:: 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;