:: JORDAN1 semantic presentation begin Lm1: ( 0 in [.0,1.] & 1 in [.0,1.] ) proof A1: 0 in { r where r is Real : ( 0 <= r & r <= 1 ) } ; 1 in { s where s is Real : ( 0 <= s & s <= 1 ) } ; hence ( 0 in [.0,1.] & 1 in [.0,1.] ) by A1, RCOMP_1:def_1; ::_thesis: verum end; theorem Th1: :: JORDAN1:1 for GX being non empty TopSpace st ( for x, y being Point of GX ex h being Function of I[01],GX st ( h is continuous & x = h . 0 & y = h . 1 ) ) holds GX is connected proof let GX be non empty TopSpace; ::_thesis: ( ( for x, y being Point of GX ex h being Function of I[01],GX st ( h is continuous & x = h . 0 & y = h . 1 ) ) implies GX is connected ) assume A1: for x, y being Point of GX ex h being Function of I[01],GX st ( h is continuous & x = h . 0 & y = h . 1 ) ; ::_thesis: GX is connected for x, y being Point of GX ex GY being non empty TopSpace st ( GY is connected & ex f being Function of GY,GX st ( f is continuous & x in rng f & y in rng f ) ) proof let x, y be Point of GX; ::_thesis: ex GY being non empty TopSpace st ( GY is connected & ex f being Function of GY,GX st ( f is continuous & x in rng f & y in rng f ) ) now__::_thesis:_ex_GY_being_non_empty_TopSpace_st_ (_GY_is_connected_&_ex_f_being_Function_of_GY,GX_st_ (_f_is_continuous_&_x_in_rng_f_&_y_in_rng_f_)_) consider h being Function of I[01],GX such that A2: h is continuous and A3: x = h . 0 and A4: y = h . 1 by A1; A5: 0 in dom h by Lm1, BORSUK_1:40, FUNCT_2:def_1; A6: 1 in dom h by Lm1, BORSUK_1:40, FUNCT_2:def_1; A7: x in rng h by A3, A5, FUNCT_1:def_3; y in rng h by A4, A6, FUNCT_1:def_3; hence ex GY being non empty TopSpace st ( GY is connected & ex f being Function of GY,GX st ( f is continuous & x in rng f & y in rng f ) ) by A2, A7, TREAL_1:19; ::_thesis: verum end; hence ex GY being non empty TopSpace st ( GY is connected & ex f being Function of GY,GX st ( f is continuous & x in rng f & y in rng f ) ) ; ::_thesis: verum end; hence GX is connected by TOPS_2:63; ::_thesis: verum end; theorem Th2: :: JORDAN1:2 for GX being non empty TopSpace for A being Subset of GX st ( for xa, ya being Point of GX st xa in A & ya in A & xa <> ya holds ex h being Function of I[01],(GX | A) st ( h is continuous & xa = h . 0 & ya = h . 1 ) ) holds A is connected proof let GX be non empty TopSpace; ::_thesis: for A being Subset of GX st ( for xa, ya being Point of GX st xa in A & ya in A & xa <> ya holds ex h being Function of I[01],(GX | A) st ( h is continuous & xa = h . 0 & ya = h . 1 ) ) holds A is connected let A be Subset of GX; ::_thesis: ( ( for xa, ya being Point of GX st xa in A & ya in A & xa <> ya holds ex h being Function of I[01],(GX | A) st ( h is continuous & xa = h . 0 & ya = h . 1 ) ) implies A is connected ) assume A1: for xa, ya being Point of GX st xa in A & ya in A & xa <> ya holds ex h being Function of I[01],(GX | A) st ( h is continuous & xa = h . 0 & ya = h . 1 ) ; ::_thesis: A is connected percases ( not A is empty or A is empty ) ; suppose not A is empty ; ::_thesis: A is connected then reconsider A = A as non empty Subset of GX ; A2: for xa, ya being Point of GX st xa in A & ya in A & xa = ya holds ex h being Function of I[01],(GX | A) st ( h is continuous & xa = h . 0 & ya = h . 1 ) proof let xa, ya be Point of GX; ::_thesis: ( xa in A & ya in A & xa = ya implies ex h being Function of I[01],(GX | A) st ( h is continuous & xa = h . 0 & ya = h . 1 ) ) assume that A3: xa in A and ya in A and A4: xa = ya ; ::_thesis: ex h being Function of I[01],(GX | A) st ( h is continuous & xa = h . 0 & ya = h . 1 ) reconsider xa9 = xa as Element of (GX | A) by A3, PRE_TOPC:8; reconsider h = I[01] --> xa9 as Function of I[01],(GX | A) ; take h ; ::_thesis: ( h is continuous & xa = h . 0 & ya = h . 1 ) thus ( h is continuous & xa = h . 0 & ya = h . 1 ) by A4, Lm1, BORSUK_1:40, FUNCOP_1:7; ::_thesis: verum end; for xb, yb being Point of (GX | A) ex ha being Function of I[01],(GX | A) st ( ha is continuous & xb = ha . 0 & yb = ha . 1 ) proof let xb, yb be Point of (GX | A); ::_thesis: ex ha being Function of I[01],(GX | A) st ( ha is continuous & xb = ha . 0 & yb = ha . 1 ) A5: xb in [#] (GX | A) ; A6: yb in [#] (GX | A) ; A7: xb in A by A5, PRE_TOPC:def_5; A8: yb in A by A6, PRE_TOPC:def_5; percases ( xb <> yb or xb = yb ) ; suppose xb <> yb ; ::_thesis: ex ha being Function of I[01],(GX | A) st ( ha is continuous & xb = ha . 0 & yb = ha . 1 ) hence ex ha being Function of I[01],(GX | A) st ( ha is continuous & xb = ha . 0 & yb = ha . 1 ) by A1, A7, A8; ::_thesis: verum end; suppose xb = yb ; ::_thesis: ex ha being Function of I[01],(GX | A) st ( ha is continuous & xb = ha . 0 & yb = ha . 1 ) hence ex ha being Function of I[01],(GX | A) st ( ha is continuous & xb = ha . 0 & yb = ha . 1 ) by A2, A7; ::_thesis: verum end; end; end; then GX | A is connected by Th1; hence A is connected by CONNSP_1:def_3; ::_thesis: verum end; suppose A is empty ; ::_thesis: A is connected then reconsider D = A as empty Subset of GX ; let A1, B1 be Subset of (GX | A); :: according to CONNSP_1:def_2,CONNSP_1:def_3 ::_thesis: ( not [#] (GX | A) = A1 \/ B1 or not A1,B1 are_separated or A1 = {} (GX | A) or B1 = {} (GX | A) ) assume that [#] (GX | A) = A1 \/ B1 and A1,B1 are_separated ; ::_thesis: ( A1 = {} (GX | A) or B1 = {} (GX | A) ) [#] (GX | D) = D ; hence ( A1 = {} (GX | A) or B1 = {} (GX | A) ) ; ::_thesis: verum end; end; end; theorem :: JORDAN1:3 for GX being non empty TopSpace for A0, A1 being Subset of GX st A0 is connected & A1 is connected & A0 meets A1 holds A0 \/ A1 is connected by CONNSP_1:1, CONNSP_1:17; theorem Th4: :: JORDAN1:4 for GX being non empty TopSpace for A0, A1, A2 being Subset of GX st A0 is connected & A1 is connected & A2 is connected & A0 meets A1 & A1 meets A2 holds (A0 \/ A1) \/ A2 is connected proof let GX be non empty TopSpace; ::_thesis: for A0, A1, A2 being Subset of GX st A0 is connected & A1 is connected & A2 is connected & A0 meets A1 & A1 meets A2 holds (A0 \/ A1) \/ A2 is connected let A0, A1, A2 be Subset of GX; ::_thesis: ( A0 is connected & A1 is connected & A2 is connected & A0 meets A1 & A1 meets A2 implies (A0 \/ A1) \/ A2 is connected ) assume that A1: A0 is connected and A2: A1 is connected and A3: A2 is connected and A4: A0 meets A1 and A5: A1 meets A2 ; ::_thesis: (A0 \/ A1) \/ A2 is connected A6: A1 /\ A2 <> {} by A5, XBOOLE_0:def_7; A7: A0 \/ A1 is connected by A1, A2, A4, CONNSP_1:1, CONNSP_1:17; (A0 \/ A1) /\ A2 = (A0 /\ A2) \/ (A1 /\ A2) by XBOOLE_1:23; then A0 \/ A1 meets A2 by A6, XBOOLE_0:def_7; hence (A0 \/ A1) \/ A2 is connected by A3, A7, CONNSP_1:1, CONNSP_1:17; ::_thesis: verum end; theorem Th5: :: JORDAN1:5 for GX being non empty TopSpace for A0, A1, A2, A3 being Subset of GX st A0 is connected & A1 is connected & A2 is connected & A3 is connected & A0 meets A1 & A1 meets A2 & A2 meets A3 holds ((A0 \/ A1) \/ A2) \/ A3 is connected proof let GX be non empty TopSpace; ::_thesis: for A0, A1, A2, A3 being Subset of GX st A0 is connected & A1 is connected & A2 is connected & A3 is connected & A0 meets A1 & A1 meets A2 & A2 meets A3 holds ((A0 \/ A1) \/ A2) \/ A3 is connected let A0, A1, A2, A3 be Subset of GX; ::_thesis: ( A0 is connected & A1 is connected & A2 is connected & A3 is connected & A0 meets A1 & A1 meets A2 & A2 meets A3 implies ((A0 \/ A1) \/ A2) \/ A3 is connected ) assume that A1: A0 is connected and A2: A1 is connected and A3: A2 is connected and A4: A3 is connected and A5: A0 meets A1 and A6: A1 meets A2 and A7: A2 meets A3 ; ::_thesis: ((A0 \/ A1) \/ A2) \/ A3 is connected A8: A2 /\ A3 <> {} by A7, XBOOLE_0:def_7; A9: (A0 \/ A1) \/ A2 is connected by A1, A2, A3, A5, A6, Th4; ((A0 \/ A1) \/ A2) /\ A3 = ((A0 \/ A1) /\ A3) \/ (A2 /\ A3) by XBOOLE_1:23; then (A0 \/ A1) \/ A2 meets A3 by A8, XBOOLE_0:def_7; hence ((A0 \/ A1) \/ A2) \/ A3 is connected by A4, A9, CONNSP_1:1, CONNSP_1:17; ::_thesis: verum end; begin definition let V be RealLinearSpace; let P be Subset of V; redefine attr P is convex means :: JORDAN1:def 1 for w1, w2 being Element of V st w1 in P & w2 in P holds LSeg (w1,w2) c= P; compatibility ( P is convex iff for w1, w2 being Element of V st w1 in P & w2 in P holds LSeg (w1,w2) c= P ) by RLTOPSP1:22; end; :: deftheorem defines convex JORDAN1:def_1_:_ for V being RealLinearSpace for P being Subset of V holds ( P is convex iff for w1, w2 being Element of V st w1 in P & w2 in P holds LSeg (w1,w2) c= P ); registration let n be Nat; cluster convex -> connected for Element of K6( the carrier of (TOP-REAL n)); coherence for b1 being Subset of (TOP-REAL n) st b1 is convex holds b1 is connected proof let P be Subset of (TOP-REAL n); ::_thesis: ( P is convex implies P is connected ) assume A1: for w3, w4 being Point of (TOP-REAL n) st w3 in P & w4 in P holds LSeg (w3,w4) c= P ; :: according to JORDAN1:def_1 ::_thesis: P is connected for w1, w2 being Point of (TOP-REAL n) st w1 in P & w2 in P & w1 <> w2 holds ex h being Function of I[01],((TOP-REAL n) | P) st ( h is continuous & w1 = h . 0 & w2 = h . 1 ) proof let w1, w2 be Point of (TOP-REAL n); ::_thesis: ( w1 in P & w2 in P & w1 <> w2 implies ex h being Function of I[01],((TOP-REAL n) | P) st ( h is continuous & w1 = h . 0 & w2 = h . 1 ) ) assume that A2: w1 in P and A3: w2 in P and A4: w1 <> w2 ; ::_thesis: ex h being Function of I[01],((TOP-REAL n) | P) st ( h is continuous & w1 = h . 0 & w2 = h . 1 ) A5: LSeg (w1,w2) c= P by A1, A2, A3; LSeg (w1,w2) is_an_arc_of w1,w2 by A4, TOPREAL1:9; then consider f being Function of I[01],((TOP-REAL n) | (LSeg (w1,w2))) such that A6: f is being_homeomorphism and A7: f . 0 = w1 and A8: f . 1 = w2 by TOPREAL1:def_1; A9: f is continuous by A6, TOPS_2:def_5; A10: rng f = [#] ((TOP-REAL n) | (LSeg (w1,w2))) by A6, TOPS_2:def_5; then A11: rng f c= P by A5, PRE_TOPC:def_5; then [#] ((TOP-REAL n) | (LSeg (w1,w2))) c= [#] ((TOP-REAL n) | P) by A10, PRE_TOPC:def_5; then A12: (TOP-REAL n) | (LSeg (w1,w2)) is SubSpace of (TOP-REAL n) | P by TOPMETR:3; dom f = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1; then reconsider g = f as Function of [.0,1.],P by A11, FUNCT_2:2; the carrier of ((TOP-REAL n) | P) = [#] ((TOP-REAL n) | P) .= P by PRE_TOPC:def_5 ; then reconsider gt = g as Function of I[01],((TOP-REAL n) | P) by BORSUK_1:40; gt is continuous by A9, A12, PRE_TOPC:26; hence ex h being Function of I[01],((TOP-REAL n) | P) st ( h is continuous & w1 = h . 0 & w2 = h . 1 ) by A7, A8; ::_thesis: verum end; hence P is connected by Th2; ::_thesis: verum end; end; Lm2: the carrier of (TOP-REAL 2) = REAL 2 by EUCLID:22; Lm3: for s1 being Real holds { |[sb,tb]| where sb, tb is Real : sb < s1 } is Subset of (REAL 2) proof let s1 be Real; ::_thesis: { |[sb,tb]| where sb, tb is Real : sb < s1 } is Subset of (REAL 2) { |[sb,tb]| where sb, tb is Real : sb < s1 } c= REAL 2 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in { |[sb,tb]| where sb, tb is Real : sb < s1 } or y in REAL 2 ) assume y in { |[sb,tb]| where sb, tb is Real : sb < s1 } ; ::_thesis: y in REAL 2 then consider s7, t7 being Real such that A1: |[s7,t7]| = y and s7 < s1 ; |[s7,t7]| in the carrier of (TOP-REAL 2) ; hence y in REAL 2 by A1, EUCLID:22; ::_thesis: verum end; hence { |[sb,tb]| where sb, tb is Real : sb < s1 } is Subset of (REAL 2) ; ::_thesis: verum end; Lm4: for t1 being Real holds { |[sb,tb]| where sb, tb is Real : tb < t1 } is Subset of (REAL 2) proof let t1 be Real; ::_thesis: { |[sb,tb]| where sb, tb is Real : tb < t1 } is Subset of (REAL 2) { |[sd,td]| where sd, td is Real : td < t1 } c= REAL 2 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in { |[sd,td]| where sd, td is Real : td < t1 } or y in REAL 2 ) assume y in { |[sd,td]| where sd, td is Real : td < t1 } ; ::_thesis: y in REAL 2 then consider s7, t7 being Real such that A1: |[s7,t7]| = y and t7 < t1 ; |[s7,t7]| in the carrier of (TOP-REAL 2) ; hence y in REAL 2 by A1, EUCLID:22; ::_thesis: verum end; hence { |[sb,tb]| where sb, tb is Real : tb < t1 } is Subset of (REAL 2) ; ::_thesis: verum end; Lm5: for s2 being Real holds { |[sb,tb]| where sb, tb is Real : s2 < sb } is Subset of (REAL 2) proof let s2 be Real; ::_thesis: { |[sb,tb]| where sb, tb is Real : s2 < sb } is Subset of (REAL 2) { |[sb,tb]| where sb, tb is Real : s2 < sb } c= REAL 2 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in { |[sb,tb]| where sb, tb is Real : s2 < sb } or y in REAL 2 ) assume y in { |[sb,tb]| where sb, tb is Real : s2 < sb } ; ::_thesis: y in REAL 2 then consider s7, t7 being Real such that A1: |[s7,t7]| = y and s2 < s7 ; |[s7,t7]| in the carrier of (TOP-REAL 2) ; hence y in REAL 2 by A1, EUCLID:22; ::_thesis: verum end; hence { |[sb,tb]| where sb, tb is Real : s2 < sb } is Subset of (REAL 2) ; ::_thesis: verum end; Lm6: for t2 being Real holds { |[sb,tb]| where sb, tb is Real : t2 < tb } is Subset of (REAL 2) proof let t2 be Real; ::_thesis: { |[sb,tb]| where sb, tb is Real : t2 < tb } is Subset of (REAL 2) { |[sb,tb]| where sb, tb is Real : t2 < tb } c= REAL 2 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in { |[sb,tb]| where sb, tb is Real : t2 < tb } or y in REAL 2 ) assume y in { |[sb,tb]| where sb, tb is Real : t2 < tb } ; ::_thesis: y in REAL 2 then consider s7, t7 being Real such that A1: |[s7,t7]| = y and t2 < t7 ; |[s7,t7]| in the carrier of (TOP-REAL 2) ; hence y in REAL 2 by A1, EUCLID:22; ::_thesis: verum end; hence { |[sb,tb]| where sb, tb is Real : t2 < tb } is Subset of (REAL 2) ; ::_thesis: verum end; Lm7: for s1, s2, t1, t2 being Real holds { |[s,t]| where s, t is Real : ( s1 < s & s < s2 & t1 < t & t < t2 ) } is Subset of (REAL 2) proof let s1, s2, t1, t2 be Real; ::_thesis: { |[s,t]| where s, t is Real : ( s1 < s & s < s2 & t1 < t & t < t2 ) } is Subset of (REAL 2) { |[sb,tb]| where sb, tb is Real : ( s1 < sb & sb < s2 & t1 < tb & tb < t2 ) } c= REAL 2 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in { |[sb,tb]| where sb, tb is Real : ( s1 < sb & sb < s2 & t1 < tb & tb < t2 ) } or y in REAL 2 ) assume y in { |[sb,tb]| where sb, tb is Real : ( s1 < sb & sb < s2 & t1 < tb & tb < t2 ) } ; ::_thesis: y in REAL 2 then consider s7, t7 being Real such that A1: |[s7,t7]| = y and s1 < s7 and s7 < s2 and t1 < t7 and t7 < t2 ; |[s7,t7]| in the carrier of (TOP-REAL 2) ; hence y in REAL 2 by A1, EUCLID:22; ::_thesis: verum end; hence { |[s,t]| where s, t is Real : ( s1 < s & s < s2 & t1 < t & t < t2 ) } is Subset of (REAL 2) ; ::_thesis: verum end; Lm8: for s1, s2, t1, t2 being Real holds { |[s,t]| where s, t is Real : ( not s1 <= s or not s <= s2 or not t1 <= t or not t <= t2 ) } is Subset of (REAL 2) proof let s1, s2, t1, t2 be Real; ::_thesis: { |[s,t]| where s, t is Real : ( not s1 <= s or not s <= s2 or not t1 <= t or not t <= t2 ) } is Subset of (REAL 2) { |[sb,tb]| where sb, tb is Real : ( not s1 <= sb or not sb <= s2 or not t1 <= tb or not tb <= t2 ) } c= REAL 2 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in { |[sb,tb]| where sb, tb is Real : ( not s1 <= sb or not sb <= s2 or not t1 <= tb or not tb <= t2 ) } or y in REAL 2 ) assume y in { |[sb,tb]| where sb, tb is Real : ( not s1 <= sb or not sb <= s2 or not t1 <= tb or not tb <= t2 ) } ; ::_thesis: y in REAL 2 then consider s7, t7 being Real such that A1: |[s7,t7]| = y and ( not s1 <= s7 or not s7 <= s2 or not t1 <= t7 or not t7 <= t2 ) ; |[s7,t7]| in the carrier of (TOP-REAL 2) ; hence y in REAL 2 by A1, EUCLID:22; ::_thesis: verum end; hence { |[s,t]| where s, t is Real : ( not s1 <= s or not s <= s2 or not t1 <= t or not t <= t2 ) } is Subset of (REAL 2) ; ::_thesis: verum end; theorem :: JORDAN1:6 canceled; theorem Th7: :: JORDAN1:7 for s1, s2, t1, t2 being Real holds { |[s,t]| where s, t is Real : ( s1 < s & s < s2 & t1 < t & t < t2 ) } = (( { |[s3,t3]| where s3, t3 is Real : s1 < s3 } /\ { |[s4,t4]| where s4, t4 is Real : s4 < s2 } ) /\ { |[s5,t5]| where s5, t5 is Real : t1 < t5 } ) /\ { |[s6,t6]| where s6, t6 is Real : t6 < t2 } proof let s1, s2, t1, t2 be Real; ::_thesis: { |[s,t]| where s, t is Real : ( s1 < s & s < s2 & t1 < t & t < t2 ) } = (( { |[s3,t3]| where s3, t3 is Real : s1 < s3 } /\ { |[s4,t4]| where s4, t4 is Real : s4 < s2 } ) /\ { |[s5,t5]| where s5, t5 is Real : t1 < t5 } ) /\ { |[s6,t6]| where s6, t6 is Real : t6 < t2 } now__::_thesis:_for_x_being_set_st_x_in__{__|[s,t]|_where_s,_t_is_Real_:_(_s1_<_s_&_s_<_s2_&_t1_<_t_&_t_<_t2_)__}__holds_ x_in_((_{__|[s3,t3]|_where_s3,_t3_is_Real_:_s1_<_s3__}__/\__{__|[s4,t4]|_where_s4,_t4_is_Real_:_s4_<_s2__}__)_/\__{__|[s5,t5]|_where_s5,_t5_is_Real_:_t1_<_t5__}__)_/\__{__|[s6,t6]|_where_s6,_t6_is_Real_:_t6_<_t2__}_ let x be set ; ::_thesis: ( x in { |[s,t]| where s, t is Real : ( s1 < s & s < s2 & t1 < t & t < t2 ) } implies x in (( { |[s3,t3]| where s3, t3 is Real : s1 < s3 } /\ { |[s4,t4]| where s4, t4 is Real : s4 < s2 } ) /\ { |[s5,t5]| where s5, t5 is Real : t1 < t5 } ) /\ { |[s6,t6]| where s6, t6 is Real : t6 < t2 } ) assume x in { |[s,t]| where s, t is Real : ( s1 < s & s < s2 & t1 < t & t < t2 ) } ; ::_thesis: x in (( { |[s3,t3]| where s3, t3 is Real : s1 < s3 } /\ { |[s4,t4]| where s4, t4 is Real : s4 < s2 } ) /\ { |[s5,t5]| where s5, t5 is Real : t1 < t5 } ) /\ { |[s6,t6]| where s6, t6 is Real : t6 < t2 } then A1: ex s, t being Real st ( |[s,t]| = x & s1 < s & s < s2 & t1 < t & t < t2 ) ; then A2: x in { |[s3,t3]| where s3, t3 is Real : s1 < s3 } ; x in { |[s4,t4]| where s4, t4 is Real : s4 < s2 } by A1; then A3: x in { |[s3,t3]| where s3, t3 is Real : s1 < s3 } /\ { |[s4,t4]| where s4, t4 is Real : s4 < s2 } by A2, XBOOLE_0:def_4; A4: x in { |[s5,t5]| where s5, t5 is Real : t1 < t5 } by A1; A5: x in { |[s6,t6]| where s6, t6 is Real : t6 < t2 } by A1; x in ( { |[s3,t3]| where s3, t3 is Real : s1 < s3 } /\ { |[s4,t4]| where s4, t4 is Real : s4 < s2 } ) /\ { |[s5,t5]| where s5, t5 is Real : t1 < t5 } by A3, A4, XBOOLE_0:def_4; hence x in (( { |[s3,t3]| where s3, t3 is Real : s1 < s3 } /\ { |[s4,t4]| where s4, t4 is Real : s4 < s2 } ) /\ { |[s5,t5]| where s5, t5 is Real : t1 < t5 } ) /\ { |[s6,t6]| where s6, t6 is Real : t6 < t2 } by A5, XBOOLE_0:def_4; ::_thesis: verum end; then A6: { |[s,t]| where s, t is Real : ( s1 < s & s < s2 & t1 < t & t < t2 ) } c= (( { |[s3,t3]| where s3, t3 is Real : s1 < s3 } /\ { |[s4,t4]| where s4, t4 is Real : s4 < s2 } ) /\ { |[s5,t5]| where s5, t5 is Real : t1 < t5 } ) /\ { |[s6,t6]| where s6, t6 is Real : t6 < t2 } by TARSKI:def_3; now__::_thesis:_for_x_being_set_st_x_in_((_{__|[s3,t3]|_where_s3,_t3_is_Real_:_s1_<_s3__}__/\__{__|[s4,t4]|_where_s4,_t4_is_Real_:_s4_<_s2__}__)_/\__{__|[s5,t5]|_where_s5,_t5_is_Real_:_t1_<_t5__}__)_/\__{__|[s6,t6]|_where_s6,_t6_is_Real_:_t6_<_t2__}__holds_ x_in__{__|[s,t]|_where_s,_t_is_Real_:_(_s1_<_s_&_s_<_s2_&_t1_<_t_&_t_<_t2_)__}_ let x be set ; ::_thesis: ( x in (( { |[s3,t3]| where s3, t3 is Real : s1 < s3 } /\ { |[s4,t4]| where s4, t4 is Real : s4 < s2 } ) /\ { |[s5,t5]| where s5, t5 is Real : t1 < t5 } ) /\ { |[s6,t6]| where s6, t6 is Real : t6 < t2 } implies x in { |[s,t]| where s, t is Real : ( s1 < s & s < s2 & t1 < t & t < t2 ) } ) assume A7: x in (( { |[s3,t3]| where s3, t3 is Real : s1 < s3 } /\ { |[s4,t4]| where s4, t4 is Real : s4 < s2 } ) /\ { |[s5,t5]| where s5, t5 is Real : t1 < t5 } ) /\ { |[s6,t6]| where s6, t6 is Real : t6 < t2 } ; ::_thesis: x in { |[s,t]| where s, t is Real : ( s1 < s & s < s2 & t1 < t & t < t2 ) } then A8: x in ( { |[s3,t3]| where s3, t3 is Real : s1 < s3 } /\ { |[s4,t4]| where s4, t4 is Real : s4 < s2 } ) /\ { |[s5,t5]| where s5, t5 is Real : t1 < t5 } by XBOOLE_0:def_4; then A9: x in { |[s3,t3]| where s3, t3 is Real : s1 < s3 } /\ { |[s4,t4]| where s4, t4 is Real : s4 < s2 } by XBOOLE_0:def_4; A10: x in { |[s6,t6]| where s6, t6 is Real : t6 < t2 } by A7, XBOOLE_0:def_4; A11: x in { |[s3,t3]| where s3, t3 is Real : s1 < s3 } by A9, XBOOLE_0:def_4; A12: x in { |[s4,t4]| where s4, t4 is Real : s4 < s2 } by A9, XBOOLE_0:def_4; A13: x in { |[s5,t5]| where s5, t5 is Real : t1 < t5 } by A8, XBOOLE_0:def_4; A14: ex sa, ta being Real st ( |[sa,ta]| = x & s1 < sa ) by A11; A15: ex sb, tb being Real st ( |[sb,tb]| = x & sb < s2 ) by A12; A16: ex sc, tc being Real st ( |[sc,tc]| = x & t1 < tc ) by A13; A17: ex sd, td being Real st ( |[sd,td]| = x & td < t2 ) by A10; consider sa, ta being Real such that A18: |[sa,ta]| = x and A19: s1 < sa by A11; reconsider p = x as Point of (TOP-REAL 2) by A14; A20: p `1 = sa by A18, EUCLID:52; A21: p `2 = ta by A18, EUCLID:52; A22: sa < s2 by A15, A20, EUCLID:52; A23: t1 < ta by A16, A21, EUCLID:52; ta < t2 by A17, A21, EUCLID:52; hence x in { |[s,t]| where s, t is Real : ( s1 < s & s < s2 & t1 < t & t < t2 ) } by A18, A19, A22, A23; ::_thesis: verum end; then (( { |[s3,t3]| where s3, t3 is Real : s1 < s3 } /\ { |[s4,t4]| where s4, t4 is Real : s4 < s2 } ) /\ { |[s5,t5]| where s5, t5 is Real : t1 < t5 } ) /\ { |[s6,t6]| where s6, t6 is Real : t6 < t2 } c= { |[s,t]| where s, t is Real : ( s1 < s & s < s2 & t1 < t & t < t2 ) } by TARSKI:def_3; hence { |[s,t]| where s, t is Real : ( s1 < s & s < s2 & t1 < t & t < t2 ) } = (( { |[s3,t3]| where s3, t3 is Real : s1 < s3 } /\ { |[s4,t4]| where s4, t4 is Real : s4 < s2 } ) /\ { |[s5,t5]| where s5, t5 is Real : t1 < t5 } ) /\ { |[s6,t6]| where s6, t6 is Real : t6 < t2 } by A6, XBOOLE_0:def_10; ::_thesis: verum end; theorem Th8: :: JORDAN1:8 for s1, s2, t1, t2 being Real holds { |[s,t]| where s, t is Real : ( not s1 <= s or not s <= s2 or not t1 <= t or not t <= t2 ) } = (( { |[s3,t3]| where s3, t3 is Real : s3 < s1 } \/ { |[s4,t4]| where s4, t4 is Real : t4 < t1 } ) \/ { |[s5,t5]| where s5, t5 is Real : s2 < s5 } ) \/ { |[s6,t6]| where s6, t6 is Real : t2 < t6 } proof let s1, s2, t1, t2 be Real; ::_thesis: { |[s,t]| where s, t is Real : ( not s1 <= s or not s <= s2 or not t1 <= t or not t <= t2 ) } = (( { |[s3,t3]| where s3, t3 is Real : s3 < s1 } \/ { |[s4,t4]| where s4, t4 is Real : t4 < t1 } ) \/ { |[s5,t5]| where s5, t5 is Real : s2 < s5 } ) \/ { |[s6,t6]| where s6, t6 is Real : t2 < t6 } now__::_thesis:_for_x_being_set_st_x_in__{__|[s,t]|_where_s,_t_is_Real_:_(_not_s1_<=_s_or_not_s_<=_s2_or_not_t1_<=_t_or_not_t_<=_t2_)__}__holds_ x_in_((_{__|[s3,t3]|_where_s3,_t3_is_Real_:_s3_<_s1__}__\/__{__|[s4,t4]|_where_s4,_t4_is_Real_:_t4_<_t1__}__)_\/__{__|[s5,t5]|_where_s5,_t5_is_Real_:_s2_<_s5__}__)_\/__{__|[s6,t6]|_where_s6,_t6_is_Real_:_t2_<_t6__}_ let x be set ; ::_thesis: ( x in { |[s,t]| where s, t is Real : ( not s1 <= s or not s <= s2 or not t1 <= t or not t <= t2 ) } implies x in (( { |[s3,t3]| where s3, t3 is Real : s3 < s1 } \/ { |[s4,t4]| where s4, t4 is Real : t4 < t1 } ) \/ { |[s5,t5]| where s5, t5 is Real : s2 < s5 } ) \/ { |[s6,t6]| where s6, t6 is Real : t2 < t6 } ) assume x in { |[s,t]| where s, t is Real : ( not s1 <= s or not s <= s2 or not t1 <= t or not t <= t2 ) } ; ::_thesis: x in (( { |[s3,t3]| where s3, t3 is Real : s3 < s1 } \/ { |[s4,t4]| where s4, t4 is Real : t4 < t1 } ) \/ { |[s5,t5]| where s5, t5 is Real : s2 < s5 } ) \/ { |[s6,t6]| where s6, t6 is Real : t2 < t6 } then ex s, t being Real st ( |[s,t]| = x & ( not s1 <= s or not s <= s2 or not t1 <= t or not t <= t2 ) ) ; then ( x in { |[s3,t3]| where s3, t3 is Real : s3 < s1 } or x in { |[s4,t4]| where s4, t4 is Real : t4 < t1 } or x in { |[s5,t5]| where s5, t5 is Real : s2 < s5 } or x in { |[s6,t6]| where s6, t6 is Real : t2 < t6 } ) ; then ( x in { |[s3,t3]| where s3, t3 is Real : s3 < s1 } \/ { |[s4,t4]| where s4, t4 is Real : t4 < t1 } or x in { |[s5,t5]| where s5, t5 is Real : s2 < s5 } or x in { |[s6,t6]| where s6, t6 is Real : t2 < t6 } ) by XBOOLE_0:def_3; then ( x in ( { |[s3,t3]| where s3, t3 is Real : s3 < s1 } \/ { |[s4,t4]| where s4, t4 is Real : t4 < t1 } ) \/ { |[s5,t5]| where s5, t5 is Real : s2 < s5 } or x in { |[s6,t6]| where s6, t6 is Real : t2 < t6 } ) by XBOOLE_0:def_3; hence x in (( { |[s3,t3]| where s3, t3 is Real : s3 < s1 } \/ { |[s4,t4]| where s4, t4 is Real : t4 < t1 } ) \/ { |[s5,t5]| where s5, t5 is Real : s2 < s5 } ) \/ { |[s6,t6]| where s6, t6 is Real : t2 < t6 } by XBOOLE_0:def_3; ::_thesis: verum end; then A1: { |[s,t]| where s, t is Real : ( not s1 <= s or not s <= s2 or not t1 <= t or not t <= t2 ) } c= (( { |[s3,t3]| where s3, t3 is Real : s3 < s1 } \/ { |[s4,t4]| where s4, t4 is Real : t4 < t1 } ) \/ { |[s5,t5]| where s5, t5 is Real : s2 < s5 } ) \/ { |[s6,t6]| where s6, t6 is Real : t2 < t6 } by TARSKI:def_3; now__::_thesis:_for_x_being_set_st_x_in_((_{__|[s3,t3]|_where_s3,_t3_is_Real_:_s3_<_s1__}__\/__{__|[s4,t4]|_where_s4,_t4_is_Real_:_t4_<_t1__}__)_\/__{__|[s5,t5]|_where_s5,_t5_is_Real_:_s2_<_s5__}__)_\/__{__|[s6,t6]|_where_s6,_t6_is_Real_:_t2_<_t6__}__holds_ x_in__{__|[s,t]|_where_s,_t_is_Real_:_(_not_s1_<=_s_or_not_s_<=_s2_or_not_t1_<=_t_or_not_t_<=_t2_)__}_ let x be set ; ::_thesis: ( x in (( { |[s3,t3]| where s3, t3 is Real : s3 < s1 } \/ { |[s4,t4]| where s4, t4 is Real : t4 < t1 } ) \/ { |[s5,t5]| where s5, t5 is Real : s2 < s5 } ) \/ { |[s6,t6]| where s6, t6 is Real : t2 < t6 } implies x in { |[s,t]| where s, t is Real : ( not s1 <= s or not s <= s2 or not t1 <= t or not t <= t2 ) } ) assume x in (( { |[s3,t3]| where s3, t3 is Real : s3 < s1 } \/ { |[s4,t4]| where s4, t4 is Real : t4 < t1 } ) \/ { |[s5,t5]| where s5, t5 is Real : s2 < s5 } ) \/ { |[s6,t6]| where s6, t6 is Real : t2 < t6 } ; ::_thesis: x in { |[s,t]| where s, t is Real : ( not s1 <= s or not s <= s2 or not t1 <= t or not t <= t2 ) } then ( x in ( { |[s3,t3]| where s3, t3 is Real : s3 < s1 } \/ { |[s4,t4]| where s4, t4 is Real : t4 < t1 } ) \/ { |[s5,t5]| where s5, t5 is Real : s2 < s5 } or x in { |[s6,t6]| where s6, t6 is Real : t2 < t6 } ) by XBOOLE_0:def_3; then ( x in { |[s3,t3]| where s3, t3 is Real : s3 < s1 } \/ { |[s4,t4]| where s4, t4 is Real : t4 < t1 } or x in { |[s5,t5]| where s5, t5 is Real : s2 < s5 } or x in { |[s6,t6]| where s6, t6 is Real : t2 < t6 } ) by XBOOLE_0:def_3; then ( x in { |[s3,t3]| where s3, t3 is Real : s3 < s1 } or x in { |[s4,t4]| where s4, t4 is Real : t4 < t1 } or x in { |[s5,t5]| where s5, t5 is Real : s2 < s5 } or x in { |[s6,t6]| where s6, t6 is Real : t2 < t6 } ) by XBOOLE_0:def_3; then ( ex sa, ta being Real st ( |[sa,ta]| = x & sa < s1 ) or ex sc, tc being Real st ( |[sc,tc]| = x & tc < t1 ) or ex sb, tb being Real st ( |[sb,tb]| = x & s2 < sb ) or ex sd, td being Real st ( |[sd,td]| = x & t2 < td ) ) ; hence x in { |[s,t]| where s, t is Real : ( not s1 <= s or not s <= s2 or not t1 <= t or not t <= t2 ) } ; ::_thesis: verum end; then (( { |[s3,t3]| where s3, t3 is Real : s3 < s1 } \/ { |[s4,t4]| where s4, t4 is Real : t4 < t1 } ) \/ { |[s5,t5]| where s5, t5 is Real : s2 < s5 } ) \/ { |[s6,t6]| where s6, t6 is Real : t2 < t6 } c= { |[s,t]| where s, t is Real : ( not s1 <= s or not s <= s2 or not t1 <= t or not t <= t2 ) } by TARSKI:def_3; hence { |[s,t]| where s, t is Real : ( not s1 <= s or not s <= s2 or not t1 <= t or not t <= t2 ) } = (( { |[s3,t3]| where s3, t3 is Real : s3 < s1 } \/ { |[s4,t4]| where s4, t4 is Real : t4 < t1 } ) \/ { |[s5,t5]| where s5, t5 is Real : s2 < s5 } ) \/ { |[s6,t6]| where s6, t6 is Real : t2 < t6 } by A1, XBOOLE_0:def_10; ::_thesis: verum end; theorem Th9: :: JORDAN1:9 for s1, t1, s2, t2 being Real for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where s, t is Real : ( s1 < s & s < s2 & t1 < t & t < t2 ) } holds P is convex proof let s1, t1, s2, t2 be Real; ::_thesis: for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where s, t is Real : ( s1 < s & s < s2 & t1 < t & t < t2 ) } holds P is convex let P be Subset of (TOP-REAL 2); ::_thesis: ( P = { |[s,t]| where s, t is Real : ( s1 < s & s < s2 & t1 < t & t < t2 ) } implies P is convex ) assume A1: P = { |[s,t]| where s, t is Real : ( s1 < s & s < s2 & t1 < t & t < t2 ) } ; ::_thesis: P is convex let w1 be Point of (TOP-REAL 2); :: according to JORDAN1:def_1 ::_thesis: for w2 being Element of (TOP-REAL 2) st w1 in P & w2 in P holds LSeg (w1,w2) c= P let w2 be Point of (TOP-REAL 2); ::_thesis: ( w1 in P & w2 in P implies LSeg (w1,w2) c= P ) assume that A2: w1 in P and A3: w2 in P ; ::_thesis: LSeg (w1,w2) c= P let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (w1,w2) or x in P ) assume A4: x in LSeg (w1,w2) ; ::_thesis: x in P consider s3, t3 being Real such that A5: |[s3,t3]| = w1 and A6: s1 < s3 and A7: s3 < s2 and A8: t1 < t3 and A9: t3 < t2 by A1, A2; A10: w1 `1 = s3 by A5, EUCLID:52; A11: w1 `2 = t3 by A5, EUCLID:52; consider s4, t4 being Real such that A12: |[s4,t4]| = w2 and A13: s1 < s4 and A14: s4 < s2 and A15: t1 < t4 and A16: t4 < t2 by A1, A3; A17: w2 `1 = s4 by A12, EUCLID:52; A18: w2 `2 = t4 by A12, EUCLID:52; consider l being Real such that A19: x = ((1 - l) * w1) + (l * w2) and A20: 0 <= l and A21: l <= 1 by A4; set w = ((1 - l) * w1) + (l * w2); A22: ((1 - l) * w1) + (l * w2) = |[((((1 - l) * w1) `1) + ((l * w2) `1)),((((1 - l) * w1) `2) + ((l * w2) `2))]| by EUCLID:55; A23: (1 - l) * w1 = |[((1 - l) * (w1 `1)),((1 - l) * (w1 `2))]| by EUCLID:57; A24: l * w2 = |[(l * (w2 `1)),(l * (w2 `2))]| by EUCLID:57; A25: ((1 - l) * w1) `1 = (1 - l) * (w1 `1) by A23, EUCLID:52; A26: ((1 - l) * w1) `2 = (1 - l) * (w1 `2) by A23, EUCLID:52; A27: (l * w2) `1 = l * (w2 `1) by A24, EUCLID:52; A28: (l * w2) `2 = l * (w2 `2) by A24, EUCLID:52; A29: (((1 - l) * w1) + (l * w2)) `1 = ((1 - l) * (w1 `1)) + (l * (w2 `1)) by A22, A25, A27, EUCLID:52; A30: (((1 - l) * w1) + (l * w2)) `2 = ((1 - l) * (w1 `2)) + (l * (w2 `2)) by A22, A26, A28, EUCLID:52; A31: s1 < (((1 - l) * w1) + (l * w2)) `1 by A6, A10, A13, A17, A20, A21, A29, XREAL_1:175; A32: (((1 - l) * w1) + (l * w2)) `1 < s2 by A7, A10, A14, A17, A20, A21, A29, XREAL_1:176; A33: t1 < (((1 - l) * w1) + (l * w2)) `2 by A8, A11, A15, A18, A20, A21, A30, XREAL_1:175; A34: (((1 - l) * w1) + (l * w2)) `2 < t2 by A9, A11, A16, A18, A20, A21, A30, XREAL_1:176; ((1 - l) * w1) + (l * w2) = |[((((1 - l) * w1) + (l * w2)) `1),((((1 - l) * w1) + (l * w2)) `2)]| by EUCLID:53; hence x in P by A1, A19, A31, A32, A33, A34; ::_thesis: verum end; theorem :: JORDAN1:10 canceled; theorem Th11: :: JORDAN1:11 for s1 being Real for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where s, t is Real : s1 < s } holds P is convex proof let s1 be Real; ::_thesis: for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where s, t is Real : s1 < s } holds P is convex let P be Subset of (TOP-REAL 2); ::_thesis: ( P = { |[s,t]| where s, t is Real : s1 < s } implies P is convex ) assume A1: P = { |[s,t]| where s, t is Real : s1 < s } ; ::_thesis: P is convex let w1 be Point of (TOP-REAL 2); :: according to JORDAN1:def_1 ::_thesis: for w2 being Element of (TOP-REAL 2) st w1 in P & w2 in P holds LSeg (w1,w2) c= P let w2 be Point of (TOP-REAL 2); ::_thesis: ( w1 in P & w2 in P implies LSeg (w1,w2) c= P ) assume that A2: w1 in P and A3: w2 in P ; ::_thesis: LSeg (w1,w2) c= P let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (w1,w2) or x in P ) assume A4: x in LSeg (w1,w2) ; ::_thesis: x in P consider s3, t3 being Real such that A5: |[s3,t3]| = w1 and A6: s1 < s3 by A1, A2; A7: w1 `1 = s3 by A5, EUCLID:52; consider s4, t4 being Real such that A8: |[s4,t4]| = w2 and A9: s1 < s4 by A1, A3; A10: w2 `1 = s4 by A8, EUCLID:52; consider l being Real such that A11: x = ((1 - l) * w1) + (l * w2) and A12: 0 <= l and A13: l <= 1 by A4; set w = ((1 - l) * w1) + (l * w2); A14: ((1 - l) * w1) + (l * w2) = |[((((1 - l) * w1) `1) + ((l * w2) `1)),((((1 - l) * w1) `2) + ((l * w2) `2))]| by EUCLID:55; A15: (1 - l) * w1 = |[((1 - l) * (w1 `1)),((1 - l) * (w1 `2))]| by EUCLID:57; A16: l * w2 = |[(l * (w2 `1)),(l * (w2 `2))]| by EUCLID:57; A17: ((1 - l) * w1) `1 = (1 - l) * (w1 `1) by A15, EUCLID:52; (l * w2) `1 = l * (w2 `1) by A16, EUCLID:52; then (((1 - l) * w1) + (l * w2)) `1 = ((1 - l) * (w1 `1)) + (l * (w2 `1)) by A14, A17, EUCLID:52; then A18: s1 < (((1 - l) * w1) + (l * w2)) `1 by A6, A7, A9, A10, A12, A13, XREAL_1:175; ((1 - l) * w1) + (l * w2) = |[((((1 - l) * w1) + (l * w2)) `1),((((1 - l) * w1) + (l * w2)) `2)]| by EUCLID:53; hence x in P by A1, A11, A18; ::_thesis: verum end; theorem :: JORDAN1:12 canceled; theorem Th13: :: JORDAN1:13 for s2 being Real for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where s, t is Real : s < s2 } holds P is convex proof let s2 be Real; ::_thesis: for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where s, t is Real : s < s2 } holds P is convex let P be Subset of (TOP-REAL 2); ::_thesis: ( P = { |[s,t]| where s, t is Real : s < s2 } implies P is convex ) assume A1: P = { |[s,t]| where s, t is Real : s < s2 } ; ::_thesis: P is convex let w1 be Point of (TOP-REAL 2); :: according to JORDAN1:def_1 ::_thesis: for w2 being Element of (TOP-REAL 2) st w1 in P & w2 in P holds LSeg (w1,w2) c= P let w2 be Point of (TOP-REAL 2); ::_thesis: ( w1 in P & w2 in P implies LSeg (w1,w2) c= P ) assume that A2: w1 in P and A3: w2 in P ; ::_thesis: LSeg (w1,w2) c= P let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (w1,w2) or x in P ) assume A4: x in LSeg (w1,w2) ; ::_thesis: x in P consider s3, t3 being Real such that A5: |[s3,t3]| = w1 and A6: s3 < s2 by A1, A2; A7: w1 `1 = s3 by A5, EUCLID:52; consider s4, t4 being Real such that A8: |[s4,t4]| = w2 and A9: s4 < s2 by A1, A3; A10: w2 `1 = s4 by A8, EUCLID:52; consider l being Real such that A11: x = ((1 - l) * w1) + (l * w2) and A12: 0 <= l and A13: l <= 1 by A4; set w = ((1 - l) * w1) + (l * w2); A14: ((1 - l) * w1) + (l * w2) = |[((((1 - l) * w1) `1) + ((l * w2) `1)),((((1 - l) * w1) `2) + ((l * w2) `2))]| by EUCLID:55; A15: (1 - l) * w1 = |[((1 - l) * (w1 `1)),((1 - l) * (w1 `2))]| by EUCLID:57; A16: l * w2 = |[(l * (w2 `1)),(l * (w2 `2))]| by EUCLID:57; A17: ((1 - l) * w1) `1 = (1 - l) * (w1 `1) by A15, EUCLID:52; (l * w2) `1 = l * (w2 `1) by A16, EUCLID:52; then (((1 - l) * w1) + (l * w2)) `1 = ((1 - l) * (w1 `1)) + (l * (w2 `1)) by A14, A17, EUCLID:52; then A18: s2 > (((1 - l) * w1) + (l * w2)) `1 by A6, A7, A9, A10, A12, A13, XREAL_1:176; ((1 - l) * w1) + (l * w2) = |[((((1 - l) * w1) + (l * w2)) `1),((((1 - l) * w1) + (l * w2)) `2)]| by EUCLID:53; hence x in P by A1, A11, A18; ::_thesis: verum end; theorem :: JORDAN1:14 canceled; theorem Th15: :: JORDAN1:15 for t1 being Real for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where s, t is Real : t1 < t } holds P is convex proof let t1 be Real; ::_thesis: for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where s, t is Real : t1 < t } holds P is convex let P be Subset of (TOP-REAL 2); ::_thesis: ( P = { |[s,t]| where s, t is Real : t1 < t } implies P is convex ) assume A1: P = { |[s,t]| where s, t is Real : t1 < t } ; ::_thesis: P is convex let w1 be Point of (TOP-REAL 2); :: according to JORDAN1:def_1 ::_thesis: for w2 being Element of (TOP-REAL 2) st w1 in P & w2 in P holds LSeg (w1,w2) c= P let w2 be Point of (TOP-REAL 2); ::_thesis: ( w1 in P & w2 in P implies LSeg (w1,w2) c= P ) assume that A2: w1 in P and A3: w2 in P ; ::_thesis: LSeg (w1,w2) c= P let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (w1,w2) or x in P ) assume A4: x in LSeg (w1,w2) ; ::_thesis: x in P consider s3, t3 being Real such that A5: |[s3,t3]| = w1 and A6: t1 < t3 by A1, A2; A7: w1 `2 = t3 by A5, EUCLID:52; consider s4, t4 being Real such that A8: |[s4,t4]| = w2 and A9: t1 < t4 by A1, A3; A10: w2 `2 = t4 by A8, EUCLID:52; consider l being Real such that A11: x = ((1 - l) * w1) + (l * w2) and A12: 0 <= l and A13: l <= 1 by A4; set w = ((1 - l) * w1) + (l * w2); A14: ((1 - l) * w1) + (l * w2) = |[((((1 - l) * w1) `1) + ((l * w2) `1)),((((1 - l) * w1) `2) + ((l * w2) `2))]| by EUCLID:55; A15: (1 - l) * w1 = |[((1 - l) * (w1 `1)),((1 - l) * (w1 `2))]| by EUCLID:57; A16: l * w2 = |[(l * (w2 `1)),(l * (w2 `2))]| by EUCLID:57; A17: ((1 - l) * w1) `2 = (1 - l) * (w1 `2) by A15, EUCLID:52; (l * w2) `2 = l * (w2 `2) by A16, EUCLID:52; then (((1 - l) * w1) + (l * w2)) `2 = ((1 - l) * (w1 `2)) + (l * (w2 `2)) by A14, A17, EUCLID:52; then A18: t1 < (((1 - l) * w1) + (l * w2)) `2 by A6, A7, A9, A10, A12, A13, XREAL_1:175; ((1 - l) * w1) + (l * w2) = |[((((1 - l) * w1) + (l * w2)) `1),((((1 - l) * w1) + (l * w2)) `2)]| by EUCLID:53; hence x in P by A1, A11, A18; ::_thesis: verum end; theorem :: JORDAN1:16 canceled; theorem Th17: :: JORDAN1:17 for t2 being Real for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where s, t is Real : t < t2 } holds P is convex proof let t2 be Real; ::_thesis: for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where s, t is Real : t < t2 } holds P is convex let P be Subset of (TOP-REAL 2); ::_thesis: ( P = { |[s,t]| where s, t is Real : t < t2 } implies P is convex ) assume A1: P = { |[s,t]| where s, t is Real : t < t2 } ; ::_thesis: P is convex let w1 be Point of (TOP-REAL 2); :: according to JORDAN1:def_1 ::_thesis: for w2 being Element of (TOP-REAL 2) st w1 in P & w2 in P holds LSeg (w1,w2) c= P let w2 be Point of (TOP-REAL 2); ::_thesis: ( w1 in P & w2 in P implies LSeg (w1,w2) c= P ) assume that A2: w1 in P and A3: w2 in P ; ::_thesis: LSeg (w1,w2) c= P let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (w1,w2) or x in P ) assume A4: x in LSeg (w1,w2) ; ::_thesis: x in P consider s3, t3 being Real such that A5: |[s3,t3]| = w1 and A6: t3 < t2 by A1, A2; A7: w1 `2 = t3 by A5, EUCLID:52; consider s4, t4 being Real such that A8: |[s4,t4]| = w2 and A9: t4 < t2 by A1, A3; A10: w2 `2 = t4 by A8, EUCLID:52; consider l being Real such that A11: x = ((1 - l) * w1) + (l * w2) and A12: 0 <= l and A13: l <= 1 by A4; set w = ((1 - l) * w1) + (l * w2); A14: ((1 - l) * w1) + (l * w2) = |[((((1 - l) * w1) `1) + ((l * w2) `1)),((((1 - l) * w1) `2) + ((l * w2) `2))]| by EUCLID:55; A15: (1 - l) * w1 = |[((1 - l) * (w1 `1)),((1 - l) * (w1 `2))]| by EUCLID:57; A16: l * w2 = |[(l * (w2 `1)),(l * (w2 `2))]| by EUCLID:57; A17: ((1 - l) * w1) `2 = (1 - l) * (w1 `2) by A15, EUCLID:52; (l * w2) `2 = l * (w2 `2) by A16, EUCLID:52; then (((1 - l) * w1) + (l * w2)) `2 = ((1 - l) * (w1 `2)) + (l * (w2 `2)) by A14, A17, EUCLID:52; then A18: t2 > (((1 - l) * w1) + (l * w2)) `2 by A6, A7, A9, A10, A12, A13, XREAL_1:176; ((1 - l) * w1) + (l * w2) = |[((((1 - l) * w1) + (l * w2)) `1),((((1 - l) * w1) + (l * w2)) `2)]| by EUCLID:53; hence x in P by A1, A11, A18; ::_thesis: verum end; theorem :: JORDAN1:18 canceled; theorem Th19: :: JORDAN1:19 for s1, t1, s2, t2 being Real for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where s, t is Real : ( not s1 <= s or not s <= s2 or not t1 <= t or not t <= t2 ) } holds P is connected proof let s1, t1, s2, t2 be Real; ::_thesis: for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where s, t is Real : ( not s1 <= s or not s <= s2 or not t1 <= t or not t <= t2 ) } holds P is connected let P be Subset of (TOP-REAL 2); ::_thesis: ( P = { |[s,t]| where s, t is Real : ( not s1 <= s or not s <= s2 or not t1 <= t or not t <= t2 ) } implies P is connected ) assume P = { |[s,t]| where s, t is Real : ( not s1 <= s or not s <= s2 or not t1 <= t or not t <= t2 ) } ; ::_thesis: P is connected then A1: P = (( { |[s3,t3]| where s3, t3 is Real : s3 < s1 } \/ { |[s4,t4]| where s4, t4 is Real : t4 < t1 } ) \/ { |[s5,t5]| where s5, t5 is Real : s2 < s5 } ) \/ { |[s6,t6]| where s6, t6 is Real : t2 < t6 } by Th8; reconsider A0 = { |[s,t]| where s, t is Real : s < s1 } , A1 = { |[s,t]| where s, t is Real : t < t1 } , A2 = { |[s,t]| where s, t is Real : s2 < s } , A3 = { |[s,t]| where s, t is Real : t2 < t } as Subset of (TOP-REAL 2) by Lm2, Lm3, Lm4, Lm5, Lm6; A2: s1 - 1 < s1 by XREAL_1:44; A3: t1 - 1 < t1 by XREAL_1:44; A4: |[(s1 - 1),(t1 - 1)]| in A0 by A2; |[(s1 - 1),(t1 - 1)]| in A1 by A3; then A0 /\ A1 <> {} by A4, XBOOLE_0:def_4; then A5: A0 meets A1 by XBOOLE_0:def_7; A6: s2 < s2 + 1 by XREAL_1:29; A7: |[(s2 + 1),(t1 - 1)]| in A1 by A3; |[(s2 + 1),(t1 - 1)]| in A2 by A6; then A1 /\ A2 <> {} by A7, XBOOLE_0:def_4; then A8: A1 meets A2 by XBOOLE_0:def_7; A9: t2 < t2 + 1 by XREAL_1:29; A10: |[(s2 + 1),(t2 + 1)]| in A2 by A6; |[(s2 + 1),(t2 + 1)]| in A3 by A9; then A2 /\ A3 <> {} by A10, XBOOLE_0:def_4; then A11: A2 meets A3 by XBOOLE_0:def_7; A12: A0 is convex by Th13; A13: A1 is convex by Th17; A14: A2 is convex by Th11; A3 is convex by Th15; hence P is connected by A1, A5, A8, A11, A12, A13, A14, Th5; ::_thesis: verum end; Lm9: TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopSpaceMetr (Euclid 2) by EUCLID:def_8; theorem Th20: :: JORDAN1:20 for s1 being Real for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where s, t is Real : s1 < s } holds P is open proof let s1 be Real; ::_thesis: for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where s, t is Real : s1 < s } holds P is open let P be Subset of (TOP-REAL 2); ::_thesis: ( P = { |[s,t]| where s, t is Real : s1 < s } implies P is open ) assume A1: P = { |[s,t]| where s, t is Real : s1 < s } ; ::_thesis: P is open reconsider PP = P as Subset of (TopSpaceMetr (Euclid 2)) by Lm9; for pe being Point of (Euclid 2) st pe in P holds ex r being real number st ( r > 0 & Ball (pe,r) c= P ) proof let pe be Point of (Euclid 2); ::_thesis: ( pe in P implies ex r being real number st ( r > 0 & Ball (pe,r) c= P ) ) assume pe in P ; ::_thesis: ex r being real number st ( r > 0 & Ball (pe,r) c= P ) then consider s, t being Real such that A2: |[s,t]| = pe and A3: s1 < s by A1; set r = (s - s1) / 2; A4: s - s1 > 0 by A3, XREAL_1:50; then A5: (s - s1) / 2 > 0 by XREAL_1:139; Ball (pe,((s - s1) / 2)) c= P proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Ball (pe,((s - s1) / 2)) or x in P ) assume x in Ball (pe,((s - s1) / 2)) ; ::_thesis: x in P then x in { q where q is Element of (Euclid 2) : dist (pe,q) < (s - s1) / 2 } by METRIC_1:17; then consider q being Element of (Euclid 2) such that A6: q = x and A7: dist (pe,q) < (s - s1) / 2 ; reconsider ppe = pe, pq = q as Point of (TOP-REAL 2) by EUCLID:22; (Pitag_dist 2) . (pe,q) = dist (pe,q) by METRIC_1:def_1; then A8: sqrt ((((ppe `1) - (pq `1)) ^2) + (((ppe `2) - (pq `2)) ^2)) < (s - s1) / 2 by A7, TOPREAL3:7; A9: 0 <= ((ppe `1) - (pq `1)) ^2 by XREAL_1:63; 0 <= ((ppe `2) - (pq `2)) ^2 by XREAL_1:63; then A10: 0 + 0 <= (((ppe `1) - (pq `1)) ^2) + (((ppe `2) - (pq `2)) ^2) by A9, XREAL_1:7; then 0 <= sqrt ((((ppe `1) - (pq `1)) ^2) + (((ppe `2) - (pq `2)) ^2)) by SQUARE_1:def_2; then (sqrt ((((ppe `1) - (pq `1)) ^2) + (((ppe `2) - (pq `2)) ^2))) ^2 < ((s - s1) / 2) ^2 by A8, SQUARE_1:16; then A11: (((ppe `1) - (pq `1)) ^2) + (((ppe `2) - (pq `2)) ^2) < ((s - s1) / 2) ^2 by A10, SQUARE_1:def_2; ((ppe `1) - (pq `1)) ^2 <= (((ppe `1) - (pq `1)) ^2) + (((ppe `2) - (pq `2)) ^2) by XREAL_1:31, XREAL_1:63; then ((ppe `1) - (pq `1)) ^2 < ((s - s1) / 2) ^2 by A11, XXREAL_0:2; then (ppe `1) - (pq `1) < (s - s1) / 2 by A5, SQUARE_1:15; then ppe `1 < (pq `1) + ((s - s1) / 2) by XREAL_1:19; then (ppe `1) - ((s - s1) / 2) < pq `1 by XREAL_1:19; then A12: s - ((s - s1) / 2) < pq `1 by A2, EUCLID:52; s - ((s - s1) / 2) = ((s - s1) / 2) + s1 ; then A13: s1 < s - ((s - s1) / 2) by A4, XREAL_1:29, XREAL_1:139; A14: |[(pq `1),(pq `2)]| = x by A6, EUCLID:53; s1 < pq `1 by A12, A13, XXREAL_0:2; hence x in P by A1, A14; ::_thesis: verum end; hence ex r being real number st ( r > 0 & Ball (pe,r) c= P ) by A4, XREAL_1:139; ::_thesis: verum end; then PP is open by TOPMETR:15; hence P is open by Lm9, PRE_TOPC:30; ::_thesis: verum end; theorem Th21: :: JORDAN1:21 for s1 being Real for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where s, t is Real : s1 > s } holds P is open proof let s1 be Real; ::_thesis: for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where s, t is Real : s1 > s } holds P is open let P be Subset of (TOP-REAL 2); ::_thesis: ( P = { |[s,t]| where s, t is Real : s1 > s } implies P is open ) assume A1: P = { |[s,t]| where s, t is Real : s1 > s } ; ::_thesis: P is open reconsider PP = P as Subset of (TopSpaceMetr (Euclid 2)) by Lm9; for pe being Point of (Euclid 2) st pe in P holds ex r being real number st ( r > 0 & Ball (pe,r) c= P ) proof let pe be Point of (Euclid 2); ::_thesis: ( pe in P implies ex r being real number st ( r > 0 & Ball (pe,r) c= P ) ) assume pe in P ; ::_thesis: ex r being real number st ( r > 0 & Ball (pe,r) c= P ) then consider s, t being Real such that A2: |[s,t]| = pe and A3: s1 > s by A1; set r = (s1 - s) / 2; A4: s1 - s > 0 by A3, XREAL_1:50; then A5: (s1 - s) / 2 > 0 by XREAL_1:139; Ball (pe,((s1 - s) / 2)) c= P proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Ball (pe,((s1 - s) / 2)) or x in P ) assume x in Ball (pe,((s1 - s) / 2)) ; ::_thesis: x in P then x in { q where q is Element of (Euclid 2) : dist (pe,q) < (s1 - s) / 2 } by METRIC_1:17; then consider q being Element of (Euclid 2) such that A6: q = x and A7: dist (pe,q) < (s1 - s) / 2 ; reconsider ppe = pe, pq = q as Point of (TOP-REAL 2) by EUCLID:22; (Pitag_dist 2) . (pe,q) = dist (pe,q) by METRIC_1:def_1; then A8: sqrt ((((ppe `1) - (pq `1)) ^2) + (((ppe `2) - (pq `2)) ^2)) < (s1 - s) / 2 by A7, TOPREAL3:7; A9: 0 <= ((ppe `1) - (pq `1)) ^2 by XREAL_1:63; 0 <= ((ppe `2) - (pq `2)) ^2 by XREAL_1:63; then A10: 0 + 0 <= (((ppe `1) - (pq `1)) ^2) + (((ppe `2) - (pq `2)) ^2) by A9, XREAL_1:7; then 0 <= sqrt ((((ppe `1) - (pq `1)) ^2) + (((ppe `2) - (pq `2)) ^2)) by SQUARE_1:def_2; then (sqrt ((((ppe `1) - (pq `1)) ^2) + (((ppe `2) - (pq `2)) ^2))) ^2 < ((s1 - s) / 2) ^2 by A8, SQUARE_1:16; then A11: (((ppe `1) - (pq `1)) ^2) + (((ppe `2) - (pq `2)) ^2) < ((s1 - s) / 2) ^2 by A10, SQUARE_1:def_2; ((ppe `1) - (pq `1)) ^2 <= (((ppe `1) - (pq `1)) ^2) + (((ppe `2) - (pq `2)) ^2) by XREAL_1:31, XREAL_1:63; then ((pq `1) - (ppe `1)) ^2 < ((s1 - s) / 2) ^2 by A11, XXREAL_0:2; then (pq `1) - (ppe `1) < (s1 - s) / 2 by A5, SQUARE_1:15; then (ppe `1) + ((s1 - s) / 2) > pq `1 by XREAL_1:19; then A12: s + ((s1 - s) / 2) > pq `1 by A2, EUCLID:52; s + ((s1 - s) / 2) = s1 - ((s1 - s) / 2) ; then A13: s1 > s + ((s1 - s) / 2) by A4, XREAL_1:44, XREAL_1:139; A14: |[(pq `1),(pq `2)]| = x by A6, EUCLID:53; s1 > pq `1 by A12, A13, XXREAL_0:2; hence x in P by A1, A14; ::_thesis: verum end; hence ex r being real number st ( r > 0 & Ball (pe,r) c= P ) by A4, XREAL_1:139; ::_thesis: verum end; then PP is open by TOPMETR:15; hence P is open by Lm9, PRE_TOPC:30; ::_thesis: verum end; theorem Th22: :: JORDAN1:22 for s1 being Real for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where s, t is Real : s1 < t } holds P is open proof let s1 be Real; ::_thesis: for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where s, t is Real : s1 < t } holds P is open let P be Subset of (TOP-REAL 2); ::_thesis: ( P = { |[s,t]| where s, t is Real : s1 < t } implies P is open ) assume A1: P = { |[s,t]| where s, t is Real : s1 < t } ; ::_thesis: P is open reconsider PP = P as Subset of (TopSpaceMetr (Euclid 2)) by Lm9; for pe being Point of (Euclid 2) st pe in P holds ex r being real number st ( r > 0 & Ball (pe,r) c= P ) proof let pe be Point of (Euclid 2); ::_thesis: ( pe in P implies ex r being real number st ( r > 0 & Ball (pe,r) c= P ) ) assume pe in P ; ::_thesis: ex r being real number st ( r > 0 & Ball (pe,r) c= P ) then consider s, t being Real such that A2: |[s,t]| = pe and A3: s1 < t by A1; set r = (t - s1) / 2; A4: t - s1 > 0 by A3, XREAL_1:50; then A5: (t - s1) / 2 > 0 by XREAL_1:139; Ball (pe,((t - s1) / 2)) c= P proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Ball (pe,((t - s1) / 2)) or x in P ) assume x in Ball (pe,((t - s1) / 2)) ; ::_thesis: x in P then x in { q where q is Element of (Euclid 2) : dist (pe,q) < (t - s1) / 2 } by METRIC_1:17; then consider q being Element of (Euclid 2) such that A6: q = x and A7: dist (pe,q) < (t - s1) / 2 ; reconsider ppe = pe, pq = q as Point of (TOP-REAL 2) by EUCLID:22; (Pitag_dist 2) . (pe,q) = dist (pe,q) by METRIC_1:def_1; then A8: sqrt ((((ppe `1) - (pq `1)) ^2) + (((ppe `2) - (pq `2)) ^2)) < (t - s1) / 2 by A7, TOPREAL3:7; A9: 0 <= ((ppe `1) - (pq `1)) ^2 by XREAL_1:63; 0 <= ((ppe `2) - (pq `2)) ^2 by XREAL_1:63; then A10: 0 + 0 <= (((ppe `1) - (pq `1)) ^2) + (((ppe `2) - (pq `2)) ^2) by A9, XREAL_1:7; then 0 <= sqrt ((((ppe `1) - (pq `1)) ^2) + (((ppe `2) - (pq `2)) ^2)) by SQUARE_1:def_2; then (sqrt ((((ppe `1) - (pq `1)) ^2) + (((ppe `2) - (pq `2)) ^2))) ^2 < ((t - s1) / 2) ^2 by A8, SQUARE_1:16; then A11: (((ppe `1) - (pq `1)) ^2) + (((ppe `2) - (pq `2)) ^2) < ((t - s1) / 2) ^2 by A10, SQUARE_1:def_2; ((ppe `2) - (pq `2)) ^2 <= (((ppe `2) - (pq `2)) ^2) + (((ppe `1) - (pq `1)) ^2) by XREAL_1:31, XREAL_1:63; then ((ppe `2) - (pq `2)) ^2 < ((t - s1) / 2) ^2 by A11, XXREAL_0:2; then (ppe `2) - (pq `2) < (t - s1) / 2 by A5, SQUARE_1:15; then ppe `2 < (pq `2) + ((t - s1) / 2) by XREAL_1:19; then (ppe `2) - ((t - s1) / 2) < pq `2 by XREAL_1:19; then A12: t - ((t - s1) / 2) < pq `2 by A2, EUCLID:52; t - ((t - s1) / 2) = ((t - s1) / 2) + s1 ; then A13: s1 < t - ((t - s1) / 2) by A4, XREAL_1:29, XREAL_1:139; A14: |[(pq `1),(pq `2)]| = x by A6, EUCLID:53; s1 < pq `2 by A12, A13, XXREAL_0:2; hence x in P by A1, A14; ::_thesis: verum end; hence ex r being real number st ( r > 0 & Ball (pe,r) c= P ) by A4, XREAL_1:139; ::_thesis: verum end; then PP is open by TOPMETR:15; hence P is open by Lm9, PRE_TOPC:30; ::_thesis: verum end; theorem Th23: :: JORDAN1:23 for s1 being Real for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where s, t is Real : s1 > t } holds P is open proof let s1 be Real; ::_thesis: for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where s, t is Real : s1 > t } holds P is open let P be Subset of (TOP-REAL 2); ::_thesis: ( P = { |[s,t]| where s, t is Real : s1 > t } implies P is open ) assume A1: P = { |[s,t]| where s, t is Real : s1 > t } ; ::_thesis: P is open reconsider PP = P as Subset of (TopSpaceMetr (Euclid 2)) by Lm9; for pe being Point of (Euclid 2) st pe in P holds ex r being real number st ( r > 0 & Ball (pe,r) c= P ) proof let pe be Point of (Euclid 2); ::_thesis: ( pe in P implies ex r being real number st ( r > 0 & Ball (pe,r) c= P ) ) assume pe in P ; ::_thesis: ex r being real number st ( r > 0 & Ball (pe,r) c= P ) then consider s, t being Real such that A2: |[s,t]| = pe and A3: s1 > t by A1; set r = (s1 - t) / 2; A4: s1 - t > 0 by A3, XREAL_1:50; then A5: (s1 - t) / 2 > 0 by XREAL_1:139; Ball (pe,((s1 - t) / 2)) c= P proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Ball (pe,((s1 - t) / 2)) or x in P ) assume x in Ball (pe,((s1 - t) / 2)) ; ::_thesis: x in P then x in { q where q is Element of (Euclid 2) : dist (pe,q) < (s1 - t) / 2 } by METRIC_1:17; then consider q being Element of (Euclid 2) such that A6: q = x and A7: dist (pe,q) < (s1 - t) / 2 ; reconsider ppe = pe, pq = q as Point of (TOP-REAL 2) by EUCLID:22; (Pitag_dist 2) . (pe,q) = dist (pe,q) by METRIC_1:def_1; then A8: sqrt ((((ppe `1) - (pq `1)) ^2) + (((ppe `2) - (pq `2)) ^2)) < (s1 - t) / 2 by A7, TOPREAL3:7; A9: 0 <= ((ppe `1) - (pq `1)) ^2 by XREAL_1:63; 0 <= ((ppe `2) - (pq `2)) ^2 by XREAL_1:63; then A10: 0 + 0 <= (((ppe `1) - (pq `1)) ^2) + (((ppe `2) - (pq `2)) ^2) by A9, XREAL_1:7; then 0 <= sqrt ((((ppe `1) - (pq `1)) ^2) + (((ppe `2) - (pq `2)) ^2)) by SQUARE_1:def_2; then (sqrt ((((ppe `1) - (pq `1)) ^2) + (((ppe `2) - (pq `2)) ^2))) ^2 < ((s1 - t) / 2) ^2 by A8, SQUARE_1:16; then A11: (((ppe `1) - (pq `1)) ^2) + (((ppe `2) - (pq `2)) ^2) < ((s1 - t) / 2) ^2 by A10, SQUARE_1:def_2; ((ppe `2) - (pq `2)) ^2 <= (((ppe `1) - (pq `1)) ^2) + (((ppe `2) - (pq `2)) ^2) by XREAL_1:31, XREAL_1:63; then ((pq `2) - (ppe `2)) ^2 < ((s1 - t) / 2) ^2 by A11, XXREAL_0:2; then (pq `2) - (ppe `2) < (s1 - t) / 2 by A5, SQUARE_1:15; then (ppe `2) + ((s1 - t) / 2) > pq `2 by XREAL_1:19; then A12: t + ((s1 - t) / 2) > pq `2 by A2, EUCLID:52; t + ((s1 - t) / 2) = s1 - ((s1 - t) / 2) ; then A13: s1 > t + ((s1 - t) / 2) by A4, XREAL_1:44, XREAL_1:139; A14: |[(pq `1),(pq `2)]| = x by A6, EUCLID:53; s1 > pq `2 by A12, A13, XXREAL_0:2; hence x in P by A1, A14; ::_thesis: verum end; hence ex r being real number st ( r > 0 & Ball (pe,r) c= P ) by A4, XREAL_1:139; ::_thesis: verum end; then PP is open by TOPMETR:15; hence P is open by Lm9, PRE_TOPC:30; ::_thesis: verum end; theorem Th24: :: JORDAN1:24 for s1, t1, s2, t2 being Real for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where s, t is Real : ( s1 < s & s < s2 & t1 < t & t < t2 ) } holds P is open proof let s1, t1, s2, t2 be Real; ::_thesis: for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where s, t is Real : ( s1 < s & s < s2 & t1 < t & t < t2 ) } holds P is open let P be Subset of (TOP-REAL 2); ::_thesis: ( P = { |[s,t]| where s, t is Real : ( s1 < s & s < s2 & t1 < t & t < t2 ) } implies P is open ) assume A1: P = { |[s,t]| where s, t is Real : ( s1 < s & s < s2 & t1 < t & t < t2 ) } ; ::_thesis: P is open reconsider P1 = { |[s,t]| where s, t is Real : s1 < s } as Subset of (TOP-REAL 2) by Lm2, Lm5; reconsider P2 = { |[s,t]| where s, t is Real : s < s2 } as Subset of (TOP-REAL 2) by Lm2, Lm3; reconsider P3 = { |[s,t]| where s, t is Real : t1 < t } as Subset of (TOP-REAL 2) by Lm2, Lm6; reconsider P4 = { |[s,t]| where s, t is Real : t < t2 } as Subset of (TOP-REAL 2) by Lm2, Lm4; A2: P = ((P1 /\ P2) /\ P3) /\ P4 by A1, Th7; A3: P1 is open by Th20; P2 is open by Th21; then A4: P1 /\ P2 is open by A3, TOPS_1:11; A5: P3 is open by Th22; A6: P4 is open by Th23; (P1 /\ P2) /\ P3 is open by A4, A5, TOPS_1:11; hence P is open by A2, A6, TOPS_1:11; ::_thesis: verum end; theorem Th25: :: JORDAN1:25 for s1, t1, s2, t2 being Real for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where s, t is Real : ( not s1 <= s or not s <= s2 or not t1 <= t or not t <= t2 ) } holds P is open proof let s1, t1, s2, t2 be Real; ::_thesis: for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where s, t is Real : ( not s1 <= s or not s <= s2 or not t1 <= t or not t <= t2 ) } holds P is open let P be Subset of (TOP-REAL 2); ::_thesis: ( P = { |[s,t]| where s, t is Real : ( not s1 <= s or not s <= s2 or not t1 <= t or not t <= t2 ) } implies P is open ) assume P = { |[s,t]| where s, t is Real : ( not s1 <= s or not s <= s2 or not t1 <= t or not t <= t2 ) } ; ::_thesis: P is open then A1: P = (( { |[s3,t3]| where s3, t3 is Real : s3 < s1 } \/ { |[s4,t4]| where s4, t4 is Real : t4 < t1 } ) \/ { |[s5,t5]| where s5, t5 is Real : s2 < s5 } ) \/ { |[s6,t6]| where s6, t6 is Real : t2 < t6 } by Th8; reconsider A0 = { |[s,t]| where s, t is Real : s < s1 } as Subset of (TOP-REAL 2) by Lm2, Lm3; reconsider A1 = { |[s,t]| where s, t is Real : t < t1 } as Subset of (TOP-REAL 2) by Lm2, Lm4; reconsider A2 = { |[s,t]| where s, t is Real : s2 < s } as Subset of (TOP-REAL 2) by Lm2, Lm5; reconsider A3 = { |[s,t]| where s, t is Real : t2 < t } as Subset of (TOP-REAL 2) by Lm2, Lm6; A2: A0 is open by Th21; A1 is open by Th23; then A3: A0 \/ A1 is open by A2, TOPS_1:10; A2 is open by Th20; then A4: (A0 \/ A1) \/ A2 is open by A3, TOPS_1:10; A3 is open by Th22; hence P is open by A1, A4, TOPS_1:10; ::_thesis: verum end; theorem Th26: :: JORDAN1:26 for s1, t1, s2, t2 being Real for P, Q being Subset of (TOP-REAL 2) st P = { |[sa,ta]| where sa, ta is Real : ( s1 < sa & sa < s2 & t1 < ta & ta < t2 ) } & Q = { |[sb,tb]| where sb, tb is Real : ( not s1 <= sb or not sb <= s2 or not t1 <= tb or not tb <= t2 ) } holds P misses Q proof let s1, t1, s2, t2 be Real; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st P = { |[sa,ta]| where sa, ta is Real : ( s1 < sa & sa < s2 & t1 < ta & ta < t2 ) } & Q = { |[sb,tb]| where sb, tb is Real : ( not s1 <= sb or not sb <= s2 or not t1 <= tb or not tb <= t2 ) } holds P misses Q let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( P = { |[sa,ta]| where sa, ta is Real : ( s1 < sa & sa < s2 & t1 < ta & ta < t2 ) } & Q = { |[sb,tb]| where sb, tb is Real : ( not s1 <= sb or not sb <= s2 or not t1 <= tb or not tb <= t2 ) } implies P misses Q ) assume that A1: P = { |[sa,ta]| where sa, ta is Real : ( s1 < sa & sa < s2 & t1 < ta & ta < t2 ) } and A2: Q = { |[sb,tb]| where sb, tb is Real : ( not s1 <= sb or not sb <= s2 or not t1 <= tb or not tb <= t2 ) } ; ::_thesis: P misses Q assume not P misses Q ; ::_thesis: contradiction then consider x being set such that A3: x in P and A4: x in Q by XBOOLE_0:3; consider sa, ta being Real such that A5: |[sa,ta]| = x and A6: s1 < sa and A7: sa < s2 and A8: t1 < ta and A9: ta < t2 by A1, A3; A10: ex sb, tb being Real st ( |[sb,tb]| = x & ( not s1 <= sb or not sb <= s2 or not t1 <= tb or not tb <= t2 ) ) by A2, A4; set p = |[sa,ta]|; A11: |[sa,ta]| `1 = sa by EUCLID:52; |[sa,ta]| `2 = ta by EUCLID:52; hence contradiction by A5, A6, A7, A8, A9, A10, A11, EUCLID:52; ::_thesis: verum end; theorem Th27: :: JORDAN1:27 for s1, s2, t1, t2 being Real holds { p where p is Point of (TOP-REAL 2) : ( s1 < p `1 & p `1 < s2 & t1 < p `2 & p `2 < t2 ) } = { |[sa,ta]| where sa, ta is Real : ( s1 < sa & sa < s2 & t1 < ta & ta < t2 ) } proof let s1, s2, t1, t2 be Real; ::_thesis: { p where p is Point of (TOP-REAL 2) : ( s1 < p `1 & p `1 < s2 & t1 < p `2 & p `2 < t2 ) } = { |[sa,ta]| where sa, ta is Real : ( s1 < sa & sa < s2 & t1 < ta & ta < t2 ) } now__::_thesis:_for_x_being_set_holds_ (_x_in__{__p_where_p_is_Point_of_(TOP-REAL_2)_:_(_s1_<_p_`1_&_p_`1_<_s2_&_t1_<_p_`2_&_p_`2_<_t2_)__}__iff_x_in__{__|[sa,ta]|_where_sa,_ta_is_Real_:_(_s1_<_sa_&_sa_<_s2_&_t1_<_ta_&_ta_<_t2_)__}__) let x be set ; ::_thesis: ( x in { p where p is Point of (TOP-REAL 2) : ( s1 < p `1 & p `1 < s2 & t1 < p `2 & p `2 < t2 ) } iff x in { |[sa,ta]| where sa, ta is Real : ( s1 < sa & sa < s2 & t1 < ta & ta < t2 ) } ) A1: now__::_thesis:_(_x_in__{__p_where_p_is_Point_of_(TOP-REAL_2)_:_(_s1_<_p_`1_&_p_`1_<_s2_&_t1_<_p_`2_&_p_`2_<_t2_)__}__implies_x_in__{__|[s1a,t1a]|_where_s1a,_t1a_is_Real_:_(_s1_<_s1a_&_s1a_<_s2_&_t1_<_t1a_&_t1a_<_t2_)__}__) assume x in { p where p is Point of (TOP-REAL 2) : ( s1 < p `1 & p `1 < s2 & t1 < p `2 & p `2 < t2 ) } ; ::_thesis: x in { |[s1a,t1a]| where s1a, t1a is Real : ( s1 < s1a & s1a < s2 & t1 < t1a & t1a < t2 ) } then consider pp being Point of (TOP-REAL 2) such that A2: pp = x and A3: s1 < pp `1 and A4: pp `1 < s2 and A5: t1 < pp `2 and A6: pp `2 < t2 ; |[(pp `1),(pp `2)]| = x by A2, EUCLID:53; hence x in { |[s1a,t1a]| where s1a, t1a is Real : ( s1 < s1a & s1a < s2 & t1 < t1a & t1a < t2 ) } by A3, A4, A5, A6; ::_thesis: verum end; now__::_thesis:_(_x_in__{__|[sa,ta]|_where_sa,_ta_is_Real_:_(_s1_<_sa_&_sa_<_s2_&_t1_<_ta_&_ta_<_t2_)__}__implies_x_in__{__p_where_p_is_Point_of_(TOP-REAL_2)_:_(_s1_<_p_`1_&_p_`1_<_s2_&_t1_<_p_`2_&_p_`2_<_t2_)__}__) assume x in { |[sa,ta]| where sa, ta is Real : ( s1 < sa & sa < s2 & t1 < ta & ta < t2 ) } ; ::_thesis: x in { p where p is Point of (TOP-REAL 2) : ( s1 < p `1 & p `1 < s2 & t1 < p `2 & p `2 < t2 ) } then consider sa, ta being Real such that A7: |[sa,ta]| = x and A8: s1 < sa and A9: sa < s2 and A10: t1 < ta and A11: ta < t2 ; set pa = |[sa,ta]|; A12: s1 < |[sa,ta]| `1 by A8, EUCLID:52; A13: |[sa,ta]| `1 < s2 by A9, EUCLID:52; A14: t1 < |[sa,ta]| `2 by A10, EUCLID:52; |[sa,ta]| `2 < t2 by A11, EUCLID:52; hence x in { p where p is Point of (TOP-REAL 2) : ( s1 < p `1 & p `1 < s2 & t1 < p `2 & p `2 < t2 ) } by A7, A12, A13, A14; ::_thesis: verum end; hence ( x in { p where p is Point of (TOP-REAL 2) : ( s1 < p `1 & p `1 < s2 & t1 < p `2 & p `2 < t2 ) } iff x in { |[sa,ta]| where sa, ta is Real : ( s1 < sa & sa < s2 & t1 < ta & ta < t2 ) } ) by A1; ::_thesis: verum end; hence { p where p is Point of (TOP-REAL 2) : ( s1 < p `1 & p `1 < s2 & t1 < p `2 & p `2 < t2 ) } = { |[sa,ta]| where sa, ta is Real : ( s1 < sa & sa < s2 & t1 < ta & ta < t2 ) } by TARSKI:1; ::_thesis: verum end; theorem Th28: :: JORDAN1:28 for s1, s2, t1, t2 being Real holds { qc where qc is Point of (TOP-REAL 2) : ( not s1 <= qc `1 or not qc `1 <= s2 or not t1 <= qc `2 or not qc `2 <= t2 ) } = { |[sb,tb]| where sb, tb is Real : ( not s1 <= sb or not sb <= s2 or not t1 <= tb or not tb <= t2 ) } proof let s1, s2, t1, t2 be Real; ::_thesis: { qc where qc is Point of (TOP-REAL 2) : ( not s1 <= qc `1 or not qc `1 <= s2 or not t1 <= qc `2 or not qc `2 <= t2 ) } = { |[sb,tb]| where sb, tb is Real : ( not s1 <= sb or not sb <= s2 or not t1 <= tb or not tb <= t2 ) } now__::_thesis:_for_x_being_set_holds_ (_x_in__{__qc_where_qc_is_Point_of_(TOP-REAL_2)_:_(_not_s1_<=_qc_`1_or_not_qc_`1_<=_s2_or_not_t1_<=_qc_`2_or_not_qc_`2_<=_t2_)__}__iff_x_in__{__|[sb,tb]|_where_sb,_tb_is_Real_:_(_not_s1_<=_sb_or_not_sb_<=_s2_or_not_t1_<=_tb_or_not_tb_<=_t2_)__}__) let x be set ; ::_thesis: ( x in { qc where qc is Point of (TOP-REAL 2) : ( not s1 <= qc `1 or not qc `1 <= s2 or not t1 <= qc `2 or not qc `2 <= t2 ) } iff x in { |[sb,tb]| where sb, tb is Real : ( not s1 <= sb or not sb <= s2 or not t1 <= tb or not tb <= t2 ) } ) A1: now__::_thesis:_(_x_in__{__qc_where_qc_is_Point_of_(TOP-REAL_2)_:_(_not_s1_<=_qc_`1_or_not_qc_`1_<=_s2_or_not_t1_<=_qc_`2_or_not_qc_`2_<=_t2_)__}__implies_x_in__{__|[s2a,t2a]|_where_s2a,_t2a_is_Real_:_(_not_s1_<=_s2a_or_not_s2a_<=_s2_or_not_t1_<=_t2a_or_not_t2a_<=_t2_)__}__) assume x in { qc where qc is Point of (TOP-REAL 2) : ( not s1 <= qc `1 or not qc `1 <= s2 or not t1 <= qc `2 or not qc `2 <= t2 ) } ; ::_thesis: x in { |[s2a,t2a]| where s2a, t2a is Real : ( not s1 <= s2a or not s2a <= s2 or not t1 <= t2a or not t2a <= t2 ) } then consider q being Point of (TOP-REAL 2) such that A2: q = x and A3: ( not s1 <= q `1 or not q `1 <= s2 or not t1 <= q `2 or not q `2 <= t2 ) ; |[(q `1),(q `2)]| = x by A2, EUCLID:53; hence x in { |[s2a,t2a]| where s2a, t2a is Real : ( not s1 <= s2a or not s2a <= s2 or not t1 <= t2a or not t2a <= t2 ) } by A3; ::_thesis: verum end; now__::_thesis:_(_x_in__{__|[sb,tb]|_where_sb,_tb_is_Real_:_(_not_s1_<=_sb_or_not_sb_<=_s2_or_not_t1_<=_tb_or_not_tb_<=_t2_)__}__implies_x_in__{__qc_where_qc_is_Point_of_(TOP-REAL_2)_:_(_not_s1_<=_qc_`1_or_not_qc_`1_<=_s2_or_not_t1_<=_qc_`2_or_not_qc_`2_<=_t2_)__}__) assume x in { |[sb,tb]| where sb, tb is Real : ( not s1 <= sb or not sb <= s2 or not t1 <= tb or not tb <= t2 ) } ; ::_thesis: x in { qc where qc is Point of (TOP-REAL 2) : ( not s1 <= qc `1 or not qc `1 <= s2 or not t1 <= qc `2 or not qc `2 <= t2 ) } then consider sb, tb being Real such that A4: |[sb,tb]| = x and A5: ( not s1 <= sb or not sb <= s2 or not t1 <= tb or not tb <= t2 ) ; set qa = |[sb,tb]|; ( not s1 <= |[sb,tb]| `1 or not |[sb,tb]| `1 <= s2 or not t1 <= |[sb,tb]| `2 or not |[sb,tb]| `2 <= t2 ) by A5, EUCLID:52; hence x in { qc where qc is Point of (TOP-REAL 2) : ( not s1 <= qc `1 or not qc `1 <= s2 or not t1 <= qc `2 or not qc `2 <= t2 ) } by A4; ::_thesis: verum end; hence ( x in { qc where qc is Point of (TOP-REAL 2) : ( not s1 <= qc `1 or not qc `1 <= s2 or not t1 <= qc `2 or not qc `2 <= t2 ) } iff x in { |[sb,tb]| where sb, tb is Real : ( not s1 <= sb or not sb <= s2 or not t1 <= tb or not tb <= t2 ) } ) by A1; ::_thesis: verum end; hence { qc where qc is Point of (TOP-REAL 2) : ( not s1 <= qc `1 or not qc `1 <= s2 or not t1 <= qc `2 or not qc `2 <= t2 ) } = { |[sb,tb]| where sb, tb is Real : ( not s1 <= sb or not sb <= s2 or not t1 <= tb or not tb <= t2 ) } by TARSKI:1; ::_thesis: verum end; theorem Th29: :: JORDAN1:29 for s1, s2, t1, t2 being Real holds { p0 where p0 is Point of (TOP-REAL 2) : ( s1 < p0 `1 & p0 `1 < s2 & t1 < p0 `2 & p0 `2 < t2 ) } is Subset of (TOP-REAL 2) proof let s1, s2, t1, t2 be Real; ::_thesis: { p0 where p0 is Point of (TOP-REAL 2) : ( s1 < p0 `1 & p0 `1 < s2 & t1 < p0 `2 & p0 `2 < t2 ) } is Subset of (TOP-REAL 2) { |[sc,tc]| where sc, tc is Real : ( s1 < sc & sc < s2 & t1 < tc & tc < t2 ) } is Subset of (TOP-REAL 2) by Lm2, Lm7; hence { p0 where p0 is Point of (TOP-REAL 2) : ( s1 < p0 `1 & p0 `1 < s2 & t1 < p0 `2 & p0 `2 < t2 ) } is Subset of (TOP-REAL 2) by Th27; ::_thesis: verum end; theorem Th30: :: JORDAN1:30 for s1, s2, t1, t2 being Real holds { pq where pq is Point of (TOP-REAL 2) : ( not s1 <= pq `1 or not pq `1 <= s2 or not t1 <= pq `2 or not pq `2 <= t2 ) } is Subset of (TOP-REAL 2) proof let s1, s2, t1, t2 be Real; ::_thesis: { pq where pq is Point of (TOP-REAL 2) : ( not s1 <= pq `1 or not pq `1 <= s2 or not t1 <= pq `2 or not pq `2 <= t2 ) } is Subset of (TOP-REAL 2) { |[sb,tb]| where sb, tb is Real : ( not s1 <= sb or not sb <= s2 or not t1 <= tb or not tb <= t2 ) } is Subset of (TOP-REAL 2) by Lm2, Lm8; hence { pq where pq is Point of (TOP-REAL 2) : ( not s1 <= pq `1 or not pq `1 <= s2 or not t1 <= pq `2 or not pq `2 <= t2 ) } is Subset of (TOP-REAL 2) by Th28; ::_thesis: verum end; theorem Th31: :: JORDAN1:31 for s1, t1, s2, t2 being Real for P being Subset of (TOP-REAL 2) st P = { p0 where p0 is Point of (TOP-REAL 2) : ( s1 < p0 `1 & p0 `1 < s2 & t1 < p0 `2 & p0 `2 < t2 ) } holds P is convex proof let s1, t1, s2, t2 be Real; ::_thesis: for P being Subset of (TOP-REAL 2) st P = { p0 where p0 is Point of (TOP-REAL 2) : ( s1 < p0 `1 & p0 `1 < s2 & t1 < p0 `2 & p0 `2 < t2 ) } holds P is convex let P be Subset of (TOP-REAL 2); ::_thesis: ( P = { p0 where p0 is Point of (TOP-REAL 2) : ( s1 < p0 `1 & p0 `1 < s2 & t1 < p0 `2 & p0 `2 < t2 ) } implies P is convex ) assume P = { p0 where p0 is Point of (TOP-REAL 2) : ( s1 < p0 `1 & p0 `1 < s2 & t1 < p0 `2 & p0 `2 < t2 ) } ; ::_thesis: P is convex then P = { |[sa,ta]| where sa, ta is Real : ( s1 < sa & sa < s2 & t1 < ta & ta < t2 ) } by Th27; hence P is convex by Th9; ::_thesis: verum end; theorem Th32: :: JORDAN1:32 for s1, t1, s2, t2 being Real for P being Subset of (TOP-REAL 2) st P = { pq where pq is Point of (TOP-REAL 2) : ( not s1 <= pq `1 or not pq `1 <= s2 or not t1 <= pq `2 or not pq `2 <= t2 ) } holds P is connected proof let s1, t1, s2, t2 be Real; ::_thesis: for P being Subset of (TOP-REAL 2) st P = { pq where pq is Point of (TOP-REAL 2) : ( not s1 <= pq `1 or not pq `1 <= s2 or not t1 <= pq `2 or not pq `2 <= t2 ) } holds P is connected let P be Subset of (TOP-REAL 2); ::_thesis: ( P = { pq where pq is Point of (TOP-REAL 2) : ( not s1 <= pq `1 or not pq `1 <= s2 or not t1 <= pq `2 or not pq `2 <= t2 ) } implies P is connected ) assume P = { pq where pq is Point of (TOP-REAL 2) : ( not s1 <= pq `1 or not pq `1 <= s2 or not t1 <= pq `2 or not pq `2 <= t2 ) } ; ::_thesis: P is connected then P = { |[sb,tb]| where sb, tb is Real : ( not s1 <= sb or not sb <= s2 or not t1 <= tb or not tb <= t2 ) } by Th28; hence P is connected by Th19; ::_thesis: verum end; theorem Th33: :: JORDAN1:33 for s1, t1, s2, t2 being Real for P being Subset of (TOP-REAL 2) st P = { p0 where p0 is Point of (TOP-REAL 2) : ( s1 < p0 `1 & p0 `1 < s2 & t1 < p0 `2 & p0 `2 < t2 ) } holds P is open proof let s1, t1, s2, t2 be Real; ::_thesis: for P being Subset of (TOP-REAL 2) st P = { p0 where p0 is Point of (TOP-REAL 2) : ( s1 < p0 `1 & p0 `1 < s2 & t1 < p0 `2 & p0 `2 < t2 ) } holds P is open let P be Subset of (TOP-REAL 2); ::_thesis: ( P = { p0 where p0 is Point of (TOP-REAL 2) : ( s1 < p0 `1 & p0 `1 < s2 & t1 < p0 `2 & p0 `2 < t2 ) } implies P is open ) assume P = { p0 where p0 is Point of (TOP-REAL 2) : ( s1 < p0 `1 & p0 `1 < s2 & t1 < p0 `2 & p0 `2 < t2 ) } ; ::_thesis: P is open then P = { |[sa,ta]| where sa, ta is Real : ( s1 < sa & sa < s2 & t1 < ta & ta < t2 ) } by Th27; hence P is open by Th24; ::_thesis: verum end; theorem Th34: :: JORDAN1:34 for s1, t1, s2, t2 being Real for P being Subset of (TOP-REAL 2) st P = { pq where pq is Point of (TOP-REAL 2) : ( not s1 <= pq `1 or not pq `1 <= s2 or not t1 <= pq `2 or not pq `2 <= t2 ) } holds P is open proof let s1, t1, s2, t2 be Real; ::_thesis: for P being Subset of (TOP-REAL 2) st P = { pq where pq is Point of (TOP-REAL 2) : ( not s1 <= pq `1 or not pq `1 <= s2 or not t1 <= pq `2 or not pq `2 <= t2 ) } holds P is open let P be Subset of (TOP-REAL 2); ::_thesis: ( P = { pq where pq is Point of (TOP-REAL 2) : ( not s1 <= pq `1 or not pq `1 <= s2 or not t1 <= pq `2 or not pq `2 <= t2 ) } implies P is open ) assume P = { pq where pq is Point of (TOP-REAL 2) : ( not s1 <= pq `1 or not pq `1 <= s2 or not t1 <= pq `2 or not pq `2 <= t2 ) } ; ::_thesis: P is open then P = { |[sb,tb]| where sb, tb is Real : ( not s1 <= sb or not sb <= s2 or not t1 <= tb or not tb <= t2 ) } by Th28; hence P is open by Th25; ::_thesis: verum end; theorem Th35: :: JORDAN1:35 for s1, t1, s2, t2 being Real for P, Q being Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : ( s1 < p `1 & p `1 < s2 & t1 < p `2 & p `2 < t2 ) } & Q = { qc where qc is Point of (TOP-REAL 2) : ( not s1 <= qc `1 or not qc `1 <= s2 or not t1 <= qc `2 or not qc `2 <= t2 ) } holds P misses Q proof let s1, t1, s2, t2 be Real; ::_thesis: for P, Q being Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : ( s1 < p `1 & p `1 < s2 & t1 < p `2 & p `2 < t2 ) } & Q = { qc where qc is Point of (TOP-REAL 2) : ( not s1 <= qc `1 or not qc `1 <= s2 or not t1 <= qc `2 or not qc `2 <= t2 ) } holds P misses Q let P, Q be Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : ( s1 < p `1 & p `1 < s2 & t1 < p `2 & p `2 < t2 ) } & Q = { qc where qc is Point of (TOP-REAL 2) : ( not s1 <= qc `1 or not qc `1 <= s2 or not t1 <= qc `2 or not qc `2 <= t2 ) } implies P misses Q ) assume that A1: P = { p where p is Point of (TOP-REAL 2) : ( s1 < p `1 & p `1 < s2 & t1 < p `2 & p `2 < t2 ) } and A2: Q = { qc where qc is Point of (TOP-REAL 2) : ( not s1 <= qc `1 or not qc `1 <= s2 or not t1 <= qc `2 or not qc `2 <= t2 ) } ; ::_thesis: P misses Q A3: P = { |[sa,ta]| where sa, ta is Real : ( s1 < sa & sa < s2 & t1 < ta & ta < t2 ) } by A1, Th27; Q = { |[sb,tb]| where sb, tb is Real : ( not s1 <= sb or not sb <= s2 or not t1 <= tb or not tb <= t2 ) } by A2, Th28; hence P misses Q by A3, Th26; ::_thesis: verum end; theorem Th36: :: JORDAN1:36 for s1, t1, s2, t2 being Real for P, P1, P2 being Subset of (TOP-REAL 2) st s1 < s2 & t1 < t2 & P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P1 = { pa where pa is Point of (TOP-REAL 2) : ( s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) } & P2 = { pb where pb is Point of (TOP-REAL 2) : ( not s1 <= pb `1 or not pb `1 <= s2 or not t1 <= pb `2 or not pb `2 <= t2 ) } holds ( P ` = P1 \/ P2 & P ` <> {} & P1 misses P2 & ( for P1A, P2B being Subset of ((TOP-REAL 2) | (P `)) st P1A = P1 & P2B = P2 holds ( P1A is a_component & P2B is a_component ) ) ) proof let s1, t1, s2, t2 be Real; ::_thesis: for P, P1, P2 being Subset of (TOP-REAL 2) st s1 < s2 & t1 < t2 & P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P1 = { pa where pa is Point of (TOP-REAL 2) : ( s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) } & P2 = { pb where pb is Point of (TOP-REAL 2) : ( not s1 <= pb `1 or not pb `1 <= s2 or not t1 <= pb `2 or not pb `2 <= t2 ) } holds ( P ` = P1 \/ P2 & P ` <> {} & P1 misses P2 & ( for P1A, P2B being Subset of ((TOP-REAL 2) | (P `)) st P1A = P1 & P2B = P2 holds ( P1A is a_component & P2B is a_component ) ) ) let P, P1, P2 be Subset of (TOP-REAL 2); ::_thesis: ( s1 < s2 & t1 < t2 & P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P1 = { pa where pa is Point of (TOP-REAL 2) : ( s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) } & P2 = { pb where pb is Point of (TOP-REAL 2) : ( not s1 <= pb `1 or not pb `1 <= s2 or not t1 <= pb `2 or not pb `2 <= t2 ) } implies ( P ` = P1 \/ P2 & P ` <> {} & P1 misses P2 & ( for P1A, P2B being Subset of ((TOP-REAL 2) | (P `)) st P1A = P1 & P2B = P2 holds ( P1A is a_component & P2B is a_component ) ) ) ) assume that A1: s1 < s2 and A2: t1 < t2 and A3: P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } and A4: P1 = { pa where pa is Point of (TOP-REAL 2) : ( s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) } and A5: P2 = { pc where pc is Point of (TOP-REAL 2) : ( not s1 <= pc `1 or not pc `1 <= s2 or not t1 <= pc `2 or not pc `2 <= t2 ) } ; ::_thesis: ( P ` = P1 \/ P2 & P ` <> {} & P1 misses P2 & ( for P1A, P2B being Subset of ((TOP-REAL 2) | (P `)) st P1A = P1 & P2B = P2 holds ( P1A is a_component & P2B is a_component ) ) ) now__::_thesis:_for_x_being_set_st_x_in_P_`_holds_ x_in_P1_\/_P2 let x be set ; ::_thesis: ( x in P ` implies x in P1 \/ P2 ) assume A6: x in P ` ; ::_thesis: x in P1 \/ P2 then A7: not x in P by XBOOLE_0:def_5; reconsider pd = x as Point of (TOP-REAL 2) by A6; ( not ( pd `1 = s1 & pd `2 <= t2 & pd `2 >= t1 ) & not ( pd `1 <= s2 & pd `1 >= s1 & pd `2 = t2 ) & not ( pd `1 <= s2 & pd `1 >= s1 & pd `2 = t1 ) & not ( pd `1 = s2 & pd `2 <= t2 & pd `2 >= t1 ) ) by A3, A7; then ( ( s1 < pd `1 & pd `1 < s2 & t1 < pd `2 & pd `2 < t2 ) or not s1 <= pd `1 or not pd `1 <= s2 or not t1 <= pd `2 or not pd `2 <= t2 ) by XXREAL_0:1; then ( x in P1 or x in P2 ) by A4, A5; hence x in P1 \/ P2 by XBOOLE_0:def_3; ::_thesis: verum end; then A8: P ` c= P1 \/ P2 by TARSKI:def_3; now__::_thesis:_for_x_being_set_st_x_in_P1_\/_P2_holds_ x_in_P_` let x be set ; ::_thesis: ( x in P1 \/ P2 implies x in P ` ) assume A9: x in P1 \/ P2 ; ::_thesis: x in P ` now__::_thesis:_x_in_P_` percases ( x in P1 or x in P2 ) by A9, XBOOLE_0:def_3; supposeA10: x in P1 ; ::_thesis: x in P ` then A11: ex pa being Point of (TOP-REAL 2) st ( pa = x & s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) by A4; now__::_thesis:_not_x_in_P assume x in P ; ::_thesis: contradiction then ex pb being Point of (TOP-REAL 2) st ( pb = x & ( ( pb `1 = s1 & pb `2 <= t2 & pb `2 >= t1 ) or ( pb `1 <= s2 & pb `1 >= s1 & pb `2 = t2 ) or ( pb `1 <= s2 & pb `1 >= s1 & pb `2 = t1 ) or ( pb `1 = s2 & pb `2 <= t2 & pb `2 >= t1 ) ) ) by A3; hence contradiction by A11; ::_thesis: verum end; hence x in P ` by A10, SUBSET_1:29; ::_thesis: verum end; suppose x in P2 ; ::_thesis: x in P ` then consider pc being Point of (TOP-REAL 2) such that A12: pc = x and A13: ( not s1 <= pc `1 or not pc `1 <= s2 or not t1 <= pc `2 or not pc `2 <= t2 ) by A5; now__::_thesis:_not_pc_in_P assume pc in P ; ::_thesis: contradiction then ex p being Point of (TOP-REAL 2) st ( p = pc & ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) ) by A3; hence contradiction by A1, A2, A13; ::_thesis: verum end; hence x in P ` by A12, SUBSET_1:29; ::_thesis: verum end; end; end; hence x in P ` ; ::_thesis: verum end; then A14: P1 \/ P2 c= P ` by TARSKI:def_3; hence A15: P ` = P1 \/ P2 by A8, XBOOLE_0:def_10; ::_thesis: ( P ` <> {} & P1 misses P2 & ( for P1A, P2B being Subset of ((TOP-REAL 2) | (P `)) st P1A = P1 & P2B = P2 holds ( P1A is a_component & P2B is a_component ) ) ) set s3 = (s1 + s2) / 2; set t3 = (t1 + t2) / 2; A16: s1 + s1 < s1 + s2 by A1, XREAL_1:6; A17: t1 + t1 < t1 + t2 by A2, XREAL_1:6; A18: (s1 + s1) / 2 < (s1 + s2) / 2 by A16, XREAL_1:74; A19: (t1 + t1) / 2 < (t1 + t2) / 2 by A17, XREAL_1:74; A20: s1 + s2 < s2 + s2 by A1, XREAL_1:6; A21: t1 + t2 < t2 + t2 by A2, XREAL_1:6; A22: (s1 + s2) / 2 < (s2 + s2) / 2 by A20, XREAL_1:74; A23: (t1 + t2) / 2 < (t2 + t2) / 2 by A21, XREAL_1:74; set pp = |[((s1 + s2) / 2),((t1 + t2) / 2)]|; A24: |[((s1 + s2) / 2),((t1 + t2) / 2)]| `1 = (s1 + s2) / 2 by EUCLID:52; |[((s1 + s2) / 2),((t1 + t2) / 2)]| `2 = (t1 + t2) / 2 by EUCLID:52; then A25: |[((s1 + s2) / 2),((t1 + t2) / 2)]| in { pp2 where pp2 is Point of (TOP-REAL 2) : ( s1 < pp2 `1 & pp2 `1 < s2 & t1 < pp2 `2 & pp2 `2 < t2 ) } by A18, A19, A22, A23, A24; hence P ` <> {} by A4, A14; ::_thesis: ( P1 misses P2 & ( for P1A, P2B being Subset of ((TOP-REAL 2) | (P `)) st P1A = P1 & P2B = P2 holds ( P1A is a_component & P2B is a_component ) ) ) set P9 = P ` ; P1 misses P2 by A4, A5, Th35; hence A26: P1 /\ P2 = {} by XBOOLE_0:def_7; :: according to XBOOLE_0:def_7 ::_thesis: for P1A, P2B being Subset of ((TOP-REAL 2) | (P `)) st P1A = P1 & P2B = P2 holds ( P1A is a_component & P2B is a_component ) now__::_thesis:_for_P1A,_P2B_being_Subset_of_((TOP-REAL_2)_|_(P_`))_st_P1A_=_P1_&_P2B_=_P2_holds_ (_P1A_is_a_component_&_P1A_is_a_component_&_P2B_is_a_component_) let P1A, P2B be Subset of ((TOP-REAL 2) | (P `)); ::_thesis: ( P1A = P1 & P2B = P2 implies ( P1A is a_component & P1A is a_component & P2B is a_component ) ) assume that A27: P1A = P1 and A28: P2B = P2 ; ::_thesis: ( P1A is a_component & P1A is a_component & P2B is a_component ) P1 is convex by A4, Th31; then A29: P1A is connected by A27, CONNSP_1:23; A30: P2 is connected by A5, Th32; A31: P2 = { |[sa,ta]| where sa, ta is Real : ( not s1 <= sa or not sa <= s2 or not t1 <= ta or not ta <= t2 ) } by A5, Th28; reconsider A0 = { |[s3a,t3a]| where s3a, t3a is Real : s3a < s1 } as Subset of (TOP-REAL 2) by Lm2, Lm3; reconsider A1 = { |[s4,t4]| where s4, t4 is Real : t4 < t1 } as Subset of (TOP-REAL 2) by Lm2, Lm4; reconsider A2 = { |[s5,t5]| where s5, t5 is Real : s2 < s5 } as Subset of (TOP-REAL 2) by Lm2, Lm5; reconsider A3 = { |[s6,t6]| where s6, t6 is Real : t2 < t6 } as Subset of (TOP-REAL 2) by Lm2, Lm6; A32: P2 = ((A0 \/ A1) \/ A2) \/ A3 by A31, Th8; t2 + 1 > t2 by XREAL_1:29; then A33: |[(s2 + 1),(t2 + 1)]| in A3 ; A34: P2B is connected by A28, A30, CONNSP_1:23; A35: for CP being Subset of ((TOP-REAL 2) | (P `)) st CP is connected & P1A c= CP holds P1A = CP proof let CP be Subset of ((TOP-REAL 2) | (P `)); ::_thesis: ( CP is connected & P1A c= CP implies P1A = CP ) assume CP is connected ; ::_thesis: ( not P1A c= CP or P1A = CP ) then A36: ((TOP-REAL 2) | (P `)) | CP is connected by CONNSP_1:def_3; now__::_thesis:_(_P1A_c=_CP_&_P1A_c=_CP_implies_P1A_=_CP_) assume A37: P1A c= CP ; ::_thesis: ( not P1A c= CP or P1A = CP ) P1A /\ CP c= CP by XBOOLE_1:17; then reconsider AP = P1A /\ CP as Subset of (((TOP-REAL 2) | (P `)) | CP) by PRE_TOPC:8; A38: P1 /\ (P `) = P1A by A15, A27, XBOOLE_1:21; P1 is open by A4, Th33; then A39: P1 in the topology of (TOP-REAL 2) by PRE_TOPC:def_2; A40: P ` = [#] ((TOP-REAL 2) | (P `)) by PRE_TOPC:def_5; P1 /\ ([#] ((TOP-REAL 2) | (P `))) = P1A by A38, PRE_TOPC:def_5; then A41: P1A in the topology of ((TOP-REAL 2) | (P `)) by A39, PRE_TOPC:def_4; A42: CP = [#] (((TOP-REAL 2) | (P `)) | CP) by PRE_TOPC:def_5; A43: AP <> {} (((TOP-REAL 2) | (P `)) | CP) by A4, A25, A27, A37, XBOOLE_1:28; AP in the topology of (((TOP-REAL 2) | (P `)) | CP) by A41, A42, PRE_TOPC:def_4; then A44: AP is open by PRE_TOPC:def_2; P2B /\ CP c= CP by XBOOLE_1:17; then reconsider BP = P2B /\ CP as Subset of (((TOP-REAL 2) | (P `)) | CP) by PRE_TOPC:8; A45: P2 /\ (P `) = P2B by A15, A28, XBOOLE_1:21; P2 is open by A5, Th34; then A46: P2 in the topology of (TOP-REAL 2) by PRE_TOPC:def_2; P2 /\ ([#] ((TOP-REAL 2) | (P `))) = P2B by A45, PRE_TOPC:def_5; then A47: P2B in the topology of ((TOP-REAL 2) | (P `)) by A46, PRE_TOPC:def_4; CP = [#] (((TOP-REAL 2) | (P `)) | CP) by PRE_TOPC:def_5; then BP in the topology of (((TOP-REAL 2) | (P `)) | CP) by A47, PRE_TOPC:def_4; then A48: BP is open by PRE_TOPC:def_2; A49: CP = (P1A \/ P2B) /\ CP by A15, A27, A28, A40, XBOOLE_1:28 .= AP \/ BP by XBOOLE_1:23 ; now__::_thesis:_not_BP_<>_{} assume A50: BP <> {} ; ::_thesis: contradiction A51: AP /\ BP = (P1A /\ (P2B /\ CP)) /\ CP by XBOOLE_1:16 .= ((P1A /\ P2B) /\ CP) /\ CP by XBOOLE_1:16 .= (P1A /\ P2B) /\ (CP /\ CP) by XBOOLE_1:16 .= (P1A /\ P2B) /\ CP ; P1 misses P2 by A4, A5, Th35; then P1 /\ P2 = {} by XBOOLE_0:def_7; then AP misses BP by A27, A28, A51, XBOOLE_0:def_7; hence contradiction by A36, A42, A43, A44, A48, A49, A50, CONNSP_1:11; ::_thesis: verum end; hence ( not P1A c= CP or P1A = CP ) by A49, XBOOLE_1:28; ::_thesis: verum end; hence ( not P1A c= CP or P1A = CP ) ; ::_thesis: verum end; hence P1A is a_component by A29, CONNSP_1:def_5; ::_thesis: ( P1A is a_component & P2B is a_component ) for CP being Subset of ((TOP-REAL 2) | (P `)) st CP is connected & P2B c= CP holds P2B = CP proof let CP be Subset of ((TOP-REAL 2) | (P `)); ::_thesis: ( CP is connected & P2B c= CP implies P2B = CP ) assume CP is connected ; ::_thesis: ( not P2B c= CP or P2B = CP ) then A52: ((TOP-REAL 2) | (P `)) | CP is connected by CONNSP_1:def_3; assume A53: P2B c= CP ; ::_thesis: P2B = CP P2B /\ CP c= CP by XBOOLE_1:17; then reconsider BP = P2B /\ CP as Subset of (((TOP-REAL 2) | (P `)) | CP) by PRE_TOPC:8; A54: P2 /\ (P `) = P2B by A15, A28, XBOOLE_1:21; P2 is open by A5, Th34; then A55: P2 in the topology of (TOP-REAL 2) by PRE_TOPC:def_2; A56: P ` = [#] ((TOP-REAL 2) | (P `)) by PRE_TOPC:def_5; P2 /\ ([#] ((TOP-REAL 2) | (P `))) = P2B by A54, PRE_TOPC:def_5; then A57: P2B in the topology of ((TOP-REAL 2) | (P `)) by A55, PRE_TOPC:def_4; A58: CP = [#] (((TOP-REAL 2) | (P `)) | CP) by PRE_TOPC:def_5; A59: BP <> {} (((TOP-REAL 2) | (P `)) | CP) by A28, A32, A33, A53, XBOOLE_1:28; BP in the topology of (((TOP-REAL 2) | (P `)) | CP) by A57, A58, PRE_TOPC:def_4; then A60: BP is open by PRE_TOPC:def_2; P1A /\ CP c= CP by XBOOLE_1:17; then reconsider AP = P1A /\ CP as Subset of (((TOP-REAL 2) | (P `)) | CP) by PRE_TOPC:8; A61: P1 /\ (P `) = P1A by A15, A27, XBOOLE_1:21; P1 is open by A4, Th33; then A62: P1 in the topology of (TOP-REAL 2) by PRE_TOPC:def_2; P1 /\ ([#] ((TOP-REAL 2) | (P `))) = P1A by A61, PRE_TOPC:def_5; then A63: P1A in the topology of ((TOP-REAL 2) | (P `)) by A62, PRE_TOPC:def_4; CP = [#] (((TOP-REAL 2) | (P `)) | CP) by PRE_TOPC:def_5; then AP in the topology of (((TOP-REAL 2) | (P `)) | CP) by A63, PRE_TOPC:def_4; then A64: AP is open by PRE_TOPC:def_2; A65: CP = (P1A \/ P2B) /\ CP by A15, A27, A28, A56, XBOOLE_1:28 .= AP \/ BP by XBOOLE_1:23 ; now__::_thesis:_not_AP_<>_{} assume A66: AP <> {} ; ::_thesis: contradiction AP /\ BP = (P1A /\ (P2B /\ CP)) /\ CP by XBOOLE_1:16 .= ((P1A /\ P2B) /\ CP) /\ CP by XBOOLE_1:16 .= (P1A /\ P2B) /\ (CP /\ CP) by XBOOLE_1:16 .= (P1A /\ P2B) /\ CP ; then AP misses BP by A26, A27, A28, XBOOLE_0:def_7; hence contradiction by A52, A58, A59, A60, A64, A65, A66, CONNSP_1:11; ::_thesis: verum end; hence P2B = CP by A53, A65, XBOOLE_1:28; ::_thesis: verum end; hence ( P1A is a_component & P2B is a_component ) by A29, A34, A35, CONNSP_1:def_5; ::_thesis: verum end; hence for P1A, P2B being Subset of ((TOP-REAL 2) | (P `)) st P1A = P1 & P2B = P2 holds ( P1A is a_component & P2B is a_component ) ; ::_thesis: verum end; Lm10: for s1, t1, s2, t2 being Real for P, P1 being Subset of (TOP-REAL 2) st s1 < s2 & t1 < t2 & P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P1 = { pa where pa is Point of (TOP-REAL 2) : ( s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) } holds Cl P1 = P \/ P1 proof let s1, t1, s2, t2 be Real; ::_thesis: for P, P1 being Subset of (TOP-REAL 2) st s1 < s2 & t1 < t2 & P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P1 = { pa where pa is Point of (TOP-REAL 2) : ( s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) } holds Cl P1 = P \/ P1 let P, P1 be Subset of (TOP-REAL 2); ::_thesis: ( s1 < s2 & t1 < t2 & P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P1 = { pa where pa is Point of (TOP-REAL 2) : ( s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) } implies Cl P1 = P \/ P1 ) assume that A1: s1 < s2 and A2: t1 < t2 and A3: P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } and A4: P1 = { pa where pa is Point of (TOP-REAL 2) : ( s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) } ; ::_thesis: Cl P1 = P \/ P1 reconsider P2 = { pb where pb is Point of (TOP-REAL 2) : ( not s1 <= pb `1 or not pb `1 <= s2 or not t1 <= pb `2 or not pb `2 <= t2 ) } as Subset of (TOP-REAL 2) by Th30; A5: P1 c= Cl P1 by PRE_TOPC:18; reconsider PP = P ` as Subset of (TOP-REAL 2) ; A6: PP = P1 \/ P2 by A1, A2, A3, A4, Th36; P1 misses P2 by A1, A2, A3, A4, Th36; then A7: P1 c= P2 ` by SUBSET_1:23; P = (P1 \/ P2) ` by A6; then A8: P = (([#] (TOP-REAL 2)) \ P1) /\ (([#] (TOP-REAL 2)) \ P2) by XBOOLE_1:53; [#] (TOP-REAL 2) = P \/ (P1 \/ P2) by A6, PRE_TOPC:2; then A9: [#] (TOP-REAL 2) = (P \/ P1) \/ P2 by XBOOLE_1:4; now__::_thesis:_for_x_being_set_st_x_in_P2_`_holds_ x_in_P_\/_P1 let x be set ; ::_thesis: ( x in P2 ` implies x in P \/ P1 ) assume A10: x in P2 ` ; ::_thesis: x in P \/ P1 then not x in P2 by XBOOLE_0:def_5; hence x in P \/ P1 by A9, A10, XBOOLE_0:def_3; ::_thesis: verum end; then A11: P2 ` c= P \/ P1 by TARSKI:def_3; now__::_thesis:_for_x_being_set_st_x_in_P_\/_P1_holds_ x_in_P2_` let x be set ; ::_thesis: ( x in P \/ P1 implies x in P2 ` ) assume x in P \/ P1 ; ::_thesis: x in P2 ` then ( x in P or x in P1 ) by XBOOLE_0:def_3; hence x in P2 ` by A7, A8, XBOOLE_0:def_4; ::_thesis: verum end; then P \/ P1 c= P2 ` by TARSKI:def_3; then A12: P2 ` = P \/ P1 by A11, XBOOLE_0:def_10; A13: P2 is open by Th34; ([#] (TOP-REAL 2)) \ (P2 `) = (P2 `) ` .= P2 ; then A14: P \/ P1 is closed by A12, A13, PRE_TOPC:def_3; A15: P1 c= P \/ P1 by XBOOLE_1:7; thus Cl P1 c= P \/ P1 :: according to XBOOLE_0:def_10 ::_thesis: P \/ P1 c= Cl P1 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Cl P1 or x in P \/ P1 ) assume x in Cl P1 ; ::_thesis: x in P \/ P1 hence x in P \/ P1 by A14, A15, PRE_TOPC:15; ::_thesis: verum end; P c= Cl P1 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in P or x in Cl P1 ) assume x in P ; ::_thesis: x in Cl P1 then consider p being Point of (TOP-REAL 2) such that A16: p = x and A17: ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) by A3; reconsider q = p as Point of (Euclid 2) by EUCLID:22; now__::_thesis:_x_in_Cl_P1 percases ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) by A17; supposeA18: ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) ; ::_thesis: x in Cl P1 now__::_thesis:_p_in_Cl_P1 percases ( ( p `1 = s1 & p `2 < t2 & p `2 > t1 ) or ( p `1 = s1 & p `2 = t1 ) or ( p `1 = s1 & p `2 = t2 ) ) by A18, XXREAL_0:1; supposeA19: ( p `1 = s1 & p `2 < t2 & p `2 > t1 ) ; ::_thesis: p in Cl P1 for Q being Subset of (TOP-REAL 2) st Q is open & p in Q holds P1 meets Q proof let Q be Subset of (TOP-REAL 2); ::_thesis: ( Q is open & p in Q implies P1 meets Q ) assume that A20: Q is open and A21: p in Q ; ::_thesis: P1 meets Q reconsider QQ = Q as Subset of (TopSpaceMetr (Euclid 2)) by Lm9; QQ is open by A20, Lm9, PRE_TOPC:30; then consider r being real number such that A22: r > 0 and A23: Ball (q,r) c= Q by A21, TOPMETR:15; reconsider r = r as Real by XREAL_0:def_1; A24: r / 2 > 0 by A22, XREAL_1:215; A25: r / 2 < r by A22, XREAL_1:216; set r2 = min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))); A26: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= r / 2 by XXREAL_0:17; A27: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= min (((s2 - s1) / 2),((t2 - t1) / 2)) by XXREAL_0:17; min (((s2 - s1) / 2),((t2 - t1) / 2)) <= (s2 - s1) / 2 by XXREAL_0:17; then A28: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= (s2 - s1) / 2 by A27, XXREAL_0:2; A29: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) < r by A25, A26, XXREAL_0:2; A30: s2 - s1 > 0 by A1, XREAL_1:50; A31: t2 - t1 > 0 by A2, XREAL_1:50; A32: (s2 - s1) / 2 > 0 by A30, XREAL_1:215; (t2 - t1) / 2 > 0 by A31, XREAL_1:215; then 0 < min (((s2 - s1) / 2),((t2 - t1) / 2)) by A32, XXREAL_0:15; then A33: 0 < min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) by A24, XXREAL_0:15; set pa = |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]|; A34: |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `1 = (p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))) by EUCLID:52; A35: |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `2 = p `2 by EUCLID:52; (s2 - s1) / 2 < s2 - s1 by A30, XREAL_1:216; then A36: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) < s2 - (p `1) by A19, A28, XXREAL_0:2; A37: s1 < |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `1 by A19, A33, A34, XREAL_1:29; |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `1 < s2 by A34, A36, XREAL_1:20; then A38: |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| in P1 by A4, A19, A35, A37; reconsider qa = |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| as Point of (Euclid 2) by EUCLID:22; A39: 0 <= ((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `1)) ^2 by XREAL_1:63; 0 <= ((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `2)) ^2 by XREAL_1:63; then A40: 0 + 0 <= (((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `1)) ^2) + (((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `2)) ^2) by A39, XREAL_1:7; ((|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `1) - (p `1)) ^2 = ((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `1)) ^2 ; then (((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `1)) ^2) + (((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `2)) ^2) < r ^2 by A29, A33, A34, A35, SQUARE_1:16; then (sqrt ((((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `1)) ^2) + (((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `2)) ^2))) ^2 < r ^2 by A40, SQUARE_1:def_2; then A41: sqrt ((((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `1)) ^2) + (((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `2)) ^2)) < r by A22, SQUARE_1:15; (Pitag_dist 2) . (q,qa) = dist (q,qa) by METRIC_1:def_1; then dist (q,qa) < r by A41, TOPREAL3:7; then |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| in Ball (q,r) by METRIC_1:11; hence P1 meets Q by A23, A38, XBOOLE_0:3; ::_thesis: verum end; hence p in Cl P1 by PRE_TOPC:def_7; ::_thesis: verum end; supposeA42: ( p `1 = s1 & p `2 = t1 ) ; ::_thesis: p in Cl P1 for Q being Subset of (TOP-REAL 2) st Q is open & p in Q holds P1 meets Q proof let Q be Subset of (TOP-REAL 2); ::_thesis: ( Q is open & p in Q implies P1 meets Q ) assume that A43: Q is open and A44: p in Q ; ::_thesis: P1 meets Q reconsider QQ = Q as Subset of (TopSpaceMetr (Euclid 2)) by Lm9; QQ is open by A43, Lm9, PRE_TOPC:30; then consider r being real number such that A45: r > 0 and A46: Ball (q,r) c= Q by A44, TOPMETR:15; reconsider r = r as Real by XREAL_0:def_1; A47: r / 2 > 0 by A45, XREAL_1:215; set r2 = min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))); A48: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= min (((s2 - s1) / 2),((t2 - t1) / 2)) by XXREAL_0:17; A49: min (((s2 - s1) / 2),((t2 - t1) / 2)) <= (s2 - s1) / 2 by XXREAL_0:17; A50: min (((s2 - s1) / 2),((t2 - t1) / 2)) <= (t2 - t1) / 2 by XXREAL_0:17; A51: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= (s2 - s1) / 2 by A48, A49, XXREAL_0:2; A52: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= (t2 - t1) / 2 by A48, A50, XXREAL_0:2; A53: s2 - s1 > 0 by A1, XREAL_1:50; A54: t2 - t1 > 0 by A2, XREAL_1:50; A55: (s2 - s1) / 2 > 0 by A53, XREAL_1:215; (t2 - t1) / 2 > 0 by A54, XREAL_1:215; then 0 < min (((s2 - s1) / 2),((t2 - t1) / 2)) by A55, XXREAL_0:15; then A56: 0 < min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) by A47, XXREAL_0:15; set pa = |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]|; A57: |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1 = (p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))) by EUCLID:52; A58: |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 = (p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))) by EUCLID:52; A59: |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1 > s1 by A42, A56, A57, XREAL_1:29; A60: |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 > t1 by A42, A56, A58, XREAL_1:29; (s2 - s1) / 2 < s2 - s1 by A53, XREAL_1:216; then min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) < s2 - (p `1) by A42, A51, XXREAL_0:2; then A61: |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1 < s2 by A57, XREAL_1:20; (t2 - t1) / 2 < t2 - t1 by A54, XREAL_1:216; then min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) < t2 - (p `2) by A42, A52, XXREAL_0:2; then |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 < t2 by A58, XREAL_1:20; then A62: |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| in P1 by A4, A59, A60, A61; reconsider qa = |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| as Point of (Euclid 2) by EUCLID:22; A63: 0 <= ((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2 by XREAL_1:63; 0 <= ((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2 by XREAL_1:63; then A64: 0 + 0 <= (((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2) by A63, XREAL_1:7; ((|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1) - (p `1)) ^2 = ((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2 ; then ((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2 <= (r / 2) ^2 by A56, A57, SQUARE_1:15, XXREAL_0:17; then A65: (((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2) <= ((r / 2) ^2) + ((r / 2) ^2) by A57, A58, XREAL_1:7; r ^2 = (((r / 2) ^2) + ((r / 2) ^2)) + ((2 * (r / 2)) * (r / 2)) ; then r ^2 > ((r / 2) ^2) + ((r / 2) ^2) by A45, A47, XREAL_1:29, XREAL_1:129; then (((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2) < r ^2 by A65, XXREAL_0:2; then (sqrt ((((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2))) ^2 < r ^2 by A64, SQUARE_1:def_2; then A66: sqrt ((((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2)) < r by A45, SQUARE_1:15; (Pitag_dist 2) . (q,qa) = dist (q,qa) by METRIC_1:def_1; then dist (q,qa) < r by A66, TOPREAL3:7; then |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| in Ball (q,r) by METRIC_1:11; hence P1 meets Q by A46, A62, XBOOLE_0:3; ::_thesis: verum end; hence p in Cl P1 by PRE_TOPC:def_7; ::_thesis: verum end; supposeA67: ( p `1 = s1 & p `2 = t2 ) ; ::_thesis: p in Cl P1 for Q being Subset of (TOP-REAL 2) st Q is open & p in Q holds P1 meets Q proof let Q be Subset of (TOP-REAL 2); ::_thesis: ( Q is open & p in Q implies P1 meets Q ) assume that A68: Q is open and A69: p in Q ; ::_thesis: P1 meets Q reconsider QQ = Q as Subset of (TopSpaceMetr (Euclid 2)) by Lm9; QQ is open by A68, Lm9, PRE_TOPC:30; then consider r being real number such that A70: r > 0 and A71: Ball (q,r) c= Q by A69, TOPMETR:15; reconsider r = r as Real by XREAL_0:def_1; A72: r / 2 > 0 by A70, XREAL_1:215; set r2 = min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))); A73: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= min (((s2 - s1) / 2),((t2 - t1) / 2)) by XXREAL_0:17; A74: min (((s2 - s1) / 2),((t2 - t1) / 2)) <= (s2 - s1) / 2 by XXREAL_0:17; A75: min (((s2 - s1) / 2),((t2 - t1) / 2)) <= (t2 - t1) / 2 by XXREAL_0:17; A76: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= (s2 - s1) / 2 by A73, A74, XXREAL_0:2; A77: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= (t2 - t1) / 2 by A73, A75, XXREAL_0:2; A78: s2 - s1 > 0 by A1, XREAL_1:50; A79: t2 - t1 > 0 by A2, XREAL_1:50; A80: (s2 - s1) / 2 > 0 by A78, XREAL_1:215; (t2 - t1) / 2 > 0 by A79, XREAL_1:215; then 0 < min (((s2 - s1) / 2),((t2 - t1) / 2)) by A80, XXREAL_0:15; then A81: 0 < min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) by A72, XXREAL_0:15; set pa = |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]|; A82: |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1 = (p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))) by EUCLID:52; A83: |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 = (p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))) by EUCLID:52; A84: |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1 > s1 by A67, A81, A82, XREAL_1:29; A85: |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 < t2 by A67, A81, A83, XREAL_1:44; (s2 - s1) / 2 < s2 - s1 by A78, XREAL_1:216; then min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) < s2 - (p `1) by A67, A76, XXREAL_0:2; then A86: |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1 < s2 by A82, XREAL_1:20; (t2 - t1) / 2 < t2 - t1 by A79, XREAL_1:216; then min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) < (p `2) - t1 by A67, A77, XXREAL_0:2; then |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 > t1 by A83, XREAL_1:12; then A87: |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| in P1 by A4, A84, A85, A86; reconsider qa = |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| as Point of (Euclid 2) by EUCLID:22; A88: 0 <= ((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2 by XREAL_1:63; 0 <= ((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2 by XREAL_1:63; then A89: 0 + 0 <= (((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2) by A88, XREAL_1:7; (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))) ^2 <= (r / 2) ^2 by A81, SQUARE_1:15, XXREAL_0:17; then A90: (((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2) <= ((r / 2) ^2) + ((r / 2) ^2) by A82, A83, XREAL_1:7; r ^2 = (((r / 2) ^2) + ((r / 2) ^2)) + ((2 * (r / 2)) * (r / 2)) ; then r ^2 > ((r / 2) ^2) + ((r / 2) ^2) by A70, A72, XREAL_1:29, XREAL_1:129; then (((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2) < r ^2 by A90, XXREAL_0:2; then (sqrt ((((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2))) ^2 < r ^2 by A89, SQUARE_1:def_2; then A91: sqrt ((((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2)) < r by A70, SQUARE_1:15; (Pitag_dist 2) . (q,qa) = dist (q,qa) by METRIC_1:def_1; then dist (q,qa) < r by A91, TOPREAL3:7; then |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| in Ball (q,r) by METRIC_1:11; hence P1 meets Q by A71, A87, XBOOLE_0:3; ::_thesis: verum end; hence p in Cl P1 by PRE_TOPC:def_7; ::_thesis: verum end; end; end; hence x in Cl P1 by A16; ::_thesis: verum end; supposeA92: ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) ; ::_thesis: x in Cl P1 now__::_thesis:_p_in_Cl_P1 percases ( ( p `2 = t2 & p `1 < s2 & p `1 > s1 ) or ( p `2 = t2 & p `1 = s1 ) or ( p `2 = t2 & p `1 = s2 ) ) by A92, XXREAL_0:1; supposeA93: ( p `2 = t2 & p `1 < s2 & p `1 > s1 ) ; ::_thesis: p in Cl P1 for Q being Subset of (TOP-REAL 2) st Q is open & p in Q holds P1 meets Q proof let Q be Subset of (TOP-REAL 2); ::_thesis: ( Q is open & p in Q implies P1 meets Q ) assume that A94: Q is open and A95: p in Q ; ::_thesis: P1 meets Q reconsider QQ = Q as Subset of (TopSpaceMetr (Euclid 2)) by Lm9; QQ is open by A94, Lm9, PRE_TOPC:30; then consider r being real number such that A96: r > 0 and A97: Ball (q,r) c= Q by A95, TOPMETR:15; reconsider r = r as Real by XREAL_0:def_1; A98: r / 2 > 0 by A96, XREAL_1:215; A99: r / 2 < r by A96, XREAL_1:216; set r2 = min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))); A100: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= r / 2 by XXREAL_0:17; A101: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= min (((s2 - s1) / 2),((t2 - t1) / 2)) by XXREAL_0:17; min (((s2 - s1) / 2),((t2 - t1) / 2)) <= (t2 - t1) / 2 by XXREAL_0:17; then A102: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= (t2 - t1) / 2 by A101, XXREAL_0:2; A103: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) < r by A99, A100, XXREAL_0:2; A104: s2 - s1 > 0 by A1, XREAL_1:50; A105: t2 - t1 > 0 by A2, XREAL_1:50; A106: (s2 - s1) / 2 > 0 by A104, XREAL_1:215; (t2 - t1) / 2 > 0 by A105, XREAL_1:215; then 0 < min (((s2 - s1) / 2),((t2 - t1) / 2)) by A106, XXREAL_0:15; then A107: 0 < min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) by A98, XXREAL_0:15; set pa = |[(p `1),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]|; A108: |[(p `1),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1 = p `1 by EUCLID:52; A109: |[(p `1),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 = (p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))) by EUCLID:52; (t2 - t1) / 2 < t2 - t1 by A105, XREAL_1:216; then min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) < (p `2) - t1 by A93, A102, XXREAL_0:2; then A110: t1 < |[(p `1),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 by A109, XREAL_1:12; |[(p `1),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 < t2 by A93, A107, A109, XREAL_1:44; then A111: |[(p `1),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| in P1 by A4, A93, A108, A110; reconsider qa = |[(p `1),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| as Point of (Euclid 2) by EUCLID:22; A112: 0 <= ((p `1) - (|[(p `1),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2 by XREAL_1:63; 0 <= ((p `2) - (|[(p `1),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2 by XREAL_1:63; then A113: 0 + 0 <= (((p `1) - (|[(p `1),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[(p `1),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2) by A112, XREAL_1:7; (((p `1) - (|[(p `1),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[(p `1),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2) < r ^2 by A103, A107, A108, A109, SQUARE_1:16; then (sqrt ((((p `1) - (|[(p `1),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[(p `1),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2))) ^2 < r ^2 by A113, SQUARE_1:def_2; then A114: sqrt ((((p `1) - (|[(p `1),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[(p `1),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2)) < r by A96, SQUARE_1:15; (Pitag_dist 2) . (q,qa) = dist (q,qa) by METRIC_1:def_1; then dist (q,qa) < r by A114, TOPREAL3:7; then |[(p `1),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| in Ball (q,r) by METRIC_1:11; hence P1 meets Q by A97, A111, XBOOLE_0:3; ::_thesis: verum end; hence p in Cl P1 by PRE_TOPC:def_7; ::_thesis: verum end; supposeA115: ( p `2 = t2 & p `1 = s1 ) ; ::_thesis: p in Cl P1 for Q being Subset of (TOP-REAL 2) st Q is open & p in Q holds P1 meets Q proof let Q be Subset of (TOP-REAL 2); ::_thesis: ( Q is open & p in Q implies P1 meets Q ) assume that A116: Q is open and A117: p in Q ; ::_thesis: P1 meets Q reconsider QQ = Q as Subset of (TopSpaceMetr (Euclid 2)) by Lm9; QQ is open by A116, Lm9, PRE_TOPC:30; then consider r being real number such that A118: r > 0 and A119: Ball (q,r) c= Q by A117, TOPMETR:15; reconsider r = r as Real by XREAL_0:def_1; A120: r / 2 > 0 by A118, XREAL_1:215; set r2 = min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))); A121: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= min (((s2 - s1) / 2),((t2 - t1) / 2)) by XXREAL_0:17; A122: min (((s2 - s1) / 2),((t2 - t1) / 2)) <= (s2 - s1) / 2 by XXREAL_0:17; A123: min (((s2 - s1) / 2),((t2 - t1) / 2)) <= (t2 - t1) / 2 by XXREAL_0:17; A124: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= (s2 - s1) / 2 by A121, A122, XXREAL_0:2; A125: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= (t2 - t1) / 2 by A121, A123, XXREAL_0:2; A126: s2 - s1 > 0 by A1, XREAL_1:50; A127: t2 - t1 > 0 by A2, XREAL_1:50; A128: (s2 - s1) / 2 > 0 by A126, XREAL_1:215; (t2 - t1) / 2 > 0 by A127, XREAL_1:215; then 0 < min (((s2 - s1) / 2),((t2 - t1) / 2)) by A128, XXREAL_0:15; then A129: 0 < min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) by A120, XXREAL_0:15; set pa = |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]|; A130: |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1 = (p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))) by EUCLID:52; A131: |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 = (p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))) by EUCLID:52; A132: |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1 > s1 by A115, A129, A130, XREAL_1:29; A133: |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 < t2 by A115, A129, A131, XREAL_1:44; (t2 - t1) / 2 < t2 - t1 by A127, XREAL_1:216; then min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) < (p `2) - t1 by A115, A125, XXREAL_0:2; then A134: |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 > t1 by A131, XREAL_1:12; (s2 - s1) / 2 < s2 - s1 by A126, XREAL_1:216; then min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) < s2 - (p `1) by A115, A124, XXREAL_0:2; then |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1 < s2 by A130, XREAL_1:20; then A135: |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| in P1 by A4, A132, A133, A134; reconsider qa = |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| as Point of (Euclid 2) by EUCLID:22; A136: 0 <= ((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2 by XREAL_1:63; 0 <= ((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2 by XREAL_1:63; then A137: 0 + 0 <= (((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2) by A136, XREAL_1:7; ((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2 <= (r / 2) ^2 by A129, A131, SQUARE_1:15, XXREAL_0:17; then A138: (((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2) <= ((r / 2) ^2) + ((r / 2) ^2) by A130, A131, XREAL_1:7; r ^2 = (((r / 2) ^2) + ((r / 2) ^2)) + ((2 * (r / 2)) * (r / 2)) ; then r ^2 > ((r / 2) ^2) + ((r / 2) ^2) by A118, A120, XREAL_1:29, XREAL_1:129; then (((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2) < r ^2 by A138, XXREAL_0:2; then (sqrt ((((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2))) ^2 < r ^2 by A137, SQUARE_1:def_2; then A139: sqrt ((((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2)) < r by A118, SQUARE_1:15; (Pitag_dist 2) . (q,qa) = dist (q,qa) by METRIC_1:def_1; then dist (q,qa) < r by A139, TOPREAL3:7; then |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| in Ball (q,r) by METRIC_1:11; hence P1 meets Q by A119, A135, XBOOLE_0:3; ::_thesis: verum end; hence p in Cl P1 by PRE_TOPC:def_7; ::_thesis: verum end; supposeA140: ( p `2 = t2 & p `1 = s2 ) ; ::_thesis: p in Cl P1 for Q being Subset of (TOP-REAL 2) st Q is open & p in Q holds P1 meets Q proof let Q be Subset of (TOP-REAL 2); ::_thesis: ( Q is open & p in Q implies P1 meets Q ) assume that A141: Q is open and A142: p in Q ; ::_thesis: P1 meets Q reconsider QQ = Q as Subset of (TopSpaceMetr (Euclid 2)) by Lm9; QQ is open by A141, Lm9, PRE_TOPC:30; then consider r being real number such that A143: r > 0 and A144: Ball (q,r) c= Q by A142, TOPMETR:15; reconsider r = r as Real by XREAL_0:def_1; A145: r / 2 > 0 by A143, XREAL_1:215; set r2 = min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))); A146: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= min (((s2 - s1) / 2),((t2 - t1) / 2)) by XXREAL_0:17; A147: min (((s2 - s1) / 2),((t2 - t1) / 2)) <= (s2 - s1) / 2 by XXREAL_0:17; A148: min (((s2 - s1) / 2),((t2 - t1) / 2)) <= (t2 - t1) / 2 by XXREAL_0:17; A149: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= (s2 - s1) / 2 by A146, A147, XXREAL_0:2; A150: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= (t2 - t1) / 2 by A146, A148, XXREAL_0:2; A151: s2 - s1 > 0 by A1, XREAL_1:50; A152: t2 - t1 > 0 by A2, XREAL_1:50; A153: (s2 - s1) / 2 > 0 by A151, XREAL_1:215; (t2 - t1) / 2 > 0 by A152, XREAL_1:215; then 0 < min (((s2 - s1) / 2),((t2 - t1) / 2)) by A153, XXREAL_0:15; then A154: 0 < min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) by A145, XXREAL_0:15; set pa = |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]|; A155: |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1 = (p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))) by EUCLID:52; A156: |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 = (p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))) by EUCLID:52; A157: |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1 < s2 by A140, A154, A155, XREAL_1:44; A158: |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 < t2 by A140, A154, A156, XREAL_1:44; (s2 - s1) / 2 < s2 - s1 by A151, XREAL_1:216; then min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) < (p `1) - s1 by A140, A149, XXREAL_0:2; then A159: |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1 > s1 by A155, XREAL_1:12; (t2 - t1) / 2 < t2 - t1 by A152, XREAL_1:216; then min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) < (p `2) - t1 by A140, A150, XXREAL_0:2; then |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 > t1 by A156, XREAL_1:12; then A160: |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| in P1 by A4, A157, A158, A159; reconsider qa = |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| as Point of (Euclid 2) by EUCLID:22; A161: 0 <= ((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2 by XREAL_1:63; 0 <= ((p `2) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2 by XREAL_1:63; then A162: 0 + 0 <= (((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2) by A161, XREAL_1:7; (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))) ^2 <= (r / 2) ^2 by A154, SQUARE_1:15, XXREAL_0:17; then A163: (((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2) <= ((r / 2) ^2) + ((r / 2) ^2) by A155, A156, XREAL_1:7; r ^2 = (((r / 2) ^2) + ((r / 2) ^2)) + ((2 * (r / 2)) * (r / 2)) ; then r ^2 > ((r / 2) ^2) + ((r / 2) ^2) by A143, A145, XREAL_1:29, XREAL_1:129; then (((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2) < r ^2 by A163, XXREAL_0:2; then (sqrt ((((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2))) ^2 < r ^2 by A162, SQUARE_1:def_2; then A164: sqrt ((((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2)) < r by A143, SQUARE_1:15; (Pitag_dist 2) . (q,qa) = dist (q,qa) by METRIC_1:def_1; then dist (q,qa) < r by A164, TOPREAL3:7; then |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| in Ball (q,r) by METRIC_1:11; hence P1 meets Q by A144, A160, XBOOLE_0:3; ::_thesis: verum end; hence p in Cl P1 by PRE_TOPC:def_7; ::_thesis: verum end; end; end; hence x in Cl P1 by A16; ::_thesis: verum end; supposeA165: ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) ; ::_thesis: x in Cl P1 now__::_thesis:_p_in_Cl_P1 percases ( ( p `2 = t1 & p `1 < s2 & p `1 > s1 ) or ( p `2 = t1 & p `1 = s1 ) or ( p `2 = t1 & p `1 = s2 ) ) by A165, XXREAL_0:1; supposeA166: ( p `2 = t1 & p `1 < s2 & p `1 > s1 ) ; ::_thesis: p in Cl P1 for Q being Subset of (TOP-REAL 2) st Q is open & p in Q holds P1 meets Q proof let Q be Subset of (TOP-REAL 2); ::_thesis: ( Q is open & p in Q implies P1 meets Q ) assume that A167: Q is open and A168: p in Q ; ::_thesis: P1 meets Q reconsider QQ = Q as Subset of (TopSpaceMetr (Euclid 2)) by Lm9; QQ is open by A167, Lm9, PRE_TOPC:30; then consider r being real number such that A169: r > 0 and A170: Ball (q,r) c= Q by A168, TOPMETR:15; reconsider r = r as Real by XREAL_0:def_1; A171: r / 2 > 0 by A169, XREAL_1:215; A172: r / 2 < r by A169, XREAL_1:216; set r2 = min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))); A173: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= r / 2 by XXREAL_0:17; A174: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= min (((s2 - s1) / 2),((t2 - t1) / 2)) by XXREAL_0:17; min (((s2 - s1) / 2),((t2 - t1) / 2)) <= (t2 - t1) / 2 by XXREAL_0:17; then A175: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= (t2 - t1) / 2 by A174, XXREAL_0:2; A176: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) < r by A172, A173, XXREAL_0:2; A177: s2 - s1 > 0 by A1, XREAL_1:50; A178: t2 - t1 > 0 by A2, XREAL_1:50; A179: (s2 - s1) / 2 > 0 by A177, XREAL_1:215; (t2 - t1) / 2 > 0 by A178, XREAL_1:215; then 0 < min (((s2 - s1) / 2),((t2 - t1) / 2)) by A179, XXREAL_0:15; then A180: 0 < min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) by A171, XXREAL_0:15; set pa = |[(p `1),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]|; A181: |[(p `1),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 = (p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))) by EUCLID:52; A182: |[(p `1),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1 = p `1 by EUCLID:52; (t2 - t1) / 2 < t2 - t1 by A178, XREAL_1:216; then A183: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) < t2 - (p `2) by A166, A175, XXREAL_0:2; A184: t1 < |[(p `1),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 by A166, A180, A181, XREAL_1:29; |[(p `1),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 < t2 by A181, A183, XREAL_1:20; then A185: |[(p `1),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| in P1 by A4, A166, A182, A184; reconsider qa = |[(p `1),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| as Point of (Euclid 2) by EUCLID:22; A186: 0 <= ((p `1) - (|[(p `1),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2 by XREAL_1:63; 0 <= ((p `2) - (|[(p `1),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2 by XREAL_1:63; then A187: 0 + 0 <= (((p `1) - (|[(p `1),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[(p `1),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2) by A186, XREAL_1:7; ((|[(p `1),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2) - (p `2)) ^2 = ((p `2) - (|[(p `1),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2 ; then (((p `1) - (|[(p `1),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[(p `1),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2) < r ^2 by A176, A180, A181, A182, SQUARE_1:16; then (sqrt ((((p `1) - (|[(p `1),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[(p `1),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2))) ^2 < r ^2 by A187, SQUARE_1:def_2; then A188: sqrt ((((p `1) - (|[(p `1),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[(p `1),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2)) < r by A169, SQUARE_1:15; (Pitag_dist 2) . (q,qa) = dist (q,qa) by METRIC_1:def_1; then dist (q,qa) < r by A188, TOPREAL3:7; then |[(p `1),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| in Ball (q,r) by METRIC_1:11; hence P1 meets Q by A170, A185, XBOOLE_0:3; ::_thesis: verum end; hence p in Cl P1 by PRE_TOPC:def_7; ::_thesis: verum end; supposeA189: ( p `2 = t1 & p `1 = s1 ) ; ::_thesis: p in Cl P1 for Q being Subset of (TOP-REAL 2) st Q is open & p in Q holds P1 meets Q proof let Q be Subset of (TOP-REAL 2); ::_thesis: ( Q is open & p in Q implies P1 meets Q ) assume that A190: Q is open and A191: p in Q ; ::_thesis: P1 meets Q reconsider QQ = Q as Subset of (TopSpaceMetr (Euclid 2)) by Lm9; QQ is open by A190, Lm9, PRE_TOPC:30; then consider r being real number such that A192: r > 0 and A193: Ball (q,r) c= Q by A191, TOPMETR:15; reconsider r = r as Real by XREAL_0:def_1; A194: r / 2 > 0 by A192, XREAL_1:215; set r2 = min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))); A195: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= min (((s2 - s1) / 2),((t2 - t1) / 2)) by XXREAL_0:17; A196: min (((s2 - s1) / 2),((t2 - t1) / 2)) <= (s2 - s1) / 2 by XXREAL_0:17; A197: min (((s2 - s1) / 2),((t2 - t1) / 2)) <= (t2 - t1) / 2 by XXREAL_0:17; A198: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= (s2 - s1) / 2 by A195, A196, XXREAL_0:2; A199: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= (t2 - t1) / 2 by A195, A197, XXREAL_0:2; A200: s2 - s1 > 0 by A1, XREAL_1:50; A201: t2 - t1 > 0 by A2, XREAL_1:50; A202: (s2 - s1) / 2 > 0 by A200, XREAL_1:215; (t2 - t1) / 2 > 0 by A201, XREAL_1:215; then 0 < min (((s2 - s1) / 2),((t2 - t1) / 2)) by A202, XXREAL_0:15; then A203: 0 < min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) by A194, XXREAL_0:15; set pa = |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]|; A204: |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1 = (p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))) by EUCLID:52; A205: |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 = (p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))) by EUCLID:52; A206: |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1 > s1 by A189, A203, A204, XREAL_1:29; A207: |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 > t1 by A189, A203, A205, XREAL_1:29; (s2 - s1) / 2 < s2 - s1 by A200, XREAL_1:216; then min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) < s2 - (p `1) by A189, A198, XXREAL_0:2; then A208: |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1 < s2 by A204, XREAL_1:20; (t2 - t1) / 2 < t2 - t1 by A201, XREAL_1:216; then min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) < t2 - (p `2) by A189, A199, XXREAL_0:2; then |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 < t2 by A205, XREAL_1:20; then A209: |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| in P1 by A4, A206, A207, A208; reconsider qa = |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| as Point of (Euclid 2) by EUCLID:22; A210: 0 <= ((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2 by XREAL_1:63; 0 <= ((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2 by XREAL_1:63; then A211: 0 + 0 <= (((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2) by A210, XREAL_1:7; ((|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1) - (p `1)) ^2 = ((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2 ; then ((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2 <= (r / 2) ^2 by A203, A204, SQUARE_1:15, XXREAL_0:17; then A212: (((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2) <= ((r / 2) ^2) + ((r / 2) ^2) by A204, A205, XREAL_1:7; r ^2 = (((r / 2) ^2) + ((r / 2) ^2)) + ((2 * (r / 2)) * (r / 2)) ; then r ^2 > ((r / 2) ^2) + ((r / 2) ^2) by A192, A194, XREAL_1:29, XREAL_1:129; then (((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2) < r ^2 by A212, XXREAL_0:2; then (sqrt ((((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2))) ^2 < r ^2 by A211, SQUARE_1:def_2; then A213: sqrt ((((p `1) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2)) < r by A192, SQUARE_1:15; (Pitag_dist 2) . (q,qa) = dist (q,qa) by METRIC_1:def_1; then dist (q,qa) < r by A213, TOPREAL3:7; then |[((p `1) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| in Ball (q,r) by METRIC_1:11; hence P1 meets Q by A193, A209, XBOOLE_0:3; ::_thesis: verum end; hence p in Cl P1 by PRE_TOPC:def_7; ::_thesis: verum end; supposeA214: ( p `2 = t1 & p `1 = s2 ) ; ::_thesis: p in Cl P1 for Q being Subset of (TOP-REAL 2) st Q is open & p in Q holds P1 meets Q proof let Q be Subset of (TOP-REAL 2); ::_thesis: ( Q is open & p in Q implies P1 meets Q ) assume that A215: Q is open and A216: p in Q ; ::_thesis: P1 meets Q reconsider QQ = Q as Subset of (TopSpaceMetr (Euclid 2)) by Lm9; QQ is open by A215, Lm9, PRE_TOPC:30; then consider r being real number such that A217: r > 0 and A218: Ball (q,r) c= Q by A216, TOPMETR:15; reconsider r = r as Real by XREAL_0:def_1; A219: r / 2 > 0 by A217, XREAL_1:215; set r2 = min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))); A220: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= min (((s2 - s1) / 2),((t2 - t1) / 2)) by XXREAL_0:17; A221: min (((s2 - s1) / 2),((t2 - t1) / 2)) <= (s2 - s1) / 2 by XXREAL_0:17; A222: min (((s2 - s1) / 2),((t2 - t1) / 2)) <= (t2 - t1) / 2 by XXREAL_0:17; A223: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= (s2 - s1) / 2 by A220, A221, XXREAL_0:2; A224: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= (t2 - t1) / 2 by A220, A222, XXREAL_0:2; A225: s2 - s1 > 0 by A1, XREAL_1:50; A226: t2 - t1 > 0 by A2, XREAL_1:50; A227: (s2 - s1) / 2 > 0 by A225, XREAL_1:215; (t2 - t1) / 2 > 0 by A226, XREAL_1:215; then 0 < min (((s2 - s1) / 2),((t2 - t1) / 2)) by A227, XXREAL_0:15; then A228: 0 < min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) by A219, XXREAL_0:15; set pa = |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]|; A229: |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1 = (p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))) by EUCLID:52; A230: |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 = (p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))) by EUCLID:52; then A231: |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 > t1 by A214, A228, XREAL_1:29; A232: |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1 < s2 by A214, A228, A229, XREAL_1:44; (t2 - t1) / 2 < t2 - t1 by A226, XREAL_1:216; then min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) < t2 - (p `2) by A214, A224, XXREAL_0:2; then A233: |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 < t2 by A230, XREAL_1:20; (s2 - s1) / 2 < s2 - s1 by A225, XREAL_1:216; then min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) < (p `1) - s1 by A214, A223, XXREAL_0:2; then |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1 > s1 by A229, XREAL_1:12; then A234: |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| in P1 by A4, A231, A232, A233; reconsider qa = |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| as Point of (Euclid 2) by EUCLID:22; A235: 0 <= ((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2 by XREAL_1:63; 0 <= ((p `2) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2 by XREAL_1:63; then A236: 0 + 0 <= (((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2) by A235, XREAL_1:7; (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))) ^2 <= (r / 2) ^2 by A228, SQUARE_1:15, XXREAL_0:17; then A237: (((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2) <= ((r / 2) ^2) + ((r / 2) ^2) by A229, A230, XREAL_1:7; r ^2 = (((r / 2) ^2) + ((r / 2) ^2)) + ((2 * (r / 2)) * (r / 2)) ; then r ^2 > ((r / 2) ^2) + ((r / 2) ^2) by A217, A219, XREAL_1:29, XREAL_1:129; then (((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2) < r ^2 by A237, XXREAL_0:2; then (sqrt ((((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2))) ^2 < r ^2 by A236, SQUARE_1:def_2; then A238: sqrt ((((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2)) < r by A217, SQUARE_1:15; (Pitag_dist 2) . (q,qa) = dist (q,qa) by METRIC_1:def_1; then dist (q,qa) < r by A238, TOPREAL3:7; then |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| in Ball (q,r) by METRIC_1:11; hence P1 meets Q by A218, A234, XBOOLE_0:3; ::_thesis: verum end; hence p in Cl P1 by PRE_TOPC:def_7; ::_thesis: verum end; end; end; hence x in Cl P1 by A16; ::_thesis: verum end; supposeA239: ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ; ::_thesis: x in Cl P1 now__::_thesis:_p_in_Cl_P1 percases ( ( p `1 = s2 & p `2 < t2 & p `2 > t1 ) or ( p `1 = s2 & p `2 = t1 ) or ( p `1 = s2 & p `2 = t2 ) ) by A239, XXREAL_0:1; supposeA240: ( p `1 = s2 & p `2 < t2 & p `2 > t1 ) ; ::_thesis: p in Cl P1 for Q being Subset of (TOP-REAL 2) st Q is open & p in Q holds P1 meets Q proof let Q be Subset of (TOP-REAL 2); ::_thesis: ( Q is open & p in Q implies P1 meets Q ) assume that A241: Q is open and A242: p in Q ; ::_thesis: P1 meets Q reconsider QQ = Q as Subset of (TopSpaceMetr (Euclid 2)) by Lm9; QQ is open by A241, Lm9, PRE_TOPC:30; then consider r being real number such that A243: r > 0 and A244: Ball (q,r) c= Q by A242, TOPMETR:15; reconsider r = r as Real by XREAL_0:def_1; A245: r / 2 > 0 by A243, XREAL_1:215; A246: r / 2 < r by A243, XREAL_1:216; set r2 = min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))); A247: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= r / 2 by XXREAL_0:17; A248: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= min (((s2 - s1) / 2),((t2 - t1) / 2)) by XXREAL_0:17; min (((s2 - s1) / 2),((t2 - t1) / 2)) <= (s2 - s1) / 2 by XXREAL_0:17; then A249: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= (s2 - s1) / 2 by A248, XXREAL_0:2; A250: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) < r by A246, A247, XXREAL_0:2; A251: s2 - s1 > 0 by A1, XREAL_1:50; A252: t2 - t1 > 0 by A2, XREAL_1:50; A253: (s2 - s1) / 2 > 0 by A251, XREAL_1:215; (t2 - t1) / 2 > 0 by A252, XREAL_1:215; then 0 < min (((s2 - s1) / 2),((t2 - t1) / 2)) by A253, XXREAL_0:15; then A254: 0 < min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) by A245, XXREAL_0:15; set pa = |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]|; A255: |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `2 = p `2 by EUCLID:52; A256: |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `1 = (p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))) by EUCLID:52; (s2 - s1) / 2 < s2 - s1 by A251, XREAL_1:216; then min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) < (p `1) - s1 by A240, A249, XXREAL_0:2; then A257: s1 < |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `1 by A256, XREAL_1:12; |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `1 < s2 by A240, A254, A256, XREAL_1:44; then A258: |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| in P1 by A4, A240, A255, A257; reconsider qa = |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| as Point of (Euclid 2) by EUCLID:22; A259: 0 <= ((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `1)) ^2 by XREAL_1:63; 0 <= ((p `2) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `2)) ^2 by XREAL_1:63; then A260: 0 + 0 <= (((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `1)) ^2) + (((p `2) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `2)) ^2) by A259, XREAL_1:7; (((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `1)) ^2) + (((p `2) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `2)) ^2) < r ^2 by A250, A254, A255, A256, SQUARE_1:16; then (sqrt ((((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `1)) ^2) + (((p `2) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `2)) ^2))) ^2 < r ^2 by A260, SQUARE_1:def_2; then A261: sqrt ((((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `1)) ^2) + (((p `2) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| `2)) ^2)) < r by A243, SQUARE_1:15; (Pitag_dist 2) . (q,qa) = dist (q,qa) by METRIC_1:def_1; then dist (q,qa) < r by A261, TOPREAL3:7; then |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),(p `2)]| in Ball (q,r) by METRIC_1:11; hence P1 meets Q by A244, A258, XBOOLE_0:3; ::_thesis: verum end; hence p in Cl P1 by PRE_TOPC:def_7; ::_thesis: verum end; supposeA262: ( p `1 = s2 & p `2 = t1 ) ; ::_thesis: p in Cl P1 for Q being Subset of (TOP-REAL 2) st Q is open & p in Q holds P1 meets Q proof let Q be Subset of (TOP-REAL 2); ::_thesis: ( Q is open & p in Q implies P1 meets Q ) assume that A263: Q is open and A264: p in Q ; ::_thesis: P1 meets Q reconsider QQ = Q as Subset of (TopSpaceMetr (Euclid 2)) by Lm9; QQ is open by A263, Lm9, PRE_TOPC:30; then consider r being real number such that A265: r > 0 and A266: Ball (q,r) c= Q by A264, TOPMETR:15; reconsider r = r as Real by XREAL_0:def_1; A267: r / 2 > 0 by A265, XREAL_1:215; set r2 = min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))); A268: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= min (((s2 - s1) / 2),((t2 - t1) / 2)) by XXREAL_0:17; A269: min (((s2 - s1) / 2),((t2 - t1) / 2)) <= (s2 - s1) / 2 by XXREAL_0:17; A270: min (((s2 - s1) / 2),((t2 - t1) / 2)) <= (t2 - t1) / 2 by XXREAL_0:17; A271: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= (s2 - s1) / 2 by A268, A269, XXREAL_0:2; A272: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= (t2 - t1) / 2 by A268, A270, XXREAL_0:2; A273: s2 - s1 > 0 by A1, XREAL_1:50; A274: t2 - t1 > 0 by A2, XREAL_1:50; A275: (s2 - s1) / 2 > 0 by A273, XREAL_1:215; (t2 - t1) / 2 > 0 by A274, XREAL_1:215; then 0 < min (((s2 - s1) / 2),((t2 - t1) / 2)) by A275, XXREAL_0:15; then A276: 0 < min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) by A267, XXREAL_0:15; set pa = |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]|; A277: |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 = (p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))) by EUCLID:52; A278: |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1 = (p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))) by EUCLID:52; A279: |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 > t1 by A262, A276, A277, XREAL_1:29; A280: |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1 < s2 by A262, A276, A278, XREAL_1:44; (s2 - s1) / 2 < s2 - s1 by A273, XREAL_1:216; then min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) < (p `1) - s1 by A262, A271, XXREAL_0:2; then A281: |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1 > s1 by A278, XREAL_1:12; (t2 - t1) / 2 < t2 - t1 by A274, XREAL_1:216; then min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) < t2 - (p `2) by A262, A272, XXREAL_0:2; then |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 < t2 by A277, XREAL_1:20; then A282: |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| in P1 by A4, A279, A280, A281; reconsider qa = |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| as Point of (Euclid 2) by EUCLID:22; A283: 0 <= ((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2 by XREAL_1:63; 0 <= ((p `2) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2 by XREAL_1:63; then A284: 0 + 0 <= (((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2) by A283, XREAL_1:7; ((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2 <= (r / 2) ^2 by A276, A278, SQUARE_1:15, XXREAL_0:17; then A285: (((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2) <= ((r / 2) ^2) + ((r / 2) ^2) by A277, A278, XREAL_1:7; r ^2 = (((r / 2) ^2) + ((r / 2) ^2)) + ((2 * (r / 2)) * (r / 2)) ; then r ^2 > ((r / 2) ^2) + ((r / 2) ^2) by A265, A267, XREAL_1:29, XREAL_1:129; then (((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2) < r ^2 by A285, XXREAL_0:2; then (sqrt ((((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2))) ^2 < r ^2 by A284, SQUARE_1:def_2; then A286: sqrt ((((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2)) < r by A265, SQUARE_1:15; (Pitag_dist 2) . (q,qa) = dist (q,qa) by METRIC_1:def_1; then dist (q,qa) < r by A286, TOPREAL3:7; then |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) + (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| in Ball (q,r) by METRIC_1:11; hence P1 meets Q by A266, A282, XBOOLE_0:3; ::_thesis: verum end; hence p in Cl P1 by PRE_TOPC:def_7; ::_thesis: verum end; supposeA287: ( p `1 = s2 & p `2 = t2 ) ; ::_thesis: p in Cl P1 for Q being Subset of (TOP-REAL 2) st Q is open & p in Q holds P1 meets Q proof let Q be Subset of (TOP-REAL 2); ::_thesis: ( Q is open & p in Q implies P1 meets Q ) assume that A288: Q is open and A289: p in Q ; ::_thesis: P1 meets Q reconsider QQ = Q as Subset of (TopSpaceMetr (Euclid 2)) by Lm9; QQ is open by A288, Lm9, PRE_TOPC:30; then consider r being real number such that A290: r > 0 and A291: Ball (q,r) c= Q by A289, TOPMETR:15; reconsider r = r as Real by XREAL_0:def_1; A292: r / 2 > 0 by A290, XREAL_1:215; set r2 = min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))); A293: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= min (((s2 - s1) / 2),((t2 - t1) / 2)) by XXREAL_0:17; A294: min (((s2 - s1) / 2),((t2 - t1) / 2)) <= (s2 - s1) / 2 by XXREAL_0:17; A295: min (((s2 - s1) / 2),((t2 - t1) / 2)) <= (t2 - t1) / 2 by XXREAL_0:17; A296: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= (s2 - s1) / 2 by A293, A294, XXREAL_0:2; A297: min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) <= (t2 - t1) / 2 by A293, A295, XXREAL_0:2; A298: s2 - s1 > 0 by A1, XREAL_1:50; A299: t2 - t1 > 0 by A2, XREAL_1:50; A300: (s2 - s1) / 2 > 0 by A298, XREAL_1:215; (t2 - t1) / 2 > 0 by A299, XREAL_1:215; then 0 < min (((s2 - s1) / 2),((t2 - t1) / 2)) by A300, XXREAL_0:15; then A301: 0 < min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) by A292, XXREAL_0:15; set pa = |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]|; A302: |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1 = (p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))) by EUCLID:52; A303: |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 = (p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))) by EUCLID:52; A304: |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1 < s2 by A287, A301, A302, XREAL_1:44; A305: |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 < t2 by A287, A301, A303, XREAL_1:44; (s2 - s1) / 2 < s2 - s1 by A298, XREAL_1:216; then min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) < (p `1) - s1 by A287, A296, XXREAL_0:2; then A306: |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1 > s1 by A302, XREAL_1:12; (t2 - t1) / 2 < t2 - t1 by A299, XREAL_1:216; then min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))) < (p `2) - t1 by A287, A297, XXREAL_0:2; then |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2 > t1 by A303, XREAL_1:12; then A307: |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| in P1 by A4, A304, A305, A306; reconsider qa = |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| as Point of (Euclid 2) by EUCLID:22; A308: 0 <= ((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2 by XREAL_1:63; 0 <= ((p `2) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2 by XREAL_1:63; then A309: 0 + 0 <= (((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2) by A308, XREAL_1:7; (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))) ^2 <= (r / 2) ^2 by A301, SQUARE_1:15, XXREAL_0:17; then A310: (((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2) <= ((r / 2) ^2) + ((r / 2) ^2) by A302, A303, XREAL_1:7; r ^2 = (((r / 2) ^2) + ((r / 2) ^2)) + ((2 * (r / 2)) * (r / 2)) ; then r ^2 > ((r / 2) ^2) + ((r / 2) ^2) by A290, A292, XREAL_1:29, XREAL_1:129; then (((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2) < r ^2 by A310, XXREAL_0:2; then (sqrt ((((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2))) ^2 < r ^2 by A309, SQUARE_1:def_2; then A311: sqrt ((((p `1) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `1)) ^2) + (((p `2) - (|[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| `2)) ^2)) < r by A290, SQUARE_1:15; (Pitag_dist 2) . (q,qa) = dist (q,qa) by METRIC_1:def_1; then dist (q,qa) < r by A311, TOPREAL3:7; then |[((p `1) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2)))))),((p `2) - (min ((r / 2),(min (((s2 - s1) / 2),((t2 - t1) / 2))))))]| in Ball (q,r) by METRIC_1:11; hence P1 meets Q by A291, A307, XBOOLE_0:3; ::_thesis: verum end; hence p in Cl P1 by PRE_TOPC:def_7; ::_thesis: verum end; end; end; hence x in Cl P1 by A16; ::_thesis: verum end; end; end; hence x in Cl P1 ; ::_thesis: verum end; hence P \/ P1 c= Cl P1 by A5, XBOOLE_1:8; ::_thesis: verum end; Lm11: for s1, t1, s2, t2 being Real for P, P2 being Subset of (TOP-REAL 2) st s1 < s2 & t1 < t2 & P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P2 = { pb where pb is Point of (TOP-REAL 2) : ( not s1 <= pb `1 or not pb `1 <= s2 or not t1 <= pb `2 or not pb `2 <= t2 ) } holds Cl P2 = P \/ P2 proof let s1, t1, s2, t2 be Real; ::_thesis: for P, P2 being Subset of (TOP-REAL 2) st s1 < s2 & t1 < t2 & P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P2 = { pb where pb is Point of (TOP-REAL 2) : ( not s1 <= pb `1 or not pb `1 <= s2 or not t1 <= pb `2 or not pb `2 <= t2 ) } holds Cl P2 = P \/ P2 let P, P2 be Subset of (TOP-REAL 2); ::_thesis: ( s1 < s2 & t1 < t2 & P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P2 = { pb where pb is Point of (TOP-REAL 2) : ( not s1 <= pb `1 or not pb `1 <= s2 or not t1 <= pb `2 or not pb `2 <= t2 ) } implies Cl P2 = P \/ P2 ) assume that A1: s1 < s2 and A2: t1 < t2 and A3: P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } and A4: P2 = { pb where pb is Point of (TOP-REAL 2) : ( not s1 <= pb `1 or not pb `1 <= s2 or not t1 <= pb `2 or not pb `2 <= t2 ) } ; ::_thesis: Cl P2 = P \/ P2 A5: P2 c= Cl P2 by PRE_TOPC:18; reconsider P1 = { pa where pa is Point of (TOP-REAL 2) : ( s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) } as Subset of (TOP-REAL 2) by Th29; reconsider PP = P ` as Subset of (TOP-REAL 2) ; A6: PP = P1 \/ P2 by A1, A2, A3, A4, Th36; P1 misses P2 by A1, A2, A3, A4, Th36; then A7: P2 c= P1 ` by SUBSET_1:23; P = (P1 \/ P2) ` by A6; then A8: P = (([#] (TOP-REAL 2)) \ P1) /\ (([#] (TOP-REAL 2)) \ P2) by XBOOLE_1:53; A9: [#] (TOP-REAL 2) = P \/ (P2 \/ P1) by A6, PRE_TOPC:2 .= (P \/ P2) \/ P1 by XBOOLE_1:4 ; now__::_thesis:_for_x_being_set_st_x_in_P1_`_holds_ x_in_P_\/_P2 let x be set ; ::_thesis: ( x in P1 ` implies x in P \/ P2 ) assume A10: x in P1 ` ; ::_thesis: x in P \/ P2 then not x in P1 by XBOOLE_0:def_5; hence x in P \/ P2 by A9, A10, XBOOLE_0:def_3; ::_thesis: verum end; then A11: P1 ` c= P \/ P2 by TARSKI:def_3; now__::_thesis:_for_x_being_set_st_x_in_P_\/_P2_holds_ x_in_P1_` let x be set ; ::_thesis: ( x in P \/ P2 implies x in P1 ` ) assume x in P \/ P2 ; ::_thesis: x in P1 ` then ( x in P or x in P2 ) by XBOOLE_0:def_3; hence x in P1 ` by A7, A8, XBOOLE_0:def_4; ::_thesis: verum end; then P \/ P2 c= P1 ` by TARSKI:def_3; then A12: P1 ` = P \/ P2 by A11, XBOOLE_0:def_10; A13: P1 is open by Th33; ([#] (TOP-REAL 2)) \ (P1 `) = (P1 `) ` .= P1 ; then A14: P \/ P2 is closed by A12, A13, PRE_TOPC:def_3; A15: P2 c= P \/ P2 by XBOOLE_1:7; thus Cl P2 c= P \/ P2 :: according to XBOOLE_0:def_10 ::_thesis: P \/ P2 c= Cl P2 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Cl P2 or x in P \/ P2 ) assume x in Cl P2 ; ::_thesis: x in P \/ P2 hence x in P \/ P2 by A14, A15, PRE_TOPC:15; ::_thesis: verum end; P c= Cl P2 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in P or x in Cl P2 ) assume x in P ; ::_thesis: x in Cl P2 then consider p being Point of (TOP-REAL 2) such that A16: p = x and A17: ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) by A3; reconsider q = p as Point of (Euclid 2) by EUCLID:22; now__::_thesis:_x_in_Cl_P2 percases ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) by A17; supposeA18: ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) ; ::_thesis: x in Cl P2 for Q being Subset of (TOP-REAL 2) st Q is open & p in Q holds P2 meets Q proof let Q be Subset of (TOP-REAL 2); ::_thesis: ( Q is open & p in Q implies P2 meets Q ) assume that A19: Q is open and A20: p in Q ; ::_thesis: P2 meets Q reconsider QQ = Q as Subset of (TopSpaceMetr (Euclid 2)) by Lm9; QQ is open by A19, Lm9, PRE_TOPC:30; then consider r being real number such that A21: r > 0 and A22: Ball (q,r) c= Q by A20, TOPMETR:15; reconsider r = r as Real by XREAL_0:def_1; set pa = |[((p `1) - (r / 2)),(p `2)]|; A23: |[((p `1) - (r / 2)),(p `2)]| `1 = (p `1) - (r / 2) by EUCLID:52; A24: |[((p `1) - (r / 2)),(p `2)]| `2 = p `2 by EUCLID:52; A25: r / 2 > 0 by A21, XREAL_1:215; ( not s1 <= |[((p `1) - (r / 2)),(p `2)]| `1 or not |[((p `1) - (r / 2)),(p `2)]| `1 <= s2 or not t1 <= |[((p `1) - (r / 2)),(p `2)]| `2 or not |[((p `1) - (r / 2)),(p `2)]| `2 <= t2 ) by A18, A21, A23, XREAL_1:44, XREAL_1:215; then A26: |[((p `1) - (r / 2)),(p `2)]| in P2 by A4; reconsider qa = |[((p `1) - (r / 2)),(p `2)]| as Point of (Euclid 2) by EUCLID:22; A27: 0 <= ((p `1) - (|[((p `1) - (r / 2)),(p `2)]| `1)) ^2 by XREAL_1:63; 0 <= ((p `2) - (|[((p `1) - (r / 2)),(p `2)]| `2)) ^2 by XREAL_1:63; then A28: 0 + 0 <= (((p `1) - (|[((p `1) - (r / 2)),(p `2)]| `1)) ^2) + (((p `2) - (|[((p `1) - (r / 2)),(p `2)]| `2)) ^2) by A27, XREAL_1:7; (p `1) - (|[((p `1) - (r / 2)),(p `2)]| `1) < r by A21, A23, XREAL_1:216; then (((p `1) - (|[((p `1) - (r / 2)),(p `2)]| `1)) ^2) + (((p `2) - (|[((p `1) - (r / 2)),(p `2)]| `2)) ^2) < r ^2 by A23, A24, A25, SQUARE_1:16; then (sqrt ((((p `1) - (|[((p `1) - (r / 2)),(p `2)]| `1)) ^2) + (((p `2) - (|[((p `1) - (r / 2)),(p `2)]| `2)) ^2))) ^2 < r ^2 by A28, SQUARE_1:def_2; then A29: sqrt ((((p `1) - (|[((p `1) - (r / 2)),(p `2)]| `1)) ^2) + (((p `2) - (|[((p `1) - (r / 2)),(p `2)]| `2)) ^2)) < r by A21, SQUARE_1:15; (Pitag_dist 2) . (q,qa) = dist (q,qa) by METRIC_1:def_1; then dist (q,qa) < r by A29, TOPREAL3:7; then |[((p `1) - (r / 2)),(p `2)]| in Ball (q,r) by METRIC_1:11; then P2 /\ Q <> {} by A22, A26, XBOOLE_0:def_4; hence P2 meets Q by XBOOLE_0:def_7; ::_thesis: verum end; hence x in Cl P2 by A16, PRE_TOPC:def_7; ::_thesis: verum end; supposeA30: ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) ; ::_thesis: x in Cl P2 for Q being Subset of (TOP-REAL 2) st Q is open & p in Q holds P2 meets Q proof let Q be Subset of (TOP-REAL 2); ::_thesis: ( Q is open & p in Q implies P2 meets Q ) assume that A31: Q is open and A32: p in Q ; ::_thesis: P2 meets Q reconsider QQ = Q as Subset of (TopSpaceMetr (Euclid 2)) by Lm9; QQ is open by A31, Lm9, PRE_TOPC:30; then consider r being real number such that A33: r > 0 and A34: Ball (q,r) c= Q by A32, TOPMETR:15; reconsider r = r as Real by XREAL_0:def_1; set pa = |[(p `1),((p `2) + (r / 2))]|; A35: |[(p `1),((p `2) + (r / 2))]| `1 = p `1 by EUCLID:52; A36: |[(p `1),((p `2) + (r / 2))]| `2 = (p `2) + (r / 2) by EUCLID:52; A37: r / 2 > 0 by A33, XREAL_1:215; ( not s1 <= |[(p `1),((p `2) + (r / 2))]| `1 or not |[(p `1),((p `2) + (r / 2))]| `1 <= s2 or not t1 <= |[(p `1),((p `2) + (r / 2))]| `2 or not |[(p `1),((p `2) + (r / 2))]| `2 <= t2 ) by A30, A33, A36, XREAL_1:29, XREAL_1:215; then A38: |[(p `1),((p `2) + (r / 2))]| in P2 by A4; reconsider qa = |[(p `1),((p `2) + (r / 2))]| as Point of (Euclid 2) by EUCLID:22; A39: 0 <= ((p `1) - (|[(p `1),((p `2) + (r / 2))]| `1)) ^2 by XREAL_1:63; 0 <= ((p `2) - (|[(p `1),((p `2) + (r / 2))]| `2)) ^2 by XREAL_1:63; then A40: 0 + 0 <= (((p `1) - (|[(p `1),((p `2) + (r / 2))]| `1)) ^2) + (((p `2) - (|[(p `1),((p `2) + (r / 2))]| `2)) ^2) by A39, XREAL_1:7; A41: (|[(p `1),((p `2) + (r / 2))]| `2) - (p `2) < r by A33, A36, XREAL_1:216; ((|[(p `1),((p `2) + (r / 2))]| `2) - (p `2)) ^2 = ((p `2) - (|[(p `1),((p `2) + (r / 2))]| `2)) ^2 ; then (((p `1) - (|[(p `1),((p `2) + (r / 2))]| `1)) ^2) + (((p `2) - (|[(p `1),((p `2) + (r / 2))]| `2)) ^2) < r ^2 by A35, A36, A37, A41, SQUARE_1:16; then (sqrt ((((p `1) - (|[(p `1),((p `2) + (r / 2))]| `1)) ^2) + (((p `2) - (|[(p `1),((p `2) + (r / 2))]| `2)) ^2))) ^2 < r ^2 by A40, SQUARE_1:def_2; then A42: sqrt ((((p `1) - (|[(p `1),((p `2) + (r / 2))]| `1)) ^2) + (((p `2) - (|[(p `1),((p `2) + (r / 2))]| `2)) ^2)) < r by A33, SQUARE_1:15; (Pitag_dist 2) . (q,qa) = dist (q,qa) by METRIC_1:def_1; then dist (q,qa) < r by A42, TOPREAL3:7; then |[(p `1),((p `2) + (r / 2))]| in Ball (q,r) by METRIC_1:11; then P2 /\ Q <> {} by A34, A38, XBOOLE_0:def_4; hence P2 meets Q by XBOOLE_0:def_7; ::_thesis: verum end; hence x in Cl P2 by A16, PRE_TOPC:def_7; ::_thesis: verum end; supposeA43: ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) ; ::_thesis: x in Cl P2 for Q being Subset of (TOP-REAL 2) st Q is open & p in Q holds P2 meets Q proof let Q be Subset of (TOP-REAL 2); ::_thesis: ( Q is open & p in Q implies P2 meets Q ) assume that A44: Q is open and A45: p in Q ; ::_thesis: P2 meets Q reconsider QQ = Q as Subset of (TopSpaceMetr (Euclid 2)) by Lm9; QQ is open by A44, Lm9, PRE_TOPC:30; then consider r being real number such that A46: r > 0 and A47: Ball (q,r) c= Q by A45, TOPMETR:15; reconsider r = r as Real by XREAL_0:def_1; set pa = |[(p `1),((p `2) - (r / 2))]|; A48: |[(p `1),((p `2) - (r / 2))]| `1 = p `1 by EUCLID:52; A49: |[(p `1),((p `2) - (r / 2))]| `2 = (p `2) - (r / 2) by EUCLID:52; A50: r / 2 > 0 by A46, XREAL_1:215; ( not s1 <= |[(p `1),((p `2) - (r / 2))]| `1 or not |[(p `1),((p `2) - (r / 2))]| `1 <= s2 or not t1 <= |[(p `1),((p `2) - (r / 2))]| `2 or not |[(p `1),((p `2) - (r / 2))]| `2 <= t2 ) by A43, A46, A49, XREAL_1:44, XREAL_1:215; then A51: |[(p `1),((p `2) - (r / 2))]| in P2 by A4; reconsider qa = |[(p `1),((p `2) - (r / 2))]| as Point of (Euclid 2) by EUCLID:22; A52: 0 <= ((p `1) - (|[(p `1),((p `2) - (r / 2))]| `1)) ^2 by XREAL_1:63; 0 <= ((p `2) - (|[(p `1),((p `2) - (r / 2))]| `2)) ^2 by XREAL_1:63; then A53: 0 + 0 <= (((p `1) - (|[(p `1),((p `2) - (r / 2))]| `1)) ^2) + (((p `2) - (|[(p `1),((p `2) - (r / 2))]| `2)) ^2) by A52, XREAL_1:7; (p `2) - (|[(p `1),((p `2) - (r / 2))]| `2) < r by A46, A49, XREAL_1:216; then (((p `1) - (|[(p `1),((p `2) - (r / 2))]| `1)) ^2) + (((p `2) - (|[(p `1),((p `2) - (r / 2))]| `2)) ^2) < r ^2 by A48, A49, A50, SQUARE_1:16; then (sqrt ((((p `1) - (|[(p `1),((p `2) - (r / 2))]| `1)) ^2) + (((p `2) - (|[(p `1),((p `2) - (r / 2))]| `2)) ^2))) ^2 < r ^2 by A53, SQUARE_1:def_2; then A54: sqrt ((((p `1) - (|[(p `1),((p `2) - (r / 2))]| `1)) ^2) + (((p `2) - (|[(p `1),((p `2) - (r / 2))]| `2)) ^2)) < r by A46, SQUARE_1:15; (Pitag_dist 2) . (q,qa) = dist (q,qa) by METRIC_1:def_1; then dist (q,qa) < r by A54, TOPREAL3:7; then |[(p `1),((p `2) - (r / 2))]| in Ball (q,r) by METRIC_1:11; hence P2 meets Q by A47, A51, XBOOLE_0:3; ::_thesis: verum end; hence x in Cl P2 by A16, PRE_TOPC:def_7; ::_thesis: verum end; supposeA55: ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ; ::_thesis: x in Cl P2 for Q being Subset of (TOP-REAL 2) st Q is open & p in Q holds P2 meets Q proof let Q be Subset of (TOP-REAL 2); ::_thesis: ( Q is open & p in Q implies P2 meets Q ) assume that A56: Q is open and A57: p in Q ; ::_thesis: P2 meets Q reconsider QQ = Q as Subset of (TopSpaceMetr (Euclid 2)) by Lm9; QQ is open by A56, Lm9, PRE_TOPC:30; then consider r being real number such that A58: r > 0 and A59: Ball (q,r) c= Q by A57, TOPMETR:15; reconsider r = r as Real by XREAL_0:def_1; set pa = |[((p `1) + (r / 2)),(p `2)]|; A60: |[((p `1) + (r / 2)),(p `2)]| `1 = (p `1) + (r / 2) by EUCLID:52; A61: |[((p `1) + (r / 2)),(p `2)]| `2 = p `2 by EUCLID:52; A62: r / 2 > 0 by A58, XREAL_1:215; ( not s1 <= |[((p `1) + (r / 2)),(p `2)]| `1 or not |[((p `1) + (r / 2)),(p `2)]| `1 <= s2 or not t1 <= |[((p `1) + (r / 2)),(p `2)]| `2 or not |[((p `1) + (r / 2)),(p `2)]| `2 <= t2 ) by A55, A58, A60, XREAL_1:29, XREAL_1:215; then A63: |[((p `1) + (r / 2)),(p `2)]| in P2 by A4; reconsider qa = |[((p `1) + (r / 2)),(p `2)]| as Point of (Euclid 2) by EUCLID:22; A64: 0 <= ((p `1) - (|[((p `1) + (r / 2)),(p `2)]| `1)) ^2 by XREAL_1:63; 0 <= ((p `2) - (|[((p `1) + (r / 2)),(p `2)]| `2)) ^2 by XREAL_1:63; then A65: 0 + 0 <= (((p `1) - (|[((p `1) + (r / 2)),(p `2)]| `1)) ^2) + (((p `2) - (|[((p `1) + (r / 2)),(p `2)]| `2)) ^2) by A64, XREAL_1:7; A66: (|[((p `1) + (r / 2)),(p `2)]| `1) - (p `1) < r by A58, A60, XREAL_1:216; ((|[((p `1) + (r / 2)),(p `2)]| `1) - (p `1)) ^2 = ((p `1) - (|[((p `1) + (r / 2)),(p `2)]| `1)) ^2 ; then (((p `1) - (|[((p `1) + (r / 2)),(p `2)]| `1)) ^2) + (((p `2) - (|[((p `1) + (r / 2)),(p `2)]| `2)) ^2) < r ^2 by A60, A61, A62, A66, SQUARE_1:16; then (sqrt ((((p `1) - (|[((p `1) + (r / 2)),(p `2)]| `1)) ^2) + (((p `2) - (|[((p `1) + (r / 2)),(p `2)]| `2)) ^2))) ^2 < r ^2 by A65, SQUARE_1:def_2; then A67: sqrt ((((p `1) - (|[((p `1) + (r / 2)),(p `2)]| `1)) ^2) + (((p `2) - (|[((p `1) + (r / 2)),(p `2)]| `2)) ^2)) < r by A58, SQUARE_1:15; (Pitag_dist 2) . (q,qa) = dist (q,qa) by METRIC_1:def_1; then dist (q,qa) < r by A67, TOPREAL3:7; then |[((p `1) + (r / 2)),(p `2)]| in Ball (q,r) by METRIC_1:11; hence P2 meets Q by A59, A63, XBOOLE_0:3; ::_thesis: verum end; hence x in Cl P2 by A16, PRE_TOPC:def_7; ::_thesis: verum end; end; end; hence x in Cl P2 ; ::_thesis: verum end; hence P \/ P2 c= Cl P2 by A5, XBOOLE_1:8; ::_thesis: verum end; theorem Th37: :: JORDAN1:37 for s1, t1, s2, t2 being Real for P, P1, P2 being Subset of (TOP-REAL 2) st s1 < s2 & t1 < t2 & P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P1 = { pa where pa is Point of (TOP-REAL 2) : ( s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) } & P2 = { pb where pb is Point of (TOP-REAL 2) : ( not s1 <= pb `1 or not pb `1 <= s2 or not t1 <= pb `2 or not pb `2 <= t2 ) } holds ( P = (Cl P1) \ P1 & P = (Cl P2) \ P2 ) proof let s1, t1, s2, t2 be Real; ::_thesis: for P, P1, P2 being Subset of (TOP-REAL 2) st s1 < s2 & t1 < t2 & P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P1 = { pa where pa is Point of (TOP-REAL 2) : ( s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) } & P2 = { pb where pb is Point of (TOP-REAL 2) : ( not s1 <= pb `1 or not pb `1 <= s2 or not t1 <= pb `2 or not pb `2 <= t2 ) } holds ( P = (Cl P1) \ P1 & P = (Cl P2) \ P2 ) let P, P1, P2 be Subset of (TOP-REAL 2); ::_thesis: ( s1 < s2 & t1 < t2 & P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P1 = { pa where pa is Point of (TOP-REAL 2) : ( s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) } & P2 = { pb where pb is Point of (TOP-REAL 2) : ( not s1 <= pb `1 or not pb `1 <= s2 or not t1 <= pb `2 or not pb `2 <= t2 ) } implies ( P = (Cl P1) \ P1 & P = (Cl P2) \ P2 ) ) assume that A1: s1 < s2 and A2: t1 < t2 and A3: P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } and A4: P1 = { pa where pa is Point of (TOP-REAL 2) : ( s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) } and A5: P2 = { pb where pb is Point of (TOP-REAL 2) : ( not s1 <= pb `1 or not pb `1 <= s2 or not t1 <= pb `2 or not pb `2 <= t2 ) } ; ::_thesis: ( P = (Cl P1) \ P1 & P = (Cl P2) \ P2 ) reconsider PP = P ` as Subset of (TOP-REAL 2) ; PP = P1 \/ P2 by A1, A2, A3, A4, A5, Th36; then P = (P1 \/ P2) ` ; then A6: P = (([#] (TOP-REAL 2)) \ P1) /\ (([#] (TOP-REAL 2)) \ P2) by XBOOLE_1:53; then A7: P c= ([#] (TOP-REAL 2)) \ P2 by XBOOLE_1:17; A8: Cl P2 = P \/ P2 by A1, A2, A3, A5, Lm11; A9: (P \/ P2) \ P2 c= P proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (P \/ P2) \ P2 or x in P ) assume A10: x in (P \/ P2) \ P2 ; ::_thesis: x in P then A11: x in P \/ P2 by XBOOLE_0:def_5; not x in P2 by A10, XBOOLE_0:def_5; hence x in P by A11, XBOOLE_0:def_3; ::_thesis: verum end; P c= Cl P2 by A8, XBOOLE_1:7; then P c= (Cl P2) /\ (P2 `) by A7, XBOOLE_1:19; then A12: P c= (Cl P2) \ P2 by SUBSET_1:13; A13: P c= ([#] (TOP-REAL 2)) \ P1 by A6, XBOOLE_1:17; A14: Cl P1 = P \/ P1 by A1, A2, A3, A4, Lm10; A15: (P \/ P1) \ P1 c= P proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (P \/ P1) \ P1 or x in P ) assume A16: x in (P \/ P1) \ P1 ; ::_thesis: x in P then A17: x in P \/ P1 by XBOOLE_0:def_5; not x in P1 by A16, XBOOLE_0:def_5; hence x in P by A17, XBOOLE_0:def_3; ::_thesis: verum end; P c= Cl P1 by A14, XBOOLE_1:7; then P c= (Cl P1) /\ (P1 `) by A13, XBOOLE_1:19; then P c= (Cl P1) \ P1 by SUBSET_1:13; hence ( P = (Cl P1) \ P1 & P = (Cl P2) \ P2 ) by A8, A9, A12, A14, A15, XBOOLE_0:def_10; ::_thesis: verum end; theorem Th38: :: JORDAN1:38 for s1, s2, t1, t2 being Real for P, P1 being Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P1 = { pa where pa is Point of (TOP-REAL 2) : ( s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) } holds P1 c= [#] ((TOP-REAL 2) | (P `)) proof let s1, s2, t1, t2 be Real; ::_thesis: for P, P1 being Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P1 = { pa where pa is Point of (TOP-REAL 2) : ( s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) } holds P1 c= [#] ((TOP-REAL 2) | (P `)) let P, P1 be Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P1 = { pa where pa is Point of (TOP-REAL 2) : ( s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) } implies P1 c= [#] ((TOP-REAL 2) | (P `)) ) assume that A1: P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } and A2: P1 = { pa where pa is Point of (TOP-REAL 2) : ( s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) } ; ::_thesis: P1 c= [#] ((TOP-REAL 2) | (P `)) let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in P1 or x in [#] ((TOP-REAL 2) | (P `)) ) assume A3: x in P1 ; ::_thesis: x in [#] ((TOP-REAL 2) | (P `)) then A4: ex pa being Point of (TOP-REAL 2) st ( pa = x & s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) by A2; now__::_thesis:_not_x_in__{__p_where_p_is_Point_of_(TOP-REAL_2)_:_(_(_p_`1_=_s1_&_p_`2_<=_t2_&_p_`2_>=_t1_)_or_(_p_`1_<=_s2_&_p_`1_>=_s1_&_p_`2_=_t2_)_or_(_p_`1_<=_s2_&_p_`1_>=_s1_&_p_`2_=_t1_)_or_(_p_`1_=_s2_&_p_`2_<=_t2_&_p_`2_>=_t1_)_)__}_ assume x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } ; ::_thesis: contradiction then ex p being Point of (TOP-REAL 2) st ( p = x & ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) ) ; hence contradiction by A4; ::_thesis: verum end; then x in P ` by A1, A3, SUBSET_1:29; hence x in [#] ((TOP-REAL 2) | (P `)) by PRE_TOPC:def_5; ::_thesis: verum end; theorem :: JORDAN1:39 for s1, s2, t1, t2 being Real for P, P1 being Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P1 = { pa where pa is Point of (TOP-REAL 2) : ( s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) } holds P1 is Subset of ((TOP-REAL 2) | (P `)) proof let s1, s2, t1, t2 be Real; ::_thesis: for P, P1 being Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P1 = { pa where pa is Point of (TOP-REAL 2) : ( s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) } holds P1 is Subset of ((TOP-REAL 2) | (P `)) let P, P1 be Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P1 = { pa where pa is Point of (TOP-REAL 2) : ( s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) } implies P1 is Subset of ((TOP-REAL 2) | (P `)) ) assume that A1: P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } and A2: P1 = { pa where pa is Point of (TOP-REAL 2) : ( s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) } ; ::_thesis: P1 is Subset of ((TOP-REAL 2) | (P `)) P1 c= [#] ((TOP-REAL 2) | (P `)) by A1, A2, Th38; hence P1 is Subset of ((TOP-REAL 2) | (P `)) ; ::_thesis: verum end; theorem Th40: :: JORDAN1:40 for s1, s2, t1, t2 being Real for P, P2 being Subset of (TOP-REAL 2) st s1 < s2 & t1 < t2 & P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P2 = { pb where pb is Point of (TOP-REAL 2) : ( not s1 <= pb `1 or not pb `1 <= s2 or not t1 <= pb `2 or not pb `2 <= t2 ) } holds P2 c= [#] ((TOP-REAL 2) | (P `)) proof let s1, s2, t1, t2 be Real; ::_thesis: for P, P2 being Subset of (TOP-REAL 2) st s1 < s2 & t1 < t2 & P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P2 = { pb where pb is Point of (TOP-REAL 2) : ( not s1 <= pb `1 or not pb `1 <= s2 or not t1 <= pb `2 or not pb `2 <= t2 ) } holds P2 c= [#] ((TOP-REAL 2) | (P `)) let P, P2 be Subset of (TOP-REAL 2); ::_thesis: ( s1 < s2 & t1 < t2 & P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P2 = { pb where pb is Point of (TOP-REAL 2) : ( not s1 <= pb `1 or not pb `1 <= s2 or not t1 <= pb `2 or not pb `2 <= t2 ) } implies P2 c= [#] ((TOP-REAL 2) | (P `)) ) assume that A1: s1 < s2 and A2: t1 < t2 and A3: P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } and A4: P2 = { pb where pb is Point of (TOP-REAL 2) : ( not s1 <= pb `1 or not pb `1 <= s2 or not t1 <= pb `2 or not pb `2 <= t2 ) } ; ::_thesis: P2 c= [#] ((TOP-REAL 2) | (P `)) let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in P2 or x in [#] ((TOP-REAL 2) | (P `)) ) assume A5: x in P2 ; ::_thesis: x in [#] ((TOP-REAL 2) | (P `)) then A6: ex pa being Point of (TOP-REAL 2) st ( pa = x & ( not s1 <= pa `1 or not pa `1 <= s2 or not t1 <= pa `2 or not pa `2 <= t2 ) ) by A4; now__::_thesis:_not_x_in__{__p_where_p_is_Point_of_(TOP-REAL_2)_:_(_(_p_`1_=_s1_&_p_`2_<=_t2_&_p_`2_>=_t1_)_or_(_p_`1_<=_s2_&_p_`1_>=_s1_&_p_`2_=_t2_)_or_(_p_`1_<=_s2_&_p_`1_>=_s1_&_p_`2_=_t1_)_or_(_p_`1_=_s2_&_p_`2_<=_t2_&_p_`2_>=_t1_)_)__}_ assume x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } ; ::_thesis: contradiction then ex p being Point of (TOP-REAL 2) st ( p = x & ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) ) ; hence contradiction by A1, A2, A6; ::_thesis: verum end; then x in P ` by A3, A5, SUBSET_1:29; hence x in [#] ((TOP-REAL 2) | (P `)) by PRE_TOPC:def_5; ::_thesis: verum end; theorem :: JORDAN1:41 for s1, s2, t1, t2 being Real for P, P2 being Subset of (TOP-REAL 2) st s1 < s2 & t1 < t2 & P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P2 = { pb where pb is Point of (TOP-REAL 2) : ( not s1 <= pb `1 or not pb `1 <= s2 or not t1 <= pb `2 or not pb `2 <= t2 ) } holds P2 is Subset of ((TOP-REAL 2) | (P `)) proof let s1, s2, t1, t2 be Real; ::_thesis: for P, P2 being Subset of (TOP-REAL 2) st s1 < s2 & t1 < t2 & P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P2 = { pb where pb is Point of (TOP-REAL 2) : ( not s1 <= pb `1 or not pb `1 <= s2 or not t1 <= pb `2 or not pb `2 <= t2 ) } holds P2 is Subset of ((TOP-REAL 2) | (P `)) let P, P2 be Subset of (TOP-REAL 2); ::_thesis: ( s1 < s2 & t1 < t2 & P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P2 = { pb where pb is Point of (TOP-REAL 2) : ( not s1 <= pb `1 or not pb `1 <= s2 or not t1 <= pb `2 or not pb `2 <= t2 ) } implies P2 is Subset of ((TOP-REAL 2) | (P `)) ) assume that A1: s1 < s2 and A2: t1 < t2 and A3: P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } and A4: P2 = { pb where pb is Point of (TOP-REAL 2) : ( not s1 <= pb `1 or not pb `1 <= s2 or not t1 <= pb `2 or not pb `2 <= t2 ) } ; ::_thesis: P2 is Subset of ((TOP-REAL 2) | (P `)) P2 c= [#] ((TOP-REAL 2) | (P `)) by A1, A2, A3, A4, Th40; hence P2 is Subset of ((TOP-REAL 2) | (P `)) ; ::_thesis: verum end; begin definition let S be Subset of (TOP-REAL 2); attrS is Jordan means :Def2: :: JORDAN1:def 2 ( S ` <> {} & ex A1, A2 being Subset of (TOP-REAL 2) st ( S ` = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 & ( for C1, C2 being Subset of ((TOP-REAL 2) | (S `)) st C1 = A1 & C2 = A2 holds ( C1 is a_component & C2 is a_component ) ) ) ); end; :: deftheorem Def2 defines Jordan JORDAN1:def_2_:_ for S being Subset of (TOP-REAL 2) holds ( S is Jordan iff ( S ` <> {} & ex A1, A2 being Subset of (TOP-REAL 2) st ( S ` = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 & ( for C1, C2 being Subset of ((TOP-REAL 2) | (S `)) st C1 = A1 & C2 = A2 holds ( C1 is a_component & C2 is a_component ) ) ) ) ); theorem :: JORDAN1:42 for S being Subset of (TOP-REAL 2) st S is Jordan holds ( S ` <> {} & ex A1, A2 being Subset of (TOP-REAL 2) ex C1, C2 being Subset of ((TOP-REAL 2) | (S `)) st ( S ` = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 & C1 = A1 & C2 = A2 & C1 is a_component & C2 is a_component & ( for C3 being Subset of ((TOP-REAL 2) | (S `)) holds ( not C3 is a_component or C3 = C1 or C3 = C2 ) ) ) ) proof let S be Subset of (TOP-REAL 2); ::_thesis: ( S is Jordan implies ( S ` <> {} & ex A1, A2 being Subset of (TOP-REAL 2) ex C1, C2 being Subset of ((TOP-REAL 2) | (S `)) st ( S ` = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 & C1 = A1 & C2 = A2 & C1 is a_component & C2 is a_component & ( for C3 being Subset of ((TOP-REAL 2) | (S `)) holds ( not C3 is a_component or C3 = C1 or C3 = C2 ) ) ) ) ) assume A1: S is Jordan ; ::_thesis: ( S ` <> {} & ex A1, A2 being Subset of (TOP-REAL 2) ex C1, C2 being Subset of ((TOP-REAL 2) | (S `)) st ( S ` = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 & C1 = A1 & C2 = A2 & C1 is a_component & C2 is a_component & ( for C3 being Subset of ((TOP-REAL 2) | (S `)) holds ( not C3 is a_component or C3 = C1 or C3 = C2 ) ) ) ) then reconsider S9 = S ` as non empty Subset of (TOP-REAL 2) by Def2; consider A1, A2 being Subset of (TOP-REAL 2) such that A2: S ` = A1 \/ A2 and A3: A1 misses A2 and A4: (Cl A1) \ A1 = (Cl A2) \ A2 and A5: for C1, C2 being Subset of ((TOP-REAL 2) | (S `)) st C1 = A1 & C2 = A2 holds ( C1 is a_component & C2 is a_component ) by A1, Def2; A6: A1 c= S ` by A2, XBOOLE_1:7; A7: A2 c= S ` by A2, XBOOLE_1:7; A8: [#] ((TOP-REAL 2) | (S `)) = S ` by PRE_TOPC:def_5; A1 c= [#] ((TOP-REAL 2) | (S `)) by A6, PRE_TOPC:def_5; then reconsider G0A = A1, G0B = A2 as Subset of ((TOP-REAL 2) | S9) by A7, PRE_TOPC:def_5; A9: G0A = A1 ; G0B = A2 ; then A10: G0A is a_component by A5; A11: G0B is a_component by A5, A9; now__::_thesis:_for_C3_being_Subset_of_((TOP-REAL_2)_|_S9)_holds_ (_not_C3_is_a_component_or_C3_=_G0A_or_C3_=_G0B_) let C3 be Subset of ((TOP-REAL 2) | S9); ::_thesis: ( not C3 is a_component or C3 = G0A or C3 = G0B ) assume A12: C3 is a_component ; ::_thesis: ( C3 = G0A or C3 = G0B ) then A13: C3 <> {} ((TOP-REAL 2) | S9) by CONNSP_1:32; C3 /\ (G0A \/ G0B) = C3 by A2, A8, XBOOLE_1:28; then A14: (C3 /\ G0A) \/ (C3 /\ G0B) <> {} by A13, XBOOLE_1:23; now__::_thesis:_(_C3_=_G0A_or_C3_=_G0B_) percases ( C3 /\ G0A <> {} or C3 /\ A2 <> {} ) by A14; suppose C3 /\ G0A <> {} ; ::_thesis: ( C3 = G0A or C3 = G0B ) then A15: C3 meets G0A by XBOOLE_0:def_7; A16: C3 is connected by A12, CONNSP_1:def_5; A17: G0A is connected by A10, CONNSP_1:def_5; then A18: C3 \/ G0A is connected by A15, A16, CONNSP_1:1, CONNSP_1:17; G0A c= C3 \/ G0A by XBOOLE_1:7; then G0A = C3 \/ G0A by A10, A18, CONNSP_1:def_5; then C3 c= G0A by XBOOLE_1:7; hence ( C3 = G0A or C3 = G0B ) by A12, A17, CONNSP_1:def_5; ::_thesis: verum end; suppose C3 /\ A2 <> {} ; ::_thesis: ( C3 = G0A or C3 = G0B ) then A19: C3 meets G0B by XBOOLE_0:def_7; A20: C3 is connected by A12, CONNSP_1:def_5; A21: G0B is connected by A11, CONNSP_1:def_5; then A22: C3 \/ G0B is connected by A19, A20, CONNSP_1:1, CONNSP_1:17; G0B c= C3 \/ G0B by XBOOLE_1:7; then G0B = C3 \/ G0B by A11, A22, CONNSP_1:def_5; then C3 c= G0B by XBOOLE_1:7; hence ( C3 = G0A or C3 = G0B ) by A12, A21, CONNSP_1:def_5; ::_thesis: verum end; end; end; hence ( C3 = G0A or C3 = G0B ) ; ::_thesis: verum end; hence ( S ` <> {} & ex A1, A2 being Subset of (TOP-REAL 2) ex C1, C2 being Subset of ((TOP-REAL 2) | (S `)) st ( S ` = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 & C1 = A1 & C2 = A2 & C1 is a_component & C2 is a_component & ( for C3 being Subset of ((TOP-REAL 2) | (S `)) holds ( not C3 is a_component or C3 = C1 or C3 = C2 ) ) ) ) by A2, A3, A4, A10, A11; ::_thesis: verum end; theorem :: JORDAN1:43 for s1, s2, t1, t2 being Real for P being Subset of (TOP-REAL 2) st s1 < s2 & t1 < t2 & P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } holds P is Jordan proof let s1, s2, t1, t2 be Real; ::_thesis: for P being Subset of (TOP-REAL 2) st s1 < s2 & t1 < t2 & P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } holds P is Jordan let P be Subset of (TOP-REAL 2); ::_thesis: ( s1 < s2 & t1 < t2 & P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } implies P is Jordan ) assume that A1: s1 < s2 and A2: t1 < t2 and A3: P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } ; ::_thesis: P is Jordan reconsider P1 = { pa where pa is Point of (TOP-REAL 2) : ( s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) } , P2 = { pc where pc is Point of (TOP-REAL 2) : ( not s1 <= pc `1 or not pc `1 <= s2 or not t1 <= pc `2 or not pc `2 <= t2 ) } as Subset of (TOP-REAL 2) by Th29, Th30; reconsider PC = P ` as Subset of (TOP-REAL 2) ; A4: P1 = { pa where pa is Point of (TOP-REAL 2) : ( s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) } ; A5: P2 = { pc where pc is Point of (TOP-REAL 2) : ( not s1 <= pc `1 or not pc `1 <= s2 or not t1 <= pc `2 or not pc `2 <= t2 ) } ; A6: PC = P1 \/ P2 by A1, A2, A3, Th36; A7: PC <> {} by A1, A2, A3, A4, A5, Th36; A8: P1 misses P2 by A1, A2, A3, Th36; A9: P = (Cl P1) \ P1 by A1, A2, A3, A5, Th37; A10: P = (Cl P2) \ P2 by A1, A2, A3, A4, Th37; for P1A, P2B being Subset of ((TOP-REAL 2) | (P `)) st P1A = P1 & P2B = P2 holds ( P1A is a_component & P2B is a_component ) by A1, A2, A3, Th36; hence P is Jordan by A6, A7, A8, A9, A10, Def2; ::_thesis: verum end; theorem :: JORDAN1:44 for s1, t1, s2, t2 being Real for P, P2 being Subset of (TOP-REAL 2) st s1 < s2 & t1 < t2 & P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P2 = { pb where pb is Point of (TOP-REAL 2) : ( not s1 <= pb `1 or not pb `1 <= s2 or not t1 <= pb `2 or not pb `2 <= t2 ) } holds Cl P2 = P \/ P2 by Lm11; theorem :: JORDAN1:45 for s1, t1, s2, t2 being Real for P, P1 being Subset of (TOP-REAL 2) st s1 < s2 & t1 < t2 & P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P1 = { pa where pa is Point of (TOP-REAL 2) : ( s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) } holds Cl P1 = P \/ P1 by Lm10; theorem :: JORDAN1:46 for p, q being Point of (TOP-REAL 2) holds (LSeg (p,q)) \ {p,q} is convex proof let p, q, w1, w2 be Point of (TOP-REAL 2); :: according to JORDAN1:def_1 ::_thesis: ( w1 in (LSeg (p,q)) \ {p,q} & w2 in (LSeg (p,q)) \ {p,q} implies LSeg (w1,w2) c= (LSeg (p,q)) \ {p,q} ) set P = (LSeg (p,q)) \ {p,q}; assume that A1: w1 in (LSeg (p,q)) \ {p,q} and A2: w2 in (LSeg (p,q)) \ {p,q} ; ::_thesis: LSeg (w1,w2) c= (LSeg (p,q)) \ {p,q} A3: w1 in LSeg (p,q) by A1, XBOOLE_0:def_5; A4: w2 in LSeg (p,q) by A2, XBOOLE_0:def_5; A5: not w1 in {p,q} by A1, XBOOLE_0:def_5; A6: not w2 in {p,q} by A2, XBOOLE_0:def_5; A7: w1 <> p by A5, TARSKI:def_2; A8: w2 <> p by A6, TARSKI:def_2; A9: w1 <> q by A5, TARSKI:def_2; A10: w2 <> q by A6, TARSKI:def_2; A11: not p in LSeg (w1,w2) by A3, A4, A7, A8, SPPOL_1:7, TOPREAL1:6; not q in LSeg (w1,w2) by A3, A4, A9, A10, SPPOL_1:7, TOPREAL1:6; then LSeg (w1,w2) misses {p,q} by A11, ZFMISC_1:51; hence LSeg (w1,w2) c= (LSeg (p,q)) \ {p,q} by A3, A4, TOPREAL1:6, XBOOLE_1:86; ::_thesis: verum end; Lm12: for x0, y0 being Real for P being Subset of (TOP-REAL 2) st P = { |[x,y0]| where x is Real : x <= x0 } holds P is convex proof let x0, y0 be Real; ::_thesis: for P being Subset of (TOP-REAL 2) st P = { |[x,y0]| where x is Real : x <= x0 } holds P is convex let P be Subset of (TOP-REAL 2); ::_thesis: ( P = { |[x,y0]| where x is Real : x <= x0 } implies P is convex ) assume A1: P = { |[x,y0]| where x is Real : x <= x0 } ; ::_thesis: P is convex let w1, w2 be Point of (TOP-REAL 2); :: according to JORDAN1:def_1 ::_thesis: ( w1 in P & w2 in P implies LSeg (w1,w2) c= P ) assume that A2: w1 in P and A3: w2 in P ; ::_thesis: LSeg (w1,w2) c= P consider x1 being Real such that A4: w1 = |[x1,y0]| and A5: x1 <= x0 by A1, A2; consider x2 being Real such that A6: w2 = |[x2,y0]| and A7: x2 <= x0 by A1, A3; let v be set ; :: according to TARSKI:def_3 ::_thesis: ( not v in LSeg (w1,w2) or v in P ) assume A8: v in LSeg (w1,w2) ; ::_thesis: v in P then reconsider v1 = v as Point of (TOP-REAL 2) ; consider l being Real such that A9: v1 = ((1 - l) * w1) + (l * w2) and A10: 0 <= l and A11: l <= 1 by A8; A12: v1 = |[((1 - l) * x1),((1 - l) * y0)]| + (l * |[x2,y0]|) by A4, A6, A9, EUCLID:58 .= |[((1 - l) * x1),((1 - l) * y0)]| + |[(l * x2),(l * y0)]| by EUCLID:58 .= |[(((1 - l) * x1) + (l * x2)),(((1 - l) * y0) + (l * y0))]| by EUCLID:56 .= |[(((1 - l) * x1) + (l * x2)),(1 * y0)]| ; ((1 - l) * x1) + (l * x2) <= x0 by A5, A7, A10, A11, XREAL_1:174; hence v in P by A1, A12; ::_thesis: verum end; Lm13: for x0, y0 being Real for P being Subset of (TOP-REAL 2) st P = { |[x0,y]| where y is Real : y <= y0 } holds P is convex proof let x0, y0 be Real; ::_thesis: for P being Subset of (TOP-REAL 2) st P = { |[x0,y]| where y is Real : y <= y0 } holds P is convex let P be Subset of (TOP-REAL 2); ::_thesis: ( P = { |[x0,y]| where y is Real : y <= y0 } implies P is convex ) assume A1: P = { |[x0,y]| where y is Real : y <= y0 } ; ::_thesis: P is convex let w1, w2 be Point of (TOP-REAL 2); :: according to JORDAN1:def_1 ::_thesis: ( w1 in P & w2 in P implies LSeg (w1,w2) c= P ) assume that A2: w1 in P and A3: w2 in P ; ::_thesis: LSeg (w1,w2) c= P consider y1 being Real such that A4: w1 = |[x0,y1]| and A5: y1 <= y0 by A1, A2; consider y2 being Real such that A6: w2 = |[x0,y2]| and A7: y2 <= y0 by A1, A3; let v be set ; :: according to TARSKI:def_3 ::_thesis: ( not v in LSeg (w1,w2) or v in P ) assume A8: v in LSeg (w1,w2) ; ::_thesis: v in P then reconsider v1 = v as Point of (TOP-REAL 2) ; consider l being Real such that A9: v1 = ((1 - l) * w1) + (l * w2) and A10: 0 <= l and A11: l <= 1 by A8; A12: v1 = |[((1 - l) * x0),((1 - l) * y1)]| + (l * |[x0,y2]|) by A4, A6, A9, EUCLID:58 .= |[((1 - l) * x0),((1 - l) * y1)]| + |[(l * x0),(l * y2)]| by EUCLID:58 .= |[(((1 - l) * x0) + (l * x0)),(((1 - l) * y1) + (l * y2))]| by EUCLID:56 .= |[(1 * x0),(((1 - l) * y1) + (l * y2))]| ; ((1 - l) * y1) + (l * y2) <= y0 by A5, A7, A10, A11, XREAL_1:174; hence v in P by A1, A12; ::_thesis: verum end; Lm14: for x0, y0 being Real for P being Subset of (TOP-REAL 2) st P = { |[x,y0]| where x is Real : x >= x0 } holds P is convex proof let x0, y0 be Real; ::_thesis: for P being Subset of (TOP-REAL 2) st P = { |[x,y0]| where x is Real : x >= x0 } holds P is convex let P be Subset of (TOP-REAL 2); ::_thesis: ( P = { |[x,y0]| where x is Real : x >= x0 } implies P is convex ) assume A1: P = { |[x,y0]| where x is Real : x >= x0 } ; ::_thesis: P is convex let w1, w2 be Point of (TOP-REAL 2); :: according to JORDAN1:def_1 ::_thesis: ( w1 in P & w2 in P implies LSeg (w1,w2) c= P ) assume that A2: w1 in P and A3: w2 in P ; ::_thesis: LSeg (w1,w2) c= P consider x1 being Real such that A4: w1 = |[x1,y0]| and A5: x1 >= x0 by A1, A2; consider x2 being Real such that A6: w2 = |[x2,y0]| and A7: x2 >= x0 by A1, A3; let v be set ; :: according to TARSKI:def_3 ::_thesis: ( not v in LSeg (w1,w2) or v in P ) assume A8: v in LSeg (w1,w2) ; ::_thesis: v in P then reconsider v1 = v as Point of (TOP-REAL 2) ; v1 in { (((1 - l) * w2) + (l * w1)) where l is Real : ( 0 <= l & l <= 1 ) } by A8, RLTOPSP1:def_2; then consider l being Real such that A9: v1 = ((1 - l) * w2) + (l * w1) and A10: 0 <= l and A11: l <= 1 ; A12: v1 = |[((1 - l) * x2),((1 - l) * y0)]| + (l * |[x1,y0]|) by A4, A6, A9, EUCLID:58 .= |[((1 - l) * x2),((1 - l) * y0)]| + |[(l * x1),(l * y0)]| by EUCLID:58 .= |[(((1 - l) * x2) + (l * x1)),(((1 - l) * y0) + (l * y0))]| by EUCLID:56 .= |[(((1 - l) * x2) + (l * x1)),(1 * y0)]| ; ((1 - l) * x2) + (l * x1) >= x0 by A5, A7, A10, A11, XREAL_1:173; hence v in P by A1, A12; ::_thesis: verum end; Lm15: for x0, y0 being Real for P being Subset of (TOP-REAL 2) st P = { |[x0,y]| where y is Real : y >= y0 } holds P is convex proof let x0, y0 be Real; ::_thesis: for P being Subset of (TOP-REAL 2) st P = { |[x0,y]| where y is Real : y >= y0 } holds P is convex let P be Subset of (TOP-REAL 2); ::_thesis: ( P = { |[x0,y]| where y is Real : y >= y0 } implies P is convex ) assume A1: P = { |[x0,y]| where y is Real : y >= y0 } ; ::_thesis: P is convex let w1, w2 be Point of (TOP-REAL 2); :: according to JORDAN1:def_1 ::_thesis: ( w1 in P & w2 in P implies LSeg (w1,w2) c= P ) assume that A2: w1 in P and A3: w2 in P ; ::_thesis: LSeg (w1,w2) c= P consider x1 being Real such that A4: w1 = |[x0,x1]| and A5: x1 >= y0 by A1, A2; consider x2 being Real such that A6: w2 = |[x0,x2]| and A7: x2 >= y0 by A1, A3; let v be set ; :: according to TARSKI:def_3 ::_thesis: ( not v in LSeg (w1,w2) or v in P ) assume A8: v in LSeg (w1,w2) ; ::_thesis: v in P then reconsider v1 = v as Point of (TOP-REAL 2) ; v1 in { (((1 - l) * w2) + (l * w1)) where l is Real : ( 0 <= l & l <= 1 ) } by A8, RLTOPSP1:def_2; then consider l being Real such that A9: v1 = ((1 - l) * w2) + (l * w1) and A10: 0 <= l and A11: l <= 1 ; A12: v1 = |[((1 - l) * x0),((1 - l) * x2)]| + (l * |[x0,x1]|) by A4, A6, A9, EUCLID:58 .= |[((1 - l) * x0),((1 - l) * x2)]| + |[(l * x0),(l * x1)]| by EUCLID:58 .= |[(((1 - l) * x0) + (l * x0)),(((1 - l) * x2) + (l * x1))]| by EUCLID:56 .= |[(1 * x0),(((1 - l) * x2) + (l * x1))]| ; ((1 - l) * x2) + (l * x1) >= y0 by A5, A7, A10, A11, XREAL_1:173; hence v in P by A1, A12; ::_thesis: verum end; registration let p be Point of (TOP-REAL 2); cluster north_halfline p -> convex ; coherence north_halfline p is convex proof north_halfline p = { |[(p `1),r]| where r is Element of REAL : r >= p `2 } by TOPREAL1:31; hence north_halfline p is convex by Lm15; ::_thesis: verum end; cluster east_halfline p -> convex ; coherence east_halfline p is convex proof east_halfline p = { |[r,(p `2)]| where r is Element of REAL : r >= p `1 } by TOPREAL1:33; hence east_halfline p is convex by Lm14; ::_thesis: verum end; cluster south_halfline p -> convex ; coherence south_halfline p is convex proof south_halfline p = { |[(p `1),r]| where r is Element of REAL : r <= p `2 } by TOPREAL1:35; hence south_halfline p is convex by Lm13; ::_thesis: verum end; cluster west_halfline p -> convex ; coherence west_halfline p is convex proof west_halfline p = { |[r,(p `2)]| where r is Element of REAL : r <= p `1 } by TOPREAL1:37; hence west_halfline p is convex by Lm12; ::_thesis: verum end; end;