
end;
then A252: {|[0,1]|} <> (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by ZFMISC_1:31;
 (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) c= (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A8, XBOOLE_1:27;
then  (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {} by A252, TOPREAL1:15, ZFMISC_1:33;
hence  P1 /\ P2 = {p1,p2} by A230, A250, ENUMSET1:1; ::_thesis: verum
end;
end;
end;
hence  P1 /\ P2 = {p1,p2} ; ::_thesis: verum
end;
end;
end;

Lm32:  for p1, p2 being Point of (TOP-REAL 2) st p1 <> p2 & p2 in R^2-unit_square & p1 in LSeg (|[0,0]|,|[1,0]|) holds
ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 <> p2 & p2 in R^2-unit_square & p1 in LSeg (|[0,0]|,|[1,0]|) implies ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) )
assume that
A1: p1 <> p2 and
A2: p2 in R^2-unit_square and
A3: p1 in LSeg (|[0,0]|,|[1,0]|) ; ::_thesis: ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A4: ( p2 in (LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|)) or p2 in (LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)) ) by A2, TOPREAL1:def_2, XBOOLE_0:def_3;
A5: (LSeg (|[1,0]|,p1)) /\ (LSeg (|[1,0]|,|[1,1]|)) c= (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) by A3, Lm24, TOPREAL1:6, XBOOLE_1:26;
 |[0,0]| in LSeg (p1,|[0,0]|) by RLTOPSP1:68;
then A6: |[0,0]| in (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by Lm20, XBOOLE_0:def_4;
 |[1,0]| in LSeg (|[1,0]|,p1) by RLTOPSP1:68;
then A7: (LSeg (|[1,0]|,p1)) /\ (LSeg (|[1,0]|,|[1,1]|)) <> {} by Lm25, XBOOLE_0:def_4;
A8: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) c= (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A3, Lm21, TOPREAL1:6, XBOOLE_1:26;
A9: (LSeg (p1,|[0,0]|)) /\ (LSeg (p1,|[1,0]|)) = {p1} by A3, TOPREAL1:8;
A10: LSeg (|[0,0]|,p1) c= LSeg (|[0,0]|,|[1,0]|) by A3, Lm21, TOPREAL1:6;
 (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by TOPREAL1:19, XBOOLE_0:def_7;
then A11: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by A10, XBOOLE_1:3, XBOOLE_1:26;
A12: LSeg (|[1,0]|,p1) c= LSeg (|[0,0]|,|[1,0]|) by A3, Lm24, TOPREAL1:6;
 (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by TOPREAL1:19, XBOOLE_0:def_7;
then A13: (LSeg (|[1,0]|,p1)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by A12, XBOOLE_1:3, XBOOLE_1:26;
consider p being Point of (TOP-REAL 2) such that
A14: p = p1 and
A15: p `1 <= 1 and
A16: p `1 >= 0 and
A17: p `2 = 0 by A3, TOPREAL1:13;
percases ( p2 in LSeg (|[0,0]|,|[0,1]|) or p2 in LSeg (|[0,1]|,|[1,1]|) or p2 in LSeg (|[0,0]|,|[1,0]|) or p2 in LSeg (|[1,0]|,|[1,1]|) ) by A4, XBOOLE_0:def_3;
supposeA18: p2 in LSeg (|[0,0]|,|[0,1]|) ; ::_thesis: ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A19: LSeg (|[0,1]|,|[1,1]|) is_an_arc_of |[1,1]|,|[0,1]| by Lm6, Lm10, TOPREAL1:9;
 LSeg (|[1,0]|,|[1,1]|) is_an_arc_of |[1,0]|,|[1,1]| by Lm9, Lm11, TOPREAL1:9;
then A20: (LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|)) is_an_arc_of |[1,0]|,|[0,1]| by A19, TOPREAL1:2, TOPREAL1:18;
 (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by TOPREAL1:19, XBOOLE_0:def_7;
then A21: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by A10, XBOOLE_1:3, XBOOLE_1:26;
take P1 = (LSeg (p1,|[0,0]|)) \/ (LSeg (|[0,0]|,p2)); ::_thesis: ex P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
take P2 = (LSeg (p1,|[1,0]|)) \/ (((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,1]|,p2))); ::_thesis: ( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A22: (LSeg (p1,|[0,0]|)) \/ (LSeg (p1,|[1,0]|)) = LSeg (|[0,0]|,|[1,0]|) by A3, TOPREAL1:5;
 |[0,1]| in LSeg (|[0,1]|,p2) by RLTOPSP1:68;
then A23: |[0,1]| in (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)) by Lm23, XBOOLE_0:def_4;
A24: |[0,0]| in LSeg (|[0,0]|,p2) by RLTOPSP1:68;
 |[0,0]| in LSeg (p1,|[0,0]|) by RLTOPSP1:68;
then A25: |[0,0]| in (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,0]|,p2)) by A24, XBOOLE_0:def_4;
A26: LSeg (|[0,0]|,p2) c= LSeg (|[0,0]|,|[0,1]|) by A18, Lm20, TOPREAL1:6;
then  (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,0]|,p2)) c= (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A10, XBOOLE_1:27;
then A27: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,0]|,p2)) = {|[0,0]|} by A25, TOPREAL1:17, ZFMISC_1:33;
A28: ex q being Point of (TOP-REAL 2) st
( q = p2 & q `1 = 0 & q `2 <= 1 & q `2 >= 0 ) by A18, TOPREAL1:13;
now__::_thesis:_not_|[0,0]|_in_(LSeg_(p1,|[1,0]|))_/\_(LSeg_(|[0,1]|,p2))
A29: p2 `2 <= |[0,1]| `2 by A28, EUCLID:52;
assume A30: |[0,0]| in (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,1]|,p2)) ; ::_thesis: contradiction
then A31: |[0,0]| in LSeg (p1,|[1,0]|) by XBOOLE_0:def_4;
 |[0,0]| in LSeg (p2,|[0,1]|) by A30, XBOOLE_0:def_4;
then A32: 0 = p2 `2 by A28, A29, Lm5, TOPREAL1:4;
 p1 `1 <= |[1,0]| `1 by A14, A15, EUCLID:52;
then  0 = p1 `1 by A14, A16, A31, Lm4, TOPREAL1:3;
then p1 = |[0,0]| by A14, A17, EUCLID:53
.= p2 by A28, A32, EUCLID:53 ;
hence  contradiction by A1; ::_thesis: verum
end;
then A33: {|[0,0]|} <> (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,1]|,p2)) by ZFMISC_1:31;
 ( p1 <> |[0,0]| or |[0,0]| <> p2 ) by A1;
hence  P1 is_an_arc_of p1,p2 by A27, TOPREAL1:12; ::_thesis: ( P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A34: {p1} = (LSeg (p1,|[0,0]|)) /\ (LSeg (p1,|[1,0]|)) by A3, TOPREAL1:8;
 (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by TOPREAL1:20, XBOOLE_0:def_7;
then A35: (LSeg (|[0,0]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by A26, XBOOLE_1:3, XBOOLE_1:26;
A36: LSeg (p2,|[0,1]|) c= LSeg (|[0,0]|,|[0,1]|) by A18, Lm22, TOPREAL1:6;
then A37: (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)) c= (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by XBOOLE_1:27;
A38: (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)) = {} by A36, Lm3, XBOOLE_1:3, XBOOLE_1:26;
((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) /\ (LSeg (|[0,1]|,p2)) = ((LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,1]|,p2))) \/ ((LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,1]|,p2))) by XBOOLE_1:23
.= {|[0,1]|} by A38, A37, A23, TOPREAL1:15, ZFMISC_1:33 ;
then A39: ((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,1]|,p2)) is_an_arc_of |[1,0]|,p2 by A20, TOPREAL1:10;
A40: {p2} = (LSeg (|[0,0]|,p2)) /\ (LSeg (|[0,1]|,p2)) by A18, TOPREAL1:8;
A41: (LSeg (|[0,1]|,p2)) \/ (LSeg (|[0,0]|,p2)) = LSeg (|[0,0]|,|[0,1]|) by A18, TOPREAL1:5;
 (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,1]|,p2)) c= (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A12, A36, XBOOLE_1:27;
then A42: (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,1]|,p2)) = {} by A33, TOPREAL1:17, ZFMISC_1:33;
(LSeg (p1,|[1,0]|)) /\ (((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,1]|,p2))) = ((LSeg (p1,|[1,0]|)) /\ ((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,1]|,p2))) by XBOOLE_1:23
.= ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ ((LSeg (|[1,0]|,p1)) /\ (LSeg (|[0,1]|,|[1,1]|))) by A42, XBOOLE_1:23
.= {|[1,0]|} by A13, A5, A7, TOPREAL1:16, ZFMISC_1:33 ;
hence  P2 is_an_arc_of p1,p2 by A39, TOPREAL1:11; ::_thesis: ( R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
thus P1 \/ P2 = (LSeg (|[0,0]|,p2)) \/ ((LSeg (p1,|[0,0]|)) \/ ((LSeg (p1,|[1,0]|)) \/ (((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,1]|,p2))))) by XBOOLE_1:4
.= (LSeg (|[0,0]|,p2)) \/ ((LSeg (|[0,0]|,|[1,0]|)) \/ (((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,1]|,p2)))) by A22, XBOOLE_1:4
.= (LSeg (|[0,0]|,p2)) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ ((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|)))) \/ (LSeg (|[0,1]|,p2))) by XBOOLE_1:4
.= (LSeg (|[0,0]|,p2)) \/ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,1]|,p2))) by XBOOLE_1:4
.= (LSeg (|[0,0]|,p2)) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ ((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,1]|,p2)))) by XBOOLE_1:4
.= (((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,1]|,p2))) \/ (LSeg (|[0,0]|,p2))) \/ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) by XBOOLE_1:4
.= R^2-unit_square by A41, TOPREAL1:def_2, XBOOLE_1:4 ; ::_thesis: P1 /\ P2 = {p1,p2}
A43: P1 /\ P2 = ((LSeg (p1,|[0,0]|)) /\ ((LSeg (p1,|[1,0]|)) \/ (((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,1]|,p2))))) \/ ((LSeg (|[0,0]|,p2)) /\ ((LSeg (p1,|[1,0]|)) \/ (((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,1]|,p2))))) by XBOOLE_1:23
.= (((LSeg (p1,|[0,0]|)) /\ (LSeg (p1,|[1,0]|))) \/ ((LSeg (p1,|[0,0]|)) /\ (((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,1]|,p2))))) \/ ((LSeg (|[0,0]|,p2)) /\ ((LSeg (p1,|[1,0]|)) \/ (((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,1]|,p2))))) by XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[0,0]|)) /\ ((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2))))) \/ ((LSeg (|[0,0]|,p2)) /\ ((LSeg (p1,|[1,0]|)) \/ (((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,1]|,p2))))) by A34, XBOOLE_1:23
.= ({p1} \/ ((((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2))))) \/ ((LSeg (|[0,0]|,p2)) /\ ((LSeg (p1,|[1,0]|)) \/ (((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,1]|,p2))))) by XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2))))) \/ (((LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[1,0]|))) \/ ((LSeg (|[0,0]|,p2)) /\ (((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,1]|,p2))))) by A21, XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2))))) \/ (((LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[1,0]|))) \/ (((LSeg (|[0,0]|,p2)) /\ ((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|)))) \/ {p2})) by A40, XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2))))) \/ (((LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[1,0]|))) \/ ((((LSeg (|[0,0]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ ((LSeg (|[0,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ {p2})) by XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2))))) \/ (((LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[1,0]|))) \/ (((LSeg (|[0,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2})) by A35 ;
A44: now__::_thesis:_P1_/\_P2_=_({p1}_\/_((LSeg_(p1,|[0,0]|))_/\_(LSeg_(|[0,1]|,p2))))_\/_(((LSeg_(|[0,0]|,p2))_/\_(LSeg_(|[0,1]|,|[1,1]|)))_\/_{p2})
percases ( p1 = |[0,0]| or p1 = |[1,0]| or ( p1 <> |[1,0]| & p1 <> |[0,0]| ) ) ;
supposeA45: p1 = |[0,0]| ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2)))) \/ (((LSeg (|[0,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2})
A46: p1 in LSeg (p1,|[1,0]|) by RLTOPSP1:68;
 p1 in LSeg (|[0,0]|,p2) by A45, RLTOPSP1:68;
then A47: (LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[1,0]|)) <> {} by A46, XBOOLE_0:def_4;
 (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {|[0,0]|} /\ (LSeg (|[1,0]|,|[1,1]|)) by A45, RLTOPSP1:70;
then A48: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by Lm1, Lm12;
 (LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[1,0]|)) c= {p1} by A18, A45, Lm20, TOPREAL1:6, TOPREAL1:17, XBOOLE_1:26;
then  (LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[1,0]|)) = {p1} by A47, ZFMISC_1:33;
hence P1 /\ P2 = ({p1} \/ ({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2))))) \/ (((LSeg (|[0,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2}) by A43, A48, XBOOLE_1:4
.= (({p1} \/ {p1}) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2)))) \/ (((LSeg (|[0,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2}) by XBOOLE_1:4
.= ({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2)))) \/ (((LSeg (|[0,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2}) ;
::_thesis: verum
end;
supposeA49: p1 = |[1,0]| ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2)))) \/ (((LSeg (|[0,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2})
then A50: (LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[1,0]|)) = (LSeg (|[0,0]|,p2)) /\ {|[1,0]|} by RLTOPSP1:70;
 not |[1,0]| in LSeg (|[0,0]|,p2) by A26, Lm4, Lm6, Lm8, TOPREAL1:3;
then  (LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[1,0]|)) = {} by A50, Lm1;
hence P1 /\ P2 = (({p1} \/ {p1}) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2)))) \/ (((LSeg (|[0,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2}) by A43, A49, TOPREAL1:16, XBOOLE_1:4
.= ({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2)))) \/ (((LSeg (|[0,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2}) ;
::_thesis: verum
end;
supposeA51: ( p1 <> |[1,0]| & p1 <> |[0,0]| ) ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2)))) \/ (((LSeg (|[0,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2})
now__::_thesis:_not_|[0,0]|_in_(LSeg_(|[0,0]|,p2))_/\_(LSeg_(p1,|[1,0]|))
assume  |[0,0]| in (LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[1,0]|)) ; ::_thesis: contradiction
then A52: |[0,0]| in LSeg (p1,|[1,0]|) by XBOOLE_0:def_4;
 p1 `1 <= |[1,0]| `1 by A14, A15, EUCLID:52;
then  0 = p1 `1 by A14, A16, A52, Lm4, TOPREAL1:3;
hence  contradiction by A14, A17, A51, EUCLID:53; ::_thesis: verum
end;
then A53: {|[0,0]|} <> (LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[1,0]|)) by ZFMISC_1:31;
 (LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[1,0]|)) c= {|[0,0]|} by A12, A26, TOPREAL1:17, XBOOLE_1:27;
then A54: (LSeg (|[0,0]|,p2)) /\ (LSeg (p1,|[1,0]|)) = {} by A53, ZFMISC_1:33;
now__::_thesis:_not_|[1,0]|_in_(LSeg_(p1,|[0,0]|))_/\_(LSeg_(|[1,0]|,|[1,1]|))
assume  |[1,0]| in (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) ; ::_thesis: contradiction
then A55: |[1,0]| in LSeg (|[0,0]|,p1) by XBOOLE_0:def_4;
 |[0,0]| `1 <= p1 `1 by A14, A16, EUCLID:52;
then  |[1,0]| `1 <= p1 `1 by A55, TOPREAL1:3;
then  p1 `1 = 1 by A14, A15, Lm8, XXREAL_0:1;
hence  contradiction by A14, A17, A51, EUCLID:53; ::_thesis: verum
end;
then A56: {|[1,0]|} <> (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) by ZFMISC_1:31;
 (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) c= {|[1,0]|} by A3, Lm21, TOPREAL1:6, TOPREAL1:16, XBOOLE_1:26;
then  (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by A56, ZFMISC_1:33;
hence  P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2)))) \/ (((LSeg (|[0,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2}) by A43, A54; ::_thesis: verum
end;
end;
end;
now__::_thesis:_(_P1_/\_P2_=_{p1,p2}_&_P1_/\_P2_=_{p1,p2}_)
percases ( p2 = |[0,0]| or p2 = |[0,1]| or ( p2 <> |[0,1]| & p2 <> |[0,0]| ) ) ;
supposeA57: p2 = |[0,0]| ; ::_thesis: ( P1 /\ P2 = {p1,p2} & P1 /\ P2 = {p1,p2} )
 |[0,0]| in LSeg (p1,|[0,0]|) by RLTOPSP1:68;
then A58: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2)) <> {} by A57, Lm20, XBOOLE_0:def_4;
 (LSeg (|[0,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {|[0,0]|} /\ (LSeg (|[0,1]|,|[1,1]|)) by A57, RLTOPSP1:70;
then A59: (LSeg (|[0,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by Lm1, Lm13;
 (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2)) c= (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A10, A36, XBOOLE_1:27;
then  (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2)) = {p2} by A57, A58, TOPREAL1:17, ZFMISC_1:33;
hence P1 /\ P2 = {p1} \/ ({p2} \/ {p2}) by A44, A59, XBOOLE_1:4
.= {p1,p2} by ENUMSET1:1 ;
::_thesis: P1 /\ P2 = {p1,p2}
hence  P1 /\ P2 = {p1,p2} ; ::_thesis: verum
end;
supposeA60: p2 = |[0,1]| ; ::_thesis: ( P1 /\ P2 = {p1,p2} & P1 /\ P2 = {p1,p2} )
then A61: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2)) = (LSeg (p1,|[0,0]|)) /\ {|[0,1]|} by RLTOPSP1:70;
 not |[0,1]| in LSeg (p1,|[0,0]|) by A10, Lm5, Lm7, Lm9, TOPREAL1:4;
then A62: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2)) = {} by A61, Lm1;
hence  P1 /\ P2 = {p1,p2} by A44, A60, ENUMSET1:1, TOPREAL1:15; ::_thesis: P1 /\ P2 = {p1,p2}
thus  P1 /\ P2 = {p1,p2} by A44, A60, A62, ENUMSET1:1, TOPREAL1:15; ::_thesis: verum
end;
supposeA63: ( p2 <> |[0,1]| & p2 <> |[0,0]| ) ; ::_thesis: ( P1 /\ P2 = {p1,p2} & P1 /\ P2 = {p1,p2} )
now__::_thesis:_not_|[0,1]|_in_(LSeg_(|[0,0]|,p2))_/\_(LSeg_(|[0,1]|,|[1,1]|))
assume  |[0,1]| in (LSeg (|[0,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) ; ::_thesis: contradiction
then A64: |[0,1]| in LSeg (|[0,0]|,p2) by XBOOLE_0:def_4;
 |[0,0]| `2 <= p2 `2 by A28, EUCLID:52;
then  |[0,1]| `2 <= p2 `2 by A64, TOPREAL1:4;
then  1 = p2 `2 by A28, Lm7, XXREAL_0:1;
hence  contradiction by A28, A63, EUCLID:53; ::_thesis: verum
end;
then A65: {|[0,1]|} <> (LSeg (|[0,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) by ZFMISC_1:31;
 (LSeg (|[0,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) c= {|[0,1]|} by A18, Lm20, TOPREAL1:6, TOPREAL1:15, XBOOLE_1:26;
then A66: (LSeg (|[0,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by A65, ZFMISC_1:33;
now__::_thesis:_not_|[0,0]|_in_(LSeg_(p1,|[0,0]|))_/\_(LSeg_(|[0,1]|,p2))
assume  |[0,0]| in (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2)) ; ::_thesis: contradiction
then A67: |[0,0]| in LSeg (p2,|[0,1]|) by XBOOLE_0:def_4;
 p2 `2 <= |[0,1]| `2 by A28, EUCLID:52;
then  p2 `2 = 0 by A28, A67, Lm5, TOPREAL1:4;
hence  contradiction by A28, A63, EUCLID:53; ::_thesis: verum
end;
then A68: {|[0,0]|} <> (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2)) by ZFMISC_1:31;
 (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2)) c= (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A10, A36, XBOOLE_1:27;
then A69: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2)) = {} by A68, TOPREAL1:17, ZFMISC_1:33;
hence  P1 /\ P2 = {p1,p2} by A44, A66, ENUMSET1:1; ::_thesis: P1 /\ P2 = {p1,p2}
thus  P1 /\ P2 = {p1,p2} by A44, A69, A66, ENUMSET1:1; ::_thesis: verum
end;
end;
end;
hence  P1 /\ P2 = {p1,p2} ; ::_thesis: verum
end;
supposeA70: p2 in LSeg (|[0,1]|,|[1,1]|) ; ::_thesis: ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
then A71: LSeg (p2,|[1,1]|) c= LSeg (|[0,1]|,|[1,1]|) by Lm26, TOPREAL1:6;
then A72: (LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)) = {} by A12, Lm2, XBOOLE_1:3, XBOOLE_1:27;
A73: LSeg (p2,|[0,1]|) c= LSeg (|[0,1]|,|[1,1]|) by A70, Lm23, TOPREAL1:6;
then A74: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2)) = {} by A10, Lm2, XBOOLE_1:3, XBOOLE_1:27;
take P1 = ((LSeg (p1,|[0,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2)); ::_thesis: ex P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
take P2 = ((LSeg (p1,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2)); ::_thesis: ( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
 |[0,1]| in LSeg (|[0,1]|,p2) by RLTOPSP1:68;
then A75: (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[0,1]|,p2)) <> {} by Lm22, XBOOLE_0:def_4;
 (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[0,1]|,p2)) c= {|[0,1]|} by A70, Lm23, TOPREAL1:6, TOPREAL1:15, XBOOLE_1:26;
then  (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[0,1]|,p2)) = {|[0,1]|} by A75, ZFMISC_1:33;
then A76: (LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,p2)) is_an_arc_of |[0,0]|,p2 by Lm5, Lm7, TOPREAL1:12;
(LSeg (p1,|[0,0]|)) /\ ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,p2))) = ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,p2))) by XBOOLE_1:23
.= {|[0,0]|} by A8, A6, A74, TOPREAL1:17, ZFMISC_1:33 ;
then  (LSeg (p1,|[0,0]|)) \/ ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,p2))) is_an_arc_of p1,p2 by A76, TOPREAL1:11;
hence  P1 is_an_arc_of p1,p2 by XBOOLE_1:4; ::_thesis: ( P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
 |[1,1]| in LSeg (|[1,1]|,p2) by RLTOPSP1:68;
then A77: (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[1,1]|,p2)) <> {} by Lm27, XBOOLE_0:def_4;
 (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[1,1]|,p2)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) by A70, Lm26, TOPREAL1:6, XBOOLE_1:26;
then  (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[1,1]|,p2)) = {|[1,1]|} by A77, TOPREAL1:18, ZFMISC_1:33;
then A78: (LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[1,1]|,p2)) is_an_arc_of |[1,0]|,p2 by Lm9, Lm11, TOPREAL1:12;
(LSeg (p1,|[1,0]|)) /\ ((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[1,1]|,p2))) = ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2))) by XBOOLE_1:23
.= {|[1,0]|} by A5, A7, A72, TOPREAL1:16, ZFMISC_1:33 ;
then  (LSeg (p1,|[1,0]|)) \/ ((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[1,1]|,p2))) is_an_arc_of p1,p2 by A78, TOPREAL1:11;
hence  P2 is_an_arc_of p1,p2 by XBOOLE_1:4; ::_thesis: ( R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
thus  R^2-unit_square = ((LSeg (|[0,0]|,|[0,1]|)) \/ ((LSeg (|[0,1]|,p2)) \/ (LSeg (|[1,1]|,p2)))) \/ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) by A70, TOPREAL1:5, TOPREAL1:def_2
.= (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,p2))) \/ (LSeg (|[1,1]|,p2))) \/ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) by XBOOLE_1:4
.= ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,p2))) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))) by XBOOLE_1:4
.= ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,p2))) \/ ((LSeg (|[0,0]|,|[1,0]|)) \/ ((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[1,1]|,p2)))) by XBOOLE_1:4
.= ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,p2))) \/ (((LSeg (p1,|[0,0]|)) \/ (LSeg (p1,|[1,0]|))) \/ ((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[1,1]|,p2)))) by A3, TOPREAL1:5
.= ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,p2))) \/ ((LSeg (p1,|[0,0]|)) \/ ((LSeg (p1,|[1,0]|)) \/ ((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[1,1]|,p2))))) by XBOOLE_1:4
.= ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,p2))) \/ ((LSeg (p1,|[0,0]|)) \/ (((LSeg (p1,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2)))) by XBOOLE_1:4
.= ((LSeg (p1,|[0,0]|)) \/ ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,p2)))) \/ (((LSeg (p1,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))) by XBOOLE_1:4
.= P1 \/ P2 by XBOOLE_1:4 ; ::_thesis: P1 /\ P2 = {p1,p2}
A79: ex q being Point of (TOP-REAL 2) st
( q = p2 & q `1 <= 1 & q `1 >= 0 & q `2 = 1 ) by A70, TOPREAL1:13;
 (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by TOPREAL1:19, XBOOLE_0:def_7;
then A80: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,1]|,p2)) = {} by A10, A71, XBOOLE_1:3, XBOOLE_1:27;
A81: (LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[1,0]|)) = {} by A12, A73, Lm2, XBOOLE_1:3, XBOOLE_1:27;
A82: (LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,1]|,p2)) = {p2} by A70, TOPREAL1:8;
A83: P1 /\ P2 = (((LSeg (p1,|[0,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) /\ (((LSeg (p1,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2)))) \/ ((LSeg (|[0,1]|,p2)) /\ (((LSeg (p1,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2)))) by XBOOLE_1:23
.= (((LSeg (p1,|[0,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) /\ (((LSeg (p1,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2)))) \/ (((LSeg (|[0,1]|,p2)) /\ ((LSeg (p1,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)))) \/ {p2}) by A82, XBOOLE_1:23
.= (((LSeg (p1,|[0,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) /\ (((LSeg (p1,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2)))) \/ ((((LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[1,0]|))) \/ ((LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ {p2}) by XBOOLE_1:23
.= (((LSeg (p1,|[0,0]|)) /\ (((LSeg (p1,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2)))) \/ ((LSeg (|[0,0]|,|[0,1]|)) /\ (((LSeg (p1,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ {p2}) by A81, XBOOLE_1:23
.= ((((LSeg (p1,|[0,0]|)) /\ ((LSeg (p1,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,1]|,p2)))) \/ ((LSeg (|[0,0]|,|[0,1]|)) /\ (((LSeg (p1,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ {p2}) by XBOOLE_1:23
.= (({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (|[0,0]|,|[0,1]|)) /\ (((LSeg (p1,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ {p2}) by A9, A80, XBOOLE_1:23
.= (({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ (((LSeg (|[0,0]|,|[0,1]|)) /\ ((LSeg (p1,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,1]|,p2))))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ {p2}) by XBOOLE_1:23
.= (({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (p1,|[1,0]|))) \/ ((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,1]|,p2))))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ {p2}) by XBOOLE_1:23
.= (({p1} \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ (((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (p1,|[1,0]|))) \/ ((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,1]|,p2))))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ {p2}) by Lm3 ;
A84: now__::_thesis:_P1_/\_P2_=_({p1}_\/_((LSeg_(|[0,0]|,|[0,1]|))_/\_(LSeg_(|[1,1]|,p2))))_\/_(((LSeg_(|[0,1]|,p2))_/\_(LSeg_(|[1,0]|,|[1,1]|)))_\/_{p2})
percases ( p1 = |[0,0]| or p1 = |[1,0]| or ( p1 <> |[1,0]| & p1 <> |[0,0]| ) ) ;
supposeA85: p1 = |[0,0]| ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ {p2})
then  (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {|[0,0]|} /\ (LSeg (|[1,0]|,|[1,1]|)) by RLTOPSP1:70;
then  (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by Lm1, Lm12;
hence P1 /\ P2 = (({p1} \/ {p1}) \/ ((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ {p2}) by A83, A85, TOPREAL1:17, XBOOLE_1:4
.= ({p1} \/ ((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ {p2}) ;
::_thesis: verum
end;
supposeA86: p1 = |[1,0]| ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ {p2})
then  (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (p1,|[1,0]|)) = (LSeg (|[0,0]|,|[0,1]|)) /\ {|[1,0]|} by RLTOPSP1:70;
then  (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (p1,|[1,0]|)) = {} by Lm1, Lm16;
hence  P1 /\ P2 = ({p1} \/ ((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ {p2}) by A83, A86, TOPREAL1:16; ::_thesis: verum
end;
supposeA87: ( p1 <> |[1,0]| & p1 <> |[0,0]| ) ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ {p2})
now__::_thesis:_not_|[0,0]|_in_(LSeg_(|[0,0]|,|[0,1]|))_/\_(LSeg_(p1,|[1,0]|))
assume  |[0,0]| in (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (p1,|[1,0]|)) ; ::_thesis: contradiction
then A88: |[0,0]| in LSeg (p1,|[1,0]|) by XBOOLE_0:def_4;
 p1 `1 <= |[1,0]| `1 by A14, A15, EUCLID:52;
then  0 = p1 `1 by A14, A16, A88, Lm4, TOPREAL1:3;
hence  contradiction by A14, A17, A87, EUCLID:53; ::_thesis: verum
end;
then A89: {|[0,0]|} <> (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (p1,|[1,0]|)) by ZFMISC_1:31;
 (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (p1,|[1,0]|)) c= {|[0,0]|} by A3, Lm24, TOPREAL1:6, TOPREAL1:17, XBOOLE_1:26;
then A90: (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (p1,|[1,0]|)) = {} by A89, ZFMISC_1:33;
now__::_thesis:_not_|[1,0]|_in_(LSeg_(p1,|[0,0]|))_/\_(LSeg_(|[1,0]|,|[1,1]|))
assume  |[1,0]| in (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) ; ::_thesis: contradiction
then A91: |[1,0]| in LSeg (|[0,0]|,p1) by XBOOLE_0:def_4;
 |[0,0]| `1 <= p1 `1 by A14, A16, EUCLID:52;
then  |[1,0]| `1 <= p1 `1 by A91, TOPREAL1:3;
then  1 = p1 `1 by A14, A15, Lm8, XXREAL_0:1;
hence  contradiction by A14, A17, A87, EUCLID:53; ::_thesis: verum
end;
then A92: {|[1,0]|} <> (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) by ZFMISC_1:31;
 (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) c= {|[1,0]|} by A3, Lm21, TOPREAL1:6, TOPREAL1:16, XBOOLE_1:26;
then  (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by A92, ZFMISC_1:33;
hence  P1 /\ P2 = ({p1} \/ ((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ {p2}) by A83, A90; ::_thesis: verum
end;
end;
end;
now__::_thesis:_P1_/\_P2_=_{p1,p2}
percases ( p2 = |[0,1]| or p2 = |[1,1]| or ( p2 <> |[1,1]| & p2 <> |[0,1]| ) ) ;
supposeA93: p2 = |[0,1]| ; ::_thesis: P1 /\ P2 = {p1,p2}
then  (LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {|[0,1]|} /\ (LSeg (|[1,0]|,|[1,1]|)) by RLTOPSP1:70;
then  (LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by Lm1, Lm15;
hence P1 /\ P2 = {p1} \/ ({p2} \/ {p2}) by A84, A93, TOPREAL1:15, XBOOLE_1:4
.= {p1,p2} by ENUMSET1:1 ;
::_thesis: verum
end;
supposeA94: p2 = |[1,1]| ; ::_thesis: P1 /\ P2 = {p1,p2}
then  (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)) = (LSeg (|[0,0]|,|[0,1]|)) /\ {|[1,1]|} by RLTOPSP1:70;
then  (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)) = {} by Lm1, Lm18;
hence  P1 /\ P2 = {p1,p2} by A84, A94, ENUMSET1:1, TOPREAL1:18; ::_thesis: verum
end;
supposeA95: ( p2 <> |[1,1]| & p2 <> |[0,1]| ) ; ::_thesis: P1 /\ P2 = {p1,p2}
now__::_thesis:_not_|[1,1]|_in_(LSeg_(|[0,1]|,p2))_/\_(LSeg_(|[1,0]|,|[1,1]|))
assume  |[1,1]| in (LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) ; ::_thesis: contradiction
then A96: |[1,1]| in LSeg (|[0,1]|,p2) by XBOOLE_0:def_4;
 |[0,1]| `1 <= p2 `1 by A79, EUCLID:52;
then  |[1,1]| `1 <= p2 `1 by A96, TOPREAL1:3;
then  1 = p2 `1 by A79, Lm10, XXREAL_0:1;
hence  contradiction by A79, A95, EUCLID:53; ::_thesis: verum
end;
then A97: {|[1,1]|} <> (LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) by ZFMISC_1:31;
 (LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) c= {|[1,1]|} by A70, Lm23, TOPREAL1:6, TOPREAL1:18, XBOOLE_1:26;
then A98: (LSeg (|[0,1]|,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by A97, ZFMISC_1:33;
now__::_thesis:_not_|[0,1]|_in_(LSeg_(|[0,0]|,|[0,1]|))_/\_(LSeg_(|[1,1]|,p2))
assume  |[0,1]| in (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)) ; ::_thesis: contradiction
then A99: |[0,1]| in LSeg (p2,|[1,1]|) by XBOOLE_0:def_4;
 p2 `1 <= |[1,1]| `1 by A79, EUCLID:52;
then  p2 `1 = 0 by A79, A99, Lm6, TOPREAL1:3;
hence  contradiction by A79, A95, EUCLID:53; ::_thesis: verum
end;
then A100: {|[0,1]|} <> (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)) by ZFMISC_1:31;
 (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)) c= {|[0,1]|} by A70, Lm26, TOPREAL1:6, TOPREAL1:15, XBOOLE_1:26;
then  (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)) = {} by A100, ZFMISC_1:33;
hence  P1 /\ P2 = {p1,p2} by A84, A98, ENUMSET1:1; ::_thesis: verum
end;
end;
end;
hence  P1 /\ P2 = {p1,p2} ; ::_thesis: verum
end;
supposeA101: p2 in LSeg (|[0,0]|,|[1,0]|) ; ::_thesis: ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A102: p = |[(p `1),(p `2)]| by EUCLID:53;
A103: LSeg (p1,p2) c= LSeg (|[0,0]|,|[1,0]|) by A3, A101, TOPREAL1:6;
consider q being Point of (TOP-REAL 2) such that
A104: q = p2 and
A105: q `1 <= 1 and
A106: q `1 >= 0 and
A107: q `2 = 0 by A101, TOPREAL1:13;
A108: q = |[(q `1),(q `2)]| by EUCLID:53;
now__::_thesis:_ex_P1_being_Element_of_K19(_the_carrier_of_(TOP-REAL_2))_ex_P2_being_Element_of_K19(_the_carrier_of_(TOP-REAL_2))_st_
(_P1_is_an_arc_of_p1,p2_&_P2_is_an_arc_of_p1,p2_&_P1_\/_P2_=_R^2-unit_square_&_P1_/\_P2_=_{p1,p2}_)
percases ( p `1 < q `1 or q `1 < p `1 ) by A1, A14, A17, A104, A107, A102, A108, XXREAL_0:1;
supposeA109: p `1 < q `1 ; ::_thesis: ex P1 being Element of K19( the carrier of (TOP-REAL 2)) ex P2 being Element of K19( the carrier of (TOP-REAL 2)) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
now__::_thesis:_not_|[1,0]|_in_(LSeg_(p1,|[0,0]|))_/\_(LSeg_(|[1,0]|,|[1,1]|))
assume  |[1,0]| in (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) ; ::_thesis: contradiction
then A110: |[1,0]| in LSeg (|[0,0]|,p1) by XBOOLE_0:def_4;
 |[0,0]| `1 <= p1 `1 by A14, A16, EUCLID:52;
then  |[1,0]| `1 <= p1 `1 by A110, TOPREAL1:3;
hence  contradiction by A14, A15, A105, A109, Lm8, XXREAL_0:1; ::_thesis: verum
end;
then A111: {|[1,0]|} <> (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) by ZFMISC_1:31;
 (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) c= {|[1,0]|} by A3, Lm21, TOPREAL1:6, TOPREAL1:16, XBOOLE_1:26;
then A112: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by A111, ZFMISC_1:33;
 |[0,0]| in LSeg (p1,|[0,0]|) by RLTOPSP1:68;
then A113: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) <> {} by Lm20, XBOOLE_0:def_4;
now__::_thesis:_not_|[0,0]|_in_(LSeg_(|[0,0]|,|[0,1]|))_/\_(LSeg_(|[1,0]|,p2))
assume  |[0,0]| in (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,p2)) ; ::_thesis: contradiction
then A114: |[0,0]| in LSeg (p2,|[1,0]|) by XBOOLE_0:def_4;
 p2 `1 <= |[1,0]| `1 by A104, A105, EUCLID:52;
hence  contradiction by A16, A104, A109, A114, Lm4, TOPREAL1:3; ::_thesis: verum
end;
then A115: {|[0,0]|} <> (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,p2)) by ZFMISC_1:31;
 (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,p2)) c= {|[0,0]|} by A101, Lm24, TOPREAL1:6, TOPREAL1:17, XBOOLE_1:26;
then A116: (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,p2)) = {} by A115, ZFMISC_1:33;
 (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by TOPREAL1:19, XBOOLE_0:def_7;
then A117: (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by A103, XBOOLE_1:3, XBOOLE_1:26;
A118: (LSeg (p1,p2)) /\ (LSeg (p1,|[0,0]|)) c= {p1}
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in (LSeg (p1,p2)) /\ (LSeg (p1,|[0,0]|)) or a in {p1} )
assume A119: a in (LSeg (p1,p2)) /\ (LSeg (p1,|[0,0]|)) ; ::_thesis: a in {p1}
then reconsider p = a as Point of (TOP-REAL 2) ;
A120: p in LSeg (|[0,0]|,p1) by A119, XBOOLE_0:def_4;
 |[0,0]| `1 <= p1 `1 by A14, A16, EUCLID:52;
then A121: p `1 <= p1 `1 by A120, TOPREAL1:3;
A122: p in LSeg (p1,p2) by A119, XBOOLE_0:def_4;
then  p1 `1 <= p `1 by A14, A104, A109, TOPREAL1:3;
then A123: p1 `1 = p `1 by A121, XXREAL_0:1;
 p1 `2 <= p `2 by A14, A17, A104, A107, A122, TOPREAL1:4;
then  p `2 = 0 by A14, A17, A104, A107, A122, TOPREAL1:4;
then p = |[(p1 `1),0]| by A123, EUCLID:53
.= p1 by A14, A17, EUCLID:53 ;
hence  a in {p1} by TARSKI:def_1; ::_thesis: verum
end;
A124: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) c= (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A3, Lm21, TOPREAL1:6, XBOOLE_1:26;
take P1 = LSeg (p1,p2); ::_thesis: ex P2 being Element of K19( the carrier of (TOP-REAL 2)) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
take P2 = (LSeg (p1,|[0,0]|)) \/ ((((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,0]|,p2))); ::_thesis: ( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
A125: (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by TOPREAL1:19, XBOOLE_0:def_7;
 (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[1,0]|,p2)) c= (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by A101, Lm24, TOPREAL1:6, XBOOLE_1:26;
then A126: (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[1,0]|,p2)) = {} by A125, XBOOLE_1:3;
thus  P1 is_an_arc_of p1,p2 by A1, TOPREAL1:9; ::_thesis: ( P2 is_an_arc_of p1,p2 & P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
A127: ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) /\ (LSeg (|[1,0]|,|[1,1]|)) = ((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ ((LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|))) by XBOOLE_1:23
.= {|[1,1]|} by Lm3, TOPREAL1:18 ;
A128: (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,p2)) c= {p2}
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,p2)) or a in {p2} )
assume A129: a in (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,p2)) ; ::_thesis: a in {p2}
then reconsider p = a as Point of (TOP-REAL 2) ;
A130: p in LSeg (p2,|[1,0]|) by A129, XBOOLE_0:def_4;
 p2 `1 <= |[1,0]| `1 by A104, A105, EUCLID:52;
then A131: p2 `1 <= p `1 by A130, TOPREAL1:3;
A132: p in LSeg (p1,p2) by A129, XBOOLE_0:def_4;
then  p `1 <= p2 `1 by A14, A104, A109, TOPREAL1:3;
then A133: p2 `1 = p `1 by A131, XXREAL_0:1;
 p1 `2 <= p `2 by A14, A17, A104, A107, A132, TOPREAL1:4;
then  p `2 = 0 by A14, A17, A104, A107, A132, TOPREAL1:4;
then p = |[(p2 `1),0]| by A133, EUCLID:53
.= p2 by A104, A107, EUCLID:53 ;
hence  a in {p2} by TARSKI:def_1; ::_thesis: verum
end;
 (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by TOPREAL1:19, XBOOLE_0:def_7;
then A134: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by A10, XBOOLE_1:3, XBOOLE_1:26;
A135: now__::_thesis:_not_(LSeg_(p1,|[0,0]|))_/\_(LSeg_(|[1,0]|,p2))_<>_{}
set a = the Element of (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,p2));
assume A136: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,p2)) <> {} ; ::_thesis: contradiction
then reconsider p = the Element of (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,p2)) as Point of (TOP-REAL 2) by TARSKI:def_3;
A137: p in LSeg (|[0,0]|,p1) by A136, XBOOLE_0:def_4;
A138: p in LSeg (p2,|[1,0]|) by A136, XBOOLE_0:def_4;
 p2 `1 <= |[1,0]| `1 by A104, A105, EUCLID:52;
then A139: p2 `1 <= p `1 by A138, TOPREAL1:3;
 |[0,0]| `1 <= p1 `1 by A14, A16, EUCLID:52;
then  p `1 <= p1 `1 by A137, TOPREAL1:3;
hence  contradiction by A14, A104, A109, A139, XXREAL_0:2; ::_thesis: verum
end;
 |[1,0]| in LSeg (|[1,0]|,p2) by RLTOPSP1:68;
then A140: (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[1,0]|,p2)) <> {} by Lm25, XBOOLE_0:def_4;
 (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[1,0]|,p2)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by A101, Lm24, TOPREAL1:6, XBOOLE_1:26;
then A141: (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[1,0]|,p2)) = {|[1,0]|} by A140, TOPREAL1:16, ZFMISC_1:33;
 (LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|)) is_an_arc_of |[0,0]|,|[1,1]| by Lm5, Lm7, TOPREAL1:9, TOPREAL1:10, TOPREAL1:15;
then A142: ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,0]|,|[1,1]|)) is_an_arc_of |[0,0]|,|[1,0]| by A127, TOPREAL1:10;
(((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,0]|,|[1,1]|))) /\ (LSeg (|[1,0]|,p2)) = (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) /\ (LSeg (|[1,0]|,p2))) \/ ((LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[1,0]|,p2))) by XBOOLE_1:23
.= (((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,p2))) \/ ((LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[1,0]|,p2)))) \/ {|[1,0]|} by A141, XBOOLE_1:23
.= {|[1,0]|} by A116, A126 ;
then A143: (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,0]|,p2)) is_an_arc_of |[0,0]|,p2 by A142, TOPREAL1:10;
(LSeg (p1,|[0,0]|)) /\ ((((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,0]|,p2))) = ((LSeg (p1,|[0,0]|)) /\ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,0]|,|[1,1]|)))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,p2))) by XBOOLE_1:23
.= ((LSeg (p1,|[0,0]|)) /\ ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|))) by A135, XBOOLE_1:23
.= ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|))) by A112, XBOOLE_1:23
.= {|[0,0]|} by A134, A124, A113, TOPREAL1:17, ZFMISC_1:33 ;
hence  P2 is_an_arc_of p1,p2 by A143, TOPREAL1:11; ::_thesis: ( P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
A144: p1 in LSeg (p1,|[0,0]|) by RLTOPSP1:68;
 p1 in LSeg (p1,p2) by RLTOPSP1:68;
then  p1 in (LSeg (p1,p2)) /\ (LSeg (p1,|[0,0]|)) by A144, XBOOLE_0:def_4;
then  {p1} c= (LSeg (p1,p2)) /\ (LSeg (p1,|[0,0]|)) by ZFMISC_1:31;
then A145: (LSeg (p1,p2)) /\ (LSeg (p1,|[0,0]|)) = {p1} by A118, XBOOLE_0:def_10;
thus P1 \/ P2 = ((LSeg (|[0,0]|,p1)) \/ (LSeg (p1,p2))) \/ ((((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,0]|,p2))) by XBOOLE_1:4
.= (((LSeg (|[0,0]|,p1)) \/ (LSeg (p1,p2))) \/ (LSeg (p2,|[1,0]|))) \/ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,0]|,|[1,1]|))) by XBOOLE_1:4
.= (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[0,0]|,|[1,0]|)) by A3, A101, TOPREAL1:7
.= R^2-unit_square by TOPREAL1:def_2, XBOOLE_1:4 ; ::_thesis: P1 /\ P2 = {p1,p2}
A146: p2 in LSeg (|[1,0]|,p2) by RLTOPSP1:68;
 p2 in LSeg (p1,p2) by RLTOPSP1:68;
then  p2 in (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,p2)) by A146, XBOOLE_0:def_4;
then  {p2} c= (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,p2)) by ZFMISC_1:31;
then A147: (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,p2)) = {p2} by A128, XBOOLE_0:def_10;
A148: P1 /\ P2 = ((LSeg (p1,p2)) /\ (LSeg (p1,|[0,0]|))) \/ ((LSeg (p1,p2)) /\ ((((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[1,0]|,p2)))) by XBOOLE_1:23
.= {p1} \/ (((LSeg (p1,p2)) /\ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,0]|,|[1,1]|)))) \/ {p2}) by A145, A147, XBOOLE_1:23
.= {p1} \/ ((((LSeg (p1,p2)) /\ ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ {p2}) by XBOOLE_1:23
.= {p1} \/ (((((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ ((LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ {p2}) by XBOOLE_1:23
.= {p1} \/ (((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ (((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ {p2})) by A117, XBOOLE_1:4
.= ({p1} \/ ((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)))) \/ (((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ {p2}) by XBOOLE_1:4 ;
A149: now__::_thesis:_P1_/\_P2_=_{p1}_\/_(((LSeg_(p1,p2))_/\_(LSeg_(|[1,0]|,|[1,1]|)))_\/_{p2})
percases ( p1 = |[0,0]| or p1 <> |[0,0]| ) ;
supposeA150: p1 = |[0,0]| ; ::_thesis: P1 /\ P2 = {p1} \/ (((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ {p2})
 p1 in LSeg (p1,p2) by RLTOPSP1:68;
then A151: (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) <> {} by A150, Lm20, XBOOLE_0:def_4;
 (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) c= (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A3, A101, TOPREAL1:6, XBOOLE_1:26;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {p1} by A150, A151, TOPREAL1:17, ZFMISC_1:33;
hence  P1 /\ P2 = {p1} \/ (((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ {p2}) by A148; ::_thesis: verum
end;
supposeA152: p1 <> |[0,0]| ; ::_thesis: P1 /\ P2 = {p1} \/ (((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ {p2})
now__::_thesis:_not_|[0,0]|_in_(LSeg_(p1,p2))_/\_(LSeg_(|[0,0]|,|[0,1]|))
assume  |[0,0]| in (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) ; ::_thesis: contradiction
then  |[0,0]| in LSeg (p1,p2) by XBOOLE_0:def_4;
then  p1 `1 = 0 by A14, A16, A104, A109, Lm4, TOPREAL1:3;
hence  contradiction by A14, A17, A152, EUCLID:53; ::_thesis: verum
end;
then A153: {|[0,0]|} <> (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) by ZFMISC_1:31;
 (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) c= (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A3, A101, TOPREAL1:6, XBOOLE_1:26;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {} by A153, TOPREAL1:17, ZFMISC_1:33;
hence  P1 /\ P2 = {p1} \/ (((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ {p2}) by A148; ::_thesis: verum
end;
end;
end;
now__::_thesis:_P1_/\_P2_=_{p1,p2}
percases ( p2 = |[1,0]| or p2 <> |[1,0]| ) ;
supposeA154: p2 = |[1,0]| ; ::_thesis: P1 /\ P2 = {p1,p2}
 p2 in LSeg (p1,p2) by RLTOPSP1:68;
then A155: (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) <> {} by A154, Lm25, XBOOLE_0:def_4;
 (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) c= {p2} by A3, A101, A154, TOPREAL1:6, TOPREAL1:16, XBOOLE_1:26;
then  (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {p2} by A155, ZFMISC_1:33;
hence  P1 /\ P2 = {p1,p2} by A149, ENUMSET1:1; ::_thesis: verum
end;
supposeA156: p2 <> |[1,0]| ; ::_thesis: P1 /\ P2 = {p1,p2}
now__::_thesis:_not_|[1,0]|_in_(LSeg_(p1,p2))_/\_(LSeg_(|[1,0]|,|[1,1]|))
assume  |[1,0]| in (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) ; ::_thesis: contradiction
then  |[1,0]| in LSeg (p1,p2) by XBOOLE_0:def_4;
then  |[1,0]| `1 <= p2 `1 by A14, A104, A109, TOPREAL1:3;
then  p2 `1 = 1 by A104, A105, Lm8, XXREAL_0:1;
hence  contradiction by A104, A107, A156, EUCLID:53; ::_thesis: verum
end;
then A157: {|[1,0]|} <> (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) by ZFMISC_1:31;
 (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) c= {|[1,0]|} by A3, A101, TOPREAL1:6, TOPREAL1:16, XBOOLE_1:26;
then  (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by A157, ZFMISC_1:33;
hence  P1 /\ P2 = {p1,p2} by A149, ENUMSET1:1; ::_thesis: verum
end;
end;
end;
hence  P1 /\ P2 = {p1,p2} ; ::_thesis: verum
end;
supposeA158: q `1 < p `1 ; ::_thesis: ex P1 being Element of K19( the carrier of (TOP-REAL 2)) ex P2 being Element of K19( the carrier of (TOP-REAL 2)) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
A159: (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,p2)) c= {p2}
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,p2)) or a in {p2} )
assume A160: a in (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,p2)) ; ::_thesis: a in {p2}
then reconsider p = a as Point of (TOP-REAL 2) ;
A161: p in LSeg (|[0,0]|,p2) by A160, XBOOLE_0:def_4;
 |[0,0]| `1 <= p2 `1 by A104, A106, EUCLID:52;
then A162: p `1 <= p2 `1 by A161, TOPREAL1:3;
A163: p in LSeg (p2,p1) by A160, XBOOLE_0:def_4;
then  p2 `1 <= p `1 by A14, A104, A158, TOPREAL1:3;
then A164: p2 `1 = p `1 by A162, XXREAL_0:1;
 p2 `2 <= p `2 by A14, A17, A104, A107, A163, TOPREAL1:4;
then  p `2 = 0 by A14, A17, A104, A107, A163, TOPREAL1:4;
then p = |[(p2 `1),0]| by A164, EUCLID:53
.= p2 by A104, A107, EUCLID:53 ;
hence  a in {p2} by TARSKI:def_1; ::_thesis: verum
end;
 |[1,0]| in LSeg (p1,|[1,0]|) by RLTOPSP1:68;
then A165: (LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) <> {} by Lm25, XBOOLE_0:def_4;
now__::_thesis:_not_|[1,0]|_in_(LSeg_(|[1,0]|,|[1,1]|))_/\_(LSeg_(|[0,0]|,p2))
assume  |[1,0]| in (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) ; ::_thesis: contradiction
then A166: |[1,0]| in LSeg (|[0,0]|,p2) by XBOOLE_0:def_4;
 |[0,0]| `1 <= p2 `1 by A104, A106, EUCLID:52;
then  |[1,0]| `1 <= p2 `1 by A166, TOPREAL1:3;
hence  contradiction by A15, A104, A105, A158, Lm8, XXREAL_0:1; ::_thesis: verum
end;
then A167: {|[1,0]|} <> (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) by ZFMISC_1:31;
 (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by A101, Lm21, TOPREAL1:6, XBOOLE_1:26;
then A168: (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) = {} by A167, TOPREAL1:16, ZFMISC_1:33;
A169: ((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) /\ (LSeg (|[0,0]|,|[0,1]|)) = ((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ ((LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|))) by XBOOLE_1:23
.= {|[0,1]|} by Lm3, TOPREAL1:15 ;
 (LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|)) is_an_arc_of |[1,0]|,|[0,1]| by Lm9, Lm11, TOPREAL1:9, TOPREAL1:10, TOPREAL1:18;
then A170: ((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,0]|,|[0,1]|)) is_an_arc_of |[1,0]|,|[0,0]| by A169, TOPREAL1:10;
now__::_thesis:_not_|[0,0]|_in_(LSeg_(p1,|[1,0]|))_/\_(LSeg_(|[0,0]|,|[0,1]|))
assume  |[0,0]| in (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) ; ::_thesis: contradiction
then A171: |[0,0]| in LSeg (p1,|[1,0]|) by XBOOLE_0:def_4;
 p1 `1 <= |[1,0]| `1 by A14, A15, EUCLID:52;
hence  contradiction by A14, A106, A158, A171, Lm4, TOPREAL1:3; ::_thesis: verum
end;
then A172: {|[0,0]|} <> (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by ZFMISC_1:31;
 (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) c= (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A3, Lm24, TOPREAL1:6, XBOOLE_1:26;
then A173: (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {} by A172, TOPREAL1:17, ZFMISC_1:33;
 |[0,0]| in LSeg (|[0,0]|,p2) by RLTOPSP1:68;
then A174: (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[0,0]|,p2)) <> {} by Lm20, XBOOLE_0:def_4;
 (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[0,0]|,p2)) c= {|[0,0]|} by A101, Lm21, TOPREAL1:6, TOPREAL1:17, XBOOLE_1:26;
then A175: (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[0,0]|,p2)) = {|[0,0]|} by A174, ZFMISC_1:33;
A176: (LSeg (p1,p2)) /\ (LSeg (p1,|[1,0]|)) c= {p1}
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in (LSeg (p1,p2)) /\ (LSeg (p1,|[1,0]|)) or a in {p1} )
assume A177: a in (LSeg (p1,p2)) /\ (LSeg (p1,|[1,0]|)) ; ::_thesis: a in {p1}
then reconsider p = a as Point of (TOP-REAL 2) ;
A178: p in LSeg (p1,|[1,0]|) by A177, XBOOLE_0:def_4;
 p1 `1 <= |[1,0]| `1 by A14, A15, EUCLID:52;
then A179: p1 `1 <= p `1 by A178, TOPREAL1:3;
A180: p in LSeg (p2,p1) by A177, XBOOLE_0:def_4;
then  p `1 <= p1 `1 by A14, A104, A158, TOPREAL1:3;
then A181: p1 `1 = p `1 by A179, XXREAL_0:1;
 p2 `2 <= p `2 by A14, A17, A104, A107, A180, TOPREAL1:4;
then  p `2 = 0 by A14, A17, A104, A107, A180, TOPREAL1:4;
then p = |[(p1 `1),0]| by A181, EUCLID:53
.= p1 by A14, A17, EUCLID:53 ;
hence  a in {p1} by TARSKI:def_1; ::_thesis: verum
end;
 (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by TOPREAL1:19, XBOOLE_0:def_7;
then A182: (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by A12, XBOOLE_1:3, XBOOLE_1:26;
A183: now__::_thesis:_not_(LSeg_(p1,|[1,0]|))_/\_(LSeg_(|[0,0]|,p2))_<>_{}
set a = the Element of (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2));
assume A184: (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2)) <> {} ; ::_thesis: contradiction
then reconsider p = the Element of (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2)) as Point of (TOP-REAL 2) by TARSKI:def_3;
A185: p in LSeg (p1,|[1,0]|) by A184, XBOOLE_0:def_4;
A186: p in LSeg (|[0,0]|,p2) by A184, XBOOLE_0:def_4;
 |[0,0]| `1 <= p2 `1 by A104, A106, EUCLID:52;
then A187: p `1 <= p2 `1 by A186, TOPREAL1:3;
 p1 `1 <= |[1,0]| `1 by A14, A15, EUCLID:52;
then  p1 `1 <= p `1 by A185, TOPREAL1:3;
hence  contradiction by A14, A104, A158, A187, XXREAL_0:2; ::_thesis: verum
end;
take P1 = LSeg (p1,p2); ::_thesis: ex P2 being Element of K19( the carrier of (TOP-REAL 2)) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
take P2 = (LSeg (p1,|[1,0]|)) \/ ((((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,p2))); ::_thesis: ( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
A188: (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by TOPREAL1:19, XBOOLE_0:def_7;
 (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) c= (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by A101, Lm21, TOPREAL1:6, XBOOLE_1:26;
then A189: (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) = {} by A188, XBOOLE_1:3;
A190: (LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) c= {|[1,0]|} by A3, Lm24, TOPREAL1:6, TOPREAL1:16, XBOOLE_1:26;
(((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,0]|,|[0,1]|))) /\ (LSeg (|[0,0]|,p2)) = (((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) /\ (LSeg (|[0,0]|,p2))) \/ ((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[0,0]|,p2))) by XBOOLE_1:23
.= (((LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2))) \/ ((LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)))) \/ {|[0,0]|} by A175, XBOOLE_1:23
.= {|[0,0]|} by A168, A189 ;
then A191: (((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,p2)) is_an_arc_of |[1,0]|,p2 by A170, TOPREAL1:10;
thus  P1 is_an_arc_of p1,p2 by A1, TOPREAL1:9; ::_thesis: ( P2 is_an_arc_of p1,p2 & P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
A192: p2 in LSeg (|[0,0]|,p2) by RLTOPSP1:68;
(LSeg (p1,|[1,0]|)) /\ ((((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,p2))) = ((LSeg (p1,|[1,0]|)) /\ (((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,0]|,|[0,1]|)))) \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2))) by XBOOLE_1:23
.= ((LSeg (p1,|[1,0]|)) /\ ((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|))) by A183, XBOOLE_1:23
.= ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|))) by A173, XBOOLE_1:23
.= {|[1,0]|} by A182, A190, A165, ZFMISC_1:33 ;
hence  P2 is_an_arc_of p1,p2 by A191, TOPREAL1:11; ::_thesis: ( P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
A193: p1 in LSeg (p1,|[1,0]|) by RLTOPSP1:68;
thus P1 \/ P2 = ((LSeg (p2,p1)) \/ (LSeg (p1,|[1,0]|))) \/ ((((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,p2))) by XBOOLE_1:4
.= ((LSeg (|[0,0]|,p2)) \/ ((LSeg (p2,p1)) \/ (LSeg (p1,|[1,0]|)))) \/ (((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,0]|,|[0,1]|))) by XBOOLE_1:4
.= (LSeg (|[0,0]|,|[1,0]|)) \/ (((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,0]|,|[0,1]|))) by A3, A101, TOPREAL1:7
.= (LSeg (|[0,0]|,|[1,0]|)) \/ ((LSeg (|[1,0]|,|[1,1]|)) \/ ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|)))) by XBOOLE_1:4
.= R^2-unit_square by TOPREAL1:def_2, XBOOLE_1:4 ; ::_thesis: P1 /\ P2 = {p1,p2}
 (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by TOPREAL1:19, XBOOLE_0:def_7;
then A194: (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by A103, XBOOLE_1:3, XBOOLE_1:26;
 p2 in LSeg (p1,p2) by RLTOPSP1:68;
then  p2 in (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,p2)) by A192, XBOOLE_0:def_4;
then  {p2} c= (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,p2)) by ZFMISC_1:31;
then A195: (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,p2)) = {p2} by A159, XBOOLE_0:def_10;
 p1 in LSeg (p1,p2) by RLTOPSP1:68;
then  p1 in (LSeg (p1,p2)) /\ (LSeg (p1,|[1,0]|)) by A193, XBOOLE_0:def_4;
then  {p1} c= (LSeg (p1,p2)) /\ (LSeg (p1,|[1,0]|)) by ZFMISC_1:31;
then  (LSeg (p1,p2)) /\ (LSeg (p1,|[1,0]|)) = {p1} by A176, XBOOLE_0:def_10;
then A196: P1 /\ P2 = {p1} \/ ((LSeg (p1,p2)) /\ ((((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,p2)))) by XBOOLE_1:23
.= {p1} \/ (((LSeg (p1,p2)) /\ (((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,0]|,|[0,1]|)))) \/ {p2}) by A195, XBOOLE_1:23
.= {p1} \/ ((((LSeg (p1,p2)) /\ ((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)))) \/ {p2}) by XBOOLE_1:23
.= {p1} \/ (((((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ ((LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)))) \/ {p2}) by XBOOLE_1:23
.= {p1} \/ (((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|))) \/ (((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2})) by A194, XBOOLE_1:4
.= ({p1} \/ ((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ (((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2}) by XBOOLE_1:4 ;
A197: now__::_thesis:_P1_/\_P2_=_({p1}_\/_((LSeg_(p1,p2))_/\_(LSeg_(|[1,0]|,|[1,1]|))))_\/_{p2}
percases ( p2 = |[0,0]| or p2 <> |[0,0]| ) ;
supposeA198: p2 = |[0,0]| ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ {p2}
 p2 in LSeg (p1,p2) by RLTOPSP1:68;
then A199: (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) <> {} by A198, Lm20, XBOOLE_0:def_4;
 (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) c= (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A3, A101, TOPREAL1:6, XBOOLE_1:26;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {p2} by A198, A199, TOPREAL1:17, ZFMISC_1:33;
hence  P1 /\ P2 = ({p1} \/ ((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ {p2} by A196; ::_thesis: verum
end;
supposeA200: p2 <> |[0,0]| ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ {p2}
now__::_thesis:_not_|[0,0]|_in_(LSeg_(p1,p2))_/\_(LSeg_(|[0,0]|,|[0,1]|))
assume  |[0,0]| in (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) ; ::_thesis: contradiction
then  |[0,0]| in LSeg (p2,p1) by XBOOLE_0:def_4;
then  p2 `1 = 0 by A14, A104, A106, A158, Lm4, TOPREAL1:3;
hence  contradiction by A104, A107, A200, EUCLID:53; ::_thesis: verum
end;
then A201: {|[0,0]|} <> (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) by ZFMISC_1:31;
 (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) c= (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A3, A101, TOPREAL1:6, XBOOLE_1:26;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {} by A201, TOPREAL1:17, ZFMISC_1:33;
hence  P1 /\ P2 = ({p1} \/ ((LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)))) \/ {p2} by A196; ::_thesis: verum
end;
end;
end;
now__::_thesis:_P1_/\_P2_=_{p1,p2}
percases ( p1 = |[1,0]| or p1 <> |[1,0]| ) ;
supposeA202: p1 = |[1,0]| ; ::_thesis: P1 /\ P2 = {p1,p2}
 p1 in LSeg (p1,p2) by RLTOPSP1:68;
then A203: (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) <> {} by A202, Lm25, XBOOLE_0:def_4;
 (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) c= {p1} by A3, A101, A202, TOPREAL1:6, TOPREAL1:16, XBOOLE_1:26;
then  (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {p1} by A203, ZFMISC_1:33;
hence  P1 /\ P2 = {p1,p2} by A197, ENUMSET1:1; ::_thesis: verum
end;
supposeA204: p1 <> |[1,0]| ; ::_thesis: P1 /\ P2 = {p1,p2}
now__::_thesis:_not_|[1,0]|_in_(LSeg_(p1,p2))_/\_(LSeg_(|[1,0]|,|[1,1]|))
assume  |[1,0]| in (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) ; ::_thesis: contradiction
then  |[1,0]| in LSeg (p2,p1) by XBOOLE_0:def_4;
then  |[1,0]| `1 <= p1 `1 by A14, A104, A158, TOPREAL1:3;
then  p1 `1 = 1 by A14, A15, Lm8, XXREAL_0:1;
hence  contradiction by A14, A17, A204, EUCLID:53; ::_thesis: verum
end;
then A205: {|[1,0]|} <> (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) by ZFMISC_1:31;
 (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) c= {|[1,0]|} by A3, A101, TOPREAL1:6, TOPREAL1:16, XBOOLE_1:26;
then  (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by A205, ZFMISC_1:33;
hence  P1 /\ P2 = {p1,p2} by A197, ENUMSET1:1; ::_thesis: verum
end;
end;
end;
hence  P1 /\ P2 = {p1,p2} ; ::_thesis: verum
end;
end;
end;
hence  ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) ; ::_thesis: verum
end;
supposeA206: p2 in LSeg (|[1,0]|,|[1,1]|) ; ::_thesis: ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
then A207: ex q being Point of (TOP-REAL 2) st
( q = p2 & q `1 = 1 & q `2 <= 1 & q `2 >= 0 ) by TOPREAL1:13;
now__::_thesis:_not_|[1,0]|_in_(LSeg_(p1,|[0,0]|))_/\_(LSeg_(|[1,1]|,p2))
assume A208: |[1,0]| in (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,1]|,p2)) ; ::_thesis: contradiction
then A209: |[1,0]| in LSeg (|[0,0]|,p1) by XBOOLE_0:def_4;
 |[0,0]| `1 <= p1 `1 by A14, A16, EUCLID:52;
then  |[1,0]| `1 <= p1 `1 by A209, TOPREAL1:3;
then  1 = p1 `1 by A14, A15, Lm8, XXREAL_0:1;
then A210: p1 = |[1,0]| by A14, A17, EUCLID:53;
A211: p2 `2 <= |[1,1]| `2 by A207, EUCLID:52;
 |[1,0]| in LSeg (p2,|[1,1]|) by A208, XBOOLE_0:def_4;
then  0 = p2 `2 by A207, A211, Lm9, TOPREAL1:4;
hence  contradiction by A1, A207, A210, EUCLID:53; ::_thesis: verum
end;
then A212: {|[1,0]|} <> (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,1]|,p2)) by ZFMISC_1:31;
A213: LSeg (|[0,1]|,|[1,1]|) is_an_arc_of |[0,1]|,|[1,1]| by Lm6, Lm10, TOPREAL1:9;
 LSeg (|[0,0]|,|[0,1]|) is_an_arc_of |[0,0]|,|[0,1]| by Lm5, Lm7, TOPREAL1:9;
then A214: (LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|)) is_an_arc_of |[0,0]|,|[1,1]| by A213, TOPREAL1:2, TOPREAL1:15;
take P1 = (LSeg (p1,|[1,0]|)) \/ (LSeg (|[1,0]|,p2)); ::_thesis: ex P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
take P2 = (LSeg (p1,|[0,0]|)) \/ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))); ::_thesis: ( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A215: LSeg (|[0,0]|,|[1,0]|) = (LSeg (p1,|[1,0]|)) \/ (LSeg (p1,|[0,0]|)) by A3, TOPREAL1:5;
 |[1,1]| in LSeg (|[1,1]|,p2) by RLTOPSP1:68;
then A216: (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[1,1]|,p2)) <> {} by Lm26, XBOOLE_0:def_4;
A217: |[1,0]| in LSeg (|[1,0]|,p2) by RLTOPSP1:68;
 |[1,0]| in LSeg (p1,|[1,0]|) by RLTOPSP1:68;
then A218: (LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,0]|,p2)) <> {} by A217, XBOOLE_0:def_4;
A219: LSeg (p2,|[1,0]|) c= LSeg (|[1,0]|,|[1,1]|) by A206, Lm25, TOPREAL1:6;
then  (LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,0]|,p2)) c= (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) by A12, XBOOLE_1:27;
then A220: (LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,0]|,p2)) = {|[1,0]|} by A218, TOPREAL1:16, ZFMISC_1:33;
 ( p1 <> |[1,0]| or p2 <> |[1,0]| ) by A1;
hence  P1 is_an_arc_of p1,p2 by A220, TOPREAL1:12; ::_thesis: ( P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A221: (LSeg (p1,|[1,0]|)) /\ (LSeg (p1,|[0,0]|)) = {p1} by A3, TOPREAL1:8;
 (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by TOPREAL1:20, XBOOLE_0:def_7;
then A222: (LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {} by A219, XBOOLE_1:3, XBOOLE_1:26;
A223: LSeg (p2,|[1,1]|) c= LSeg (|[1,0]|,|[1,1]|) by A206, Lm27, TOPREAL1:6;
then A224: (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[1,1]|,p2)) c= {|[1,1]|} by TOPREAL1:18, XBOOLE_1:27;
A225: (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)) = {} by A223, Lm3, XBOOLE_1:3, XBOOLE_1:26;
((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) /\ (LSeg (|[1,1]|,p2)) = ((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,1]|,p2))) \/ ((LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[1,1]|,p2))) by XBOOLE_1:23
.= {|[1,1]|} by A225, A224, A216, ZFMISC_1:33 ;
then A226: ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2)) is_an_arc_of |[0,0]|,p2 by A214, TOPREAL1:10;
A227: (LSeg (|[1,0]|,p2)) /\ (LSeg (|[1,1]|,p2)) = {p2} by A206, TOPREAL1:8;
A228: LSeg (|[1,0]|,|[1,1]|) = (LSeg (|[1,1]|,p2)) \/ (LSeg (|[1,0]|,p2)) by A206, TOPREAL1:5;
 (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,1]|,p2)) c= {|[1,0]|} by A10, A223, TOPREAL1:16, XBOOLE_1:27;
then A229: (LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,1]|,p2)) = {} by A212, ZFMISC_1:33;
(LSeg (p1,|[0,0]|)) /\ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))) = ((LSeg (p1,|[0,0]|)) /\ ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[1,1]|,p2))) by XBOOLE_1:23
.= ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ ((LSeg (p1,|[0,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|))) by A229, XBOOLE_1:23
.= {|[0,0]|} by A8, A6, A11, TOPREAL1:17, ZFMISC_1:33 ;
hence  P2 is_an_arc_of p1,p2 by A226, TOPREAL1:11; ::_thesis: ( R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
thus P1 \/ P2 = (LSeg (|[1,0]|,p2)) \/ ((LSeg (p1,|[1,0]|)) \/ ((LSeg (p1,|[0,0]|)) \/ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))))) by XBOOLE_1:4
.= (LSeg (|[1,0]|,p2)) \/ ((LSeg (|[0,0]|,|[1,0]|)) \/ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2)))) by A215, XBOOLE_1:4
.= ((((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,1]|,p2))) \/ (LSeg (|[1,0]|,p2)) by XBOOLE_1:4
.= (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,|[1,1]|)) by A228, XBOOLE_1:4
.= R^2-unit_square by TOPREAL1:def_2, XBOOLE_1:4 ; ::_thesis: P1 /\ P2 = {p1,p2}
A230: P1 /\ P2 = ((LSeg (p1,|[1,0]|)) /\ ((LSeg (p1,|[0,0]|)) \/ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))))) \/ ((LSeg (|[1,0]|,p2)) /\ ((LSeg (p1,|[0,0]|)) \/ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))))) by XBOOLE_1:23
.= (((LSeg (p1,|[1,0]|)) /\ (LSeg (p1,|[0,0]|))) \/ ((LSeg (p1,|[1,0]|)) /\ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))))) \/ ((LSeg (|[1,0]|,p2)) /\ ((LSeg (p1,|[0,0]|)) \/ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))))) by XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[1,0]|)) /\ ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2))))) \/ ((LSeg (|[1,0]|,p2)) /\ ((LSeg (p1,|[0,0]|)) \/ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))))) by A221, XBOOLE_1:23
.= ({p1} \/ ((((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ ((LSeg (|[1,0]|,p1)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2))))) \/ ((LSeg (|[1,0]|,p2)) /\ ((LSeg (p1,|[0,0]|)) \/ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))))) by XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2))))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[0,0]|))) \/ ((LSeg (|[1,0]|,p2)) /\ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))))) by A13, XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2))))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[0,0]|))) \/ (((LSeg (|[1,0]|,p2)) /\ ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|)))) \/ {p2})) by A227, XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2))))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[0,0]|))) \/ ((((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ ((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ {p2})) by XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2))))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[0,0]|))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2})) by A222 ;
A231: now__::_thesis:_P1_/\_P2_=_({p1}_\/_((LSeg_(p1,|[1,0]|))_/\_(LSeg_(|[1,1]|,p2))))_\/_(((LSeg_(|[1,0]|,p2))_/\_(LSeg_(|[0,1]|,|[1,1]|)))_\/_{p2})
percases ( p1 = |[0,0]| or p1 = |[1,0]| or ( p1 <> |[1,0]| & p1 <> |[0,0]| ) ) ;
supposeA232: p1 = |[0,0]| ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2})
then A233: (LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[0,0]|)) = (LSeg (|[1,0]|,p2)) /\ {|[0,0]|} by RLTOPSP1:70;
 not |[0,0]| in LSeg (|[1,0]|,p2) by A219, Lm4, Lm8, Lm10, TOPREAL1:3;
then  (LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[0,0]|)) = {} by A233, Lm1;
hence P1 /\ P2 = (({p1} \/ {p1}) \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2}) by A230, A232, TOPREAL1:17, XBOOLE_1:4
.= ({p1} \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2}) ;
::_thesis: verum
end;
supposeA234: p1 = |[1,0]| ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2})
A235: p1 in LSeg (p1,|[0,0]|) by RLTOPSP1:68;
 p1 in LSeg (|[1,0]|,p2) by A234, RLTOPSP1:68;
then A236: {} <> (LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[0,0]|)) by A235, XBOOLE_0:def_4;
 (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {|[1,0]|} /\ (LSeg (|[0,0]|,|[0,1]|)) by A234, RLTOPSP1:70;
then A237: (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {} by Lm1, Lm16;
 (LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[0,0]|)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by A10, A219, XBOOLE_1:27;
then  (LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[0,0]|)) = {p1} by A234, A236, TOPREAL1:16, ZFMISC_1:33;
hence P1 /\ P2 = ({p1} \/ ({p1} \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2))))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2}) by A230, A237, XBOOLE_1:4
.= (({p1} \/ {p1}) \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2}) by XBOOLE_1:4
.= ({p1} \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2}) ;
::_thesis: verum
end;
supposeA238: ( p1 <> |[1,0]| & p1 <> |[0,0]| ) ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2})
now__::_thesis:_not_|[1,0]|_in_(LSeg_(|[1,0]|,p2))_/\_(LSeg_(p1,|[0,0]|))
assume  |[1,0]| in (LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[0,0]|)) ; ::_thesis: contradiction
then A239: |[1,0]| in LSeg (|[0,0]|,p1) by XBOOLE_0:def_4;
 |[0,0]| `1 <= p1 `1 by A14, A16, EUCLID:52;
then  |[1,0]| `1 <= p1 `1 by A239, TOPREAL1:3;
then  p1 `1 = 1 by A14, A15, Lm8, XXREAL_0:1;
hence  contradiction by A14, A17, A238, EUCLID:53; ::_thesis: verum
end;
then A240: {|[1,0]|} <> (LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[0,0]|)) by ZFMISC_1:31;
 (LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[0,0]|)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by A10, A219, XBOOLE_1:27;
then A241: (LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[0,0]|)) = {} by A240, TOPREAL1:16, ZFMISC_1:33;
now__::_thesis:_not_|[0,0]|_in_(LSeg_(p1,|[1,0]|))_/\_(LSeg_(|[0,0]|,|[0,1]|))
assume  |[0,0]| in (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) ; ::_thesis: contradiction
then A242: |[0,0]| in LSeg (p1,|[1,0]|) by XBOOLE_0:def_4;
 p1 `1 <= |[1,0]| `1 by A14, A15, EUCLID:52;
then  p1 `1 = 0 by A14, A16, A242, Lm4, TOPREAL1:3;
hence  contradiction by A14, A17, A238, EUCLID:53; ::_thesis: verum
end;
then A243: {|[0,0]|} <> (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by ZFMISC_1:31;
 (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) c= (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A3, Lm24, TOPREAL1:6, XBOOLE_1:26;
then  (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {} by A243, TOPREAL1:17, ZFMISC_1:33;
hence  P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2}) by A230, A241; ::_thesis: verum
end;
end;
end;
now__::_thesis:_P1_/\_P2_=_{p1,p2}
percases ( p2 = |[1,0]| or p2 = |[1,1]| or ( p2 <> |[1,1]| & p2 <> |[1,0]| ) ) ;
supposeA244: p2 = |[1,0]| ; ::_thesis: P1 /\ P2 = {p1,p2}
 |[1,0]| in LSeg (p1,|[1,0]|) by RLTOPSP1:68;
then A245: {} <> (LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)) by A244, Lm25, XBOOLE_0:def_4;
 (LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {|[1,0]|} /\ (LSeg (|[0,1]|,|[1,1]|)) by A244, RLTOPSP1:70;
then A246: (LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by Lm1, Lm17;
 (LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)) c= {p2} by A12, A244, TOPREAL1:16, XBOOLE_1:27;
then  (LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)) = {p2} by A245, ZFMISC_1:33;
hence P1 /\ P2 = {p1} \/ ({p2} \/ {p2}) by A231, A246, XBOOLE_1:4
.= {p1,p2} by ENUMSET1:1 ;
::_thesis: verum
end;
supposeA247: p2 = |[1,1]| ; ::_thesis: P1 /\ P2 = {p1,p2}
then A248: (LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)) = (LSeg (p1,|[1,0]|)) /\ {|[1,1]|} by RLTOPSP1:70;
 not |[1,1]| in LSeg (p1,|[1,0]|) by A12, Lm5, Lm9, Lm11, TOPREAL1:4;
then  (LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)) = {} by A248, Lm1;
hence  P1 /\ P2 = {p1,p2} by A231, A247, ENUMSET1:1, TOPREAL1:18; ::_thesis: verum
end;
supposeA249: ( p2 <> |[1,1]| & p2 <> |[1,0]| ) ; ::_thesis: P1 /\ P2 = {p1,p2}
now__::_thesis:_not_|[1,1]|_in_(LSeg_(|[1,0]|,p2))_/\_(LSeg_(|[0,1]|,|[1,1]|))
assume  |[1,1]| in (LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) ; ::_thesis: contradiction
then A250: |[1,1]| in LSeg (|[1,0]|,p2) by XBOOLE_0:def_4;
 |[1,0]| `2 <= p2 `2 by A207, EUCLID:52;
then  |[1,1]| `2 <= p2 `2 by A250, TOPREAL1:4;
then  1 = p2 `2 by A207, Lm11, XXREAL_0:1;
hence  contradiction by A207, A249, EUCLID:53; ::_thesis: verum
end;
then A251: {|[1,1]|} <> (LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) by ZFMISC_1:31;
 (LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) by A206, Lm25, TOPREAL1:6, XBOOLE_1:26;
then A252: (LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by A251, TOPREAL1:18, ZFMISC_1:33;
now__::_thesis:_not_|[1,0]|_in_(LSeg_(p1,|[1,0]|))_/\_(LSeg_(|[1,1]|,p2))
assume  |[1,0]| in (LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)) ; ::_thesis: contradiction
then A253: |[1,0]| in LSeg (p2,|[1,1]|) by XBOOLE_0:def_4;
 p2 `2 <= |[1,1]| `2 by A207, EUCLID:52;
then  p2 `2 = 0 by A207, A253, Lm9, TOPREAL1:4;
hence  contradiction by A207, A249, EUCLID:53; ::_thesis: verum
end;
then A254: {|[1,0]|} <> (LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)) by ZFMISC_1:31;
 (LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)) c= {|[1,0]|} by A12, A223, TOPREAL1:16, XBOOLE_1:27;
then  (LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)) = {} by A254, ZFMISC_1:33;
hence  P1 /\ P2 = {p1,p2} by A231, A252, ENUMSET1:1; ::_thesis: verum
end;
end;
end;
hence  P1 /\ P2 = {p1,p2} ; ::_thesis: verum
end;
end;
end;

Lm33:  for p1, p2 being Point of (TOP-REAL 2) st p1 <> p2 & p2 in R^2-unit_square & p1 in LSeg (|[1,0]|,|[1,1]|) holds
ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 <> p2 & p2 in R^2-unit_square & p1 in LSeg (|[1,0]|,|[1,1]|) implies ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) )
assume that
A1: p1 <> p2 and
A2: p2 in R^2-unit_square and
A3: p1 in LSeg (|[1,0]|,|[1,1]|) ; ::_thesis: ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A4: ( p2 in (LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|)) or p2 in (LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)) ) by A2, TOPREAL1:def_2, XBOOLE_0:def_3;
A5: LSeg (p1,|[1,1]|) c= LSeg (|[1,0]|,|[1,1]|) by A3, Lm27, TOPREAL1:6;
 |[1,1]| in LSeg (p1,|[1,1]|) by RLTOPSP1:68;
then A6: {} <> (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) by Lm26, XBOOLE_0:def_4;
 |[1,0]| in LSeg (p1,|[1,0]|) by RLTOPSP1:68;
then A7: {} <> (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by Lm24, XBOOLE_0:def_4;
A8: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) by A3, Lm27, TOPREAL1:6, XBOOLE_1:26;
A9: (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by A3, Lm25, TOPREAL1:6, XBOOLE_1:26;
 (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by TOPREAL1:20, XBOOLE_0:def_7;
then A10: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {} by A5, XBOOLE_1:3, XBOOLE_1:26;
A11: LSeg (p1,|[1,0]|) c= LSeg (|[1,0]|,|[1,1]|) by A3, Lm25, TOPREAL1:6;
 (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by TOPREAL1:20, XBOOLE_0:def_7;
then A12: (LSeg (|[1,0]|,p1)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {} by A11, XBOOLE_1:3, XBOOLE_1:26;
consider p being Point of (TOP-REAL 2) such that
A13: p = p1 and
A14: p `1 = 1 and
A15: p `2 <= 1 and
A16: p `2 >= 0 by A3, TOPREAL1:13;
percases ( p2 in LSeg (|[0,0]|,|[0,1]|) or p2 in LSeg (|[0,1]|,|[1,1]|) or p2 in LSeg (|[0,0]|,|[1,0]|) or p2 in LSeg (|[1,0]|,|[1,1]|) ) by A4, XBOOLE_0:def_3;
supposeA17: p2 in LSeg (|[0,0]|,|[0,1]|) ; ::_thesis: ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
 |[0,0]| in LSeg (|[0,0]|,p2) by RLTOPSP1:68;
then A18: (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,0]|,p2)) <> {} by Lm21, XBOOLE_0:def_4;
 (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,0]|,p2)) c= (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A17, Lm20, TOPREAL1:6, XBOOLE_1:26;
then  (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,0]|,p2)) = {|[0,0]|} by A18, TOPREAL1:17, ZFMISC_1:33;
then A19: (LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,p2)) is_an_arc_of |[1,0]|,p2 by Lm4, Lm8, TOPREAL1:9, TOPREAL1:10;
A20: LSeg (p2,|[0,0]|) c= LSeg (|[0,0]|,|[0,1]|) by A17, Lm20, TOPREAL1:6;
then A21: (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2)) = {} by A11, Lm3, XBOOLE_1:3, XBOOLE_1:27;
A22: LSeg (p2,|[0,1]|) c= LSeg (|[0,0]|,|[0,1]|) by A17, Lm22, TOPREAL1:6;
then A23: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)) = {} by A5, Lm3, XBOOLE_1:3, XBOOLE_1:27;
 (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by TOPREAL1:20, XBOOLE_0:def_7;
then A24: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) = {} by A5, A20, XBOOLE_1:3, XBOOLE_1:27;
 (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by TOPREAL1:20, XBOOLE_0:def_7;
then A25: (LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[1,0]|)) = {} by A11, A22, XBOOLE_1:3, XBOOLE_1:27;
A26: (LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,p2)) = {p2} by A17, TOPREAL1:8;
 |[0,1]| in LSeg (|[0,1]|,p2) by RLTOPSP1:68;
then A27: |[0,1]| in (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)) by Lm23, XBOOLE_0:def_4;
 (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)) c= (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A17, Lm22, TOPREAL1:6, XBOOLE_1:26;
then  (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)) = {|[0,1]|} by A27, TOPREAL1:15, ZFMISC_1:33;
then A28: (LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,1]|,p2)) is_an_arc_of |[1,1]|,p2 by Lm6, Lm10, TOPREAL1:9, TOPREAL1:10;
take P1 = ((LSeg (p1,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,1]|,p2)); ::_thesis: ex P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
take P2 = ((LSeg (p1,|[1,0]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[0,0]|,p2)); ::_thesis: ( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A29: (LSeg (p1,|[1,1]|)) \/ (LSeg (p1,|[1,0]|)) = LSeg (|[1,0]|,|[1,1]|) by A3, TOPREAL1:5;
(LSeg (p1,|[1,1]|)) /\ ((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,1]|,p2))) = ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2))) by XBOOLE_1:23
.= {|[1,1]|} by A8, A6, A23, TOPREAL1:18, ZFMISC_1:33 ;
then  (LSeg (p1,|[1,1]|)) \/ ((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,1]|,p2))) is_an_arc_of p1,p2 by A28, TOPREAL1:11;
hence  P1 is_an_arc_of p1,p2 by XBOOLE_1:4; ::_thesis: ( P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A30: ex q being Point of (TOP-REAL 2) st
( q = p2 & q `1 = 0 & q `2 <= 1 & q `2 >= 0 ) by A17, TOPREAL1:13;
(LSeg (p1,|[1,0]|)) /\ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,p2))) = ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2))) by XBOOLE_1:23
.= {|[1,0]|} by A9, A7, A21, TOPREAL1:16, ZFMISC_1:33 ;
then  (LSeg (p1,|[1,0]|)) \/ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,p2))) is_an_arc_of p1,p2 by A19, TOPREAL1:11;
hence  P2 is_an_arc_of p1,p2 by XBOOLE_1:4; ::_thesis: ( R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
thus  R^2-unit_square = (((LSeg (|[0,0]|,p2)) \/ (LSeg (|[0,1]|,p2))) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) by A17, TOPREAL1:5, TOPREAL1:def_2
.= ((LSeg (|[0,0]|,p2)) \/ ((LSeg (|[0,1]|,p2)) \/ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) by XBOOLE_1:4
.= ((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,1]|,p2))) \/ (((LSeg (|[1,0]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[0,0]|,p2))) by XBOOLE_1:4
.= ((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,1]|,p2))) \/ ((LSeg (|[1,0]|,|[1,1]|)) \/ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,p2)))) by XBOOLE_1:4
.= ((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,1]|,p2))) \/ ((LSeg (p1,|[1,1]|)) \/ ((LSeg (p1,|[1,0]|)) \/ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,p2))))) by A29, XBOOLE_1:4
.= ((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,1]|,p2))) \/ ((LSeg (p1,|[1,1]|)) \/ (((LSeg (p1,|[1,0]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[0,0]|,p2)))) by XBOOLE_1:4
.= ((LSeg (p1,|[1,1]|)) \/ ((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,1]|,p2)))) \/ (((LSeg (p1,|[1,0]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[0,0]|,p2))) by XBOOLE_1:4
.= P1 \/ P2 by XBOOLE_1:4 ; ::_thesis: P1 /\ P2 = {p1,p2}
A31: (LSeg (p1,|[1,1]|)) /\ (LSeg (p1,|[1,0]|)) = {p1} by A3, TOPREAL1:8;
A32: P1 /\ P2 = (((LSeg (p1,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) /\ (((LSeg (p1,|[1,0]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[0,0]|,p2)))) \/ ((LSeg (|[0,1]|,p2)) /\ (((LSeg (p1,|[1,0]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[0,0]|,p2)))) by XBOOLE_1:23
.= (((LSeg (p1,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) /\ (((LSeg (p1,|[1,0]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[0,1]|,p2)) /\ ((LSeg (p1,|[1,0]|)) \/ (LSeg (|[0,0]|,|[1,0]|)))) \/ {p2}) by A26, XBOOLE_1:23
.= (((LSeg (p1,|[1,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) /\ (((LSeg (p1,|[1,0]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[0,0]|,p2)))) \/ ((((LSeg (|[0,1]|,p2)) /\ (LSeg (p1,|[1,0]|))) \/ ((LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)))) \/ {p2}) by XBOOLE_1:23
.= (((LSeg (p1,|[1,1]|)) /\ (((LSeg (p1,|[1,0]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[0,0]|,p2)))) \/ ((LSeg (|[0,1]|,|[1,1]|)) /\ (((LSeg (p1,|[1,0]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[0,0]|,p2))))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {p2}) by A25, XBOOLE_1:23
.= (((LSeg (p1,|[1,1]|)) /\ (((LSeg (p1,|[1,0]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[0,1]|,|[1,1]|)) /\ ((LSeg (p1,|[1,0]|)) \/ (LSeg (|[0,0]|,|[1,0]|)))) \/ ((LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2))))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {p2}) by XBOOLE_1:23
.= (((LSeg (p1,|[1,1]|)) /\ (((LSeg (p1,|[1,0]|)) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[0,0]|,p2)))) \/ ((((LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (p1,|[1,0]|))) \/ ((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2))))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {p2}) by XBOOLE_1:23
.= ((((LSeg (p1,|[1,1]|)) /\ ((LSeg (p1,|[1,0]|)) \/ (LSeg (|[0,0]|,|[1,0]|)))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (p1,|[1,0]|))) \/ ((LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2))))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {p2}) by Lm2, XBOOLE_1:23
.= (({p1} \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)))) \/ (((LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (p1,|[1,0]|))) \/ ((LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2))))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {p2}) by A24, A31, XBOOLE_1:23 ;
A33: now__::_thesis:_P1_/\_P2_=_({p1}_\/_((LSeg_(|[0,1]|,|[1,1]|))_/\_(LSeg_(|[0,0]|,p2))))_\/_(((LSeg_(|[0,1]|,p2))_/\_(LSeg_(|[0,0]|,|[1,0]|)))_\/_{p2})
percases ( p1 = |[1,0]| or p1 = |[1,1]| or ( p1 <> |[1,1]| & p1 <> |[1,0]| ) ) ;
supposeA34: p1 = |[1,0]| ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {p2})
then  (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (p1,|[1,0]|)) = (LSeg (|[0,1]|,|[1,1]|)) /\ {|[1,0]|} by RLTOPSP1:70;
then  (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (p1,|[1,0]|)) = {} by Lm1, Lm17;
hence  P1 /\ P2 = ({p1} \/ ((LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {p2}) by A32, A34, TOPREAL1:16; ::_thesis: verum
end;
supposeA35: p1 = |[1,1]| ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {p2})
then  (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {|[1,1]|} /\ (LSeg (|[0,0]|,|[1,0]|)) by RLTOPSP1:70;
then  (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {} by Lm1, Lm19;
hence P1 /\ P2 = (({p1} \/ {p1}) \/ ((LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {p2}) by A32, A35, TOPREAL1:18, XBOOLE_1:4
.= ({p1} \/ ((LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {p2}) ;
::_thesis: verum
end;
supposeA36: ( p1 <> |[1,1]| & p1 <> |[1,0]| ) ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {p2})
now__::_thesis:_not_|[1,1]|_in_(LSeg_(|[0,1]|,|[1,1]|))_/\_(LSeg_(p1,|[1,0]|))
assume  |[1,1]| in (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (p1,|[1,0]|)) ; ::_thesis: contradiction
then A37: |[1,1]| in LSeg (|[1,0]|,p1) by XBOOLE_0:def_4;
 |[1,0]| `2 <= p1 `2 by A13, A16, EUCLID:52;
then  |[1,1]| `2 <= p1 `2 by A37, TOPREAL1:4;
then  1 = p1 `2 by A13, A15, Lm11, XXREAL_0:1;
hence  contradiction by A13, A14, A36, EUCLID:53; ::_thesis: verum
end;
then A38: {|[1,1]|} <> (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (p1,|[1,0]|)) by ZFMISC_1:31;
 (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (p1,|[1,0]|)) c= {|[1,1]|} by A3, Lm25, TOPREAL1:6, TOPREAL1:18, XBOOLE_1:26;
then A39: (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (p1,|[1,0]|)) = {} by A38, ZFMISC_1:33;
now__::_thesis:_not_|[1,0]|_in_(LSeg_(p1,|[1,1]|))_/\_(LSeg_(|[0,0]|,|[1,0]|))
assume  |[1,0]| in (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) ; ::_thesis: contradiction
then A40: |[1,0]| in LSeg (p1,|[1,1]|) by XBOOLE_0:def_4;
 p1 `2 <= |[1,1]| `2 by A13, A15, EUCLID:52;
then  p1 `2 = 0 by A13, A16, A40, Lm9, TOPREAL1:4;
hence  contradiction by A13, A14, A36, EUCLID:53; ::_thesis: verum
end;
then A41: {|[1,0]|} <> (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by ZFMISC_1:31;
 (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by A3, Lm27, TOPREAL1:6, XBOOLE_1:26;
then  (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {} by A41, TOPREAL1:16, ZFMISC_1:33;
hence  P1 /\ P2 = ({p1} \/ ((LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {p2}) by A32, A39; ::_thesis: verum
end;
end;
end;
now__::_thesis:_P1_/\_P2_=_{p1,p2}
percases ( p2 = |[0,0]| or p2 = |[0,1]| or ( p2 <> |[0,1]| & p2 <> |[0,0]| ) ) ;
supposeA42: p2 = |[0,0]| ; ::_thesis: P1 /\ P2 = {p1,p2}
then  (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) = (LSeg (|[0,1]|,|[1,1]|)) /\ {|[0,0]|} by RLTOPSP1:70;
then  (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) = {} by Lm1, Lm13;
hence  P1 /\ P2 = {p1,p2} by A33, A42, ENUMSET1:1, TOPREAL1:17; ::_thesis: verum
end;
supposeA43: p2 = |[0,1]| ; ::_thesis: P1 /\ P2 = {p1,p2}
then  (LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {|[0,1]|} /\ (LSeg (|[0,0]|,|[1,0]|)) by RLTOPSP1:70;
then  (LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {} by Lm1, Lm14;
hence P1 /\ P2 = {p1} \/ ({p2} \/ {p2}) by A33, A43, TOPREAL1:15, XBOOLE_1:4
.= {p1,p2} by ENUMSET1:1 ;
::_thesis: verum
end;
supposeA44: ( p2 <> |[0,1]| & p2 <> |[0,0]| ) ; ::_thesis: P1 /\ P2 = {p1,p2}
now__::_thesis:_not_|[0,0]|_in_(LSeg_(|[0,1]|,p2))_/\_(LSeg_(|[0,0]|,|[1,0]|))
assume  |[0,0]| in (LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) ; ::_thesis: contradiction
then A45: |[0,0]| in LSeg (p2,|[0,1]|) by XBOOLE_0:def_4;
 p2 `2 <= |[0,1]| `2 by A30, EUCLID:52;
then  0 = p2 `2 by A30, A45, Lm5, TOPREAL1:4;
hence  contradiction by A30, A44, EUCLID:53; ::_thesis: verum
end;
then A46: {|[0,0]|} <> (LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) by ZFMISC_1:31;
 (LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) c= {|[0,0]|} by A17, Lm22, TOPREAL1:6, TOPREAL1:17, XBOOLE_1:26;
then A47: (LSeg (|[0,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {} by A46, ZFMISC_1:33;
now__::_thesis:_not_|[0,1]|_in_(LSeg_(|[0,1]|,|[1,1]|))_/\_(LSeg_(|[0,0]|,p2))
assume  |[0,1]| in (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) ; ::_thesis: contradiction
then A48: |[0,1]| in LSeg (|[0,0]|,p2) by XBOOLE_0:def_4;
 |[0,0]| `2 <= p2 `2 by A30, EUCLID:52;
then  |[0,1]| `2 <= p2 `2 by A48, TOPREAL1:4;
then  p2 `2 = 1 by A30, Lm7, XXREAL_0:1;
hence  contradiction by A30, A44, EUCLID:53; ::_thesis: verum
end;
then A49: {|[0,1]|} <> (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) by ZFMISC_1:31;
 (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) c= (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A17, Lm20, TOPREAL1:6, XBOOLE_1:26;
then  (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) = {} by A49, TOPREAL1:15, ZFMISC_1:33;
hence  P1 /\ P2 = {p1,p2} by A33, A47, ENUMSET1:1; ::_thesis: verum
end;
end;
end;
hence  P1 /\ P2 = {p1,p2} ; ::_thesis: verum
end;
supposeA50: p2 in LSeg (|[0,1]|,|[1,1]|) ; ::_thesis: ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
then A51: ex q being Point of (TOP-REAL 2) st
( q = p2 & q `1 <= 1 & q `1 >= 0 & q `2 = 1 ) by TOPREAL1:13;
now__::_thesis:_not_|[1,1]|_in_(LSeg_(p1,|[1,0]|))_/\_(LSeg_(|[0,1]|,p2))
A52: |[0,1]| `1 <= p2 `1 by A51, EUCLID:52;
assume A53: |[1,1]| in (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,1]|,p2)) ; ::_thesis: contradiction
then A54: |[1,1]| in LSeg (|[1,0]|,p1) by XBOOLE_0:def_4;
 |[1,1]| in LSeg (|[0,1]|,p2) by A53, XBOOLE_0:def_4;
then  |[1,1]| `1 <= p2 `1 by A52, TOPREAL1:3;
then A55: 1 = p2 `1 by A51, Lm10, XXREAL_0:1;
 |[1,0]| `2 <= p1 `2 by A13, A16, EUCLID:52;
then  |[1,1]| `2 <= p1 `2 by A54, TOPREAL1:4;
then  1 = p1 `2 by A13, A15, Lm11, XXREAL_0:1;
then p1 = |[1,1]| by A13, A14, EUCLID:53
.= p2 by A51, A55, EUCLID:53 ;
hence  contradiction by A1; ::_thesis: verum
end;
then A56: {|[1,1]|} <> (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,1]|,p2)) by ZFMISC_1:31;
A57: LSeg (|[0,0]|,|[0,1]|) is_an_arc_of |[0,0]|,|[0,1]| by Lm5, Lm7, TOPREAL1:9;
 LSeg (|[0,0]|,|[1,0]|) is_an_arc_of |[1,0]|,|[0,0]| by Lm4, Lm8, TOPREAL1:9;
then A58: (LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|)) is_an_arc_of |[1,0]|,|[0,1]| by A57, TOPREAL1:2, TOPREAL1:17;
 (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by TOPREAL1:20, XBOOLE_0:def_7;
then A59: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {} by A5, XBOOLE_1:3, XBOOLE_1:26;
take P1 = (LSeg (p1,|[1,1]|)) \/ (LSeg (|[1,1]|,p2)); ::_thesis: ex P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
take P2 = (LSeg (p1,|[1,0]|)) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2))); ::_thesis: ( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A60: (LSeg (p1,|[1,1]|)) \/ (LSeg (p1,|[1,0]|)) = LSeg (|[1,0]|,|[1,1]|) by A3, TOPREAL1:5;
 |[0,1]| in LSeg (|[0,1]|,p2) by RLTOPSP1:68;
then A61: (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[0,1]|,p2)) <> {} by Lm22, XBOOLE_0:def_4;
A62: |[1,1]| in LSeg (|[1,1]|,p2) by RLTOPSP1:68;
 |[1,1]| in LSeg (p1,|[1,1]|) by RLTOPSP1:68;
then A63: |[1,1]| in (LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,1]|,p2)) by A62, XBOOLE_0:def_4;
A64: LSeg (|[1,1]|,p2) c= LSeg (|[0,1]|,|[1,1]|) by A50, Lm26, TOPREAL1:6;
then  (LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,1]|,p2)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) by A5, XBOOLE_1:27;
then A65: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,1]|,p2)) = {|[1,1]|} by A63, TOPREAL1:18, ZFMISC_1:33;
 ( p1 <> |[1,1]| or |[1,1]| <> p2 ) by A1;
hence  P1 is_an_arc_of p1,p2 by A65, TOPREAL1:12; ::_thesis: ( P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A66: {p1} = (LSeg (p1,|[1,1]|)) /\ (LSeg (p1,|[1,0]|)) by A3, TOPREAL1:8;
 (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by TOPREAL1:19, XBOOLE_0:def_7;
then A67: (LSeg (|[1,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {} by A64, XBOOLE_1:3, XBOOLE_1:26;
A68: LSeg (p2,|[0,1]|) c= LSeg (|[0,1]|,|[1,1]|) by A50, Lm23, TOPREAL1:6;
then A69: (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[0,1]|,p2)) c= {|[0,1]|} by TOPREAL1:15, XBOOLE_1:27;
A70: (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,p2)) = {} by A68, Lm2, XBOOLE_1:3, XBOOLE_1:26;
((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) /\ (LSeg (|[0,1]|,p2)) = ((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,p2))) \/ ((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[0,1]|,p2))) by XBOOLE_1:23
.= {|[0,1]|} by A70, A69, A61, ZFMISC_1:33 ;
then A71: ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2)) is_an_arc_of |[1,0]|,p2 by A58, TOPREAL1:10;
A72: {p2} = (LSeg (|[1,1]|,p2)) /\ (LSeg (|[0,1]|,p2)) by A50, TOPREAL1:8;
A73: (LSeg (|[0,1]|,p2)) \/ (LSeg (|[1,1]|,p2)) = LSeg (|[0,1]|,|[1,1]|) by A50, TOPREAL1:5;
 (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,1]|,p2)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) by A11, A68, XBOOLE_1:27;
then A74: (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,1]|,p2)) = {} by A56, TOPREAL1:18, ZFMISC_1:33;
(LSeg (p1,|[1,0]|)) /\ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2))) = ((LSeg (p1,|[1,0]|)) /\ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|)))) \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,1]|,p2))) by XBOOLE_1:23
.= ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (|[1,0]|,p1)) /\ (LSeg (|[0,0]|,|[0,1]|))) by A74, XBOOLE_1:23
.= {|[1,0]|} by A9, A7, A12, TOPREAL1:16, ZFMISC_1:33 ;
hence  P2 is_an_arc_of p1,p2 by A71, TOPREAL1:11; ::_thesis: ( R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
thus P1 \/ P2 = (LSeg (|[1,1]|,p2)) \/ ((LSeg (p1,|[1,1]|)) \/ ((LSeg (p1,|[1,0]|)) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2))))) by XBOOLE_1:4
.= (LSeg (|[1,1]|,p2)) \/ ((LSeg (|[1,0]|,|[1,1]|)) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2)))) by A60, XBOOLE_1:4
.= (LSeg (|[1,1]|,p2)) \/ (((LSeg (|[1,0]|,|[1,1]|)) \/ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|)))) \/ (LSeg (|[0,1]|,p2))) by XBOOLE_1:4
.= (LSeg (|[1,1]|,p2)) \/ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2))) by XBOOLE_1:4
.= (LSeg (|[1,1]|,p2)) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,p2)))) by XBOOLE_1:4
.= (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,p2))) \/ (LSeg (|[1,1]|,p2))) \/ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|))) by XBOOLE_1:4
.= R^2-unit_square by A73, TOPREAL1:def_2, XBOOLE_1:4 ; ::_thesis: P1 /\ P2 = {p1,p2}
A75: P1 /\ P2 = ((LSeg (p1,|[1,1]|)) /\ ((LSeg (p1,|[1,0]|)) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2))))) \/ ((LSeg (|[1,1]|,p2)) /\ ((LSeg (p1,|[1,0]|)) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2))))) by XBOOLE_1:23
.= (((LSeg (p1,|[1,1]|)) /\ (LSeg (p1,|[1,0]|))) \/ ((LSeg (p1,|[1,1]|)) /\ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2))))) \/ ((LSeg (|[1,1]|,p2)) /\ ((LSeg (p1,|[1,0]|)) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2))))) by XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[1,1]|)) /\ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|)))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2))))) \/ ((LSeg (|[1,1]|,p2)) /\ ((LSeg (p1,|[1,0]|)) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2))))) by A66, XBOOLE_1:23
.= ({p1} \/ ((((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2))))) \/ ((LSeg (|[1,1]|,p2)) /\ ((LSeg (p1,|[1,0]|)) \/ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2))))) by XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2))))) \/ (((LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[1,0]|))) \/ ((LSeg (|[1,1]|,p2)) /\ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,p2))))) by A59, XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2))))) \/ (((LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[1,0]|))) \/ (((LSeg (|[1,1]|,p2)) /\ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|)))) \/ {p2})) by A72, XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2))))) \/ (((LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[1,0]|))) \/ ((((LSeg (|[1,1]|,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (|[1,1]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)))) \/ {p2})) by XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2))))) \/ (((LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[1,0]|))) \/ (((LSeg (|[1,1]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2})) by A67 ;
A76: now__::_thesis:_P1_/\_P2_=_({p1}_\/_((LSeg_(p1,|[1,1]|))_/\_(LSeg_(|[0,1]|,p2))))_\/_(((LSeg_(|[1,1]|,p2))_/\_(LSeg_(|[0,0]|,|[0,1]|)))_\/_{p2})
percases ( p1 = |[1,0]| or p1 = |[1,1]| or ( p1 <> |[1,1]| & p1 <> |[1,0]| ) ) ;
supposeA77: p1 = |[1,0]| ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)))) \/ (((LSeg (|[1,1]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2})
then A78: (LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[1,0]|)) = (LSeg (|[1,1]|,p2)) /\ {|[1,0]|} by RLTOPSP1:70;
 not |[1,0]| in LSeg (|[1,1]|,p2) by A64, Lm7, Lm9, Lm11, TOPREAL1:4;
then A79: (LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[1,0]|)) = {} by A78, Lm1;
thus P1 /\ P2 = (({p1} \/ {p1}) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)))) \/ (((LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[1,0]|))) \/ (((LSeg (|[1,1]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2})) by A75, A77, TOPREAL1:16, XBOOLE_1:4
.= ({p1} \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)))) \/ (((LSeg (|[1,1]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2}) by A79 ; ::_thesis: verum
end;
supposeA80: p1 = |[1,1]| ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)))) \/ (((LSeg (|[1,1]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2})
 |[1,1]| in LSeg (|[1,1]|,p2) by RLTOPSP1:68;
then A81: (LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[1,0]|)) <> {} by A80, Lm27, XBOOLE_0:def_4;
 (LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[1,0]|)) c= {p1} by A64, A80, TOPREAL1:18, XBOOLE_1:27;
then A82: (LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[1,0]|)) = {p1} by A81, ZFMISC_1:33;
 (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {|[1,1]|} /\ (LSeg (|[0,0]|,|[1,0]|)) by A80, RLTOPSP1:70;
then  (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {} by Lm1, Lm19;
hence P1 /\ P2 = ({p1} \/ ({p1} \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2))))) \/ (((LSeg (|[1,1]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2}) by A75, A82, XBOOLE_1:4
.= (({p1} \/ {p1}) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)))) \/ (((LSeg (|[1,1]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2}) by XBOOLE_1:4
.= ({p1} \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)))) \/ (((LSeg (|[1,1]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2}) ;
::_thesis: verum
end;
supposeA83: ( p1 <> |[1,1]| & p1 <> |[1,0]| ) ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)))) \/ (((LSeg (|[1,1]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2})
now__::_thesis:_not_|[1,1]|_in_(LSeg_(|[1,1]|,p2))_/\_(LSeg_(p1,|[1,0]|))
assume  |[1,1]| in (LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[1,0]|)) ; ::_thesis: contradiction
then A84: |[1,1]| in LSeg (|[1,0]|,p1) by XBOOLE_0:def_4;
 |[1,0]| `2 <= p1 `2 by A13, A16, EUCLID:52;
then  |[1,1]| `2 <= p1 `2 by A84, TOPREAL1:4;
then  1 = p1 `2 by A13, A15, Lm11, XXREAL_0:1;
hence  contradiction by A13, A14, A83, EUCLID:53; ::_thesis: verum
end;
then A85: {|[1,1]|} <> (LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[1,0]|)) by ZFMISC_1:31;
 (LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[1,0]|)) c= {|[1,1]|} by A11, A64, TOPREAL1:18, XBOOLE_1:27;
then A86: (LSeg (|[1,1]|,p2)) /\ (LSeg (p1,|[1,0]|)) = {} by A85, ZFMISC_1:33;
now__::_thesis:_not_|[1,0]|_in_(LSeg_(p1,|[1,1]|))_/\_(LSeg_(|[0,0]|,|[1,0]|))
assume  |[1,0]| in (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) ; ::_thesis: contradiction
then A87: |[1,0]| in LSeg (p1,|[1,1]|) by XBOOLE_0:def_4;
 p1 `2 <= |[1,1]| `2 by A13, A15, EUCLID:52;
then  p1 `2 = 0 by A13, A16, A87, Lm9, TOPREAL1:4;
hence  contradiction by A13, A14, A83, EUCLID:53; ::_thesis: verum
end;
then A88: {|[1,0]|} <> (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by ZFMISC_1:31;
 (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by A3, Lm27, TOPREAL1:6, XBOOLE_1:26;
then  (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {} by A88, TOPREAL1:16, ZFMISC_1:33;
hence  P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)))) \/ (((LSeg (|[1,1]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2}) by A75, A86; ::_thesis: verum
end;
end;
end;
now__::_thesis:_P1_/\_P2_=_{p1,p2}
percases ( p2 = |[0,1]| or p2 = |[1,1]| or ( p2 <> |[1,1]| & p2 <> |[0,1]| ) ) ;
supposeA89: p2 = |[0,1]| ; ::_thesis: P1 /\ P2 = {p1,p2}
then A90: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)) = (LSeg (p1,|[1,1]|)) /\ {|[0,1]|} by RLTOPSP1:70;
 not |[0,1]| in LSeg (p1,|[1,1]|) by A5, Lm6, Lm8, Lm10, TOPREAL1:3;
then  (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)) = {} by A90, Lm1;
hence  P1 /\ P2 = {p1,p2} by A76, A89, ENUMSET1:1, TOPREAL1:15; ::_thesis: verum
end;
supposeA91: p2 = |[1,1]| ; ::_thesis: P1 /\ P2 = {p1,p2}
 |[1,1]| in LSeg (p1,|[1,1]|) by RLTOPSP1:68;
then A92: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)) <> {} by A91, Lm26, XBOOLE_0:def_4;
 (LSeg (|[1,1]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {|[1,1]|} /\ (LSeg (|[0,0]|,|[0,1]|)) by A91, RLTOPSP1:70;
then A93: (LSeg (|[1,1]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {} by Lm1, Lm18;
 (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) by A5, A68, XBOOLE_1:27;
then  (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)) = {p2} by A91, A92, TOPREAL1:18, ZFMISC_1:33;
hence P1 /\ P2 = {p1} \/ ({p2} \/ {p2}) by A76, A93, XBOOLE_1:4
.= {p1,p2} by ENUMSET1:1 ;
::_thesis: verum
end;
supposeA94: ( p2 <> |[1,1]| & p2 <> |[0,1]| ) ; ::_thesis: P1 /\ P2 = {p1,p2}
now__::_thesis:_not_|[0,1]|_in_(LSeg_(|[1,1]|,p2))_/\_(LSeg_(|[0,0]|,|[0,1]|))
assume  |[0,1]| in (LSeg (|[1,1]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) ; ::_thesis: contradiction
then A95: |[0,1]| in LSeg (p2,|[1,1]|) by XBOOLE_0:def_4;
 p2 `1 <= |[1,1]| `1 by A51, EUCLID:52;
then  p2 `1 = 0 by A51, A95, Lm6, TOPREAL1:3;
hence  contradiction by A51, A94, EUCLID:53; ::_thesis: verum
end;
then A96: {|[0,1]|} <> (LSeg (|[1,1]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) by ZFMISC_1:31;
 (LSeg (|[1,1]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) c= (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A50, Lm26, TOPREAL1:6, XBOOLE_1:26;
then A97: (LSeg (|[1,1]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {} by A96, TOPREAL1:15, ZFMISC_1:33;
now__::_thesis:_not_|[1,1]|_in_(LSeg_(p1,|[1,1]|))_/\_(LSeg_(|[0,1]|,p2))
assume  |[1,1]| in (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)) ; ::_thesis: contradiction
then A98: |[1,1]| in LSeg (|[0,1]|,p2) by XBOOLE_0:def_4;
 |[0,1]| `1 <= p2 `1 by A51, EUCLID:52;
then  |[1,1]| `1 <= p2 `1 by A98, TOPREAL1:3;
then  1 = p2 `1 by A51, Lm10, XXREAL_0:1;
hence  contradiction by A51, A94, EUCLID:53; ::_thesis: verum
end;
then A99: {|[1,1]|} <> (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)) by ZFMISC_1:31;
 (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) by A5, A68, XBOOLE_1:27;
then  (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,p2)) = {} by A99, TOPREAL1:18, ZFMISC_1:33;
hence  P1 /\ P2 = {p1,p2} by A76, A97, ENUMSET1:1; ::_thesis: verum
end;
end;
end;
hence  P1 /\ P2 = {p1,p2} ; ::_thesis: verum
end;
supposeA100: p2 in LSeg (|[0,0]|,|[1,0]|) ; ::_thesis: ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
then A101: ex q being Point of (TOP-REAL 2) st
( q = p2 & q `1 <= 1 & q `1 >= 0 & q `2 = 0 ) by TOPREAL1:13;
now__::_thesis:_not_|[1,0]|_in_(LSeg_(p1,|[1,1]|))_/\_(LSeg_(|[0,0]|,p2))
A102: |[0,0]| `1 <= p2 `1 by A101, EUCLID:52;
assume A103: |[1,0]| in (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) ; ::_thesis: contradiction
then A104: |[1,0]| in LSeg (p1,|[1,1]|) by XBOOLE_0:def_4;
 |[1,0]| in LSeg (|[0,0]|,p2) by A103, XBOOLE_0:def_4;
then  |[1,0]| `1 <= p2 `1 by A102, TOPREAL1:3;
then A105: 1 = p2 `1 by A101, Lm8, XXREAL_0:1;
 p1 `2 <= |[1,1]| `2 by A13, A15, EUCLID:52;
then  0 = p1 `2 by A13, A16, A104, Lm9, TOPREAL1:4;
then  p1 = |[1,0]| by A13, A14, EUCLID:53;
hence  contradiction by A1, A101, A105, EUCLID:53; ::_thesis: verum
end;
then A106: {|[1,0]|} <> (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) by ZFMISC_1:31;
A107: LSeg (|[0,0]|,|[0,1]|) is_an_arc_of |[0,1]|,|[0,0]| by Lm5, Lm7, TOPREAL1:9;
 LSeg (|[0,1]|,|[1,1]|) is_an_arc_of |[1,1]|,|[0,1]| by Lm6, Lm10, TOPREAL1:9;
then A108: (LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|)) is_an_arc_of |[1,1]|,|[0,0]| by A107, TOPREAL1:2, TOPREAL1:15;
take P1 = (LSeg (p1,|[1,0]|)) \/ (LSeg (|[1,0]|,p2)); ::_thesis: ex P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
take P2 = (LSeg (p1,|[1,1]|)) \/ (((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,p2))); ::_thesis: ( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A109: (LSeg (p1,|[1,0]|)) \/ (LSeg (p1,|[1,1]|)) = LSeg (|[1,0]|,|[1,1]|) by A3, TOPREAL1:5;
 |[0,0]| in LSeg (|[0,0]|,p2) by RLTOPSP1:68;
then A110: (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[0,0]|,p2)) <> {} by Lm20, XBOOLE_0:def_4;
A111: |[1,0]| in LSeg (|[1,0]|,p2) by RLTOPSP1:68;
 |[1,0]| in LSeg (p1,|[1,0]|) by RLTOPSP1:68;
then A112: (LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,0]|,p2)) <> {} by A111, XBOOLE_0:def_4;
A113: LSeg (p2,|[1,0]|) c= LSeg (|[0,0]|,|[1,0]|) by A100, Lm24, TOPREAL1:6;
then  (LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,0]|,p2)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by A11, XBOOLE_1:27;
then A114: (LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,0]|,p2)) = {|[1,0]|} by A112, TOPREAL1:16, ZFMISC_1:33;
 ( p1 <> |[1,0]| or p2 <> |[1,0]| ) by A1;
hence  P1 is_an_arc_of p1,p2 by A114, TOPREAL1:12; ::_thesis: ( P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A115: (LSeg (p1,|[1,0]|)) /\ (LSeg (p1,|[1,1]|)) = {p1} by A3, TOPREAL1:8;
 (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by TOPREAL1:19, XBOOLE_0:def_7;
then A116: (LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by A113, XBOOLE_1:3, XBOOLE_1:26;
A117: LSeg (p2,|[0,0]|) c= LSeg (|[0,0]|,|[1,0]|) by A100, Lm21, TOPREAL1:6;
then A118: (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[0,0]|,p2)) c= {|[0,0]|} by TOPREAL1:17, XBOOLE_1:27;
A119: (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) = {} by A117, Lm2, XBOOLE_1:3, XBOOLE_1:26;
((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) /\ (LSeg (|[0,0]|,p2)) = ((LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[0,0]|,p2))) \/ ((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[0,0]|,p2))) by XBOOLE_1:23
.= {|[0,0]|} by A119, A118, A110, ZFMISC_1:33 ;
then A120: ((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,p2)) is_an_arc_of |[1,1]|,p2 by A108, TOPREAL1:10;
A121: (LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,p2)) = {p2} by A100, TOPREAL1:8;
A122: (LSeg (|[0,0]|,p2)) \/ (LSeg (|[1,0]|,p2)) = LSeg (|[0,0]|,|[1,0]|) by A100, TOPREAL1:5;
 (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by A5, A117, XBOOLE_1:27;
then A123: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,p2)) = {} by A106, TOPREAL1:16, ZFMISC_1:33;
(LSeg (p1,|[1,1]|)) /\ (((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,p2))) = ((LSeg (p1,|[1,1]|)) /\ ((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|)))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,p2))) by XBOOLE_1:23
.= ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|))) by A123, XBOOLE_1:23
.= {|[1,1]|} by A8, A6, A10, TOPREAL1:18, ZFMISC_1:33 ;
hence  P2 is_an_arc_of p1,p2 by A120, TOPREAL1:11; ::_thesis: ( R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
thus P1 \/ P2 = (LSeg (|[1,0]|,p2)) \/ ((LSeg (p1,|[1,0]|)) \/ ((LSeg (p1,|[1,1]|)) \/ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,0]|,p2))))) by XBOOLE_1:4
.= (LSeg (|[1,0]|,p2)) \/ ((LSeg (|[1,0]|,|[1,1]|)) \/ (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,0]|,p2)))) by A109, XBOOLE_1:4
.= ((((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[0,0]|,p2))) \/ (LSeg (|[1,0]|,p2)) by XBOOLE_1:4
.= (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,0]|,|[1,1]|))) \/ (LSeg (|[0,0]|,|[1,0]|)) by A122, XBOOLE_1:4
.= R^2-unit_square by TOPREAL1:def_2, XBOOLE_1:4 ; ::_thesis: P1 /\ P2 = {p1,p2}
A124: P1 /\ P2 = ((LSeg (p1,|[1,0]|)) /\ ((LSeg (p1,|[1,1]|)) \/ (((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,p2))))) \/ ((LSeg (|[1,0]|,p2)) /\ ((LSeg (p1,|[1,1]|)) \/ (((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,p2))))) by XBOOLE_1:23
.= (((LSeg (p1,|[1,0]|)) /\ (LSeg (p1,|[1,1]|))) \/ ((LSeg (p1,|[1,0]|)) /\ (((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,p2))))) \/ ((LSeg (|[1,0]|,p2)) /\ ((LSeg (p1,|[1,1]|)) \/ (((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,p2))))) by XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[1,0]|)) /\ ((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|)))) \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2))))) \/ ((LSeg (|[1,0]|,p2)) /\ ((LSeg (p1,|[1,1]|)) \/ (((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,p2))))) by A115, XBOOLE_1:23
.= ({p1} \/ ((((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ ((LSeg (|[1,0]|,p1)) /\ (LSeg (|[0,0]|,|[0,1]|)))) \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2))))) \/ ((LSeg (|[1,0]|,p2)) /\ ((LSeg (p1,|[1,1]|)) \/ (((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,p2))))) by XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2))))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[1,1]|))) \/ ((LSeg (|[1,0]|,p2)) /\ (((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,p2))))) by A12, XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2))))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[1,1]|))) \/ (((LSeg (|[1,0]|,p2)) /\ ((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|)))) \/ {p2})) by A121, XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2))))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[1,1]|))) \/ ((((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ ((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)))) \/ {p2})) by XBOOLE_1:23
.= ({p1} \/ (((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2))))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[1,1]|))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2})) by A116 ;
A125: now__::_thesis:_P1_/\_P2_=_({p1}_\/_((LSeg_(p1,|[1,0]|))_/\_(LSeg_(|[0,0]|,p2))))_\/_(((LSeg_(|[1,0]|,p2))_/\_(LSeg_(|[0,0]|,|[0,1]|)))_\/_{p2})
percases ( p1 = |[1,0]| or p1 = |[1,1]| or ( p1 <> |[1,1]| & p1 <> |[1,0]| ) ) ;
supposeA126: p1 = |[1,0]| ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2})
 |[1,0]| in LSeg (|[1,0]|,p2) by RLTOPSP1:68;
then A127: (LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[1,1]|)) <> {} by A126, Lm25, XBOOLE_0:def_4;
 (LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[1,1]|)) c= {p1} by A113, A126, TOPREAL1:16, XBOOLE_1:27;
then A128: (LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[1,1]|)) = {p1} by A127, ZFMISC_1:33;
 (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {|[1,0]|} /\ (LSeg (|[0,1]|,|[1,1]|)) by A126, RLTOPSP1:70;
then  (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by Lm1, Lm17;
hence P1 /\ P2 = ({p1} \/ ({p1} \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2))))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2}) by A124, A128, XBOOLE_1:4
.= (({p1} \/ {p1}) \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2}) by XBOOLE_1:4
.= ({p1} \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2}) ;
::_thesis: verum
end;
supposeA129: p1 = |[1,1]| ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2})
then A130: (LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[1,1]|)) = (LSeg (|[1,0]|,p2)) /\ {|[1,1]|} by RLTOPSP1:70;
 not |[1,1]| in LSeg (|[1,0]|,p2) by A113, Lm5, Lm9, Lm11, TOPREAL1:4;
then  (LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[1,1]|)) = {} by A130, Lm1;
hence P1 /\ P2 = (({p1} \/ {p1}) \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2}) by A124, A129, TOPREAL1:18, XBOOLE_1:4
.= ({p1} \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2}) ;
::_thesis: verum
end;
supposeA131: ( p1 <> |[1,1]| & p1 <> |[1,0]| ) ; ::_thesis: P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2})
now__::_thesis:_not_|[1,0]|_in_(LSeg_(|[1,0]|,p2))_/\_(LSeg_(p1,|[1,1]|))
assume  |[1,0]| in (LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[1,1]|)) ; ::_thesis: contradiction
then A132: |[1,0]| in LSeg (p1,|[1,1]|) by XBOOLE_0:def_4;
 p1 `2 <= |[1,1]| `2 by A13, A15, EUCLID:52;
then  p1 `2 = 0 by A13, A16, A132, Lm9, TOPREAL1:4;
hence  contradiction by A13, A14, A131, EUCLID:53; ::_thesis: verum
end;
then A133: {|[1,0]|} <> (LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[1,1]|)) by ZFMISC_1:31;
 (LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[1,1]|)) c= {|[1,0]|} by A5, A113, TOPREAL1:16, XBOOLE_1:27;
then A134: (LSeg (|[1,0]|,p2)) /\ (LSeg (p1,|[1,1]|)) = {} by A133, ZFMISC_1:33;
now__::_thesis:_not_|[1,1]|_in_(LSeg_(p1,|[1,0]|))_/\_(LSeg_(|[0,1]|,|[1,1]|))
assume  |[1,1]| in (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) ; ::_thesis: contradiction
then A135: |[1,1]| in LSeg (|[1,0]|,p1) by XBOOLE_0:def_4;
 |[1,0]| `2 <= p1 `2 by A13, A16, EUCLID:52;
then  |[1,1]| `2 <= p1 `2 by A135, TOPREAL1:4;
then  p1 `2 = 1 by A13, A15, Lm11, XXREAL_0:1;
hence  contradiction by A13, A14, A131, EUCLID:53; ::_thesis: verum
end;
then A136: (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) <> {|[1,1]|} by ZFMISC_1:31;
 (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) by A3, Lm25, TOPREAL1:6, XBOOLE_1:26;
then  (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by A136, TOPREAL1:18, ZFMISC_1:33;
hence  P1 /\ P2 = ({p1} \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2)))) \/ (((LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|))) \/ {p2}) by A124, A134; ::_thesis: verum
end;
end;
end;
now__::_thesis:_P1_/\_P2_=_{p1,p2}
percases ( p2 = |[0,0]| or p2 = |[1,0]| or ( p2 <> |[1,0]| & p2 <> |[0,0]| ) ) ;
supposeA137: p2 = |[0,0]| ; ::_thesis: P1 /\ P2 = {p1,p2}
then A138: (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2)) = (LSeg (p1,|[1,0]|)) /\ {|[0,0]|} by RLTOPSP1:70;
 not |[0,0]| in LSeg (p1,|[1,0]|) by A11, Lm4, Lm8, Lm10, TOPREAL1:3;
then  (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2)) = {} by A138, Lm1;
hence  P1 /\ P2 = {p1,p2} by A125, A137, ENUMSET1:1, TOPREAL1:17; ::_thesis: verum
end;
supposeA139: p2 = |[1,0]| ; ::_thesis: P1 /\ P2 = {p1,p2}
 |[1,0]| in LSeg (p1,|[1,0]|) by RLTOPSP1:68;
then A140: (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2)) <> {} by A139, Lm24, XBOOLE_0:def_4;
 (LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {|[1,0]|} /\ (LSeg (|[0,0]|,|[0,1]|)) by A139, RLTOPSP1:70;
then A141: (LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {} by Lm1, Lm16;
 (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by A11, A117, XBOOLE_1:27;
then  (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2)) = {p2} by A139, A140, TOPREAL1:16, ZFMISC_1:33;
hence P1 /\ P2 = {p1} \/ ({p2} \/ {p2}) by A125, A141, XBOOLE_1:4
.= {p1,p2} by ENUMSET1:1 ;
::_thesis: verum
end;
supposeA142: ( p2 <> |[1,0]| & p2 <> |[0,0]| ) ; ::_thesis: P1 /\ P2 = {p1,p2}
now__::_thesis:_not_|[0,0]|_in_(LSeg_(|[1,0]|,p2))_/\_(LSeg_(|[0,0]|,|[0,1]|))
assume  |[0,0]| in (LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) ; ::_thesis: contradiction
then A143: |[0,0]| in LSeg (p2,|[1,0]|) by XBOOLE_0:def_4;
 p2 `1 <= |[1,0]| `1 by A101, EUCLID:52;
then  p2 `1 = 0 by A101, A143, Lm4, TOPREAL1:3;
hence  contradiction by A101, A142, EUCLID:53; ::_thesis: verum
end;
then A144: {|[0,0]|} <> (LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) by ZFMISC_1:31;
 (LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) c= (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) by A100, Lm24, TOPREAL1:6, XBOOLE_1:26;
then A145: (LSeg (|[1,0]|,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {} by A144, TOPREAL1:17, ZFMISC_1:33;
now__::_thesis:_not_|[1,0]|_in_(LSeg_(p1,|[1,0]|))_/\_(LSeg_(|[0,0]|,p2))
assume  |[1,0]| in (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2)) ; ::_thesis: contradiction
then A146: |[1,0]| in LSeg (|[0,0]|,p2) by XBOOLE_0:def_4;
 |[0,0]| `1 <= p2 `1 by A101, EUCLID:52;
then  |[1,0]| `1 <= p2 `1 by A146, TOPREAL1:3;
then  p2 `1 = 1 by A101, Lm8, XXREAL_0:1;
hence  contradiction by A101, A142, EUCLID:53; ::_thesis: verum
end;
then A147: {|[1,0]|} <> (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2)) by ZFMISC_1:31;
 (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by A11, A117, XBOOLE_1:27;
then  (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,p2)) = {} by A147, TOPREAL1:16, ZFMISC_1:33;
hence  P1 /\ P2 = {p1,p2} by A125, A145, ENUMSET1:1; ::_thesis: verum
end;
end;
end;
hence  P1 /\ P2 = {p1,p2} ; ::_thesis: verum
end;
supposeA148: p2 in LSeg (|[1,0]|,|[1,1]|) ; ::_thesis: ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A149: p = |[(p `1),(p `2)]| by EUCLID:53;
A150: LSeg (p1,p2) c= LSeg (|[1,0]|,|[1,1]|) by A3, A148, TOPREAL1:6;
consider q being Point of (TOP-REAL 2) such that
A151: q = p2 and
A152: q `1 = 1 and
A153: q `2 <= 1 and
A154: q `2 >= 0 by A148, TOPREAL1:13;
A155: q = |[(q `1),(q `2)]| by EUCLID:53;
now__::_thesis:_ex_P1_being_Element_of_K19(_the_carrier_of_(TOP-REAL_2))_ex_P2_being_Element_of_K19(_the_carrier_of_(TOP-REAL_2))_st_
(_P1_is_an_arc_of_p1,p2_&_P2_is_an_arc_of_p1,p2_&_P1_\/_P2_=_R^2-unit_square_&_P1_/\_P2_=_{p1,p2}_)
percases ( p `2 < q `2 or q `2 < p `2 ) by A1, A13, A14, A151, A152, A149, A155, XXREAL_0:1;
supposeA156: p `2 < q `2 ; ::_thesis: ex P1 being Element of K19( the carrier of (TOP-REAL 2)) ex P2 being Element of K19( the carrier of (TOP-REAL 2)) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
A157: (LSeg (p1,p2)) /\ (LSeg (|[1,1]|,p2)) c= {p2}
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in (LSeg (p1,p2)) /\ (LSeg (|[1,1]|,p2)) or a in {p2} )
assume A158: a in (LSeg (p1,p2)) /\ (LSeg (|[1,1]|,p2)) ; ::_thesis: a in {p2}
then reconsider p = a as Point of (TOP-REAL 2) ;
A159: p in LSeg (p2,|[1,1]|) by A158, XBOOLE_0:def_4;
 p2 `2 <= |[1,1]| `2 by A151, A153, EUCLID:52;
then A160: p2 `2 <= p `2 by A159, TOPREAL1:4;
A161: p in LSeg (p1,p2) by A158, XBOOLE_0:def_4;
then A162: p1 `1 <= p `1 by A13, A14, A151, A152, TOPREAL1:3;
 p `2 <= p2 `2 by A13, A151, A156, A161, TOPREAL1:4;
then A163: p2 `2 = p `2 by A160, XXREAL_0:1;
 p `1 <= p2 `1 by A13, A14, A151, A152, A161, TOPREAL1:3;
then  p `1 = 1 by A13, A14, A151, A152, A162, XXREAL_0:1;
then p = |[1,(p2 `2)]| by A163, EUCLID:53
.= p2 by A151, A152, EUCLID:53 ;
hence  a in {p2} by TARSKI:def_1; ::_thesis: verum
end;
 |[1,0]| in LSeg (p1,|[1,0]|) by RLTOPSP1:68;
then A164: (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) <> {} by Lm24, XBOOLE_0:def_4;
A165: now__::_thesis:_not_(LSeg_(p1,|[1,0]|))_/\_(LSeg_(|[1,1]|,p2))_<>_{}
set a = the Element of (LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2));
assume A166: (LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)) <> {} ; ::_thesis: contradiction
then reconsider p = the Element of (LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)) as Point of (TOP-REAL 2) by TARSKI:def_3;
A167: p in LSeg (|[1,0]|,p1) by A166, XBOOLE_0:def_4;
A168: p in LSeg (p2,|[1,1]|) by A166, XBOOLE_0:def_4;
 p2 `2 <= |[1,1]| `2 by A151, A153, EUCLID:52;
then A169: p2 `2 <= p `2 by A168, TOPREAL1:4;
 |[1,0]| `2 <= p1 `2 by A13, A16, EUCLID:52;
then  p `2 <= p1 `2 by A167, TOPREAL1:4;
hence  contradiction by A13, A151, A156, A169, XXREAL_0:2; ::_thesis: verum
end;
A170: ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) /\ (LSeg (|[0,1]|,|[1,1]|)) = ((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ ((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|))) by XBOOLE_1:23
.= {|[0,1]|} by Lm2, TOPREAL1:15 ;
 (LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|)) is_an_arc_of |[1,0]|,|[0,1]| by Lm4, Lm8, TOPREAL1:9, TOPREAL1:10, TOPREAL1:17;
then A171: ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,|[1,1]|)) is_an_arc_of |[1,0]|,|[1,1]| by A170, TOPREAL1:10;
now__::_thesis:_not_|[1,1]|_in_(LSeg_(p1,|[1,0]|))_/\_(LSeg_(|[0,1]|,|[1,1]|))
assume  |[1,1]| in (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) ; ::_thesis: contradiction
then A172: |[1,1]| in LSeg (|[1,0]|,p1) by XBOOLE_0:def_4;
 |[1,0]| `2 <= p1 `2 by A13, A16, EUCLID:52;
then  |[1,1]| `2 <= p1 `2 by A172, TOPREAL1:4;
hence  contradiction by A13, A15, A153, A156, Lm11, XXREAL_0:1; ::_thesis: verum
end;
then A173: {|[1,1]|} <> (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) by ZFMISC_1:31;
 (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) by A3, Lm25, TOPREAL1:6, XBOOLE_1:26;
then A174: (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by A173, TOPREAL1:18, ZFMISC_1:33;
 (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by TOPREAL1:20, XBOOLE_0:def_7;
then A175: (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {} by A11, XBOOLE_1:3, XBOOLE_1:26;
now__::_thesis:_not_|[1,0]|_in_(LSeg_(|[0,0]|,|[1,0]|))_/\_(LSeg_(|[1,1]|,p2))
assume  |[1,0]| in (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)) ; ::_thesis: contradiction
then A176: |[1,0]| in LSeg (p2,|[1,1]|) by XBOOLE_0:def_4;
 p2 `2 <= |[1,1]| `2 by A151, A153, EUCLID:52;
hence  contradiction by A16, A151, A156, A176, Lm9, TOPREAL1:4; ::_thesis: verum
end;
then A177: {|[1,0]|} <> (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)) by ZFMISC_1:31;
 (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)) c= {|[1,0]|} by A148, Lm27, TOPREAL1:6, TOPREAL1:16, XBOOLE_1:26;
then A178: (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,1]|,p2)) = {} by A177, ZFMISC_1:33;
 (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by TOPREAL1:20, XBOOLE_0:def_7;
then A179: (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {} by A150, XBOOLE_1:3, XBOOLE_1:26;
A180: (LSeg (p1,p2)) /\ (LSeg (p1,|[1,0]|)) c= {p1}
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in (LSeg (p1,p2)) /\ (LSeg (p1,|[1,0]|)) or a in {p1} )
assume A181: a in (LSeg (p1,p2)) /\ (LSeg (p1,|[1,0]|)) ; ::_thesis: a in {p1}
then reconsider p = a as Point of (TOP-REAL 2) ;
A182: p in LSeg (|[1,0]|,p1) by A181, XBOOLE_0:def_4;
 |[1,0]| `2 <= p1 `2 by A13, A16, EUCLID:52;
then A183: p `2 <= p1 `2 by A182, TOPREAL1:4;
A184: p in LSeg (p1,p2) by A181, XBOOLE_0:def_4;
then A185: p1 `1 <= p `1 by A13, A14, A151, A152, TOPREAL1:3;
 p1 `2 <= p `2 by A13, A151, A156, A184, TOPREAL1:4;
then A186: p1 `2 = p `2 by A183, XXREAL_0:1;
 p `1 <= p2 `1 by A13, A14, A151, A152, A184, TOPREAL1:3;
then  p `1 = 1 by A13, A14, A151, A152, A185, XXREAL_0:1;
then p = |[1,(p1 `2)]| by A186, EUCLID:53
.= p1 by A13, A14, EUCLID:53 ;
hence  a in {p1} by TARSKI:def_1; ::_thesis: verum
end;
A187: (LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by A3, Lm25, TOPREAL1:6, XBOOLE_1:26;
 |[1,1]| in LSeg (|[1,1]|,p2) by RLTOPSP1:68;
then A188: (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[1,1]|,p2)) <> {} by Lm26, XBOOLE_0:def_4;
 (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[1,1]|,p2)) c= {|[1,1]|} by A148, Lm27, TOPREAL1:6, TOPREAL1:18, XBOOLE_1:26;
then A189: (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[1,1]|,p2)) = {|[1,1]|} by A188, ZFMISC_1:33;
take P1 = LSeg (p1,p2); ::_thesis: ex P2 being Element of K19( the carrier of (TOP-REAL 2)) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
take P2 = (LSeg (p1,|[1,0]|)) \/ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))); ::_thesis: ( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
A190: p1 in LSeg (p1,|[1,0]|) by RLTOPSP1:68;
thus  P1 is_an_arc_of p1,p2 by A1, TOPREAL1:9; ::_thesis: ( P2 is_an_arc_of p1,p2 & P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
A191: (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by TOPREAL1:20, XBOOLE_0:def_7;
 (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)) c= (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) by A148, Lm27, TOPREAL1:6, XBOOLE_1:26;
then A192: (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)) = {} by A191, XBOOLE_1:3;
(((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,|[1,1]|))) /\ (LSeg (|[1,1]|,p2)) = (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) /\ (LSeg (|[1,1]|,p2))) \/ ((LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[1,1]|,p2))) by XBOOLE_1:23
.= (((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,1]|,p2))) \/ ((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,1]|,p2)))) \/ {|[1,1]|} by A189, XBOOLE_1:23
.= {|[1,1]|} by A178, A192 ;
then A193: (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2)) is_an_arc_of |[1,0]|,p2 by A171, TOPREAL1:10;
(LSeg (p1,|[1,0]|)) /\ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))) = ((LSeg (p1,|[1,0]|)) /\ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,|[1,1]|)))) \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[1,1]|,p2))) by XBOOLE_1:23
.= ((LSeg (p1,|[1,0]|)) /\ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|)))) \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|))) by A165, XBOOLE_1:23
.= ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (p1,|[1,0]|)) /\ (LSeg (|[0,0]|,|[0,1]|))) by A174, XBOOLE_1:23
.= {|[1,0]|} by A187, A164, A175, TOPREAL1:16, ZFMISC_1:33 ;
hence  P2 is_an_arc_of p1,p2 by A193, TOPREAL1:11; ::_thesis: ( P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
thus P1 \/ P2 = ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2))) \/ ((LSeg (p1,|[1,0]|)) \/ (LSeg (p1,p2))) by XBOOLE_1:4
.= (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (((LSeg (|[1,0]|,p1)) \/ (LSeg (p1,p2))) \/ (LSeg (p2,|[1,1]|))) by XBOOLE_1:4
.= (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,0]|,|[1,1]|)) by A3, A148, TOPREAL1:7
.= ((LSeg (|[0,0]|,|[1,0]|)) \/ ((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|)))) \/ (LSeg (|[1,0]|,|[1,1]|)) by XBOOLE_1:4
.= R^2-unit_square by TOPREAL1:def_2, XBOOLE_1:4 ; ::_thesis: P1 /\ P2 = {p1,p2}
A194: p2 in LSeg (|[1,1]|,p2) by RLTOPSP1:68;
 p2 in LSeg (p1,p2) by RLTOPSP1:68;
then  p2 in (LSeg (p1,p2)) /\ (LSeg (|[1,1]|,p2)) by A194, XBOOLE_0:def_4;
then  {p2} c= (LSeg (p1,p2)) /\ (LSeg (|[1,1]|,p2)) by ZFMISC_1:31;
then A195: (LSeg (p1,p2)) /\ (LSeg (|[1,1]|,p2)) = {p2} by A157, XBOOLE_0:def_10;
 p1 in LSeg (p1,p2) by RLTOPSP1:68;
then  p1 in (LSeg (p1,p2)) /\ (LSeg (p1,|[1,0]|)) by A190, XBOOLE_0:def_4;
then  {p1} c= (LSeg (p1,p2)) /\ (LSeg (p1,|[1,0]|)) by ZFMISC_1:31;
then  (LSeg (p1,p2)) /\ (LSeg (p1,|[1,0]|)) = {p1} by A180, XBOOLE_0:def_10;
then A196: P1 /\ P2 = {p1} \/ ((LSeg (p1,p2)) /\ ((((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[1,1]|,p2)))) by XBOOLE_1:23
.= {p1} \/ (((LSeg (p1,p2)) /\ (((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,1]|,|[1,1]|)))) \/ {p2}) by A195, XBOOLE_1:23
.= {p1} \/ ((((LSeg (p1,p2)) /\ ((LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[0,0]|,|[0,1]|)))) \/ ((LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ {p2}) by XBOOLE_1:23
.= {p1} \/ (((((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)))) \/ ((LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ {p2}) by XBOOLE_1:23
.= {p1} \/ (((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ (((LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2})) by A179, XBOOLE_1:4
.= ({p1} \/ ((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)))) \/ (((LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2}) by XBOOLE_1:4 ;
A197: (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by A3, A148, TOPREAL1:6, XBOOLE_1:26;
A198: now__::_thesis:_P1_/\_P2_=_{p1}_\/_(((LSeg_(p1,p2))_/\_(LSeg_(|[0,1]|,|[1,1]|)))_\/_{p2})
percases ( p1 = |[1,0]| or p1 <> |[1,0]| ) ;
supposeA199: p1 = |[1,0]| ; ::_thesis: P1 /\ P2 = {p1} \/ (((LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2})
then  |[1,0]| in LSeg (p1,p2) by RLTOPSP1:68;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) <> {} by Lm24, XBOOLE_0:def_4;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {p1} by A197, A199, TOPREAL1:16, ZFMISC_1:33;
hence  P1 /\ P2 = {p1} \/ (((LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2}) by A196; ::_thesis: verum
end;
supposeA200: p1 <> |[1,0]| ; ::_thesis: P1 /\ P2 = {p1} \/ (((LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2})
now__::_thesis:_not_|[1,0]|_in_(LSeg_(p1,p2))_/\_(LSeg_(|[0,0]|,|[1,0]|))
assume  |[1,0]| in (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) ; ::_thesis: contradiction
then  |[1,0]| in LSeg (p1,p2) by XBOOLE_0:def_4;
then  p1 `2 = 0 by A13, A16, A151, A156, Lm9, TOPREAL1:4;
hence  contradiction by A13, A14, A200, EUCLID:53; ::_thesis: verum
end;
then  {|[1,0]|} <> (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) by ZFMISC_1:31;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {} by A197, TOPREAL1:16, ZFMISC_1:33;
hence  P1 /\ P2 = {p1} \/ (((LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ {p2}) by A196; ::_thesis: verum
end;
end;
end;
A201: (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) by A3, A148, TOPREAL1:6, XBOOLE_1:26;
now__::_thesis:_P1_/\_P2_=_{p1,p2}
percases ( p2 = |[1,1]| or p2 <> |[1,1]| ) ;
supposeA202: p2 = |[1,1]| ; ::_thesis: P1 /\ P2 = {p1,p2}
then  |[1,1]| in LSeg (p1,p2) by RLTOPSP1:68;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) <> {} by Lm26, XBOOLE_0:def_4;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {p2} by A201, A202, TOPREAL1:18, ZFMISC_1:33;
hence  P1 /\ P2 = {p1,p2} by A198, ENUMSET1:1; ::_thesis: verum
end;
supposeA203: p2 <> |[1,1]| ; ::_thesis: P1 /\ P2 = {p1,p2}
now__::_thesis:_not_|[1,1]|_in_(LSeg_(p1,p2))_/\_(LSeg_(|[0,1]|,|[1,1]|))
assume  |[1,1]| in (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) ; ::_thesis: contradiction
then  |[1,1]| in LSeg (p1,p2) by XBOOLE_0:def_4;
then  |[1,1]| `2 <= p2 `2 by A13, A151, A156, TOPREAL1:4;
then  p2 `2 = 1 by A151, A153, Lm11, XXREAL_0:1;
hence  contradiction by A151, A152, A203, EUCLID:53; ::_thesis: verum
end;
then  {|[1,1]|} <> (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) by ZFMISC_1:31;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by A201, TOPREAL1:18, ZFMISC_1:33;
hence  P1 /\ P2 = {p1,p2} by A198, ENUMSET1:1; ::_thesis: verum
end;
end;
end;
hence  P1 /\ P2 = {p1,p2} ; ::_thesis: verum
end;
supposeA204: q `2 < p `2 ; ::_thesis: ex P1 being Element of K19( the carrier of (TOP-REAL 2)) ex P2 being Element of K19( the carrier of (TOP-REAL 2)) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
A205: (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,p2)) c= {p2}
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,p2)) or a in {p2} )
assume A206: a in (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,p2)) ; ::_thesis: a in {p2}
then reconsider p = a as Point of (TOP-REAL 2) ;
A207: p in LSeg (|[1,0]|,p2) by A206, XBOOLE_0:def_4;
 |[1,0]| `2 <= p2 `2 by A151, A154, EUCLID:52;
then A208: p `2 <= p2 `2 by A207, TOPREAL1:4;
A209: p in LSeg (p2,p1) by A206, XBOOLE_0:def_4;
then A210: p2 `1 <= p `1 by A13, A14, A151, A152, TOPREAL1:3;
 p2 `2 <= p `2 by A13, A151, A204, A209, TOPREAL1:4;
then A211: p2 `2 = p `2 by A208, XXREAL_0:1;
 p `1 <= p1 `1 by A13, A14, A151, A152, A209, TOPREAL1:3;
then  p `1 = 1 by A13, A14, A151, A152, A210, XXREAL_0:1;
then p = |[1,(p2 `2)]| by A211, EUCLID:53
.= p2 by A151, A152, EUCLID:53 ;
hence  a in {p2} by TARSKI:def_1; ::_thesis: verum
end;
 |[1,1]| in LSeg (p1,|[1,1]|) by RLTOPSP1:68;
then A212: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) <> {} by Lm26, XBOOLE_0:def_4;
A213: now__::_thesis:_not_(LSeg_(p1,|[1,1]|))_/\_(LSeg_(|[1,0]|,p2))_<>_{}
set a = the Element of (LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,0]|,p2));
assume A214: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,0]|,p2)) <> {} ; ::_thesis: contradiction
then reconsider p = the Element of (LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,0]|,p2)) as Point of (TOP-REAL 2) by TARSKI:def_3;
A215: p in LSeg (p1,|[1,1]|) by A214, XBOOLE_0:def_4;
A216: p in LSeg (|[1,0]|,p2) by A214, XBOOLE_0:def_4;
 |[1,0]| `2 <= p2 `2 by A151, A154, EUCLID:52;
then A217: p `2 <= p2 `2 by A216, TOPREAL1:4;
 p1 `2 <= |[1,1]| `2 by A13, A15, EUCLID:52;
then  p1 `2 <= p `2 by A215, TOPREAL1:4;
hence  contradiction by A13, A151, A204, A217, XXREAL_0:2; ::_thesis: verum
end;
A218: ((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) /\ (LSeg (|[0,0]|,|[1,0]|)) = ((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ ((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|))) by XBOOLE_1:23
.= {|[0,0]|} by Lm2, TOPREAL1:17 ;
 (LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|)) is_an_arc_of |[1,1]|,|[0,0]| by Lm6, Lm10, TOPREAL1:9, TOPREAL1:10, TOPREAL1:15;
then A219: ((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,|[1,0]|)) is_an_arc_of |[1,1]|,|[1,0]| by A218, TOPREAL1:10;
now__::_thesis:_not_|[1,1]|_in_(LSeg_(|[0,1]|,|[1,1]|))_/\_(LSeg_(|[1,0]|,p2))
assume  |[1,1]| in (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[1,0]|,p2)) ; ::_thesis: contradiction
then A220: |[1,1]| in LSeg (|[1,0]|,p2) by XBOOLE_0:def_4;
 |[1,0]| `2 <= p2 `2 by A151, A154, EUCLID:52;
then  |[1,1]| `2 <= p2 `2 by A220, TOPREAL1:4;
hence  contradiction by A15, A151, A153, A204, Lm11, XXREAL_0:1; ::_thesis: verum
end;
then A221: {|[1,1]|} <> (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[1,0]|,p2)) by ZFMISC_1:31;
 (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[1,0]|,p2)) c= {|[1,1]|} by A148, Lm25, TOPREAL1:6, TOPREAL1:18, XBOOLE_1:26;
then A222: (LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[1,0]|,p2)) = {} by A221, ZFMISC_1:33;
 (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by TOPREAL1:20, XBOOLE_0:def_7;
then A223: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {} by A5, XBOOLE_1:3, XBOOLE_1:26;
now__::_thesis:_not_|[1,0]|_in_(LSeg_(p1,|[1,1]|))_/\_(LSeg_(|[0,0]|,|[1,0]|))
assume  |[1,0]| in (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) ; ::_thesis: contradiction
then A224: |[1,0]| in LSeg (p1,|[1,1]|) by XBOOLE_0:def_4;
 p1 `2 <= |[1,1]| `2 by A13, A15, EUCLID:52;
hence  contradiction by A13, A154, A204, A224, Lm9, TOPREAL1:4; ::_thesis: verum
end;
then A225: {|[1,0]|} <> (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by ZFMISC_1:31;
 (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by A3, Lm27, TOPREAL1:6, XBOOLE_1:26;
then A226: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {} by A225, TOPREAL1:16, ZFMISC_1:33;
 (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by TOPREAL1:20, XBOOLE_0:def_7;
then A227: (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)) = {} by A150, XBOOLE_1:3, XBOOLE_1:26;
A228: (LSeg (p1,p2)) /\ (LSeg (p1,|[1,1]|)) c= {p1}
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in (LSeg (p1,p2)) /\ (LSeg (p1,|[1,1]|)) or a in {p1} )
assume A229: a in (LSeg (p1,p2)) /\ (LSeg (p1,|[1,1]|)) ; ::_thesis: a in {p1}
then reconsider p = a as Point of (TOP-REAL 2) ;
A230: p in LSeg (p1,|[1,1]|) by A229, XBOOLE_0:def_4;
 p1 `2 <= |[1,1]| `2 by A13, A15, EUCLID:52;
then A231: p1 `2 <= p `2 by A230, TOPREAL1:4;
A232: p in LSeg (p2,p1) by A229, XBOOLE_0:def_4;
then A233: p2 `1 <= p `1 by A13, A14, A151, A152, TOPREAL1:3;
 p `2 <= p1 `2 by A13, A151, A204, A232, TOPREAL1:4;
then A234: p1 `2 = p `2 by A231, XXREAL_0:1;
 p `1 <= p1 `1 by A13, A14, A151, A152, A232, TOPREAL1:3;
then  p `1 = 1 by A13, A14, A151, A152, A233, XXREAL_0:1;
then p = |[1,(p1 `2)]| by A234, EUCLID:53
.= p1 by A13, A14, EUCLID:53 ;
hence  a in {p1} by TARSKI:def_1; ::_thesis: verum
end;
A235: (LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) by A3, Lm27, TOPREAL1:6, XBOOLE_1:26;
 |[1,0]| in LSeg (|[1,0]|,p2) by RLTOPSP1:68;
then A236: (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,0]|,p2)) <> {} by Lm24, XBOOLE_0:def_4;
 (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,0]|,p2)) c= {|[1,0]|} by A148, Lm25, TOPREAL1:6, TOPREAL1:16, XBOOLE_1:26;
then A237: (LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,0]|,p2)) = {|[1,0]|} by A236, ZFMISC_1:33;
take P1 = LSeg (p1,p2); ::_thesis: ex P2 being Element of K19( the carrier of (TOP-REAL 2)) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
take P2 = (LSeg (p1,|[1,1]|)) \/ ((((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,p2))); ::_thesis: ( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
A238: p1 in LSeg (p1,|[1,1]|) by RLTOPSP1:68;
thus  P1 is_an_arc_of p1,p2 by A1, TOPREAL1:9; ::_thesis: ( P2 is_an_arc_of p1,p2 & P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
A239: (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) = {} by TOPREAL1:20, XBOOLE_0:def_7;
 (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,p2)) c= (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,|[1,1]|)) by A148, Lm25, TOPREAL1:6, XBOOLE_1:26;
then A240: (LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,p2)) = {} by A239, XBOOLE_1:3;
(((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,|[1,0]|))) /\ (LSeg (|[1,0]|,p2)) = (((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) /\ (LSeg (|[1,0]|,p2))) \/ ((LSeg (|[0,0]|,|[1,0]|)) /\ (LSeg (|[1,0]|,p2))) by XBOOLE_1:23
.= (((LSeg (|[0,1]|,|[1,1]|)) /\ (LSeg (|[1,0]|,p2))) \/ ((LSeg (|[0,0]|,|[0,1]|)) /\ (LSeg (|[1,0]|,p2)))) \/ {|[1,0]|} by A237, XBOOLE_1:23
.= {|[1,0]|} by A222, A240 ;
then A241: (((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,p2)) is_an_arc_of |[1,1]|,p2 by A219, TOPREAL1:10;
(LSeg (p1,|[1,1]|)) /\ ((((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,p2))) = ((LSeg (p1,|[1,1]|)) /\ (((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,|[1,0]|)))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[1,0]|,p2))) by XBOOLE_1:23
.= ((LSeg (p1,|[1,1]|)) /\ ((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|)))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|))) by A213, XBOOLE_1:23
.= ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ ((LSeg (p1,|[1,1]|)) /\ (LSeg (|[0,0]|,|[0,1]|))) by A226, XBOOLE_1:23
.= {|[1,1]|} by A235, A212, A223, TOPREAL1:18, ZFMISC_1:33 ;
hence  P2 is_an_arc_of p1,p2 by A241, TOPREAL1:11; ::_thesis: ( P1 \/ P2 = R^2-unit_square & P1 /\ P2 = {p1,p2} )
thus P1 \/ P2 = ((((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,p2))) \/ ((LSeg (p1,|[1,1]|)) \/ (LSeg (p1,p2))) by XBOOLE_1:4
.= (((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ ((LSeg (|[1,0]|,p2)) \/ ((LSeg (p1,p2)) \/ (LSeg (p1,|[1,1]|)))) by XBOOLE_1:4
.= (((LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|))) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,|[1,1]|)) by A3, A148, TOPREAL1:7
.= R^2-unit_square by TOPREAL1:def_2, XBOOLE_1:4 ; ::_thesis: P1 /\ P2 = {p1,p2}
A242: p2 in LSeg (|[1,0]|,p2) by RLTOPSP1:68;
 p2 in LSeg (p1,p2) by RLTOPSP1:68;
then  p2 in (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,p2)) by A242, XBOOLE_0:def_4;
then  {p2} c= (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,p2)) by ZFMISC_1:31;
then A243: (LSeg (p1,p2)) /\ (LSeg (|[1,0]|,p2)) = {p2} by A205, XBOOLE_0:def_10;
 p1 in LSeg (p1,p2) by RLTOPSP1:68;
then  p1 in (LSeg (p1,p2)) /\ (LSeg (p1,|[1,1]|)) by A238, XBOOLE_0:def_4;
then  {p1} c= (LSeg (p1,p2)) /\ (LSeg (p1,|[1,1]|)) by ZFMISC_1:31;
then  (LSeg (p1,p2)) /\ (LSeg (p1,|[1,1]|)) = {p1} by A228, XBOOLE_0:def_10;
then A244: P1 /\ P2 = {p1} \/ ((LSeg (p1,p2)) /\ ((((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,|[1,0]|))) \/ (LSeg (|[1,0]|,p2)))) by XBOOLE_1:23
.= {p1} \/ (((LSeg (p1,p2)) /\ (((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|))) \/ (LSeg (|[0,0]|,|[1,0]|)))) \/ {p2}) by A243, XBOOLE_1:23
.= {p1} \/ ((((LSeg (p1,p2)) /\ ((LSeg (|[0,1]|,|[1,1]|)) \/ (LSeg (|[0,0]|,|[0,1]|)))) \/ ((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)))) \/ {p2}) by XBOOLE_1:23
.= {p1} \/ (((((LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ ((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[0,1]|)))) \/ ((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)))) \/ {p2}) by XBOOLE_1:23
.= {p1} \/ (((LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|))) \/ (((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {p2})) by A227, XBOOLE_1:4
.= ({p1} \/ ((LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)))) \/ (((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {p2}) by XBOOLE_1:4 ;
A245: (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,1]|,|[1,1]|)) by A3, A148, TOPREAL1:6, XBOOLE_1:26;
A246: now__::_thesis:_P1_/\_P2_=_{p1}_\/_(((LSeg_(p1,p2))_/\_(LSeg_(|[0,0]|,|[1,0]|)))_\/_{p2})
percases ( p1 = |[1,1]| or p1 <> |[1,1]| ) ;
supposeA247: p1 = |[1,1]| ; ::_thesis: P1 /\ P2 = {p1} \/ (((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {p2})
then  |[1,1]| in LSeg (p1,p2) by RLTOPSP1:68;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) <> {} by Lm26, XBOOLE_0:def_4;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {p1} by A245, A247, TOPREAL1:18, ZFMISC_1:33;
hence  P1 /\ P2 = {p1} \/ (((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {p2}) by A244; ::_thesis: verum
end;
supposeA248: p1 <> |[1,1]| ; ::_thesis: P1 /\ P2 = {p1} \/ (((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {p2})
now__::_thesis:_not_|[1,1]|_in_(LSeg_(p1,p2))_/\_(LSeg_(|[0,1]|,|[1,1]|))
assume  |[1,1]| in (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) ; ::_thesis: contradiction
then  |[1,1]| in LSeg (p2,p1) by XBOOLE_0:def_4;
then  |[1,1]| `2 <= p1 `2 by A13, A151, A204, TOPREAL1:4;
then  p1 `2 = 1 by A13, A15, Lm11, XXREAL_0:1;
hence  contradiction by A13, A14, A248, EUCLID:53; ::_thesis: verum
end;
then  {|[1,1]|} <> (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) by ZFMISC_1:31;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,1]|,|[1,1]|)) = {} by A245, TOPREAL1:18, ZFMISC_1:33;
hence  P1 /\ P2 = {p1} \/ (((LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|))) \/ {p2}) by A244; ::_thesis: verum
end;
end;
end;
A249: (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) c= (LSeg (|[1,0]|,|[1,1]|)) /\ (LSeg (|[0,0]|,|[1,0]|)) by A3, A148, TOPREAL1:6, XBOOLE_1:26;
now__::_thesis:_P1_/\_P2_=_{p1,p2}
percases ( p2 = |[1,0]| or p2 <> |[1,0]| ) ;
supposeA250: p2 = |[1,0]| ; ::_thesis: P1 /\ P2 = {p1,p2}
then  |[1,0]| in LSeg (p1,p2) by RLTOPSP1:68;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) <> {} by Lm24, XBOOLE_0:def_4;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {p2} by A249, A250, TOPREAL1:16, ZFMISC_1:33;
hence  P1 /\ P2 = {p1,p2} by A246, ENUMSET1:1; ::_thesis: verum
end;
supposeA251: p2 <> |[1,0]| ; ::_thesis: P1 /\ P2 = {p1,p2}
now__::_thesis:_not_|[1,0]|_in_(LSeg_(p1,p2))_/\_(LSeg_(|[0,0]|,|[1,0]|))
assume  |[1,0]| in (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) ; ::_thesis: contradiction
then  |[1,0]| in LSeg (p2,p1) by XBOOLE_0:def_4;
then  p2 `2 = 0 by A13, A151, A154, A204, Lm9, TOPREAL1:4;
hence  contradiction by A151, A152, A251, EUCLID:53; ::_thesis: verum
end;
then  {|[1,0]|} <> (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) by ZFMISC_1:31;
then  (LSeg (p1,p2)) /\ (LSeg (|[0,0]|,|[1,0]|)) = {} by A249, TOPREAL1:16, ZFMISC_1:33;
hence  P1 /\ P2 = {p1,p2} by A246, ENUMSET1:1; ::_thesis: verum
end;
end;
end;
hence  P1 /\ P2 = {p1,p2} ; ::_thesis: verum
end;
end;
end;
hence  ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) ; ::_thesis: verum
end;
end;
end;

theorem Th1: :: TOPREAL2:1
for p1, p2 being Point of (TOP-REAL 2) st p1 <> p2 & p1 in R^2-unit_square & p2 in R^2-unit_square holds
ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 <> p2 & p1 in R^2-unit_square & p2 in R^2-unit_square implies ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) )
assume that
A1: p1 <> p2 and
A2: p1 in R^2-unit_square and
A3: p2 in R^2-unit_square ; ::_thesis: ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A4: ( p1 in (LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|)) or p1 in (LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)) ) by A2, TOPREAL1:def_2, XBOOLE_0:def_3;
percases ( p1 in LSeg (|[0,0]|,|[0,1]|) or p1 in LSeg (|[0,1]|,|[1,1]|) or p1 in LSeg (|[0,0]|,|[1,0]|) or p1 in LSeg (|[1,0]|,|[1,1]|) ) by A4, XBOOLE_0:def_3;
suppose p1 in LSeg (|[0,0]|,|[0,1]|) ; ::_thesis: ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
hence  ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) by A1, A3, Lm30; ::_thesis: verum
end;
suppose p1 in LSeg (|[0,1]|,|[1,1]|) ; ::_thesis: ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
hence  ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) by A1, A3, Lm31; ::_thesis: verum
end;
suppose p1 in LSeg (|[0,0]|,|[1,0]|) ; ::_thesis: ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
hence  ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) by A1, A3, Lm32; ::_thesis: verum
end;
suppose p1 in LSeg (|[1,0]|,|[1,1]|) ; ::_thesis: ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
hence  ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) by A1, A3, Lm33; ::_thesis: verum
end;
end;
end;

theorem Th2: :: TOPREAL2:2
R^2-unit_square is compact
proof
A1: I[01] is compact by HEINE:4, TOPMETR:20;
consider P1, P2 being non empty Subset of (TOP-REAL 2) such that
A2: P1 is being_S-P_arc and
A3: P2 is being_S-P_arc and
A4: R^2-unit_square = P1 \/ P2 by TOPREAL1:27;
consider f being Function of I[01],((TOP-REAL 2) | P1) such that
A5: f is being_homeomorphism by A2, TOPREAL1:29;
A6: rng f = [#] ((TOP-REAL 2) | P1) by A5, TOPS_2:def_5;
consider f0 being Function of I[01],((TOP-REAL 2) | P2) such that
A7: f0 is being_homeomorphism by A3, TOPREAL1:29;
A8: rng f0 = [#] ((TOP-REAL 2) | P2) by A7, TOPS_2:def_5;
reconsider P2 = P2 as non empty Subset of (TOP-REAL 2) ;
 f0 is continuous by A7, TOPS_2:def_5;
then  (TOP-REAL 2) | P2 is compact by A1, A8, COMPTS_1:14;
then A9: P2 is compact by COMPTS_1:3;
reconsider P1 = P1 as non empty Subset of (TOP-REAL 2) ;
 f is continuous by A5, TOPS_2:def_5;
then  (TOP-REAL 2) | P1 is compact by A1, A6, COMPTS_1:14;
then  P1 is compact by COMPTS_1:3;
hence  R^2-unit_square is compact by A4, A9, COMPTS_1:10; ::_thesis: verum
end;

theorem Th3: :: TOPREAL2:3
for q1, q2 being Point of (TOP-REAL 2)
for Q, P being non empty Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | Q),((TOP-REAL 2) | P) st f is being_homeomorphism & Q is_an_arc_of q1,q2 holds
for p1, p2 being Point of (TOP-REAL 2) st p1 = f . q1 & p2 = f . q2 holds
P is_an_arc_of p1,p2
proof
let q1, q2 be Point of (TOP-REAL 2); ::_thesis: for Q, P being non empty Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | Q),((TOP-REAL 2) | P) st f is being_homeomorphism & Q is_an_arc_of q1,q2 holds
for p1, p2 being Point of (TOP-REAL 2) st p1 = f . q1 & p2 = f . q2 holds
P is_an_arc_of p1,p2
let Q, P be non empty Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | Q),((TOP-REAL 2) | P) st f is being_homeomorphism & Q is_an_arc_of q1,q2 holds
for p1, p2 being Point of (TOP-REAL 2) st p1 = f . q1 & p2 = f . q2 holds
P is_an_arc_of p1,p2
let f be Function of ((TOP-REAL 2) | Q),((TOP-REAL 2) | P); ::_thesis: ( f is being_homeomorphism & Q is_an_arc_of q1,q2 implies for p1, p2 being Point of (TOP-REAL 2) st p1 = f . q1 & p2 = f . q2 holds
P is_an_arc_of p1,p2 )
assume that
A1: f is being_homeomorphism and
A2: Q is_an_arc_of q1,q2 ; ::_thesis: for p1, p2 being Point of (TOP-REAL 2) st p1 = f . q1 & p2 = f . q2 holds
P is_an_arc_of p1,p2
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 = f . q1 & p2 = f . q2 implies P is_an_arc_of p1,p2 )
assume that
A3: p1 = f . q1 and
A4: p2 = f . q2 ; ::_thesis: P is_an_arc_of p1,p2
reconsider f = f as Function of ((TOP-REAL 2) | Q),((TOP-REAL 2) | P) ;
consider f1 being Function of I[01],((TOP-REAL 2) | Q) such that
A5: f1 is being_homeomorphism and
A6: f1 . 0 = q1 and
A7: f1 . 1 = q2 by A2, TOPREAL1:def_1;
set g1 = f * f1;
A8: dom f1 = the carrier of I[01] by FUNCT_2:def_1;
then  0 in dom f1 by BORSUK_1:40, XXREAL_1:1;
then A9: (f * f1) . 0 = p1 by A3, A6, FUNCT_1:13;
 1 in dom f1 by A8, BORSUK_1:40, XXREAL_1:1;
then A10: (f * f1) . 1 = p2 by A4, A7, FUNCT_1:13;
 f * f1 is being_homeomorphism by A1, A5, TOPS_2:57;
hence  P is_an_arc_of p1,p2 by A9, A10, TOPREAL1:def_1; ::_thesis: verum
end;

definition
let P be Subset of (TOP-REAL 2);
attrP is being_simple_closed_curve  means :Def1: :: TOPREAL2:def 1
 ex f being Function of ((TOP-REAL 2) | R^2-unit_square),((TOP-REAL 2) | P) st f is being_homeomorphism ;
end;

:: deftheorem Def1 defines being_simple_closed_curve TOPREAL2:def_1_:_
 for P being Subset of (TOP-REAL 2) holds
( P is being_simple_closed_curve iff ex f being Function of ((TOP-REAL 2) | R^2-unit_square),((TOP-REAL 2) | P) st f is being_homeomorphism );

registration
cluster R^2-unit_square  ->  being_simple_closed_curve ;
coherence
 R^2-unit_square is being_simple_closed_curve
proof
set T = (TOP-REAL 2) | R^2-unit_square;
take f = id ((TOP-REAL 2) | R^2-unit_square); :: according to TOPREAL2:def_1 ::_thesis: f is being_homeomorphism
thus  dom f = [#] ((TOP-REAL 2) | R^2-unit_square) by FUNCT_2:def_1; :: according to TOPS_2:def_5 ::_thesis: ( rng f = [#] ((TOP-REAL 2) | R^2-unit_square) & f is one-to-one & f is continuous & f " is continuous )
thus  rng f = [#] ((TOP-REAL 2) | R^2-unit_square) by RELAT_1:45; ::_thesis: ( f is one-to-one & f is continuous & f " is continuous )
then  ( f is onto & f is one-to-one ) by FUNCT_2:def_3;
then A1: f " = f " by TOPS_2:def_4
.= f by FUNCT_1:45 ;
thus  f is one-to-one ; ::_thesis: ( f is continuous & f " is continuous )
thus  f is continuous ::_thesis: f " is continuous
proof
let V be Subset of ((TOP-REAL 2) | R^2-unit_square); :: according to PRE_TOPC:def_6 ::_thesis: ( not V is closed or f " V is closed )
assume  V is closed ; ::_thesis: f " V is closed
hence  f " V is closed by FUNCT_2:94; ::_thesis: verum
end;
hence  f " is continuous by A1; ::_thesis: verum
end;
end;

registration
cluster functional non empty being_simple_closed_curve for Element of K19( the carrier of (TOP-REAL 2));
existence
 ex b1 being Subset of (TOP-REAL 2) st
( b1 is being_simple_closed_curve & not b1 is empty )
proof
take  R^2-unit_square ; ::_thesis: ( R^2-unit_square is being_simple_closed_curve & not R^2-unit_square is empty )
thus  ( R^2-unit_square is being_simple_closed_curve & not R^2-unit_square is empty ) ; ::_thesis: verum
end;
end;

definition
mode Simple_closed_curve is being_simple_closed_curve Subset of (TOP-REAL 2);
end;

theorem Th4: :: TOPREAL2:4
for P being non empty Subset of (TOP-REAL 2) st P is being_simple_closed_curve holds
ex p1, p2 being Point of (TOP-REAL 2) st
( p1 <> p2 & p1 in P & p2 in P )
proof
reconsider RS = R^2-unit_square as non empty Subset of (TOP-REAL 2) ;
let P be non empty Subset of (TOP-REAL 2); ::_thesis: ( P is being_simple_closed_curve implies ex p1, p2 being Point of (TOP-REAL 2) st
( p1 <> p2 & p1 in P & p2 in P ) )
A1: |[0,0]| `1 = 0 by EUCLID:52;
A2: [#] ((TOP-REAL 2) | P) c= [#] (TOP-REAL 2) by PRE_TOPC:def_4;
A3: |[1,1]| `1 = 1 by EUCLID:52;
assume  P is being_simple_closed_curve ; ::_thesis: ex p1, p2 being Point of (TOP-REAL 2) st
( p1 <> p2 & p1 in P & p2 in P )
then consider f being Function of ((TOP-REAL 2) | R^2-unit_square),((TOP-REAL 2) | P) such that
A4: f is being_homeomorphism by Def1;
A5: rng f = [#] ((TOP-REAL 2) | P) by A4, TOPS_2:def_5
.= P by PRE_TOPC:def_5 ;
reconsider f = f as Function of ((TOP-REAL 2) | RS),((TOP-REAL 2) | P) ;
A6: dom f = [#] ((TOP-REAL 2) | RS) by FUNCT_2:def_1
.= R^2-unit_square by PRE_TOPC:def_5 ;
set p1 = f . |[0,0]|;
set p2 = f . |[1,1]|;
 |[0,0]| `2 = 0 by EUCLID:52;
then A7: |[0,0]| in dom f by A1, A6, TOPREAL1:14;
then A8: f . |[0,0]| in rng f by FUNCT_1:def_3;
 |[1,1]| `2 = 1 by EUCLID:52;
then A9: |[1,1]| in dom f by A3, A6, TOPREAL1:14;
then A10: f . |[1,1]| in rng f by FUNCT_1:def_3;
 rng f = [#] ((TOP-REAL 2) | P) by A4, TOPS_2:def_5;
then reconsider p1 = f . |[0,0]|, p2 = f . |[1,1]| as Point of (TOP-REAL 2) by A2, A8, A10;
take  p1 ; ::_thesis: ex p2 being Point of (TOP-REAL 2) st
( p1 <> p2 & p1 in P & p2 in P )
take  p2 ; ::_thesis: ( p1 <> p2 & p1 in P & p2 in P )
 f is one-to-one by A4, TOPS_2:def_5;
hence  p1 <> p2 by A1, A3, A7, A9, FUNCT_1:def_4; ::_thesis: ( p1 in P & p2 in P )
thus  ( p1 in P & p2 in P ) by A5, A7, A9, FUNCT_1:def_3; ::_thesis: verum
end;

Lm34:  for p1, p2 being Point of (TOP-REAL 2)
for P, P1, P2 being non empty Subset of (TOP-REAL 2) st P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} holds
P is being_simple_closed_curve
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: for P, P1, P2 being non empty Subset of (TOP-REAL 2) st P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} holds
P is being_simple_closed_curve
reconsider RS = R^2-unit_square as non empty Subset of (TOP-REAL 2) ;
let P, P1, P2 be non empty Subset of (TOP-REAL 2); ::_thesis: ( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} implies P is being_simple_closed_curve )
assume that
A1: P1 is_an_arc_of p1,p2 and
A2: P2 is_an_arc_of p1,p2 and
A3: P = P1 \/ P2 and
A4: P1 /\ P2 = {p1,p2} ; ::_thesis: P is being_simple_closed_curve
reconsider P9 = P, P19 = P1, P29 = P2 as non empty Subset of (TOP-REAL 2) ;
A5: [#] ((TOP-REAL 2) | P1) = P1 by PRE_TOPC:def_5;
consider h1, h2 being FinSequence of (TOP-REAL 2) such that
A6: h1 is being_S-Seq and
A7: h2 is being_S-Seq and
A8: R^2-unit_square = (L~ h1) \/ (L~ h2) and
A9: (L~ h1) /\ (L~ h2) = {|[0,0]|,|[1,1]|} and
A10: h1 /. 1 = |[0,0]| and
A11: h1 /. (len h1) = |[1,1]| and
A12: h2 /. 1 = |[0,0]| and
A13: h2 /. (len h2) = |[1,1]| by TOPREAL1:24;
A14: len h2 >= 2 by A7, TOPREAL1:def_8;
 len h1 >= 2 by A6, TOPREAL1:def_8;
then reconsider Lh1 = L~ h1, Lh2 = L~ h2 as non empty Subset of (TOP-REAL 2) by A14, TOPREAL1:23;
set T1 = (TOP-REAL 2) | Lh1;
set T2 = (TOP-REAL 2) | Lh2;
set T = (TOP-REAL 2) | RS;
A15: [#] ((TOP-REAL 2) | RS) = R^2-unit_square by PRE_TOPC:def_5;
A16: [#] ((TOP-REAL 2) | Lh2) = L~ h2 by PRE_TOPC:def_5;
then A17: (TOP-REAL 2) | Lh2 is SubSpace of (TOP-REAL 2) | RS by A8, A15, TOPMETR:3, XBOOLE_1:7;
A18: [#] ((TOP-REAL 2) | Lh1) = L~ h1 by PRE_TOPC:def_5;
then A19: (TOP-REAL 2) | Lh1 is SubSpace of (TOP-REAL 2) | RS by A8, A15, TOPMETR:3, XBOOLE_1:7;
A20: [#] ((TOP-REAL 2) | P) = P by PRE_TOPC:def_5;
A21: [#] ((TOP-REAL 2) | P2) = P2 by PRE_TOPC:def_5;
then A22: (TOP-REAL 2) | P29 is SubSpace of (TOP-REAL 2) | P9 by A3, A20, TOPMETR:3, XBOOLE_1:7;
consider f2 being Function of I[01],((TOP-REAL 2) | P2) such that
A23: f2 is being_homeomorphism and
A24: f2 . 0 = p1 and
A25: f2 . 1 = p2 by A2, TOPREAL1:def_1;
A26: dom f2 = the carrier of I[01] by FUNCT_2:def_1;
 P2 c= P by A3, XBOOLE_1:7;
then  rng f2 c= the carrier of ((TOP-REAL 2) | P) by A21, A20, XBOOLE_1:1;
then reconsider ff2 = f2 as Function of I[01],((TOP-REAL 2) | P9) by A26, RELSET_1:4;
A27: dom ff2 = the carrier of I[01] by FUNCT_2:def_1;
then A28: 0 in dom ff2 by BORSUK_1:40, XXREAL_1:1;
 f2 is continuous by A23, TOPS_2:def_5;
then A29: ff2 is continuous by A22, PRE_TOPC:26;
A30: 1 in dom ff2 by A27, BORSUK_1:40, XXREAL_1:1;
A31: [#] ((TOP-REAL 2) | P) = P by PRE_TOPC:def_5;
then A32: (TOP-REAL 2) | P19 is SubSpace of (TOP-REAL 2) | P9 by A3, A5, TOPMETR:3, XBOOLE_1:7;
consider f1 being Function of I[01],((TOP-REAL 2) | P1) such that
A33: f1 is being_homeomorphism and
A34: f1 . 0 = p1 and
A35: f1 . 1 = p2 by A1, TOPREAL1:def_1;
A36: dom f1 = the carrier of I[01] by FUNCT_2:def_1;
 P1 c= P by A3, XBOOLE_1:7;
then  rng f1 c= the carrier of ((TOP-REAL 2) | P) by A5, A31, XBOOLE_1:1;
then reconsider ff1 = f1 as Function of I[01],((TOP-REAL 2) | P9) by A36, RELSET_1:4;
A37: dom f1 = the carrier of I[01] by FUNCT_2:def_1;
A38: I[01] is compact by HEINE:4, TOPMETR:20;
 f1 is continuous by A33, TOPS_2:def_5;
then A39: ff1 is continuous by A32, PRE_TOPC:26;
A40: f1 is one-to-one by A33, TOPS_2:def_5;
reconsider L1 = L~ h1, L2 = L~ h2 as non empty Subset of (TOP-REAL 2) by A9;
 L1 is_an_arc_of |[0,0]|,|[1,1]| by A6, A10, A11, TOPREAL1:25;
then consider g1 being Function of I[01],((TOP-REAL 2) | L1) such that
A41: g1 is being_homeomorphism and
A42: g1 . 0 = |[0,0]| and
A43: g1 . 1 = |[1,1]| by TOPREAL1:def_1;
 L2 is_an_arc_of |[0,0]|,|[1,1]| by A7, A12, A13, TOPREAL1:25;
then consider g2 being Function of I[01],((TOP-REAL 2) | L2) such that
A44: g2 is being_homeomorphism and
A45: g2 . 0 = |[0,0]| and
A46: g2 . 1 = |[1,1]| by TOPREAL1:def_1;
R^2-unit_square = [#] ((TOP-REAL 2) | RS) by PRE_TOPC:def_5
.= the carrier of ((TOP-REAL 2) | RS) ;
then reconsider p00 = |[0,0]|, p11 = |[1,1]| as Point of ((TOP-REAL 2) | RS) by Lm28, Lm29, TOPREAL1:14;
A47: (TOP-REAL 2) | RS is T_2 by TOPMETR:2;
set k1 = ff1 * (g1 ");
set k2 = ff2 * (g2 ");
reconsider g1 = g1 as Function of I[01],((TOP-REAL 2) | Lh1) ;
A48: g1 is one-to-one by A41, TOPS_2:def_5;
A49: dom g1 = the carrier of I[01] by FUNCT_2:def_1;
A50: rng g1 = [#] ((TOP-REAL 2) | Lh1) by A41, TOPS_2:def_5;
then  g1 is onto by FUNCT_2:def_3;
then A51: g1 " = g1 " by A48, TOPS_2:def_4;
then  rng (g1 ") = dom g1 by A48, FUNCT_1:33;
then A52: rng (ff1 * (g1 ")) = rng f1 by A37, A49, RELAT_1:28
.= P1 by A33, A5, TOPS_2:def_5 ;
A53: dom g1 = the carrier of I[01] by FUNCT_2:def_1;
then A54: 0 in dom g1 by BORSUK_1:40, XXREAL_1:1;
then A55: 0 = (g1 ") . p00 by A42, A48, A51, FUNCT_1:32;
A56: dom (g1 ") = rng g1 by A48, A51, FUNCT_1:32;
then A57: p00 in dom (g1 ") by A42, A54, FUNCT_1:def_3;
A58: 1 in dom g1 by A53, BORSUK_1:40, XXREAL_1:1;
then A59: p11 in dom (g1 ") by A43, A56, FUNCT_1:def_3;
reconsider g2 = g2 as Function of I[01],((TOP-REAL 2) | Lh2) ;
A60: g2 is one-to-one by A44, TOPS_2:def_5;
A61: rng g2 = [#] ((TOP-REAL 2) | Lh2) by A44, TOPS_2:def_5;
then  g2 is onto by FUNCT_2:def_3;
then A62: g2 " = g2 " by A60, TOPS_2:def_4;
 g2 is continuous by A44, TOPS_2:def_5;
then A63: (TOP-REAL 2) | Lh2 is compact by A38, A61, COMPTS_1:14;
A64: g2 " is continuous by A44, TOPS_2:def_5;
 g1 is continuous by A41, TOPS_2:def_5;
then A65: (TOP-REAL 2) | Lh1 is compact by A38, A50, COMPTS_1:14;
A66: f2 is one-to-one by A23, TOPS_2:def_5;
A67: dom g2 = the carrier of I[01] by FUNCT_2:def_1;
then A68: 0 in dom g2 by BORSUK_1:40, XXREAL_1:1;
then A69: p00 in rng g2 by A45, FUNCT_1:def_3;
then A70: p00 in dom (g2 ") by A60, A62, FUNCT_1:32;
 (g2 ") . p00 in dom ff2 by A45, A60, A62, A53, A67, A27, A54, FUNCT_1:32;
then A71: p00 in dom (ff2 * (g2 ")) by A70, FUNCT_1:11;
A72: dom ff1 = the carrier of I[01] by FUNCT_2:def_1;
then  (g1 ") . p00 in dom ff1 by A42, A48, A51, A53, A54, FUNCT_1:32;
then  p00 in dom (ff1 * (g1 ")) by A57, FUNCT_1:11;
then A73: (ff1 * (g1 ")) . p00 = ff1 . ((g1 ") . p00) by FUNCT_1:12
.= p1 by A34, A42, A48, A51, A54, FUNCT_1:32 ;
then A74: (ff1 * (g1 ")) . p00 = ff2 . ((g2 ") . p00) by A24, A45, A60, A62, A68, FUNCT_1:32
.= (ff2 * (g2 ")) . p00 by A71, FUNCT_1:12 ;
A75: 1 in dom g2 by A67, BORSUK_1:40, XXREAL_1:1;
then A76: 1 = (g2 ") . p11 by A46, A60, A62, FUNCT_1:32;
A77: dom (g2 ") = rng g2 by A60, A62, FUNCT_1:32;
then A78: p11 in dom (g2 ") by A46, A75, FUNCT_1:def_3;
 (g2 ") . p11 in dom ff2 by A46, A60, A62, A53, A67, A27, A58, FUNCT_1:32;
then A79: p11 in dom (ff2 * (g2 ")) by A78, FUNCT_1:11;
 (g1 ") . p11 in dom ff1 by A43, A48, A51, A53, A72, A58, FUNCT_1:32;
then  p11 in dom (ff1 * (g1 ")) by A59, FUNCT_1:11;
then A80: (ff1 * (g1 ")) . p11 = ff1 . ((g1 ") . p11) by FUNCT_1:12
.= p2 by A35, A43, A48, A51, A58, FUNCT_1:32 ;
then A81: (ff1 * (g1 ")) . p11 = ff2 . ((g2 ") . p11) by A25, A46, A60, A62, A75, FUNCT_1:32
.= (ff2 * (g2 ")) . p11 by A79, FUNCT_1:12 ;
 g1 " is continuous by A41, TOPS_2:def_5;
then reconsider h = (ff1 * (g1 ")) +* (ff2 * (g2 ")) as continuous Function of ((TOP-REAL 2) | RS),((TOP-REAL 2) | P) by A8, A9, A39, A29, A18, A16, A15, A65, A63, A47, A64, A74, A81, A19, A17, COMPTS_1:21;
A82: 1 = (g1 ") . p11 by A43, A48, A51, A58, FUNCT_1:32;
A83: rng (g2 ") = dom g2 by A60, A62, FUNCT_1:33;
then A84: rng (ff2 * (g2 ")) = rng f2 by A67, A27, RELAT_1:28
.= [#] ((TOP-REAL 2) | P2) by A23, TOPS_2:def_5
.= P2 by PRE_TOPC:def_5 ;
A85: 0 = (g2 ") . p00 by A45, A60, A62, A68, FUNCT_1:32;
now__::_thesis:_for_x1,_x2_being_set_st_x1_in_dom_(ff2_*_(g2_"))_&_x2_in_(dom_(ff1_*_(g1_")))_\_(dom_(ff2_*_(g2_")))_holds_
not_(ff2_*_(g2_"))_._x1_=_(ff1_*_(g1_"))_._x2
let x1, x2 be set ; ::_thesis: ( x1 in dom (ff2 * (g2 ")) & x2 in (dom (ff1 * (g1 "))) \ (dom (ff2 * (g2 "))) implies not (ff2 * (g2 ")) . x1 = (ff1 * (g1 ")) . x2 )
assume that
A86: x1 in dom (ff2 * (g2 ")) and
A87: x2 in (dom (ff1 * (g1 "))) \ (dom (ff2 * (g2 "))) ; ::_thesis: not (ff2 * (g2 ")) . x1 = (ff1 * (g1 ")) . x2
A88: x1 in dom (g2 ") by A86, FUNCT_1:11;
A89: (ff2 * (g2 ")) . x1 in P2 by A84, A86, FUNCT_1:def_3;
A90: x2 in dom (ff1 * (g1 ")) by A87, XBOOLE_0:def_5;
then A91: x2 in dom (g1 ") by FUNCT_1:11;
assume A92: (ff2 * (g2 ")) . x1 = (ff1 * (g1 ")) . x2 ; ::_thesis: contradiction
then  (ff2 * (g2 ")) . x1 in P1 by A52, A90, FUNCT_1:def_3;
then A93: (ff2 * (g2 ")) . x1 in P1 /\ P2 by A89, XBOOLE_0:def_4;
percases ( (ff2 * (g2 ")) . x1 = p1 or (ff2 * (g2 ")) . x1 = p2 ) by A4, A93, TARSKI:def_2;
supposeA94: (ff2 * (g2 ")) . x1 = p1 ; ::_thesis: contradiction
A95: (g1 ") . x2 in dom ff1 by A90, FUNCT_1:11;
 p1 = ff1 . ((g1 ") . x2) by A92, A90, A94, FUNCT_1:12;
then A96: (g1 ") . x2 = 0 by A34, A72, A28, A40, A95, FUNCT_1:def_4;
A97: p00 in dom (g2 ") by A60, A62, A69, FUNCT_1:32;
A98: (g2 ") . x1 in dom ff2 by A86, FUNCT_1:11;
 p1 = ff2 . ((g2 ") . x1) by A86, A94, FUNCT_1:12;
then  (g2 ") . x1 = 0 by A24, A28, A66, A98, FUNCT_1:def_4;
then A99: x1 = p00 by A60, A62, A85, A88, A97, FUNCT_1:def_4;
 p00 in dom (g1 ") by A42, A53, A28, A56, FUNCT_1:def_3;
then  x2 in dom (ff2 * (g2 ")) by A48, A51, A55, A86, A91, A99, A96, FUNCT_1:def_4;
hence  contradiction by A87, XBOOLE_0:def_5; ::_thesis: verum
end;
supposeA100: (ff2 * (g2 ")) . x1 = p2 ; ::_thesis: contradiction
A101: (g1 ") . x2 in dom ff1 by A90, FUNCT_1:11;
 p2 = ff1 . ((g1 ") . x2) by A92, A90, A100, FUNCT_1:12;
then A102: (g1 ") . x2 = 1 by A35, A72, A30, A40, A101, FUNCT_1:def_4;
A103: p11 in dom (g2 ") by A46, A67, A77, A30, FUNCT_1:def_3;
A104: (g2 ") . x1 in dom ff2 by A86, FUNCT_1:11;
 p2 = ff2 . ((g2 ") . x1) by A86, A100, FUNCT_1:12;
then  (g2 ") . x1 = 1 by A25, A30, A66, A104, FUNCT_1:def_4;
then A105: x1 = p11 by A60, A62, A76, A88, A103, FUNCT_1:def_4;
 p11 in dom (g1 ") by A43, A53, A56, A30, FUNCT_1:def_3;
then  x2 in dom (ff2 * (g2 ")) by A48, A51, A82, A86, A91, A105, A102, FUNCT_1:def_4;
hence  contradiction by A87, XBOOLE_0:def_5; ::_thesis: verum
end;
end;
end;
then A106: h is one-to-one by A48, A60, A62, A51, A40, A66, TOPMETR2:1;
A107: (TOP-REAL 2) | P9 is T_2 by TOPMETR:2;
A108: dom (ff2 * (g2 ")) = dom (g2 ") by A27, A83, RELAT_1:27;
 (ff1 * (g1 ")) .: ((dom (ff1 * (g1 "))) /\ (dom (ff2 * (g2 ")))) c= rng (ff2 * (g2 "))
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in (ff1 * (g1 ")) .: ((dom (ff1 * (g1 "))) /\ (dom (ff2 * (g2 ")))) or a in rng (ff2 * (g2 ")) )
A109: dom (ff2 * (g2 ")) = the carrier of ((TOP-REAL 2) | Lh2) by FUNCT_2:def_1;
assume  a in (ff1 * (g1 ")) .: ((dom (ff1 * (g1 "))) /\ (dom (ff2 * (g2 ")))) ; ::_thesis: a in rng (ff2 * (g2 "))
then A110: ex x being set st
( x in dom (ff1 * (g1 ")) & x in (dom (ff1 * (g1 "))) /\ (dom (ff2 * (g2 "))) & a = (ff1 * (g1 ")) . x ) by FUNCT_1:def_6;
 dom (ff1 * (g1 ")) = the carrier of ((TOP-REAL 2) | Lh1) by FUNCT_2:def_1;
then  ( a = p1 or a = p2 ) by A9, A18, A16, A73, A80, A110, A109, TARSKI:def_2;
hence  a in rng (ff2 * (g2 ")) by A70, A73, A74, A78, A80, A81, A108, FUNCT_1:def_3; ::_thesis: verum
end;
then A111: rng h = [#] ((TOP-REAL 2) | P9) by A3, A31, A52, A84, TOPMETR2:2;
A112: dom h = [#] ((TOP-REAL 2) | RS) by FUNCT_2:def_1;
reconsider h = h as Function of ((TOP-REAL 2) | R^2-unit_square),((TOP-REAL 2) | P) ;
take  h ; :: according to TOPREAL2:def_1 ::_thesis: h is being_homeomorphism
 (TOP-REAL 2) | RS is compact by Th2, COMPTS_1:3;
hence  h is being_homeomorphism by A107, A111, A112, A106, COMPTS_1:17; ::_thesis: verum
end;

theorem Th5: :: TOPREAL2:5
for P being non empty Subset of (TOP-REAL 2) holds
( P is being_simple_closed_curve iff ( ex p1, p2 being Point of (TOP-REAL 2) st
( p1 <> p2 & p1 in P & p2 in P ) & ( for p1, p2 being Point of (TOP-REAL 2) st p1 <> p2 & p1 in P & p2 in P holds
ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) ) ) )
proof
let P be non empty Subset of (TOP-REAL 2); ::_thesis: ( P is being_simple_closed_curve iff ( ex p1, p2 being Point of (TOP-REAL 2) st
( p1 <> p2 & p1 in P & p2 in P ) & ( for p1, p2 being Point of (TOP-REAL 2) st p1 <> p2 & p1 in P & p2 in P holds
ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) ) ) )
thus  ( P is being_simple_closed_curve implies ( ex p1, p2 being Point of (TOP-REAL 2) st
( p1 <> p2 & p1 in P & p2 in P ) & ( for p1, p2 being Point of (TOP-REAL 2) st p1 <> p2 & p1 in P & p2 in P holds
ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) ) ) ) ::_thesis: ( ex p1, p2 being Point of (TOP-REAL 2) st
( p1 <> p2 & p1 in P & p2 in P ) & ( for p1, p2 being Point of (TOP-REAL 2) st p1 <> p2 & p1 in P & p2 in P holds
ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) ) implies P is being_simple_closed_curve )
proof
assume A1: P is being_simple_closed_curve ; ::_thesis: ( ex p1, p2 being Point of (TOP-REAL 2) st
( p1 <> p2 & p1 in P & p2 in P ) & ( for p1, p2 being Point of (TOP-REAL 2) st p1 <> p2 & p1 in P & p2 in P holds
ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) ) )
then consider f being Function of ((TOP-REAL 2) | R^2-unit_square),((TOP-REAL 2) | P) such that
A2: f is being_homeomorphism by Def1;
A3: dom f = [#] ((TOP-REAL 2) | R^2-unit_square) by A2, TOPS_2:def_5;
A4: [#] ((TOP-REAL 2) | P) c= [#] (TOP-REAL 2) by PRE_TOPC:def_4;
A5: f is continuous by A2, TOPS_2:def_5;
thus  ex p1, p2 being Point of (TOP-REAL 2) st
( p1 <> p2 & p1 in P & p2 in P ) by A1, Th4; ::_thesis: for p1, p2 being Point of (TOP-REAL 2) st p1 <> p2 & p1 in P & p2 in P holds
ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
set RS = R^2-unit_square ;
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 <> p2 & p1 in P & p2 in P implies ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) )
assume that
A6: p1 <> p2 and
A7: p1 in P and
A8: p2 in P ; ::_thesis: ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A9: [#] ((TOP-REAL 2) | R^2-unit_square) = R^2-unit_square by PRE_TOPC:def_5;
set q1 = (f ") . p1;
set q2 = (f ") . p2;
A10: [#] ((TOP-REAL 2) | R^2-unit_square) c= [#] (TOP-REAL 2) by PRE_TOPC:def_4;
A11: I[01] is compact by HEINE:4, TOPMETR:20;
A12: f is one-to-one by A2, TOPS_2:def_5;
A13: rng f = [#] ((TOP-REAL 2) | P) by A2, TOPS_2:def_5;
then  f is onto by FUNCT_2:def_3;
then A14: f " = f " by A12, TOPS_2:def_4;
then A15: rng (f ") = dom f by A12, FUNCT_1:33;
A16: dom (f ") = rng f by A12, A14, FUNCT_1:32;
then A17: p1 in dom (f ") by A7, A13, PRE_TOPC:def_5;
A18: p2 in dom (f ") by A8, A13, A16, PRE_TOPC:def_5;
reconsider f = f as Function of ((TOP-REAL 2) | R^2-unit_square),((TOP-REAL 2) | P) ;
A19: (f ") . p1 in rng (f ") by A17, FUNCT_1:def_3;
A20: (f ") . p2 in rng (f ") by A18, FUNCT_1:def_3;
 rng (f ") = [#] ((TOP-REAL 2) | R^2-unit_square) by A15, FUNCT_2:def_1;
then reconsider q1 = (f ") . p1, q2 = (f ") . p2 as Point of (TOP-REAL 2) by A10, A19, A20;
A21: q1 <> q2 by A6, A12, A14, A17, A18, FUNCT_1:def_4;
A22: dom f = the carrier of ((TOP-REAL 2) | R^2-unit_square) by FUNCT_2:def_1;
then A23: q2 in R^2-unit_square by A15, A18, A9, FUNCT_1:def_3;
A24: p1 = f . q1 by A12, A14, A16, A17, FUNCT_1:35;
 q1 in R^2-unit_square by A15, A17, A22, A9, FUNCT_1:def_3;
then consider Q1, Q2 being non empty Subset of (TOP-REAL 2) such that
A25: Q1 is_an_arc_of q1,q2 and
A26: Q2 is_an_arc_of q1,q2 and
A27: R^2-unit_square = Q1 \/ Q2 and
A28: Q1 /\ Q2 = {q1,q2} by A21, A23, Th1;
A29: Q2 c= dom f by A22, A9, A27, XBOOLE_1:7;
set P1 = f .: Q1;
set P2 = f .: Q2;
 Q1 c= dom f by A22, A9, A27, XBOOLE_1:7;
then reconsider P1 = f .: Q1, P2 = f .: Q2 as non empty Subset of (TOP-REAL 2) by A29, A4, XBOOLE_1:1;
A30: rng (f | Q1) = P1 by RELAT_1:115
.= [#] ((TOP-REAL 2) | P1) by PRE_TOPC:def_5
.= the carrier of ((TOP-REAL 2) | P1) ;
A31: dom (f | Q1) = R^2-unit_square /\ Q1 by A22, A9, RELAT_1:61
.= Q1 by A27, XBOOLE_1:21
.= [#] ((TOP-REAL 2) | Q1) by PRE_TOPC:def_5 ;
then reconsider F1 = f | Q1 as Function of ((TOP-REAL 2) | Q1),((TOP-REAL 2) | P1) by A30, FUNCT_2:def_1, RELSET_1:4;
A32: f " P1 c= Q1 by A12, FUNCT_1:82;
 [#] ((TOP-REAL 2) | Q1) = Q1 by PRE_TOPC:def_5;
then A33: (TOP-REAL 2) | Q1 is SubSpace of (TOP-REAL 2) | R^2-unit_square by A9, A27, TOPMETR:3, XBOOLE_1:7;
 Q1 c= f " P1 by A22, A9, A27, FUNCT_1:76, XBOOLE_1:7;
then A34: f " P1 = Q1 by A32, XBOOLE_0:def_10;
 for R being Subset of ((TOP-REAL 2) | P1) st R is closed holds
F1 " R is closed
proof
let R be Subset of ((TOP-REAL 2) | P1); ::_thesis: ( R is closed implies F1 " R is closed )
assume  R is closed ; ::_thesis: F1 " R is closed
then consider S1 being Subset of (TOP-REAL 2) such that
A35: S1 is closed and
A36: R = S1 /\ ([#] ((TOP-REAL 2) | P1)) by PRE_TOPC:13;
 S1 /\ (rng f) is Subset of ((TOP-REAL 2) | P) ;
then reconsider S2 = (rng f) /\ S1 as Subset of ((TOP-REAL 2) | P) ;
 S2 is closed by A13, A35, PRE_TOPC:13;
then A37: f " S2 is closed by A5, PRE_TOPC:def_6;
F1 " R = Q1 /\ (f " R) by FUNCT_1:70
.= Q1 /\ ((f " S1) /\ (f " ([#] ((TOP-REAL 2) | P1)))) by A36, FUNCT_1:68
.= ((f " S1) /\ Q1) /\ Q1 by A34, PRE_TOPC:def_5
.= (f " S1) /\ (Q1 /\ Q1) by XBOOLE_1:16
.= (f " S1) /\ ([#] ((TOP-REAL 2) | Q1)) by PRE_TOPC:def_5
.= (f " ((rng f) /\ S1)) /\ ([#] ((TOP-REAL 2) | Q1)) by RELAT_1:133 ;
hence  F1 " R is closed by A33, A37, PRE_TOPC:13; ::_thesis: verum
end;
then A38: F1 is continuous by PRE_TOPC:def_6;
reconsider Q19 = Q1, Q29 = Q2 as non empty Subset of (TOP-REAL 2) ;
consider ff being Function of I[01],((TOP-REAL 2) | Q1) such that
A39: ff is being_homeomorphism and
 ff . 0 = q1 and
 ff . 1 = q2 by A25, TOPREAL1:def_1;
A40: rng ff = [#] ((TOP-REAL 2) | Q1) by A39, TOPS_2:def_5;
A41: rng (f | Q2) = P2 by RELAT_1:115
.= [#] ((TOP-REAL 2) | P2) by PRE_TOPC:def_5
.= the carrier of ((TOP-REAL 2) | P2) ;
A42: p2 = f . q2 by A12, A14, A16, A18, FUNCT_1:35;
A43: dom (f | Q2) = R^2-unit_square /\ Q2 by A22, A9, RELAT_1:61
.= Q2 by A27, XBOOLE_1:21
.= [#] ((TOP-REAL 2) | Q2) by PRE_TOPC:def_5 ;
then reconsider F2 = f | Q2 as Function of ((TOP-REAL 2) | Q2),((TOP-REAL 2) | P2) by A41, FUNCT_2:def_1, RELSET_1:4;
A44: f " P2 c= Q2 by A12, FUNCT_1:82;
 [#] ((TOP-REAL 2) | Q2) = Q2 by PRE_TOPC:def_5;
then A45: (TOP-REAL 2) | Q2 is SubSpace of (TOP-REAL 2) | R^2-unit_square by A9, A27, TOPMETR:3, XBOOLE_1:7;
 Q2 c= f " P2 by A22, A9, A27, FUNCT_1:76, XBOOLE_1:7;
then A46: f " P2 = Q2 by A44, XBOOLE_0:def_10;
 for R being Subset of ((TOP-REAL 2) | P2) st R is closed holds
F2 " R is closed
proof
let R be Subset of ((TOP-REAL 2) | P2); ::_thesis: ( R is closed implies F2 " R is closed )
assume  R is closed ; ::_thesis: F2 " R is closed
then consider S1 being Subset of (TOP-REAL 2) such that
A47: S1 is closed and
A48: R = S1 /\ ([#] ((TOP-REAL 2) | P2)) by PRE_TOPC:13;
 S1 /\ (rng f) is Subset of ((TOP-REAL 2) | P) ;
then reconsider S2 = (rng f) /\ S1 as Subset of ((TOP-REAL 2) | P) ;
 S2 is closed by A13, A47, PRE_TOPC:13;
then A49: f " S2 is closed by A5, PRE_TOPC:def_6;
F2 " R = Q2 /\ (f " R) by FUNCT_1:70
.= Q2 /\ ((f " S1) /\ (f " ([#] ((TOP-REAL 2) | P2)))) by A48, FUNCT_1:68
.= ((f " S1) /\ Q2) /\ Q2 by A46, PRE_TOPC:def_5
.= (f " S1) /\ (Q2 /\ Q2) by XBOOLE_1:16
.= (f " S1) /\ ([#] ((TOP-REAL 2) | Q2)) by PRE_TOPC:def_5
.= (f " ((rng f) /\ S1)) /\ ([#] ((TOP-REAL 2) | Q2)) by RELAT_1:133 ;
hence  F2 " R is closed by A45, A49, PRE_TOPC:13; ::_thesis: verum
end;
then A50: F2 is continuous by PRE_TOPC:def_6;
A51: q2 in {q1,q2} by TARSKI:def_2;
A52: q1 in {q1,q2} by TARSKI:def_2;
A53: q1 in {q1,q2} by TARSKI:def_2;
 {q1,q2} c= Q1 by A28, XBOOLE_1:17;
then A54: q1 in (dom f) /\ Q1 by A15, A19, A53, XBOOLE_0:def_4;
take  P1 ; ::_thesis: ex P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
take  P2 ; ::_thesis: ( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A55: (TOP-REAL 2) | P1 is T_2 by TOPMETR:2;
A56: q2 in {q1,q2} by TARSKI:def_2;
 {q1,q2} c= Q1 by A28, XBOOLE_1:17;
then A57: q2 in (dom f) /\ Q1 by A15, A20, A56, XBOOLE_0:def_4;
A58: p2 = f . q2 by A12, A14, A16, A18, FUNCT_1:35
.= F1 . q2 by A57, FUNCT_1:48 ;
A59: rng F1 = [#] ((TOP-REAL 2) | P1) by A30;
 ff is continuous by A39, TOPS_2:def_5;
then A60: (TOP-REAL 2) | Q19 is compact by A11, A40, COMPTS_1:14;
A61: F1 is one-to-one by A12, FUNCT_1:52;
p1 = f . q1 by A12, A14, A16, A17, FUNCT_1:35
.= F1 . q1 by A54, FUNCT_1:48 ;
hence  P1 is_an_arc_of p1,p2 by A25, A31, A59, A61, A38, A60, A55, A58, Th3, COMPTS_1:17; ::_thesis: ( P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
A62: (TOP-REAL 2) | P2 is T_2 by TOPMETR:2;
consider ff being Function of I[01],((TOP-REAL 2) | Q2) such that
A63: ff is being_homeomorphism and
 ff . 0 = q1 and
 ff . 1 = q2 by A26, TOPREAL1:def_1;
A64: rng ff = [#] ((TOP-REAL 2) | Q2) by A63, TOPS_2:def_5;
 {q1,q2} c= Q2 by A28, XBOOLE_1:17;
then  q1 in (dom f) /\ Q2 by A15, A19, A52, XBOOLE_0:def_4;
then A65: p1 = F2 . q1 by A24, FUNCT_1:48;
A66: F2 is one-to-one by A12, FUNCT_1:52;
 {q1,q2} c= Q2 by A28, XBOOLE_1:17;
then  q2 in (dom f) /\ Q2 by A15, A20, A51, XBOOLE_0:def_4;
then A67: p2 = F2 . q2 by A42, FUNCT_1:48;
 ff is continuous by A63, TOPS_2:def_5;
then A68: (TOP-REAL 2) | Q29 is compact by A11, A64, COMPTS_1:14;
 rng F2 = [#] ((TOP-REAL 2) | P2) by A41;
hence  P2 is_an_arc_of p1,p2 by A26, A43, A66, A50, A68, A62, A65, A67, Th3, COMPTS_1:17; ::_thesis: ( P = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
 [#] ((TOP-REAL 2) | P) = P by PRE_TOPC:def_5;
hence P = f .: (Q1 \/ Q2) by A13, A3, A9, A27, RELAT_1:113
.= P1 \/ P2 by RELAT_1:120 ;
::_thesis: P1 /\ P2 = {p1,p2}
thus P1 /\ P2 = f .: (Q1 /\ Q2) by A12, FUNCT_1:62
.= f .: ({q1} \/ {q2}) by A28, ENUMSET1:1
.= (Im (f,q1)) \/ (Im (f,q2)) by RELAT_1:120
.= {p1} \/ (Im (f,q2)) by A15, A19, A24, FUNCT_1:59
.= {p1} \/ {p2} by A15, A20, A42, FUNCT_1:59
.= {p1,p2} by ENUMSET1:1 ; ::_thesis: verum
end;
given p1, p2 being Point of (TOP-REAL 2) such that A69: p1 <> p2 and
A70: p1 in P and
A71: p2 in P ; ::_thesis: ( ex p1, p2 being Point of (TOP-REAL 2) st
( p1 <> p2 & p1 in P & p2 in P & ( for P1, P2 being non empty Subset of (TOP-REAL 2) holds
( not P1 is_an_arc_of p1,p2 or not P2 is_an_arc_of p1,p2 or not P = P1 \/ P2 or not P1 /\ P2 = {p1,p2} ) ) ) or P is being_simple_closed_curve )
assume  for p1, p2 being Point of (TOP-REAL 2) st p1 <> p2 & p1 in P & p2 in P holds
ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) ; ::_thesis: P is being_simple_closed_curve
then  ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) by A69, A70, A71;
hence  P is being_simple_closed_curve by Lm34; ::_thesis: verum
end;

theorem :: TOPREAL2:6
for P being non empty Subset of (TOP-REAL 2) holds
( P is being_simple_closed_curve iff ex p1, p2 being Point of (TOP-REAL 2) ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( p1 <> p2 & p1 in P & p2 in P & P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) )
proof
let P be non empty Subset of (TOP-REAL 2); ::_thesis: ( P is being_simple_closed_curve iff ex p1, p2 being Point of (TOP-REAL 2) ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( p1 <> p2 & p1 in P & p2 in P & P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) )
hereby  ::_thesis: ( ex p1, p2 being Point of (TOP-REAL 2) ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( p1 <> p2 & p1 in P & p2 in P & P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) implies P is being_simple_closed_curve )
assume A1: P is being_simple_closed_curve ; ::_thesis: ex p1, p2 being Point of (TOP-REAL 2) ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( p1 <> p2 & p1 in P & p2 in P & P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
then consider p1, p2 being Point of (TOP-REAL 2) such that
A2: p1 <> p2 and
A3: p1 in P and
A4: p2 in P by Th5;
consider P1, P2 being non empty Subset of (TOP-REAL 2) such that
A5: P1 is_an_arc_of p1,p2 and
A6: P2 is_an_arc_of p1,p2 and
A7: P = P1 \/ P2 and
A8: P1 /\ P2 = {p1,p2} by A1, A2, A3, A4, Th5;
take p1 = p1; ::_thesis: ex p2 being Point of (TOP-REAL 2) ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( p1 <> p2 & p1 in P & p2 in P & P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
take p2 = p2; ::_thesis: ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( p1 <> p2 & p1 in P & p2 in P & P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
take P1 = P1; ::_thesis: ex P2 being non empty Subset of (TOP-REAL 2) st
( p1 <> p2 & p1 in P & p2 in P & P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
take P2 = P2; ::_thesis: ( p1 <> p2 & p1 in P & p2 in P & P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} )
thus  ( p1 <> p2 & p1 in P & p2 in P & P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) by A2, A3, A4, A5, A6, A7, A8; ::_thesis: verum
end;
thus  ( ex p1, p2 being Point of (TOP-REAL 2) ex P1, P2 being non empty Subset of (TOP-REAL 2) st
( p1 <> p2 & p1 in P & p2 in P & P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} ) implies P is being_simple_closed_curve ) by Lm34; ::_thesis: verum
end;

Lm35:  for S, T being 1-sorted
for f being Function of S,T st S is empty & rng f = [#] T holds
T is empty
proof
let S, T be 1-sorted ; ::_thesis: for f being Function of S,T st S is empty & rng f = [#] T holds
T is empty
let f be Function of S,T; ::_thesis: ( S is empty & rng f = [#] T implies T is empty )
assume that
A1: S is empty and
A2: rng f = [#] T ; ::_thesis: T is empty
assume  not T is empty ; ::_thesis: contradiction
then reconsider T = T as non empty 1-sorted ;
consider y being set  such that
A3: y in the carrier of T by XBOOLE_0:def_1;
 ex x being set st
( x in dom f & f . x = y ) by A2, A3, FUNCT_1:def_3;
hence  contradiction by A1; ::_thesis: verum
end;

Lm36:  for S, T being 1-sorted
for f being Function of S,T st T is empty & dom f = [#] S holds
S is empty
proof
let S, T be 1-sorted ; ::_thesis: for f being Function of S,T st T is empty & dom f = [#] S holds
S is empty
let f be Function of S,T; ::_thesis: ( T is empty & dom f = [#] S implies S is empty )
assume that
A1: T is empty and
A2: dom f = [#] S ; ::_thesis: S is empty
assume  not S is empty ; ::_thesis: contradiction
then reconsider S = S as non empty 1-sorted ;
consider x being set  such that
A3: x in the carrier of S by XBOOLE_0:def_1;
 f . x in rng f by A2, A3, FUNCT_1:def_3;
hence  contradiction by A1; ::_thesis: verum
end;

Lm37:  for S, T being TopStruct st ex f being Function of S,T st f is being_homeomorphism holds
( S is empty iff T is empty )
proof
let S, T be TopStruct ; ::_thesis: ( ex f being Function of S,T st f is being_homeomorphism implies ( S is empty iff T is empty ) )
given f being Function of S,T such that A1: f is being_homeomorphism ; ::_thesis: ( S is empty iff T is empty )
A2: dom f = [#] S by A1, TOPS_2:def_5;
 rng f = [#] T by A1, TOPS_2:def_5;
hence  ( S is empty iff T is empty ) by A2, Lm35, Lm36; ::_thesis: verum
end;

registration
cluster being_simple_closed_curve  ->  non empty compact for Element of K19( the carrier of (TOP-REAL 2));
coherence
 for b1 being Subset of (TOP-REAL 2) st b1 is being_simple_closed_curve holds
( not b1 is empty & b1 is compact )
proof
let P be Subset of (TOP-REAL 2); ::_thesis: ( P is being_simple_closed_curve implies ( not P is empty & P is compact ) )
given f being Function of ((TOP-REAL 2) | R^2-unit_square),((TOP-REAL 2) | P) such that A1: f is being_homeomorphism ; :: according to TOPREAL2:def_1 ::_thesis: ( not P is empty & P is compact )
thus  not P is empty by A1, Lm37; ::_thesis: P is compact
A2: rng f = [#] ((TOP-REAL 2) | P) by A1, TOPS_2:def_5;
reconsider R = P as non empty Subset of (TOP-REAL 2) by A1, Lm37;
A3: f is continuous by A1, TOPS_2:def_5;
 (TOP-REAL 2) | R^2-unit_square is compact by Th2, COMPTS_1:3;
then  (TOP-REAL 2) | R is compact by A3, A2, COMPTS_1:14;
hence  P is compact by COMPTS_1:3; ::_thesis: verum
end;
end;