:: TOPGEN_5 semantic presentation

begin


theorem Th1: :: TOPGEN_5:1
for f, g being Function st f tolerates g holds
for A being set holds (f +* g) " A = (f " A) \/ (g " A)
proof
let f, g be Function; ::_thesis: ( f tolerates g implies for A being set holds (f +* g) " A = (f " A) \/ (g " A) )
assume A1: f tolerates g ; ::_thesis: for A being set holds (f +* g) " A = (f " A) \/ (g " A)
let A be set ; ::_thesis: (f +* g) " A = (f " A) \/ (g " A)
 f c= f +* g by A1, FUNCT_4:28;
then A2: f " A c= (f +* g) " A by RELAT_1:144;
thus  (f +* g) " A c= (f " A) \/ (g " A) :: according to XBOOLE_0:def_10 ::_thesis: (f " A) \/ (g " A) c= (f +* g) " A
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (f +* g) " A or x in (f " A) \/ (g " A) )
assume A3: x in (f +* g) " A ; ::_thesis: x in (f " A) \/ (g " A)
then  x in dom (f +* g) by FUNCT_1:def_7;
then  x in (dom f) \/ (dom g) by FUNCT_4:def_1;
then  ( x in dom f or x in dom g ) by XBOOLE_0:def_3;
then A4: ( ( x in dom f & (f +* g) . x = f . x ) or ( x in dom g & (f +* g) . x = g . x ) ) by A1, FUNCT_4:13, FUNCT_4:15;
 (f +* g) . x in A by A3, FUNCT_1:def_7;
then  ( x in f " A or x in g " A ) by A4, FUNCT_1:def_7;
hence  x in (f " A) \/ (g " A) by XBOOLE_0:def_3; ::_thesis: verum
end;
 g " A c= (f +* g) " A by FUNCT_4:25, RELAT_1:144;
hence  (f " A) \/ (g " A) c= (f +* g) " A by A2, XBOOLE_1:8; ::_thesis: verum
end;

theorem :: TOPGEN_5:2
for f, g being Function st dom f misses dom g holds
for A being set holds (f +* g) " A = (f " A) \/ (g " A) by Th1, PARTFUN1:56;

theorem Th3: :: TOPGEN_5:3
for x, a being set
for f being Function st a in dom f holds
(commute (x .--> f)) . a = x .--> (f . a)
proof
let x, a be set ; ::_thesis: for f being Function st a in dom f holds
(commute (x .--> f)) . a = x .--> (f . a)
let f be Function; ::_thesis: ( a in dom f implies (commute (x .--> f)) . a = x .--> (f . a) )
set g = x .--> f;
A1: dom (x .--> f) = {x} by FUNCOP_1:13;
A2: f in Funcs ((dom f),(rng f)) by FUNCT_2:def_2;
 rng (x .--> f) = {f} by FUNCOP_1:8;
then  rng (x .--> f) c= Funcs ((dom f),(rng f)) by A2, ZFMISC_1:31;
then A3: x .--> f in Funcs ({x},(Funcs ((dom f),(rng f)))) by A1, FUNCT_2:def_2;
A4: (x .--> f) . x = f by FUNCOP_1:72;
A5: x in {x} by TARSKI:def_1;
assume A6: a in dom f ; ::_thesis: (commute (x .--> f)) . a = x .--> (f . a)
then A7: ((commute (x .--> f)) . a) . x = f . a by A3, A4, A5, FUNCT_6:56;
 dom ((commute (x .--> f)) . a) = {x} by A3, A6, A4, A5, FUNCT_6:56;
hence  (commute (x .--> f)) . a = x .--> (f . a) by A7, DICKSON:1; ::_thesis: verum
end;

theorem :: TOPGEN_5:4
for b being set
for f being Function holds
( b in dom (commute f) iff ex a being set ex g being Function st
( a in dom f & g = f . a & b in dom g ) )
proof
let b be set ; ::_thesis: for f being Function holds
( b in dom (commute f) iff ex a being set ex g being Function st
( a in dom f & g = f . a & b in dom g ) )
let f be Function; ::_thesis: ( b in dom (commute f) iff ex a being set ex g being Function st
( a in dom f & g = f . a & b in dom g ) )
A1: dom (commute f) = proj2 (dom (uncurry f)) by FUNCT_5:23;
hereby  ::_thesis: ( ex a being set ex g being Function st
( a in dom f & g = f . a & b in dom g ) implies b in dom (commute f) )
assume  b in dom (commute f) ; ::_thesis: ex a being set ex g being Function st
( a in dom f & g = f . a & b in dom g )
then consider a being set  such that
A2: [a,b] in dom (uncurry f) by A1, XTUPLE_0:def_13;
consider a9 being set , g being Function, b9 being set  such that
A3: [a,b] = [a9,b9] and
A4: a9 in dom f and
A5: g = f . a9 and
A6: b9 in dom g by A2, FUNCT_5:def_2;
take a = a; ::_thesis: ex g being Function st
( a in dom f & g = f . a & b in dom g )
take g = g; ::_thesis: ( a in dom f & g = f . a & b in dom g )
thus  ( a in dom f & g = f . a & b in dom g ) by A3, A4, A5, A6, XTUPLE_0:1; ::_thesis: verum
end;
given a being set , g being Function such that A7: a in dom f and
A8: g = f . a and
A9: b in dom g ; ::_thesis: b in dom (commute f)
 [a,b] in dom (uncurry f) by A7, A8, A9, FUNCT_5:def_2;
hence  b in dom (commute f) by A1, XTUPLE_0:def_13; ::_thesis: verum
end;

theorem Th5: :: TOPGEN_5:5
for a, b being set
for f being Function holds
( a in dom ((commute f) . b) iff ex g being Function st
( a in dom f & g = f . a & b in dom g ) )
proof
let a, b be set ; ::_thesis: for f being Function holds
( a in dom ((commute f) . b) iff ex g being Function st
( a in dom f & g = f . a & b in dom g ) )
let f be Function; ::_thesis: ( a in dom ((commute f) . b) iff ex g being Function st
( a in dom f & g = f . a & b in dom g ) )
 dom (uncurry f) c= [:(dom f),(proj1 (union (rng f))):]
proof
let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in dom (uncurry f) or u in [:(dom f),(proj1 (union (rng f))):] )
assume  u in dom (uncurry f) ; ::_thesis: u in [:(dom f),(proj1 (union (rng f))):]
then consider a being set , g being Function, b being set  such that
A1: u = [a,b] and
A2: a in dom f and
A3: g = f . a and
A4: b in dom g by FUNCT_5:def_2;
 g in rng f by A2, A3, FUNCT_1:def_3;
then  g c= union (rng f) by ZFMISC_1:74;
then  proj1 g c= proj1 (union (rng f)) by XTUPLE_0:8;
hence  u in [:(dom f),(proj1 (union (rng f))):] by A1, A2, A4, ZFMISC_1:def_2; ::_thesis: verum
end;
then A5: uncurry' (commute f) = uncurry f by FUNCT_5:50;
hereby  ::_thesis: ( ex g being Function st
( a in dom f & g = f . a & b in dom g ) implies a in dom ((commute f) . b) )
assume A6: a in dom ((commute f) . b) ; ::_thesis: ex g being Function st
( a in dom f & g = f . a & b in dom g )
then  b in dom (commute f) by FUNCT_1:def_2, RELAT_1:38;
then  [a,b] in dom (uncurry f) by A5, A6, FUNCT_5:39;
then consider a9 being set , g being Function, b9 being set  such that
A7: [a,b] = [a9,b9] and
A8: a9 in dom f and
A9: g = f . a9 and
A10: b9 in dom g by FUNCT_5:def_2;
take g = g; ::_thesis: ( a in dom f & g = f . a & b in dom g )
thus  ( a in dom f & g = f . a & b in dom g ) by A7, A8, A9, A10, XTUPLE_0:1; ::_thesis: verum
end;
given g being Function such that A11: a in dom f and
A12: g = f . a and
A13: b in dom g ; ::_thesis: a in dom ((commute f) . b)
 [a,b] in dom (uncurry f) by A11, A12, A13, FUNCT_5:def_2;
hence  a in dom ((commute f) . b) by FUNCT_5:22; ::_thesis: verum
end;

theorem Th6: :: TOPGEN_5:6
for a, b being set
for f, g being Function st a in dom f & g = f . a & b in dom g holds
((commute f) . b) . a = g . b
proof
let a, b be set ; ::_thesis: for f, g being Function st a in dom f & g = f . a & b in dom g holds
((commute f) . b) . a = g . b
let f, g be Function; ::_thesis: ( a in dom f & g = f . a & b in dom g implies ((commute f) . b) . a = g . b )
assume that
A1: a in dom f and
A2: g = f . a and
A3: b in dom g ; ::_thesis: ((commute f) . b) . a = g . b
A4: [a,b] in dom (uncurry f) by A1, A2, A3, FUNCT_5:def_2;
A5: [a,b] `2 = b ;
 [a,b] `1 = a ;
then  (uncurry f) . (a,b) = g . b by A5, A4, A2, FUNCT_5:def_2;
hence  ((commute f) . b) . a = g . b by A4, FUNCT_5:22; ::_thesis: verum
end;

theorem Th7: :: TOPGEN_5:7
for a being set
for f, g, h being Function st h = f \/ g holds
(commute h) . a = ((commute f) . a) \/ ((commute g) . a)
proof
let a be set ; ::_thesis: for f, g, h being Function st h = f \/ g holds
(commute h) . a = ((commute f) . a) \/ ((commute g) . a)
let f, g, h be Function; ::_thesis: ( h = f \/ g implies (commute h) . a = ((commute f) . a) \/ ((commute g) . a) )
assume A1: h = f \/ g ; ::_thesis: (commute h) . a = ((commute f) . a) \/ ((commute g) . a)
now__::_thesis:_for_u,_v_being_set_holds_
(_(_[u,v]_in_(commute_h)_._a_implies_[u,v]_in_((commute_f)_._a)_\/_((commute_g)_._a)_)_&_(_[u,v]_in_((commute_f)_._a)_\/_((commute_g)_._a)_implies_[u,v]_in_(commute_h)_._a_)_)
let u, v be set ; ::_thesis: ( ( [u,v] in (commute h) . a implies [u,v] in ((commute f) . a) \/ ((commute g) . a) ) & ( [u,v] in ((commute f) . a) \/ ((commute g) . a) implies [b1,b2] in (commute h) . a ) )
hereby  ::_thesis: ( [u,v] in ((commute f) . a) \/ ((commute g) . a) implies [b1,b2] in (commute h) . a )
assume A2: [u,v] in (commute h) . a ; ::_thesis: [u,v] in ((commute f) . a) \/ ((commute g) . a)
then A3: ((commute h) . a) . u = v by FUNCT_1:1;
 u in dom ((commute h) . a) by A2, FUNCT_1:1;
then consider k being Function such that
A4: u in dom h and
A5: k = h . u and
A6: a in dom k by Th5;
 [u,k] in h by A4, A5, FUNCT_1:def_2;
then  ( [u,k] in f or [u,k] in g ) by A1, XBOOLE_0:def_3;
then  ( ( u in dom f & k = f . u ) or ( u in dom g & k = g . u ) ) by FUNCT_1:1;
then A7: ( ( u in dom ((commute f) . a) & ((commute f) . a) . u = k . a ) or ( u in dom ((commute g) . a) & ((commute g) . a) . u = k . a ) ) by A6, Th5, Th6;
 ((commute h) . a) . u = k . a by A4, A5, A6, Th6;
then  ( [u,v] in (commute f) . a or [u,v] in (commute g) . a ) by A7, A3, FUNCT_1:1;
hence  [u,v] in ((commute f) . a) \/ ((commute g) . a) by XBOOLE_0:def_3; ::_thesis: verum
end;
assume A8: [u,v] in ((commute f) . a) \/ ((commute g) . a) ; ::_thesis: [b1,b2] in (commute h) . a
percases ( [u,v] in (commute f) . a or [u,v] in (commute g) . a ) by A8, XBOOLE_0:def_3;
supposeA9: [u,v] in (commute f) . a ; ::_thesis: [b1,b2] in (commute h) . a
then A10: ((commute f) . a) . u = v by FUNCT_1:1;
 u in dom ((commute f) . a) by A9, FUNCT_1:1;
then consider k being Function such that
A11: u in dom f and
A12: k = f . u and
A13: a in dom k by Th5;
A14: ((commute f) . a) . u = k . a by A11, A12, A13, Th6;
 [u,k] in f by A11, A12, FUNCT_1:1;
then A15: [u,k] in h by A1, XBOOLE_0:def_3;
then A16: u in dom h by FUNCT_1:1;
A17: k = h . u by A15, FUNCT_1:1;
then A18: ((commute h) . a) . u = k . a by A16, A13, Th6;
 u in dom ((commute h) . a) by A16, A17, A13, Th5;
hence  [u,v] in (commute h) . a by A18, A14, A10, FUNCT_1:1; ::_thesis: verum
end;
supposeA19: [u,v] in (commute g) . a ; ::_thesis: [b1,b2] in (commute h) . a
then A20: ((commute g) . a) . u = v by FUNCT_1:1;
 u in dom ((commute g) . a) by A19, FUNCT_1:1;
then consider k being Function such that
A21: u in dom g and
A22: k = g . u and
A23: a in dom k by Th5;
A24: ((commute g) . a) . u = k . a by A21, A22, A23, Th6;
 [u,k] in g by A21, A22, FUNCT_1:1;
then A25: [u,k] in h by A1, XBOOLE_0:def_3;
then A26: u in dom h by FUNCT_1:1;
A27: k = h . u by A25, FUNCT_1:1;
then A28: ((commute h) . a) . u = k . a by A26, A23, Th6;
 u in dom ((commute h) . a) by A26, A27, A23, Th5;
hence  [u,v] in (commute h) . a by A28, A24, A20, FUNCT_1:1; ::_thesis: verum
end;
end;
end;
hence  (commute h) . a = ((commute f) . a) \/ ((commute g) . a) by RELAT_1:def_2; ::_thesis: verum
end;

theorem Th8: :: TOPGEN_5:8
for X, Y being set holds
( product <*X,Y*>,[:X,Y:] are_equipotent & card (product <*X,Y*>) = (card X) *` (card Y) )
proof
deffunc H1( Function) ->  set = [($1 . 1),($1 . 2)];
let X, Y be set ; ::_thesis: ( product <*X,Y*>,[:X,Y:] are_equipotent & card (product <*X,Y*>) = (card X) *` (card Y) )
consider f being Function such that
A1: ( dom f = product <*X,Y*> & ( for g being Function st g in product <*X,Y*> holds
f . g = H1(g) ) ) from FUNCT_5:sch_1();
A2: <*X,Y*> . 2 = Y by FINSEQ_1:44;
A3: dom <*X,Y*> = {1,2} by FINSEQ_1:2, FINSEQ_1:89;
A4: <*X,Y*> . 1 = X by FINSEQ_1:44;
thus  product <*X,Y*>,[:X,Y:] are_equipotent ::_thesis: card (product <*X,Y*>) = (card X) *` (card Y)
proof
take  f ; :: according to WELLORD2:def_4 ::_thesis: ( f is one-to-one & proj1 f = product <*X,Y*> & proj2 f = [:X,Y:] )
thus  f is one-to-one ::_thesis: ( proj1 f = product <*X,Y*> & proj2 f = [:X,Y:] )
proof
let x1, x2 be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not x1 in proj1 f or not x2 in proj1 f or not f . x1 = f . x2 or x1 = x2 )
assume that
A5: x1 in dom f and
A6: x2 in dom f and
A7: f . x1 = f . x2 ; ::_thesis: x1 = x2
consider g2 being Function such that
A8: x2 = g2 and
A9: dom g2 = dom <*X,Y*> and
 for y being set st y in dom <*X,Y*> holds
g2 . y in <*X,Y*> . y by A1, A6, CARD_3:def_5;
consider g1 being Function such that
A10: x1 = g1 and
A11: dom g1 = dom <*X,Y*> and
 for y being set st y in dom <*X,Y*> holds
g1 . y in <*X,Y*> . y by A5, A1, CARD_3:def_5;
A12: [(g1 . 1),(g1 . 2)] = f . x1 by A1, A5, A10
.= [(g2 . 1),(g2 . 2)] by A1, A6, A7, A8 ;
now__::_thesis:_for_z_being_set_st_z_in_{1,2}_holds_
g1_._z_=_g2_._z
let z be set ; ::_thesis: ( z in {1,2} implies g1 . z = g2 . z )
assume  z in {1,2} ; ::_thesis: g1 . z = g2 . z
then  ( z = 1 or z = 2 ) by TARSKI:def_2;
hence  g1 . z = g2 . z by A12, XTUPLE_0:1; ::_thesis: verum
end;
hence  x1 = x2 by A3, A10, A11, A8, A9, FUNCT_1:2; ::_thesis: verum
end;
thus  dom f = product <*X,Y*> by A1; ::_thesis: proj2 f = [:X,Y:]
thus  rng f c= [:X,Y:] :: according to XBOOLE_0:def_10 ::_thesis: [:X,Y:] c= proj2 f
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in rng f or z in [:X,Y:] )
assume  z in rng f ; ::_thesis: z in [:X,Y:]
then consider t being set  such that
A13: t in dom f and
A14: z = f . t by FUNCT_1:def_3;
consider g being Function such that
A15: t = g and
 dom g = dom <*X,Y*> and
A16: for y being set st y in dom <*X,Y*> holds
g . y in <*X,Y*> . y by A1, A13, CARD_3:def_5;
 1 in {1,2} by TARSKI:def_2;
then A17: g . 1 in X by A3, A4, A16;
 2 in {1,2} by TARSKI:def_2;
then A18: g . 2 in Y by A3, A2, A16;
 z = [(g . 1),(g . 2)] by A1, A13, A14, A15;
hence  z in [:X,Y:] by A17, A18, ZFMISC_1:87; ::_thesis: verum
end;
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in [:X,Y:] or z in proj2 f )
set g = <*(z `1),(z `2)*>;
A19: <*(z `1),(z `2)*> . 1 = z `1 by FINSEQ_1:44;
A20: <*(z `1),(z `2)*> . 2 = z `2 by FINSEQ_1:44;
assume A21: z in [:X,Y:] ; ::_thesis: z in proj2 f
then A22: z `2 in Y by MCART_1:10;
 z `1 in X by A21, MCART_1:10;
then A23: <*(z `1),(z `2)*> in product <*X,Y*> by A22, FINSEQ_3:124;
 z = [(z `1),(z `2)] by A21, MCART_1:21;
then  f . <*(z `1),(z `2)*> = z by A23, A1, A19, A20;
hence  z in proj2 f by A1, A23, FUNCT_1:def_3; ::_thesis: verum
end;
hence  card (product <*X,Y*>) = card [:X,Y:] by CARD_1:5
.= card [:(card X),(card Y):] by CARD_2:7
.= (card X) *` (card Y) by CARD_2:def_2 ;
::_thesis: verum
end;

scheme :: TOPGEN_5:sch 1
SCH1{ P1[ set ], F1() ->  non empty set , F2() ->  non empty set , F3() ->  non empty set , F4( set ) ->  set , F5( set ) ->  set } :
 ex f being Function of F3(),F2() st
for a being Element of F1() st a in F3() holds
( ( P1[a] implies f . a = F4(a) ) & ( P1[a] implies f . a = F5(a) ) )
provided
A1: F3() c= F1() and
A2: for a being Element of F1() st a in F3() holds
( ( P1[a] implies F4(a) in F2() ) & ( P1[a] implies F5(a) in F2() ) )
proof
A3: for a being set st a in F3() holds
( ( P1[a] implies F4(a) in F2() ) & ( P1[a] implies F5(a) in F2() ) ) by A1, A2;
consider f being Function of F3(),F2() such that
A4: for x being set st x in F3() holds
( ( P1[x] implies f . x = F4(x) ) & ( P1[x] implies f . x = F5(x) ) ) from FUNCT_2:sch_5(A3);
take  f ; ::_thesis: for a being Element of F1() st a in F3() holds
( ( P1[a] implies f . a = F4(a) ) & ( P1[a] implies f . a = F5(a) ) )
thus  for a being Element of F1() st a in F3() holds
( ( P1[a] implies f . a = F4(a) ) & ( P1[a] implies f . a = F5(a) ) ) by A4; ::_thesis: verum
end;

scheme :: TOPGEN_5:sch 2
SCH2{ P1[ set ], P2[ set ], F1() ->  non empty set , F2() ->  non empty set , F3() ->  non empty set , F4( set ) ->  set , F5( set ) ->  set , F6( set ) ->  set } :
 ex f being Function of F3(),F2() st
for a being Element of F1() st a in F3() holds
( ( P1[a] implies f . a = F4(a) ) & ( P1[a] & P2[a] implies f . a = F5(a) ) & ( P1[a] & P2[a] implies f . a = F6(a) ) )
provided
A1: F3() c= F1() and
A2: for a being Element of F1() st a in F3() holds
( ( P1[a] implies F4(a) in F2() ) & ( P1[a] & P2[a] implies F5(a) in F2() ) & ( P1[a] & P2[a] implies F6(a) in F2() ) )
proof
set D =  {  a where a is Element of F3() : ( P1[a] or P2[a] )  }  ;
percases (  {  a where a is Element of F3() : ( P1[a] or P2[a] )  }  = {} or  {  a where a is Element of F3() : ( P1[a] or P2[a] )  }  <> {} ) ;
supposeA3:  {  a where a is Element of F3() : ( P1[a] or P2[a] )  }  = {} ; ::_thesis: ex f being Function of F3(),F2() st
for a being Element of F1() st a in F3() holds
( ( P1[a] implies f . a = F4(a) ) & ( P1[a] & P2[a] implies f . a = F5(a) ) & ( P1[a] & P2[a] implies f . a = F6(a) ) )
consider f being Function such that
A4: ( dom f = F3() & ( for u being set st u in F3() holds
f . u = F6(u) ) ) from FUNCT_1:sch_3();
 rng f c= F2()
proof
let v be set ; :: according to TARSKI:def_3 ::_thesis: ( not v in rng f or v in F2() )
assume  v in rng f ; ::_thesis: v in F2()
then consider u being set  such that
A5: u in dom f and
A6: v = f . u by FUNCT_1:def_3;
A7: v = F6(u) by A4, A5, A6;
A8: not u in  {  a where a is Element of F3() : ( P1[a] or P2[a] )  }  by A3;
then A9: P2[u] by A4, A5;
 P1[u] by A8, A4, A5;
hence  v in F2() by A9, A7, A1, A2, A4, A5; ::_thesis: verum
end;
then reconsider f = f as Function of F3(),F2() by A4, FUNCT_2:2;
take  f ; ::_thesis: for a being Element of F1() st a in F3() holds
( ( P1[a] implies f . a = F4(a) ) & ( P1[a] & P2[a] implies f . a = F5(a) ) & ( P1[a] & P2[a] implies f . a = F6(a) ) )
let a be Element of F1(); ::_thesis: ( a in F3() implies ( ( P1[a] implies f . a = F4(a) ) & ( P1[a] & P2[a] implies f . a = F5(a) ) & ( P1[a] & P2[a] implies f . a = F6(a) ) ) )
assume A10: a in F3() ; ::_thesis: ( ( P1[a] implies f . a = F4(a) ) & ( P1[a] & P2[a] implies f . a = F5(a) ) & ( P1[a] & P2[a] implies f . a = F6(a) ) )
 not a in  {  a where a is Element of F3() : ( P1[a] or P2[a] )  }  by A3;
hence  ( ( P1[a] implies f . a = F4(a) ) & ( P1[a] & P2[a] implies f . a = F5(a) ) & ( P1[a] & P2[a] implies f . a = F6(a) ) ) by A4, A10; ::_thesis: verum
end;
suppose  {  a where a is Element of F3() : ( P1[a] or P2[a] )  }  <> {} ; ::_thesis: ex f being Function of F3(),F2() st
for a being Element of F1() st a in F3() holds
( ( P1[a] implies f . a = F4(a) ) & ( P1[a] & P2[a] implies f . a = F5(a) ) & ( P1[a] & P2[a] implies f . a = F6(a) ) )
then reconsider D =  {  a where a is Element of F3() : ( P1[a] or P2[a] )  }  as non empty set ;
defpred S1[ set ] means ( P1[$1] or P2[$1] );
A11: for a being set st a in D holds
( ( P1[a] implies F4(a) in F2() ) & ( P1[a] implies F5(a) in F2() ) )
proof
let a be set ; ::_thesis: ( a in D implies ( ( P1[a] implies F4(a) in F2() ) & ( P1[a] implies F5(a) in F2() ) ) )
assume  a in D ; ::_thesis: ( ( P1[a] implies F4(a) in F2() ) & ( P1[a] implies F5(a) in F2() ) )
then A12: ex b being Element of F3() st
( a = b & ( P1[b] or P2[b] ) ) ;
then  a in F3() ;
hence  ( ( P1[a] implies F4(a) in F2() ) & ( P1[a] implies F5(a) in F2() ) ) by A12, A1, A2; ::_thesis: verum
end;
consider g being Function of D,F2() such that
A13: for x being set st x in D holds
( ( P1[x] implies g . x = F4(x) ) & ( P1[x] implies g . x = F5(x) ) ) from FUNCT_2:sch_5(A11);
deffunc H1( set ) ->  set = g . $1;
A14: for a being set st a in F3() holds
( ( S1[a] implies H1(a) in F2() ) & ( not S1[a] implies F6(a) in F2() ) )
proof
let a be set ; ::_thesis: ( a in F3() implies ( ( S1[a] implies H1(a) in F2() ) & ( not S1[a] implies F6(a) in F2() ) ) )
assume A15: a in F3() ; ::_thesis: ( ( S1[a] implies H1(a) in F2() ) & ( not S1[a] implies F6(a) in F2() ) )
hereby  ::_thesis: ( not S1[a] implies F6(a) in F2() )
assume  S1[a] ; ::_thesis: H1(a) in F2()
then A16: a in D by A15;
then A17: ( P1[a] implies F5(a) in F2() ) by A11;
 ( P1[a] implies F4(a) in F2() ) by A16, A11;
hence  H1(a) in F2() by A17, A16, A13; ::_thesis: verum
end;
thus  ( not S1[a] implies F6(a) in F2() ) by A1, A2, A15; ::_thesis: verum
end;
consider f being Function of F3(),F2() such that
A18: for x being set st x in F3() holds
( ( S1[x] implies f . x = H1(x) ) & ( not S1[x] implies f . x = F6(x) ) ) from FUNCT_2:sch_5(A14);
take  f ; ::_thesis: for a being Element of F1() st a in F3() holds
( ( P1[a] implies f . a = F4(a) ) & ( P1[a] & P2[a] implies f . a = F5(a) ) & ( P1[a] & P2[a] implies f . a = F6(a) ) )
let a be Element of F1(); ::_thesis: ( a in F3() implies ( ( P1[a] implies f . a = F4(a) ) & ( P1[a] & P2[a] implies f . a = F5(a) ) & ( P1[a] & P2[a] implies f . a = F6(a) ) ) )
assume A19: a in F3() ; ::_thesis: ( ( P1[a] implies f . a = F4(a) ) & ( P1[a] & P2[a] implies f . a = F5(a) ) & ( P1[a] & P2[a] implies f . a = F6(a) ) )
then  ( S1[a] implies ( f . a = g . a & a in D ) ) by A18;
hence  ( ( P1[a] implies f . a = F4(a) ) & ( P1[a] & P2[a] implies f . a = F5(a) ) ) by A13; ::_thesis: ( P1[a] & P2[a] implies f . a = F6(a) )
thus  ( P1[a] & P2[a] implies f . a = F6(a) ) by A18, A19; ::_thesis: verum
end;
end;
end;

theorem Th9: :: TOPGEN_5:9
for a, b being real number holds |.|[a,b]|.| ^2 = (a ^2) + (b ^2)
proof
let a, b be real number ; ::_thesis: |.|[a,b]|.| ^2 = (a ^2) + (b ^2)
A1: |[a,b]| `2 = b by EUCLID:52;
 |[a,b]| `1 = a by EUCLID:52;
hence  |.|[a,b]|.| ^2 = (a ^2) + (b ^2) by A1, JGRAPH_1:29; ::_thesis: verum
end;

theorem Th10: :: TOPGEN_5:10
for X being TopSpace
for Y being non empty TopSpace
for A, B being closed Subset of X
for f being continuous Function of (X | A),Y
for g being continuous Function of (X | B),Y st f tolerates g holds
f +* g is continuous Function of (X | (A \/ B)),Y
proof
let X be TopSpace; ::_thesis: for Y being non empty TopSpace
for A, B being closed Subset of X
for f being continuous Function of (X | A),Y
for g being continuous Function of (X | B),Y st f tolerates g holds
f +* g is continuous Function of (X | (A \/ B)),Y
let Y be non empty TopSpace; ::_thesis: for A, B being closed Subset of X
for f being continuous Function of (X | A),Y
for g being continuous Function of (X | B),Y st f tolerates g holds
f +* g is continuous Function of (X | (A \/ B)),Y
let A, B be closed Subset of X; ::_thesis: for f being continuous Function of (X | A),Y
for g being continuous Function of (X | B),Y st f tolerates g holds
f +* g is continuous Function of (X | (A \/ B)),Y
let f be continuous Function of (X | A),Y; ::_thesis: for g being continuous Function of (X | B),Y st f tolerates g holds
f +* g is continuous Function of (X | (A \/ B)),Y
let g be continuous Function of (X | B),Y; ::_thesis: ( f tolerates g implies f +* g is continuous Function of (X | (A \/ B)),Y )
assume A1: f tolerates g ; ::_thesis: f +* g is continuous Function of (X | (A \/ B)),Y
A2: the carrier of (X | (A \/ B)) = A \/ B by PRE_TOPC:8;
 the carrier of (X | B) = B by PRE_TOPC:8;
then A3: dom g = B by FUNCT_2:def_1;
 the carrier of (X | A) = A by PRE_TOPC:8;
then A4: dom f = A by FUNCT_2:def_1;
A5: rng (f +* g) c= (rng f) \/ (rng g) by FUNCT_4:17;
 dom (f +* g) = (dom f) \/ (dom g) by FUNCT_4:def_1;
then reconsider h = f +* g as Function of (X | (A \/ B)),Y by A5, A2, A4, A3, FUNCT_2:2, XBOOLE_1:1;
 h is continuous
proof
let C be Subset of Y; :: according to PRE_TOPC:def_6 ::_thesis: ( not C is closed or h " C is closed )
A6: [#] (X | (A \/ B)) = A \/ B by PRE_TOPC:8;
assume A7: C is closed ; ::_thesis: h " C is closed
then  f " C is closed by PRE_TOPC:def_6;
then consider C1 being Subset of X such that
A8: C1 is closed and
A9: C1 /\ ([#] (X | A)) = f " C by PRE_TOPC:13;
 g " C is closed by A7, PRE_TOPC:def_6;
then consider C2 being Subset of X such that
A10: C2 is closed and
A11: C2 /\ ([#] (X | B)) = g " C by PRE_TOPC:13;
A12: (C1 /\ A) \/ (C2 /\ B) is closed by A8, A10;
A13: [#] (X | A) = A by PRE_TOPC:8;
A14: [#] (X | B) = B by PRE_TOPC:8;
h " C = (f " C) \/ (g " C) by A1, Th1
.= ((f " C) \/ (g " C)) /\ (A \/ B) by A13, A14, XBOOLE_1:13, XBOOLE_1:28 ;
hence  h " C is closed by A12, A9, A11, A6, A13, A14, PRE_TOPC:13; ::_thesis: verum
end;
hence  f +* g is continuous Function of (X | (A \/ B)),Y ; ::_thesis: verum
end;

theorem Th11: :: TOPGEN_5:11
for X being TopSpace
for Y being non empty TopSpace
for A, B being closed Subset of X st A misses B holds
for f being continuous Function of (X | A),Y
for g being continuous Function of (X | B),Y holds f +* g is continuous Function of (X | (A \/ B)),Y
proof
let X be TopSpace; ::_thesis: for Y being non empty TopSpace
for A, B being closed Subset of X st A misses B holds
for f being continuous Function of (X | A),Y
for g being continuous Function of (X | B),Y holds f +* g is continuous Function of (X | (A \/ B)),Y
let Y be non empty TopSpace; ::_thesis: for A, B being closed Subset of X st A misses B holds
for f being continuous Function of (X | A),Y
for g being continuous Function of (X | B),Y holds f +* g is continuous Function of (X | (A \/ B)),Y
let A, B be closed Subset of X; ::_thesis: ( A misses B implies for f being continuous Function of (X | A),Y
for g being continuous Function of (X | B),Y holds f +* g is continuous Function of (X | (A \/ B)),Y )
assume A1: A misses B ; ::_thesis: for f being continuous Function of (X | A),Y
for g being continuous Function of (X | B),Y holds f +* g is continuous Function of (X | (A \/ B)),Y
let f be continuous Function of (X | A),Y; ::_thesis: for g being continuous Function of (X | B),Y holds f +* g is continuous Function of (X | (A \/ B)),Y
let g be continuous Function of (X | B),Y; ::_thesis: f +* g is continuous Function of (X | (A \/ B)),Y
 the carrier of (X | B) = B by PRE_TOPC:8;
then A2: dom g = B by FUNCT_2:def_1;
 the carrier of (X | A) = A by PRE_TOPC:8;
then  dom f = A by FUNCT_2:def_1;
hence  f +* g is continuous Function of (X | (A \/ B)),Y by A2, A1, Th10, PARTFUN1:56; ::_thesis: verum
end;

theorem Th12: :: TOPGEN_5:12
for X being TopSpace
for Y being non empty TopSpace
for A being open closed Subset of X
for f being continuous Function of (X | A),Y
for g being continuous Function of (X | (A `)),Y holds f +* g is continuous Function of X,Y
proof
let X be TopSpace; ::_thesis: for Y being non empty TopSpace
for A being open closed Subset of X
for f being continuous Function of (X | A),Y
for g being continuous Function of (X | (A `)),Y holds f +* g is continuous Function of X,Y
let Y be non empty TopSpace; ::_thesis: for A being open closed Subset of X
for f being continuous Function of (X | A),Y
for g being continuous Function of (X | (A `)),Y holds f +* g is continuous Function of X,Y
let A be open closed Subset of X; ::_thesis: for f being continuous Function of (X | A),Y
for g being continuous Function of (X | (A `)),Y holds f +* g is continuous Function of X,Y
let f be continuous Function of (X | A),Y; ::_thesis: for g being continuous Function of (X | (A `)),Y holds f +* g is continuous Function of X,Y
let g be continuous Function of (X | (A `)),Y; ::_thesis: f +* g is continuous Function of X,Y
 A \/ (A `) = [#] X by PRE_TOPC:2;
then A1: X | (A \/ (A `)) = TopStruct(# the carrier of X, the topology of X #) by TSEP_1:93;
 A misses A ` by XBOOLE_1:79;
then A2: f +* g is continuous Function of (X | (A \/ (A `))),Y by Th11;
 TopStruct(# the carrier of Y, the topology of Y #) = TopStruct(# the carrier of Y, the topology of Y #) ;
hence  f +* g is continuous Function of X,Y by A2, A1, YELLOW12:36; ::_thesis: verum
end;

begin

theorem Th13: :: TOPGEN_5:13
for n being Element of NAT
for a being Point of (TOP-REAL n)
for r being real positive number holds a in Ball (a,r)
proof
let n be Element of NAT ; ::_thesis: for a being Point of (TOP-REAL n)
for r being real positive number holds a in Ball (a,r)
let a be Point of (TOP-REAL n); ::_thesis: for r being real positive number holds a in Ball (a,r)
let r be real positive number ; ::_thesis: a in Ball (a,r)
 a - a = 0. (TOP-REAL n) by EUCLID:42;
then  |.(a - a).| = 0 by EUCLID_2:39;
hence  a in Ball (a,r) by TOPREAL9:7; ::_thesis: verum
end;

definition
func y=0-line  ->  Subset of (TOP-REAL 2) equals :: TOPGEN_5:def 1
 {  |[x,0]| where x is Element of REAL : verum  }  ;
coherence
  {  |[x,0]| where x is Element of REAL : verum  }  is Subset of (TOP-REAL 2)
proof
set A =  {  |[x,0]| where x is Element of REAL : verum  }  ;
  {  |[x,0]| where x is Element of REAL : verum  }  c= the carrier of (TOP-REAL 2)
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in  {  |[x,0]| where x is Element of REAL : verum  }  or a in the carrier of (TOP-REAL 2) )
assume  a in  {  |[x,0]| where x is Element of REAL : verum  }  ; ::_thesis: a in the carrier of (TOP-REAL 2)
then  ex x being Element of REAL st a = |[x,0]| ;
hence  a in the carrier of (TOP-REAL 2) ; ::_thesis: verum
end;
hence   {  |[x,0]| where x is Element of REAL : verum  }  is Subset of (TOP-REAL 2) ; ::_thesis: verum
end;
func y>=0-plane  ->  Subset of (TOP-REAL 2) equals :: TOPGEN_5:def 2
 {  |[x,y]| where x, y is Element of REAL : y >= 0  }  ;
coherence
  {  |[x,y]| where x, y is Element of REAL : y >= 0  }  is Subset of (TOP-REAL 2)
proof
set A =  {  |[x,y]| where x, y is Element of REAL : y >= 0  }  ;
  {  |[x,y]| where x, y is Element of REAL : y >= 0  }  c= the carrier of (TOP-REAL 2)
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in  {  |[x,y]| where x, y is Element of REAL : y >= 0  }  or a in the carrier of (TOP-REAL 2) )
assume  a in  {  |[x,y]| where x, y is Element of REAL : y >= 0  }  ; ::_thesis: a in the carrier of (TOP-REAL 2)
then  ex x, y being Element of REAL st
( a = |[x,y]| & y >= 0 ) ;
hence  a in the carrier of (TOP-REAL 2) ; ::_thesis: verum
end;
hence   {  |[x,y]| where x, y is Element of REAL : y >= 0  }  is Subset of (TOP-REAL 2) ; ::_thesis: verum
end;
end;

:: deftheorem  defines y=0-line TOPGEN_5:def_1_:_
 y=0-line =  {  |[x,0]| where x is Element of REAL : verum  }  ;

:: deftheorem  defines y>=0-plane TOPGEN_5:def_2_:_
 y>=0-plane =  {  |[x,y]| where x, y is Element of REAL : y >= 0  }  ;

theorem :: TOPGEN_5:14
for a, b being set holds
( <*a,b*> in y=0-line iff ( a in REAL & b = 0 ) )
proof
let a, b be set ; ::_thesis: ( <*a,b*> in y=0-line iff ( a in REAL & b = 0 ) )
A1: ( <*a,b*> in y=0-line iff ex x being Element of REAL st <*a,b*> = |[x,0]| ) ;
hereby  ::_thesis: ( a in REAL & b = 0 implies <*a,b*> in y=0-line )
A2: <*a,b*> . 1 = a by FINSEQ_1:44;
assume  <*a,b*> in y=0-line ; ::_thesis: ( a in REAL & b = 0 )
then consider x, y being Element of REAL  such that
A3: <*a,b*> = |[x,0]| by A1;
 <*a,b*> . 1 = x by A3, FINSEQ_1:44;
hence  a in REAL by A2; ::_thesis: b = 0
 <*a,b*> . 2 = b by FINSEQ_1:44;
hence  b = 0 by A3, FINSEQ_1:44; ::_thesis: verum
end;
assume  a in REAL ; ::_thesis: ( not b = 0 or <*a,b*> in y=0-line )
then reconsider x = a as Real ;
 |[x,0]| = <*a,0*> ;
hence  ( not b = 0 or <*a,b*> in y=0-line ) ; ::_thesis: verum
end;

theorem Th15: :: TOPGEN_5:15
for a, b being real number holds
( |[a,b]| in y=0-line iff b = 0 )
proof
let a, b be real number ; ::_thesis: ( |[a,b]| in y=0-line iff b = 0 )
A1: a is Real by XREAL_0:def_1;
 ( |[a,b]| in y=0-line iff ex x being Element of REAL st |[a,b]| = |[x,0]| ) ;
hence  ( |[a,b]| in y=0-line iff b = 0 ) by A1, SPPOL_2:1; ::_thesis: verum
end;

theorem Th16: :: TOPGEN_5:16
card y=0-line = continuum
proof
deffunc H1( Real) ->  Element of the carrier of (TOP-REAL 2) = |[$1,0]|;
consider f being Function such that
A1: dom f = REAL and
A2: for r being Real holds f . r = H1(r) from FUNCT_1:sch_4();
 REAL , y=0-line are_equipotent
proof
take  f ; :: according to WELLORD2:def_4 ::_thesis: ( f is one-to-one & proj1 f = REAL & proj2 f = y=0-line )
thus  f is one-to-one ::_thesis: ( proj1 f = REAL & proj2 f = y=0-line )
proof
let x, y be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not x in proj1 f or not y in proj1 f or not f . x = f . y or x = y )
assume that
A3: x in dom f and
A4: y in dom f ; ::_thesis: ( not f . x = f . y or x = y )
reconsider x = x, y = y as Real by A3, A4, A1;
A5: f . y = |[y,0]| by A2;
 f . x = |[x,0]| by A2;
hence  ( not f . x = f . y or x = y ) by A5, SPPOL_2:1; ::_thesis: verum
end;
thus  dom f = REAL by A1; ::_thesis: proj2 f = y=0-line
thus  rng f c= y=0-line :: according to XBOOLE_0:def_10 ::_thesis: y=0-line c= proj2 f
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in rng f or a in y=0-line )
assume  a in rng f ; ::_thesis: a in y=0-line
then consider b being set  such that
A6: b in dom f and
A7: a = f . b by FUNCT_1:def_3;
reconsider b = b as Real by A1, A6;
 a = |[b,0]| by A2, A7;
hence  a in y=0-line ; ::_thesis: verum
end;
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in y=0-line or a in proj2 f )
assume A8: a in y=0-line ; ::_thesis: a in proj2 f
then reconsider a = a as Point of (TOP-REAL 2) ;
A9: a = |[(a `1),(a `2)]| by EUCLID:53;
then  a `2 = 0 by A8, Th15;
then  a = f . (a `1) by A2, A9;
hence  a in proj2 f by A1, FUNCT_1:def_3; ::_thesis: verum
end;
hence  card y=0-line = continuum by CARD_1:5, TOPGEN_3:def_4; ::_thesis: verum
end;

theorem :: TOPGEN_5:17
for a, b being set holds
( <*a,b*> in y>=0-plane iff ( a in REAL & ex y being Element of REAL st
( b = y & y >= 0 ) ) )
proof
let a, b be set ; ::_thesis: ( <*a,b*> in y>=0-plane iff ( a in REAL & ex y being Element of REAL st
( b = y & y >= 0 ) ) )
hereby  ::_thesis: ( a in REAL & ex y being Element of REAL st
( b = y & y >= 0 ) implies <*a,b*> in y>=0-plane )
A1: <*a,b*> . 1 = a by FINSEQ_1:44;
assume  <*a,b*> in y>=0-plane ; ::_thesis: ( a in REAL & ex y being Element of REAL st
( b = y & y >= 0 ) )
then consider x, y being Element of REAL  such that
A2: <*a,b*> = |[x,y]| and
A3: y >= 0 ;
 <*a,b*> . 1 = x by A2, FINSEQ_1:44;
hence  a in REAL by A1; ::_thesis: ex y being Element of REAL st
( b = y & y >= 0 )
take y = y; ::_thesis: ( b = y & y >= 0 )
 <*a,b*> . 2 = b by FINSEQ_1:44;
hence  ( b = y & y >= 0 ) by A3, A2, FINSEQ_1:44; ::_thesis: verum
end;
assume  a in REAL ; ::_thesis: ( for y being Element of REAL holds
( not b = y or not y >= 0 ) or <*a,b*> in y>=0-plane )
then reconsider x = a as Real ;
given y being Element of REAL  such that A4: b = y and
A5: y >= 0 ; ::_thesis: <*a,b*> in y>=0-plane
 |[x,y]| = <*a,b*> by A4;
hence  <*a,b*> in y>=0-plane by A5; ::_thesis: verum
end;

theorem Th18: :: TOPGEN_5:18
for a, b being real number holds
( |[a,b]| in y>=0-plane iff b >= 0 )
proof
let a, b be real number ; ::_thesis: ( |[a,b]| in y>=0-plane iff b >= 0 )
A1: a is Real by XREAL_0:def_1;
A2: b is Real by XREAL_0:def_1;
 ( |[a,b]| in y>=0-plane iff ex x, y being Element of REAL st
( |[a,b]| = |[x,y]| & y >= 0 ) ) ;
hence  ( |[a,b]| in y>=0-plane iff b >= 0 ) by A1, A2, SPPOL_2:1; ::_thesis: verum
end;

registration
cluster y=0-line  ->  non empty ;
coherence
 not y=0-line is empty by Th15;
cluster y>=0-plane  ->  non empty ;
coherence
 not y>=0-plane is empty by Th18;
end;

theorem Th19: :: TOPGEN_5:19
y=0-line c= y>=0-plane
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in y=0-line or x in y>=0-plane )
assume  x in y=0-line ; ::_thesis: x in y>=0-plane
then reconsider x = x as Element of y=0-line ;
A1: x = |[(x `1),(x `2)]| by EUCLID:53;
then  x `2 = 0 by Th15;
hence  x in y>=0-plane by A1; ::_thesis: verum
end;

theorem Th20: :: TOPGEN_5:20
for a, b, r being real number st r > 0 holds
( Ball (|[a,b]|,r) c= y>=0-plane iff r <= b )
proof
let a, b, r be real number ; ::_thesis: ( r > 0 implies ( Ball (|[a,b]|,r) c= y>=0-plane iff r <= b ) )
assume A1: r > 0 ; ::_thesis: ( Ball (|[a,b]|,r) c= y>=0-plane iff r <= b )
hereby  ::_thesis: ( r <= b implies Ball (|[a,b]|,r) c= y>=0-plane )
A2: |[a,b]| in Ball (|[a,b]|,r) by A1, Th13;
assume that
A3: Ball (|[a,b]|,r) c= y>=0-plane and
A4: r > b ; ::_thesis: contradiction
reconsider b = b as non negative Real by A2, A3, Th18, XREAL_0:def_1;
reconsider br = b - r as negative Real by A4, XREAL_1:49;
set y = br / 2;
reconsider r = r as positive Real by A1, XREAL_0:def_1;
 |[a,(br / 2)]| - |[a,b]| = |[(a - a),((br / 2) - b)]| by EUCLID:62;
then A5: |.(|[a,(br / 2)]| - |[a,b]|).| = |.((br / 2) - b).| by TOPREAL6:23
.= |.(b - (br / 2)).| by COMPLEX1:60
.= (b + r) / 2 by ABSVALUE:def_1 ;
 b + r < r + r by A4, XREAL_1:6;
then  (b + r) / 2 < (r + r) / 2 by XREAL_1:74;
then  |[a,(br / 2)]| in Ball (|[a,b]|,r) by A5, TOPREAL9:7;
hence  contradiction by A3, Th18; ::_thesis: verum
end;
assume A6: r <= b ; ::_thesis: Ball (|[a,b]|,r) c= y>=0-plane
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Ball (|[a,b]|,r) or x in y>=0-plane )
assume A7: x in Ball (|[a,b]|,r) ; ::_thesis: x in y>=0-plane
then reconsider z = x as Element of (TOP-REAL 2) ;
A8: |.(z - |[a,b]|).| < r by A7, TOPREAL9:7;
A9: |[((z `1) - a),((z `2) - b)]| `1 = (z `1) - a by EUCLID:52;
A10: |[((z `1) - a),((z `2) - b)]| `2 = (z `2) - b by EUCLID:52;
A11: z = |[(z `1),(z `2)]| by EUCLID:53;
then  z - |[a,b]| = |[((z `1) - a),((z `2) - b)]| by EUCLID:62;
then  |.(z - |[a,b]|).| = sqrt ((((z `1) - a) ^2) + (((z `2) - b) ^2)) by A9, A10, JGRAPH_1:30;
then  abs ((z `2) - b) <= |.(z - |[a,b]|).| by COMPLEX1:79;
then  abs ((z `2) - b) < r by A8, XXREAL_0:2;
then A12: abs (b - (z `2)) < r by COMPLEX1:60;
now__::_thesis:_not_z_`2_<_0
assume  z `2 < 0 ; ::_thesis: contradiction
then  b - (z `2) > b by XREAL_1:46;
then  b - (z `2) > r by A6, XXREAL_0:2;
hence  contradiction by A1, A12, ABSVALUE:def_1; ::_thesis: verum
end;
hence  x in y>=0-plane by A11; ::_thesis: verum
end;

theorem Th21: :: TOPGEN_5:21
for a, b, r being real number st r > 0 & b >= 0 holds
( Ball (|[a,b]|,r) misses y=0-line iff r <= b )
proof
let a, b, r be real number ; ::_thesis: ( r > 0 & b >= 0 implies ( Ball (|[a,b]|,r) misses y=0-line iff r <= b ) )
assume that
A1: r > 0 and
A2: b >= 0 ; ::_thesis: ( Ball (|[a,b]|,r) misses y=0-line iff r <= b )
hereby  ::_thesis: ( r <= b implies Ball (|[a,b]|,r) misses y=0-line )
A3: |[a,0]| in y=0-line by Th15;
assume that
A4: Ball (|[a,b]|,r) misses y=0-line and
A5: r > b ; ::_thesis: contradiction
 |[a,0]| - |[a,b]| = |[(a - a),(0 - b)]| by EUCLID:62;
then |.(|[a,0]| - |[a,b]|).| = |.(0 - b).| by TOPREAL6:23
.= |.(b - 0).| by COMPLEX1:60 ;
then  |.(|[a,0]| - |[a,b]|).| < r by A2, A5, ABSVALUE:def_1;
then  |[a,0]| in Ball (|[a,b]|,r) by TOPREAL9:7;
hence  contradiction by A3, A4, XBOOLE_0:3; ::_thesis: verum
end;
assume A6: r <= b ; ::_thesis: Ball (|[a,b]|,r) misses y=0-line
assume  Ball (|[a,b]|,r) meets y=0-line ; ::_thesis: contradiction
then consider x being set  such that
A7: x in Ball (|[a,b]|,r) and
A8: x in y=0-line by XBOOLE_0:3;
reconsider x = x as Element of (TOP-REAL 2) by A7;
A9: x = |[(x `1),(x `2)]| by EUCLID:53;
then  x `2 = 0 by A8, Th15;
then A10: x - |[a,b]| = |[((x `1) - a),(0 - b)]| by A9, EUCLID:62;
then A11: (x - |[a,b]|) `2 = 0 - b by EUCLID:52;
 (x - |[a,b]|) `1 = (x `1) - a by A10, EUCLID:52;
then  |.(x - |[a,b]|).| = sqrt ((((x `1) - a) ^2) + ((0 - b) ^2)) by A11, JGRAPH_1:30;
then  |.(x - |[a,b]|).| >= abs (0 - b) by COMPLEX1:79;
then A12: |.(x - |[a,b]|).| >= abs (b - 0) by COMPLEX1:60;
 |.(x - |[a,b]|).| < r by A7, TOPREAL9:7;
then  abs b < r by A12, XXREAL_0:2;
hence  contradiction by A1, A6, ABSVALUE:def_1; ::_thesis: verum
end;

theorem Th22: :: TOPGEN_5:22
for n being Element of NAT
for a, b being Element of (TOP-REAL n)
for r1, r2 being real positive number st |.(a - b).| <= r1 - r2 holds
Ball (b,r2) c= Ball (a,r1)
proof
let n be Element of NAT ; ::_thesis: for a, b being Element of (TOP-REAL n)
for r1, r2 being real positive number st |.(a - b).| <= r1 - r2 holds
Ball (b,r2) c= Ball (a,r1)
let a, b be Element of (TOP-REAL n); ::_thesis: for r1, r2 being real positive number st |.(a - b).| <= r1 - r2 holds
Ball (b,r2) c= Ball (a,r1)
let r1, r2 be real positive number ; ::_thesis: ( |.(a - b).| <= r1 - r2 implies Ball (b,r2) c= Ball (a,r1) )
assume  |.(a - b).| <= r1 - r2 ; ::_thesis: Ball (b,r2) c= Ball (a,r1)
then A1: |.(b - a).| <= r1 - r2 by TOPRNS_1:27;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Ball (b,r2) or x in Ball (a,r1) )
assume A2: x in Ball (b,r2) ; ::_thesis: x in Ball (a,r1)
then reconsider x = x as Element of (TOP-REAL n) ;
 |.(x - b).| < r2 by A2, TOPREAL9:7;
then A3: |.(x - b).| + |.(b - a).| < r2 + (r1 - r2) by A1, XREAL_1:8;
 |.(x - a).| <= |.(x - b).| + |.(b - a).| by TOPRNS_1:34;
then  |.(x - a).| < r2 + (r1 - r2) by A3, XXREAL_0:2;
hence  x in Ball (a,r1) by TOPREAL9:7; ::_thesis: verum
end;

theorem Th23: :: TOPGEN_5:23
for a being real number
for r1, r2 being real positive number st r1 <= r2 holds
Ball (|[a,r1]|,r1) c= Ball (|[a,r2]|,r2)
proof
let a be real number ; ::_thesis: for r1, r2 being real positive number st r1 <= r2 holds
Ball (|[a,r1]|,r1) c= Ball (|[a,r2]|,r2)
let r1, r2 be real positive number ; ::_thesis: ( r1 <= r2 implies Ball (|[a,r1]|,r1) c= Ball (|[a,r2]|,r2) )
A1: r1 - r2 is Real by XREAL_0:def_1;
assume  r1 <= r2 ; ::_thesis: Ball (|[a,r1]|,r1) c= Ball (|[a,r2]|,r2)
then A2: r2 - r1 >= 0 by XREAL_1:48;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Ball (|[a,r1]|,r1) or x in Ball (|[a,r2]|,r2) )
assume A3: x in Ball (|[a,r1]|,r1) ; ::_thesis: x in Ball (|[a,r2]|,r2)
then reconsider x = x as Element of (TOP-REAL 2) ;
A4: |.(x - |[a,r1]|).| < r1 by A3, TOPREAL9:7;
 |[a,r1]| - |[a,r2]| = |[(a - a),(r1 - r2)]| by EUCLID:62;
then  |.(|[a,r1]| - |[a,r2]|).| = |.(r1 - r2).| by A1, TOPREAL6:23;
then  |.(|[a,r1]| - |[a,r2]|).| = |.(r2 - r1).| by COMPLEX1:60;
then  |.(|[a,r1]| - |[a,r2]|).| = r2 - r1 by A2, ABSVALUE:def_1;
then A5: |.(x - |[a,r1]|).| + |.(|[a,r1]| - |[a,r2]|).| < r1 + (r2 - r1) by A4, XREAL_1:8;
 |.(x - |[a,r2]|).| <= |.(x - |[a,r1]|).| + |.(|[a,r1]| - |[a,r2]|).| by TOPRNS_1:34;
then  |.(x - |[a,r2]|).| < r2 by A5, XXREAL_0:2;
hence  x in Ball (|[a,r2]|,r2) by TOPREAL9:7; ::_thesis: verum
end;

theorem Th24: :: TOPGEN_5:24
for T1, T2 being non empty TopSpace
for B1 being Neighborhood_System of T1
for B2 being Neighborhood_System of T2 st B1 = B2 holds
TopStruct(# the carrier of T1, the topology of T1 #) = TopStruct(# the carrier of T2, the topology of T2 #)
proof
let T1, T2 be non empty TopSpace; ::_thesis: for B1 being Neighborhood_System of T1
for B2 being Neighborhood_System of T2 st B1 = B2 holds
TopStruct(# the carrier of T1, the topology of T1 #) = TopStruct(# the carrier of T2, the topology of T2 #)
let B1 be Neighborhood_System of T1; ::_thesis: for B2 being Neighborhood_System of T2 st B1 = B2 holds
TopStruct(# the carrier of T1, the topology of T1 #) = TopStruct(# the carrier of T2, the topology of T2 #)
let B2 be Neighborhood_System of T2; ::_thesis: ( B1 = B2 implies TopStruct(# the carrier of T1, the topology of T1 #) = TopStruct(# the carrier of T2, the topology of T2 #) )
A1: dom B1 = the carrier of T1 by PARTFUN1:def_2;
A2: dom B2 = the carrier of T2 by PARTFUN1:def_2;
A3: UniCl (Union B2) = the topology of T2 by YELLOW_9:22;
A4: UniCl (Union B1) = the topology of T1 by YELLOW_9:22;
assume  B1 = B2 ; ::_thesis: TopStruct(# the carrier of T1, the topology of T1 #) = TopStruct(# the carrier of T2, the topology of T2 #)
hence  TopStruct(# the carrier of T1, the topology of T1 #) = TopStruct(# the carrier of T2, the topology of T2 #) by A4, A3, A1, A2; ::_thesis: verum
end;

definition
func Niemytzki-plane  ->  non empty strict TopSpace means :Def3: :: TOPGEN_5:def 3
( the carrier of it = y>=0-plane & ex B being Neighborhood_System of it st
( ( for x being Element of REAL holds B . |[x,0]| =  {  ((Ball (|[x,r]|,r)) \/ {|[x,0]|}) where r is Element of REAL : r > 0  }  ) & ( for x, y being Element of REAL st y > 0 holds
B . |[x,y]| =  {  ((Ball (|[x,y]|,r)) /\ y>=0-plane) where r is Element of REAL : r > 0  }  ) ) );
existence
 ex b1 being non empty strict TopSpace st
( the carrier of b1 = y>=0-plane & ex B being Neighborhood_System of b1 st
( ( for x being Element of REAL holds B . |[x,0]| =  {  ((Ball (|[x,r]|,r)) \/ {|[x,0]|}) where r is Element of REAL : r > 0  }  ) & ( for x, y being Element of REAL st y > 0 holds
B . |[x,y]| =  {  ((Ball (|[x,y]|,r)) /\ y>=0-plane) where r is Element of REAL : r > 0  }  ) ) )
proof
defpred S1[ set ] means $1 in y=0-line ;
deffunc H1( Element of (TOP-REAL 2)) ->  set =  {  ((Ball (|[($1 `1),r]|,r)) \/ {$1}) where r is Element of REAL : r > 0  }  ;
set X = y>=0-plane ;
deffunc H2( Element of (TOP-REAL 2)) ->  set =  {  ((Ball ($1,r)) /\ y>=0-plane) where r is Element of REAL : r > 0  }  ;
consider B being ManySortedSet of y>=0-plane  such that
A1: for a being Element of y>=0-plane st a in y>=0-plane holds
( ( S1[a] implies B . a = H1(a) ) & ( not S1[a] implies B . a = H2(a) ) ) from PRE_CIRC:sch_2();
 B is non-empty
proof
let z be set ; :: according to PBOOLE:def_13 ::_thesis: ( not z in y>=0-plane or not B . z is empty )
assume  z in y>=0-plane ; ::_thesis: not B . z is empty
then reconsider s = z as Element of y>=0-plane ;
percases ( S1[z] or not S1[z] ) ;
supposeA2: S1[z] ; ::_thesis: not B . z is empty
set r = the positive Element of REAL ;
set a = |[(s `1), the positive Element of REAL ]|;
 B . s = H1(s) by A2, A1;
then  (Ball (|[(s `1), the positive Element of REAL ]|, the positive Element of REAL )) \/ {s} in B . s ;
hence  not B . z is empty ; ::_thesis: verum
end;
supposeA3: not S1[z] ; ::_thesis: not B . z is empty
set r = the positive Element of REAL ;
 B . s = H2(s) by A3, A1;
then  (Ball (s, the positive Element of REAL )) /\ y>=0-plane in B . s ;
hence  not B . z is empty ; ::_thesis: verum
end;
end;
end;
then reconsider B = B as non-empty ManySortedSet of y>=0-plane ;
A4: rng B c= bool (bool y>=0-plane)
proof
let w be set ; :: according to TARSKI:def_3 ::_thesis: ( not w in rng B or w in bool (bool y>=0-plane) )
assume  w in rng B ; ::_thesis: w in bool (bool y>=0-plane)
then consider a being set  such that
A5: a in dom B and
A6: w = B . a by FUNCT_1:def_3;
reconsider a = a as Element of y>=0-plane by A5;
 w c= bool y>=0-plane
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in w or z in bool y>=0-plane )
assume A7: z in w ; ::_thesis: z in bool y>=0-plane
percases ( w = H1(a) or w = H2(a) ) by A1, A6;
suppose w = H1(a) ; ::_thesis: z in bool y>=0-plane
then consider r being Element of REAL  such that
A8: z = (Ball (|[(a `1),r]|,r)) \/ {a} and
A9: r > 0 by A7;
 Ball (|[(a `1),r]|,r) c= y>=0-plane by A9, Th20;
then  z c= y>=0-plane by A8, XBOOLE_1:8;
hence  z in bool y>=0-plane ; ::_thesis: verum
end;
suppose w = H2(a) ; ::_thesis: z in bool y>=0-plane
then  ex r being Element of REAL st
( z = (Ball (a,r)) /\ y>=0-plane & r > 0 ) by A7;
then  z c= y>=0-plane by XBOOLE_1:17;
hence  z in bool y>=0-plane ; ::_thesis: verum
end;
end;
end;
hence  w in bool (bool y>=0-plane) ; ::_thesis: verum
end;
A10: for x, y, U being set st x in U & U in B . y & y in y>=0-plane holds
ex V being set st
( V in B . x & V c= U )
proof
let x, y, U be set ; ::_thesis: ( x in U & U in B . y & y in y>=0-plane implies ex V being set st
( V in B . x & V c= U ) )
assume A11: x in U ; ::_thesis: ( not U in B . y or not y in y>=0-plane or ex V being set st
( V in B . x & V c= U ) )
assume A12: U in B . y ; ::_thesis: ( not y in y>=0-plane or ex V being set st
( V in B . x & V c= U ) )
assume  y in y>=0-plane ; ::_thesis: ex V being set st
( V in B . x & V c= U )
then reconsider y = y as Element of y>=0-plane ;
percases ( S1[y] or not S1[y] ) ;
suppose S1[y] ; ::_thesis: ex V being set st
( V in B . x & V c= U )
then  B . y = H1(y) by A1;
then consider r being Element of REAL  such that
A13: U = (Ball (|[(y `1),r]|,r)) \/ {y} and
A14: r > 0 by A12;
A15: ( x in Ball (|[(y `1),r]|,r) or x = y ) by A11, A13, ZFMISC_1:136;
then reconsider z = x as Element of (TOP-REAL 2) ;
now__::_thesis:_(_x_in_Ball_(|[(y_`1),r]|,r)_implies_ex_V_being_Element_of_bool_the_carrier_of_(TOP-REAL_2)_st_
(_V_in_B_._x_&_V_c=_U_)_)
set r2 = r - |.(z - |[(y `1),r]|).|;
assume A16: x in Ball (|[(y `1),r]|,r) ; ::_thesis: ex V being Element of bool the carrier of (TOP-REAL 2) st
( V in B . x & V c= U )
take V = (Ball (z,(r - |.(z - |[(y `1),r]|).|))) /\ y>=0-plane; ::_thesis: ( V in B . x & V c= U )
 |.(z - |[(y `1),r]|).| < r by A16, TOPREAL9:7;
then reconsider r1 = r, r2 = r - |.(z - |[(y `1),r]|).| as positive Real by XREAL_1:50;
A17: r2 > 0 ;
 Ball (|[(y `1),r]|,r) misses y=0-line by A14, Th21;
then A18: not S1[x] by A16, XBOOLE_0:3;
 Ball (|[(y `1),r]|,r) c= y>=0-plane by A14, Th20;
then  B . z = H2(z) by A16, A18, A1;
hence  V in B . x by A17; ::_thesis: V c= U
A19: Ball (|[(y `1),r]|,r) c= U by A13, XBOOLE_1:7;
 r1 - r2 = |.(|[(y `1),r]| - z).| by TOPRNS_1:27;
then A20: Ball (z,r2) c= Ball (|[(y `1),r]|,r1) by Th22;
 V c= Ball (z,r2) by XBOOLE_1:17;
then  V c= Ball (|[(y `1),r]|,r1) by A20, XBOOLE_1:1;
hence  V c= U by A19, XBOOLE_1:1; ::_thesis: verum
end;
hence  ex V being set st
( V in B . x & V c= U ) by A12, A15; ::_thesis: verum
end;
suppose not S1[y] ; ::_thesis: ex V being set st
( V in B . x & V c= U )
then  B . y = H2(y) by A1;
then consider r being Element of REAL  such that
A21: U = (Ball (y,r)) /\ y>=0-plane and
 r > 0 by A12;
reconsider z = x as Element of y>=0-plane by A11, A21, XBOOLE_0:def_4;
set r2 = r - |.(z - y).|;
 z in Ball (y,r) by A11, A21, XBOOLE_0:def_4;
then  |.(z - y).| < r by TOPREAL9:7;
then reconsider r1 = r, r2 = r - |.(z - y).| as positive Real by XREAL_1:50;
A22: z = |[(z `1),(z `2)]| by EUCLID:53;
percases ( S1[z] or not S1[z] ) ;
supposeA23: S1[z] ; ::_thesis: ex V being set st
( V in B . x & V c= U )
then  z `2 = 0 by A22, Th15;
then  |[(z `1),(r2 / 2)]| - z = |[((z `1) - (z `1)),((r2 / 2) - 0)]| by A22, EUCLID:62;
then |.(|[(z `1),(r2 / 2)]| - z).| = abs (r2 / 2) by TOPREAL6:23
.= r2 / 2 by ABSVALUE:def_1 ;
then  |.(|[(z `1),(r2 / 2)]| - y).| <= (r2 / 2) + |.(z - y).| by TOPRNS_1:34;
then  |.(y - |[(z `1),(r2 / 2)]|).| <= r - (r2 / 2) by TOPRNS_1:27;
then A24: Ball (|[(z `1),(r2 / 2)]|,(r2 / 2)) c= Ball (y,r1) by Th22;
set V = (Ball (|[(z `1),(r2 / 2)]|,(r2 / 2))) \/ {z};
take  (Ball (|[(z `1),(r2 / 2)]|,(r2 / 2))) \/ {z} ; ::_thesis: ( (Ball (|[(z `1),(r2 / 2)]|,(r2 / 2))) \/ {z} in B . x & (Ball (|[(z `1),(r2 / 2)]|,(r2 / 2))) \/ {z} c= U )
 B . z = H1(z) by A23, A1;
hence  (Ball (|[(z `1),(r2 / 2)]|,(r2 / 2))) \/ {z} in B . x ; ::_thesis: (Ball (|[(z `1),(r2 / 2)]|,(r2 / 2))) \/ {z} c= U
A25: {z} c= U by A11, ZFMISC_1:31;
 Ball (|[(z `1),(r2 / 2)]|,(r2 / 2)) c= y>=0-plane by Th20;
then  Ball (|[(z `1),(r2 / 2)]|,(r2 / 2)) c= U by A24, A21, XBOOLE_1:19;
hence  (Ball (|[(z `1),(r2 / 2)]|,(r2 / 2))) \/ {z} c= U by A25, XBOOLE_1:8; ::_thesis: verum
end;
supposeA26: not S1[z] ; ::_thesis: ex V being set st
( V in B . x & V c= U )
take V = (Ball (z,r2)) /\ y>=0-plane; ::_thesis: ( V in B . x & V c= U )
 B . z = H2(z) by A26, A1;
hence  V in B . x ; ::_thesis: V c= U
 |.(y - z).| = r1 - r2 by TOPRNS_1:27;
hence  V c= U by A21, Th22, XBOOLE_1:26; ::_thesis: verum
end;
end;
end;
end;
end;
A27: for x, U1, U2 being set st x in y>=0-plane & U1 in B . x & U2 in B . x holds
ex U being set st
( U in B . x & U c= U1 /\ U2 )
proof
let x, U1, U2 be set ; ::_thesis: ( x in y>=0-plane & U1 in B . x & U2 in B . x implies ex U being set st
( U in B . x & U c= U1 /\ U2 ) )
assume  x in y>=0-plane ; ::_thesis: ( not U1 in B . x or not U2 in B . x or ex U being set st
( U in B . x & U c= U1 /\ U2 ) )
then reconsider z = x as Element of y>=0-plane ;
assume that
A28: U1 in B . x and
A29: U2 in B . x ; ::_thesis: ex U being set st
( U in B . x & U c= U1 /\ U2 )
percases ( S1[z] or not S1[z] ) ;
suppose S1[z] ; ::_thesis: ex U being set st
( U in B . x & U c= U1 /\ U2 )
then A30: B . x = H1(z) by A1;
then consider r1 being Real such that
A31: U1 = (Ball (|[(z `1),r1]|,r1)) \/ {z} and
A32: r1 > 0 by A28;
consider r2 being Real such that
A33: U2 = (Ball (|[(z `1),r2]|,r2)) \/ {z} and
A34: r2 > 0 by A29, A30;
 ( r1 <= r2 or r1 >= r2 ) ;
then consider U being set  such that
A35: ( ( r1 <= r2 & U = U1 ) or ( r1 >= r2 & U = U2 ) ) ;
A36: U c= U2 by A31, A32, A33, A35, Th23, XBOOLE_1:9;
take  U ; ::_thesis: ( U in B . x & U c= U1 /\ U2 )
thus  U in B . x by A28, A29, A35; ::_thesis: U c= U1 /\ U2
 U c= U1 by A31, A33, A34, A35, Th23, XBOOLE_1:9;
hence  U c= U1 /\ U2 by A36, XBOOLE_1:19; ::_thesis: verum
end;
suppose not S1[z] ; ::_thesis: ex U being set st
( U in B . x & U c= U1 /\ U2 )
then A37: B . x = H2(z) by A1;
then consider r1 being Real such that
A38: U1 = (Ball (z,r1)) /\ y>=0-plane and
 r1 > 0 by A28;
consider r2 being Real such that
A39: U2 = (Ball (z,r2)) /\ y>=0-plane and
 r2 > 0 by A29, A37;
 ( r1 <= r2 or r1 >= r2 ) ;
then consider U being set  such that
A40: ( ( r1 <= r2 & U = U1 ) or ( r1 >= r2 & U = U2 ) ) ;
A41: U c= U2 by A38, A39, A40, JORDAN:18, XBOOLE_1:26;
take  U ; ::_thesis: ( U in B . x & U c= U1 /\ U2 )
thus  U in B . x by A28, A29, A40; ::_thesis: U c= U1 /\ U2
 U c= U1 by A38, A39, A40, JORDAN:18, XBOOLE_1:26;
hence  U c= U1 /\ U2 by A41, XBOOLE_1:19; ::_thesis: verum
end;
end;
end;
 for x, U being set st x in y>=0-plane & U in B . x holds
x in U
proof
let x, U be set ; ::_thesis: ( x in y>=0-plane & U in B . x implies x in U )
assume  x in y>=0-plane ; ::_thesis: ( not U in B . x or x in U )
then reconsider a = x as Element of y>=0-plane ;
assume A42: U in B . x ; ::_thesis: x in U
percases ( B . x = H1(a) or B . x = H2(a) ) by A1;
supposeA43: B . x = H1(a) ; ::_thesis: x in U
A44: a in {a} by TARSKI:def_1;
 ex r being Element of REAL st
( U = (Ball (|[(a `1),r]|,r)) \/ {a} & r > 0 ) by A43, A42;
hence  x in U by A44, XBOOLE_0:def_3; ::_thesis: verum
end;
suppose B . x = H2(a) ; ::_thesis: x in U
then consider r being Element of REAL  such that
A45: U = (Ball (a,r)) /\ y>=0-plane and
A46: r > 0 by A42;
 a in Ball (a,r) by A46, Th13;
hence  x in U by A45, XBOOLE_0:def_4; ::_thesis: verum
end;
end;
end;
then consider P being Subset-Family of y>=0-plane such that
 P = Union B and
A47: for T being TopStruct st the carrier of T = y>=0-plane & the topology of T = UniCl P holds
( T is TopSpace & ( for T9 being non empty TopSpace st T9 = T holds
B is Neighborhood_System of T9 ) ) by A27, A4, A10, TOPGEN_3:3;
set T = TopStruct(# y>=0-plane,(UniCl P) #);
reconsider T = TopStruct(# y>=0-plane,(UniCl P) #) as non empty strict TopSpace by A47;
reconsider B = B as Neighborhood_System of T by A47;
take  T ; ::_thesis: ( the carrier of T = y>=0-plane & ex B being Neighborhood_System of T st
( ( for x being Element of REAL holds B . |[x,0]| =  {  ((Ball (|[x,r]|,r)) \/ {|[x,0]|}) where r is Element of REAL : r > 0  }  ) & ( for x, y being Element of REAL st y > 0 holds
B . |[x,y]| =  {  ((Ball (|[x,y]|,r)) /\ y>=0-plane) where r is Element of REAL : r > 0  }  ) ) )
thus  the carrier of T = y>=0-plane ; ::_thesis: ex B being Neighborhood_System of T st
( ( for x being Element of REAL holds B . |[x,0]| =  {  ((Ball (|[x,r]|,r)) \/ {|[x,0]|}) where r is Element of REAL : r > 0  }  ) & ( for x, y being Element of REAL st y > 0 holds
B . |[x,y]| =  {  ((Ball (|[x,y]|,r)) /\ y>=0-plane) where r is Element of REAL : r > 0  }  ) )
take  B ; ::_thesis: ( ( for x being Element of REAL holds B . |[x,0]| =  {  ((Ball (|[x,r]|,r)) \/ {|[x,0]|}) where r is Element of REAL : r > 0  }  ) & ( for x, y being Element of REAL st y > 0 holds
B . |[x,y]| =  {  ((Ball (|[x,y]|,r)) /\ y>=0-plane) where r is Element of REAL : r > 0  }  ) )
hereby  ::_thesis: for x, y being Element of REAL st y > 0 holds
B . |[x,y]| =  {  ((Ball (|[x,y]|,r)) /\ y>=0-plane) where r is Element of REAL : r > 0  }
defpred S2[ Real] means $1 > 0 ;
let x be Element of REAL ; ::_thesis: B . |[x,0]| =  {  ((Ball (|[x,r]|,r)) \/ {|[x,0]|}) where r is Element of REAL : r > 0  }
deffunc H3( Real) ->  Element of bool the carrier of (TOP-REAL 2) = (Ball (|[x,$1]|,$1)) \/ {|[x,0]|};
deffunc H4( Real) ->  Element of bool the carrier of (TOP-REAL 2) = (Ball (|[(|[x,0]| `1),$1]|,$1)) \/ {|[x,0]|};
A48: |[x,0]| in y>=0-plane ;
A49: for r being Real holds H3(r) = H4(r) by EUCLID:52;
A50:  {  H3(r) where r is Real : S2[r]  }  =  {  H4(r) where r is Real : S2[r]  }  from FRAENKEL:sch_5(A49);
 S1[|[x,0]|] ;
hence  B . |[x,0]| =  {  ((Ball (|[x,r]|,r)) \/ {|[x,0]|}) where r is Element of REAL : r > 0  }  by A48, A1, A50; ::_thesis: verum
end;
let x, y be Element of REAL ; ::_thesis: ( y > 0 implies B . |[x,y]| =  {  ((Ball (|[x,y]|,r)) /\ y>=0-plane) where r is Element of REAL : r > 0  }  )
assume A51: y > 0 ; ::_thesis: B . |[x,y]| =  {  ((Ball (|[x,y]|,r)) /\ y>=0-plane) where r is Element of REAL : r > 0  }
then A52: |[x,y]| in y>=0-plane ;
 not |[x,y]| in y=0-line by A51, Th15;
hence  B . |[x,y]| =  {  ((Ball (|[x,y]|,r)) /\ y>=0-plane) where r is Element of REAL : r > 0  }  by A52, A1; ::_thesis: verum
end;
uniqueness
 for b1, b2 being non empty strict TopSpace st the carrier of b1 = y>=0-plane & ex B being Neighborhood_System of b1 st
( ( for x being Element of REAL holds B . |[x,0]| =  {  ((Ball (|[x,r]|,r)) \/ {|[x,0]|}) where r is Element of REAL : r > 0  }  ) & ( for x, y being Element of REAL st y > 0 holds
B . |[x,y]| =  {  ((Ball (|[x,y]|,r)) /\ y>=0-plane) where r is Element of REAL : r > 0  }  ) ) & the carrier of b2 = y>=0-plane & ex B being Neighborhood_System of b2 st
( ( for x being Element of REAL holds B . |[x,0]| =  {  ((Ball (|[x,r]|,r)) \/ {|[x,0]|}) where r is Element of REAL : r > 0  }  ) & ( for x, y being Element of REAL st y > 0 holds
B . |[x,y]| =  {  ((Ball (|[x,y]|,r)) /\ y>=0-plane) where r is Element of REAL : r > 0  }  ) ) holds
b1 = b2
proof
let T1, T2 be non empty strict TopSpace; ::_thesis: ( the carrier of T1 = y>=0-plane & ex B being Neighborhood_System of T1 st
( ( for x being Element of REAL holds B . |[x,0]| =  {  ((Ball (|[x,r]|,r)) \/ {|[x,0]|}) where r is Element of REAL : r > 0  }  ) & ( for x, y being Element of REAL st y > 0 holds
B . |[x,y]| =  {  ((Ball (|[x,y]|,r)) /\ y>=0-plane) where r is Element of REAL : r > 0  }  ) ) & the carrier of T2 = y>=0-plane & ex B being Neighborhood_System of T2 st
( ( for x being Element of REAL holds B . |[x,0]| =  {  ((Ball (|[x,r]|,r)) \/ {|[x,0]|}) where r is Element of REAL : r > 0  }  ) & ( for x, y being Element of REAL st y > 0 holds
B . |[x,y]| =  {  ((Ball (|[x,y]|,r)) /\ y>=0-plane) where r is Element of REAL : r > 0  }  ) ) implies T1 = T2 )
assume that
A53: the carrier of T1 = y>=0-plane and
A54: ex B being Neighborhood_System of T1 st
( ( for x being Element of REAL holds B . |[x,0]| =  {  ((Ball (|[x,r]|,r)) \/ {|[x,0]|}) where r is Element of REAL : r > 0  }  ) & ( for x, y being Element of REAL st y > 0 holds
B . |[x,y]| =  {  ((Ball (|[x,y]|,r)) /\ y>=0-plane) where r is Element of REAL : r > 0  }  ) ) and
A55: the carrier of T2 = y>=0-plane and
A56: ex B being Neighborhood_System of T2 st
( ( for x being Element of REAL holds B . |[x,0]| =  {  ((Ball (|[x,r]|,r)) \/ {|[x,0]|}) where r is Element of REAL : r > 0  }  ) & ( for x, y being Element of REAL st y > 0 holds
B . |[x,y]| =  {  ((Ball (|[x,y]|,r)) /\ y>=0-plane) where r is Element of REAL : r > 0  }  ) ) ; ::_thesis: T1 = T2
consider B2 being Neighborhood_System of T2 such that
A57: for x being Element of REAL holds B2 . |[x,0]| =  {  ((Ball (|[x,r]|,r)) \/ {|[x,0]|}) where r is Element of REAL : r > 0  }  and
A58: for x, y being Element of REAL st y > 0 holds
B2 . |[x,y]| =  {  ((Ball (|[x,y]|,r)) /\ y>=0-plane) where r is Element of REAL : r > 0  }  by A56;
consider B1 being Neighborhood_System of T1 such that
A59: for x being Element of REAL holds B1 . |[x,0]| =  {  ((Ball (|[x,r]|,r)) \/ {|[x,0]|}) where r is Element of REAL : r > 0  }  and
A60: for x, y being Element of REAL st y > 0 holds
B1 . |[x,y]| =  {  ((Ball (|[x,y]|,r)) /\ y>=0-plane) where r is Element of REAL : r > 0  }  by A54;
now__::_thesis:_for_a_being_set_st_a_in_y>=0-plane_holds_
B1_._a_=_B2_._a
let a be set ; ::_thesis: ( a in y>=0-plane implies B1 . a = B2 . a )
assume  a in y>=0-plane ; ::_thesis: B1 . a = B2 . a
then reconsider z = a as Element of y>=0-plane ;
A61: z = |[(z `1),(z `2)]| by EUCLID:53;
then  ( z `2 = 0 or z `2 > 0 ) by Th18;
then  ( ( z `2 = 0 & B1 . z =  {  ((Ball (|[(z `1),r]|,r)) \/ {|[(z `1),0]|}) where r is Element of REAL : r > 0  }  & B2 . z =  {  ((Ball (|[(z `1),r]|,r)) \/ {|[(z `1),0]|}) where r is Element of REAL : r > 0  }  ) or ( z `2 > 0 & B1 . z =  {  ((Ball (|[(z `1),(z `2)]|,r)) /\ y>=0-plane) where r is Element of REAL : r > 0  }  & B2 . z =  {  ((Ball (|[(z `1),(z `2)]|,r)) /\ y>=0-plane) where r is Element of REAL : r > 0  }  ) ) by A59, A60, A57, A58, A61;
hence  B1 . a = B2 . a ; ::_thesis: verum
end;
hence  T1 = T2 by A53, A55, Th24, PBOOLE:3; ::_thesis: verum
end;
end;

:: deftheorem Def3 defines Niemytzki-plane TOPGEN_5:def_3_:_
 for b1 being non empty strict TopSpace holds
( b1 = Niemytzki-plane iff ( the carrier of b1 = y>=0-plane & ex B being Neighborhood_System of b1 st
( ( for x being Element of REAL holds B . |[x,0]| =  {  ((Ball (|[x,r]|,r)) \/ {|[x,0]|}) where r is Element of REAL : r > 0  }  ) & ( for x, y being Element of REAL st y > 0 holds
B . |[x,y]| =  {  ((Ball (|[x,y]|,r)) /\ y>=0-plane) where r is Element of REAL : r > 0  }  ) ) ) );

theorem Th25: :: TOPGEN_5:25
y>=0-plane \ y=0-line is open Subset of Niemytzki-plane
proof
consider BB being Neighborhood_System of Niemytzki-plane  such that
 for x being Element of REAL holds BB . |[x,0]| =  {  ((Ball (|[x,r]|,r)) \/ {|[x,0]|}) where r is Element of REAL : r > 0  }  and
A1: for x, y being Element of REAL st y > 0 holds
BB . |[x,y]| =  {  ((Ball (|[x,y]|,r)) /\ y>=0-plane) where r is Element of REAL : r > 0  }  by Def3;
A2: the carrier of Niemytzki-plane = y>=0-plane by Def3;
then reconsider A = y>=0-plane \ y=0-line as Subset of Niemytzki-plane by XBOOLE_1:36;
now__::_thesis:_for_a_being_Point_of_Niemytzki-plane_st_a_in_A_holds_
ex_B_being_Subset_of_Niemytzki-plane_st_
(_B_in_Union_BB_&_a_in_B_&_B_c=_A_)
let a be Point of Niemytzki-plane; ::_thesis: ( a in A implies ex B being Subset of Niemytzki-plane st
( B in Union BB & a in B & B c= A ) )
assume A3: a in A ; ::_thesis: ex B being Subset of Niemytzki-plane st
( B in Union BB & a in B & B c= A )
then  a in y>=0-plane by XBOOLE_0:def_5;
then consider x, y being Element of REAL  such that
A4: a = |[x,y]| and
A5: y >= 0 ;
set B = (Ball (|[x,y]|,y)) /\ y>=0-plane;
reconsider B = (Ball (|[x,y]|,y)) /\ y>=0-plane as Subset of Niemytzki-plane by A2, XBOOLE_1:17;
 not a in y=0-line by A3, XBOOLE_0:def_5;
then A6: y <> 0 by A4;
then  B in  {  ((Ball (|[x,y]|,r)) /\ y>=0-plane) where r is Element of REAL : r > 0  }  by A5;
then A7: B in BB . a by A1, A4, A5, A6;
take B = B; ::_thesis: ( B in Union BB & a in B & B c= A )
 dom BB = the carrier of Niemytzki-plane by PARTFUN1:def_2;
hence  B in Union BB by A7, CARD_5:2; ::_thesis: ( a in B & B c= A )
thus  a in B by A7, YELLOW_8:12; ::_thesis: B c= A
 Ball (|[x,y]|,y) c= y>=0-plane by A5, A6, Th20;
then  B = Ball (|[x,y]|,y) by XBOOLE_1:28;
then  B misses y=0-line by A5, A6, Th21;
hence  B c= A by A2, XBOOLE_1:86; ::_thesis: verum
end;
hence  y>=0-plane \ y=0-line is open Subset of Niemytzki-plane by YELLOW_9:31; ::_thesis: verum
end;

Lm1:  the carrier of Niemytzki-plane = y>=0-plane
by Def3;

theorem Th26: :: TOPGEN_5:26
y=0-line is closed Subset of Niemytzki-plane
proof
reconsider B = y=0-line as Subset of Niemytzki-plane by Def3, Th19;
reconsider A = y>=0-plane \ y=0-line as open Subset of Niemytzki-plane by Th25;
 B ` = A by Def3;
then  A ` = y=0-line ;
hence  y=0-line is closed Subset of Niemytzki-plane ; ::_thesis: verum
end;

theorem Th27: :: TOPGEN_5:27
for x being real number
for r being real positive number holds (Ball (|[x,r]|,r)) \/ {|[x,0]|} is open Subset of Niemytzki-plane
proof
let x be real number ; ::_thesis: for r being real positive number holds (Ball (|[x,r]|,r)) \/ {|[x,0]|} is open Subset of Niemytzki-plane
let r be real positive number ; ::_thesis: (Ball (|[x,r]|,r)) \/ {|[x,0]|} is open Subset of Niemytzki-plane
A1: r is Real by XREAL_0:def_1;
 the carrier of Niemytzki-plane = y>=0-plane by Def3;
then reconsider a = |[x,0]| as Point of Niemytzki-plane by Th18;
consider BB being Neighborhood_System of Niemytzki-plane  such that
A2: for x being Element of REAL holds BB . |[x,0]| =  {  ((Ball (|[x,q]|,q)) \/ {|[x,0]|}) where q is Element of REAL : q > 0  }  and
 for x, y being Element of REAL st y > 0 holds
BB . |[x,y]| =  {  ((Ball (|[x,y]|,q)) /\ y>=0-plane) where q is Element of REAL : q > 0  }  by Def3;
 x is Real by XREAL_0:def_1;
then  BB . |[x,0]| =  {  ((Ball (|[x,q]|,q)) \/ {|[x,0]|}) where q is Element of REAL : q > 0  }  by A2;
then  (Ball (|[x,r]|,r)) \/ {|[x,0]|} in BB . a by A1;
hence  (Ball (|[x,r]|,r)) \/ {|[x,0]|} is open Subset of Niemytzki-plane by YELLOW_8:12; ::_thesis: verum
end;

theorem Th28: :: TOPGEN_5:28
for x being real number
for y, r being real positive number holds (Ball (|[x,y]|,r)) /\ y>=0-plane is open Subset of Niemytzki-plane
proof
let x be real number ; ::_thesis: for y, r being real positive number holds (Ball (|[x,y]|,r)) /\ y>=0-plane is open Subset of Niemytzki-plane
let y, r be real positive number ; ::_thesis: (Ball (|[x,y]|,r)) /\ y>=0-plane is open Subset of Niemytzki-plane
A1: y is Real by XREAL_0:def_1;
 the carrier of Niemytzki-plane = y>=0-plane by Def3;
then reconsider a = |[x,y]| as Point of Niemytzki-plane by Th18;
A2: r is Real by XREAL_0:def_1;
consider BB being Neighborhood_System of Niemytzki-plane  such that
 for x being Element of REAL holds BB . |[x,0]| =  {  ((Ball (|[x,q]|,q)) \/ {|[x,0]|}) where q is Element of REAL : q > 0  }  and
A3: for x, y being Element of REAL st y > 0 holds
BB . |[x,y]| =  {  ((Ball (|[x,y]|,q)) /\ y>=0-plane) where q is Element of REAL : q > 0  }  by Def3;
 x is Real by XREAL_0:def_1;
then  BB . |[x,y]| =  {  ((Ball (|[x,y]|,q)) /\ y>=0-plane) where q is Element of REAL : q > 0  }  by A1, A3;
then  (Ball (|[x,y]|,r)) /\ y>=0-plane in BB . a by A2;
hence  (Ball (|[x,y]|,r)) /\ y>=0-plane is open Subset of Niemytzki-plane by YELLOW_8:12; ::_thesis: verum
end;

theorem Th29: :: TOPGEN_5:29
for x, y being real number
for r being real positive number st r <= y holds
Ball (|[x,y]|,r) is open Subset of Niemytzki-plane
proof
let x, y be real number ; ::_thesis: for r being real positive number st r <= y holds
Ball (|[x,y]|,r) is open Subset of Niemytzki-plane
let r be real positive number ; ::_thesis: ( r <= y implies Ball (|[x,y]|,r) is open Subset of Niemytzki-plane )
assume A1: r <= y ; ::_thesis: Ball (|[x,y]|,r) is open Subset of Niemytzki-plane
A2: Ball (|[x,y]|,r) c= y>=0-plane
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in Ball (|[x,y]|,r) or a in y>=0-plane )
assume A3: a in Ball (|[x,y]|,r) ; ::_thesis: a in y>=0-plane
then reconsider z = a as Element of (TOP-REAL 2) ;
A4: ( z `2 < 0 implies ( y - (z `2) > y & abs (y - (z `2)) = y - (z `2) ) ) by A1, ABSVALUE:def_1, XREAL_1:46;
A5: z = |[(z `1),(z `2)]| by EUCLID:53;
then A6: z - |[x,y]| = |[((z `1) - x),((z `2) - y)]| by EUCLID:62;
then A7: (z - |[x,y]|) `2 = (z `2) - y by EUCLID:52;
 (z - |[x,y]|) `1 = (z `1) - x by A6, EUCLID:52;
then  |.(z - |[x,y]|).| = sqrt ((((z `1) - x) ^2) + (((z `2) - y) ^2)) by A7, JGRAPH_1:30;
then  |.(z - |[x,y]|).| >= abs ((z `2) - y) by COMPLEX1:79;
then A8: |.(z - |[x,y]|).| >= abs (y - (z `2)) by COMPLEX1:60;
 |.(z - |[x,y]|).| < r by A3, TOPREAL9:7;
then  abs (y - (z `2)) < r by A8, XXREAL_0:2;
hence  a in y>=0-plane by A4, A1, A5, XXREAL_0:2; ::_thesis: verum
end;
 (Ball (|[x,y]|,r)) /\ y>=0-plane is open Subset of Niemytzki-plane by A1, Th28;
hence  Ball (|[x,y]|,r) is open Subset of Niemytzki-plane by A2, XBOOLE_1:28; ::_thesis: verum
end;

theorem Th30: :: TOPGEN_5:30
for p being Point of Niemytzki-plane
for r being real positive number ex a being Point of (TOP-REAL 2) ex U being open Subset of Niemytzki-plane st
( p in U & a in U & ( for b being Point of (TOP-REAL 2) st b in U holds
|.(b - a).| < r ) )
proof
let p be Point of Niemytzki-plane; ::_thesis: for r being real positive number ex a being Point of (TOP-REAL 2) ex U being open Subset of Niemytzki-plane st
( p in U & a in U & ( for b being Point of (TOP-REAL 2) st b in U holds
|.(b - a).| < r ) )
let r be real positive number ; ::_thesis: ex a being Point of (TOP-REAL 2) ex U being open Subset of Niemytzki-plane st
( p in U & a in U & ( for b being Point of (TOP-REAL 2) st b in U holds
|.(b - a).| < r ) )
consider BB being Neighborhood_System of Niemytzki-plane  such that
A1: for x being Element of REAL holds BB . |[x,0]| =  {  ((Ball (|[x,q]|,q)) \/ {|[x,0]|}) where q is Element of REAL : q > 0  }  and
A2: for x, y being Element of REAL st y > 0 holds
BB . |[x,y]| =  {  ((Ball (|[x,y]|,q)) /\ y>=0-plane) where q is Element of REAL : q > 0  }  by Def3;
A3: the carrier of Niemytzki-plane = y>=0-plane by Def3;
 p in the carrier of Niemytzki-plane ;
then reconsider p9 = p as Point of (TOP-REAL 2) by A3;
A4: p = |[(p9 `1),(p9 `2)]| by EUCLID:53;
percases ( p9 `2 = 0 or p9 `2 > 0 ) by A3, A4, Th18;
supposeA5: p9 `2 = 0 ; ::_thesis: ex a being Point of (TOP-REAL 2) ex U being open Subset of Niemytzki-plane st
( p in U & a in U & ( for b being Point of (TOP-REAL 2) st b in U holds
|.(b - a).| < r ) )
then  BB . p =  {  ((Ball (|[(p9 `1),q]|,q)) \/ {|[(p9 `1),0]|}) where q is Element of REAL : q > 0  }  by A1, A4;
then A6: (Ball (|[(p9 `1),(r / 2)]|,(r / 2))) \/ {|[(p9 `1),0]|} in BB . p ;
 BB . p c= the topology of Niemytzki-plane by TOPS_2:64;
then reconsider U = (Ball (|[(p9 `1),(r / 2)]|,(r / 2))) \/ {p} as open Subset of Niemytzki-plane by A6, A4, A5, PRE_TOPC:def_2;
take a = |[(p9 `1),(r / 2)]|; ::_thesis: ex U being open Subset of Niemytzki-plane st
( p in U & a in U & ( for b being Point of (TOP-REAL 2) st b in U holds
|.(b - a).| < r ) )
take  U ; ::_thesis: ( p in U & a in U & ( for b being Point of (TOP-REAL 2) st b in U holds
|.(b - a).| < r ) )
thus  p in U by ZFMISC_1:136; ::_thesis: ( a in U & ( for b being Point of (TOP-REAL 2) st b in U holds
|.(b - a).| < r ) )
A7: r / 2 < (r / 2) + (r / 2) by XREAL_1:29;
 a in Ball (a,(r / 2)) by Th13;
hence  a in U by XBOOLE_0:def_3; ::_thesis: for b being Point of (TOP-REAL 2) st b in U holds
|.(b - a).| < r
let b be Point of (TOP-REAL 2); ::_thesis: ( b in U implies |.(b - a).| < r )
assume  b in U ; ::_thesis: |.(b - a).| < r
then A8: ( b in Ball (a,(r / 2)) or b = p ) by ZFMISC_1:136;
 p9 - a = |[((p9 `1) - (p9 `1)),(0 - (r / 2))]| by A4, A5, EUCLID:62;
then |.(p9 - a).| = |.(0 - (r / 2)).| by TOPREAL6:23
.= |.((r / 2) - 0).| by COMPLEX1:60
.= r / 2 by ABSVALUE:def_1 ;
then  |.(b - a).| <= r / 2 by A8, TOPREAL9:7;
hence  |.(b - a).| < r by A7, XXREAL_0:2; ::_thesis: verum
end;
supposeA9: p9 `2 > 0 ; ::_thesis: ex a being Point of (TOP-REAL 2) ex U being open Subset of Niemytzki-plane st
( p in U & a in U & ( for b being Point of (TOP-REAL 2) st b in U holds
|.(b - a).| < r ) )
then  BB . p =  {  ((Ball (|[(p9 `1),(p9 `2)]|,q)) /\ y>=0-plane) where q is Element of REAL : q > 0  }  by A2, A4;
then A10: (Ball (p9,(r / 2))) /\ y>=0-plane in BB . p by A4;
 BB . p c= the topology of Niemytzki-plane by TOPS_2:64;
then reconsider U = (Ball (p9,(r / 2))) /\ y>=0-plane as open Subset of Niemytzki-plane by A10, PRE_TOPC:def_2;
take a = p9; ::_thesis: ex U being open Subset of Niemytzki-plane st
( p in U & a in U & ( for b being Point of (TOP-REAL 2) st b in U holds
|.(b - a).| < r ) )
take  U ; ::_thesis: ( p in U & a in U & ( for b being Point of (TOP-REAL 2) st b in U holds
|.(b - a).| < r ) )
A11: p in Ball (a,(r / 2)) by Th13;
 p in y>=0-plane by A4, A9;
hence  ( p in U & a in U ) by A11, XBOOLE_0:def_4; ::_thesis: for b being Point of (TOP-REAL 2) st b in U holds
|.(b - a).| < r
let b be Point of (TOP-REAL 2); ::_thesis: ( b in U implies |.(b - a).| < r )
assume  b in U ; ::_thesis: |.(b - a).| < r
then  b in Ball (a,(r / 2)) by XBOOLE_0:def_4;
then A12: |.(b - a).| <= r / 2 by TOPREAL9:7;
 r / 2 < (r / 2) + (r / 2) by XREAL_1:29;
hence  |.(b - a).| < r by A12, XXREAL_0:2; ::_thesis: verum
end;
end;
end;

theorem Th31: :: TOPGEN_5:31
for x, y being real number
for r being real positive number ex w, v being rational number st
( |[w,v]| in Ball (|[x,y]|,r) & |[w,v]| <> |[x,y]| )
proof
let x, y be real number ; ::_thesis: for r being real positive number ex w, v being rational number st
( |[w,v]| in Ball (|[x,y]|,r) & |[w,v]| <> |[x,y]| )
let r be real positive number ; ::_thesis: ex w, v being rational number st
( |[w,v]| in Ball (|[x,y]|,r) & |[w,v]| <> |[x,y]| )
 x < x + (r / 2) by XREAL_1:39;
then consider w being rational number  such that
A1: x < w and
A2: w < x + (r / 2) by RAT_1:7;
A3: w - x > 0 by A1, XREAL_1:50;
A4: w - x is Real by XREAL_0:def_1;
 y < y + (r / 2) by XREAL_1:39;
then consider v being rational number  such that
A5: y < v and
A6: v < y + (r / 2) by RAT_1:7;
A7: v - y > 0 by A5, XREAL_1:50;
 |[w,v]| - |[x,v]| = |[(w - x),(v - v)]| by EUCLID:62;
then  |.(|[w,v]| - |[x,v]|).| = abs (w - x) by A4, TOPREAL6:23;
then  |.(|[w,v]| - |[x,v]|).| = w - x by A3, ABSVALUE:def_1;
then A8: |.(|[w,v]| - |[x,v]|).| < (x + (r / 2)) - x by A2, XREAL_1:9;
take  w ; ::_thesis: ex v being rational number st
( |[w,v]| in Ball (|[x,y]|,r) & |[w,v]| <> |[x,y]| )
take  v ; ::_thesis: ( |[w,v]| in Ball (|[x,y]|,r) & |[w,v]| <> |[x,y]| )
A9: |[x,v]| - |[x,y]| = |[(x - x),(v - y)]| by EUCLID:62;
A10: |.(|[w,v]| - |[x,y]|).| <= |.(|[w,v]| - |[x,v]|).| + |.(|[x,v]| - |[x,y]|).| by TOPRNS_1:34;
 v - y is Real by XREAL_0:def_1;
then  |.(|[x,v]| - |[x,y]|).| = |.(v - y).| by A9, TOPREAL6:23;
then  |.(|[x,v]| - |[x,y]|).| = v - y by A7, ABSVALUE:def_1;
then  |.(|[x,v]| - |[x,y]|).| <= (y + (r / 2)) - y by A6, XREAL_1:9;
then  |.(|[w,v]| - |[x,v]|).| + |.(|[x,v]| - |[x,y]|).| < ((x + (r / 2)) - x) + ((y + (r / 2)) - y) by A8, XREAL_1:8;
then  |.(|[w,v]| - |[x,y]|).| < r by A10, XXREAL_0:2;
hence  |[w,v]| in Ball (|[x,y]|,r) by TOPREAL9:7; ::_thesis: |[w,v]| <> |[x,y]|
thus  |[w,v]| <> |[x,y]| by A5, SPPOL_2:1; ::_thesis: verum
end;

theorem Th32: :: TOPGEN_5:32
for A being Subset of Niemytzki-plane st A = (y>=0-plane \ y=0-line) /\ (product <*RAT,RAT*>) holds
for x being set holds Cl (A \ {x}) = [#] Niemytzki-plane
proof
let A be Subset of Niemytzki-plane; ::_thesis: ( A = (y>=0-plane \ y=0-line) /\ (product <*RAT,RAT*>) implies for x being set holds Cl (A \ {x}) = [#] Niemytzki-plane )
assume A1: A = (y>=0-plane \ y=0-line) /\ (product <*RAT,RAT*>) ; ::_thesis: for x being set holds Cl (A \ {x}) = [#] Niemytzki-plane
let s be set ; ::_thesis: Cl (A \ {s}) = [#] Niemytzki-plane
thus  Cl (A \ {s}) c= [#] Niemytzki-plane ; :: according to XBOOLE_0:def_10 ::_thesis: [#] Niemytzki-plane c= Cl (A \ {s})
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in [#] Niemytzki-plane or x in Cl (A \ {s}) )
assume  x in [#] Niemytzki-plane ; ::_thesis: x in Cl (A \ {s})
then reconsider a = x as Element of Niemytzki-plane ;
reconsider b = a as Element of y>=0-plane by Def3;
consider BB being Neighborhood_System of Niemytzki-plane  such that
A2: for x being Element of REAL holds BB . |[x,0]| =  {  ((Ball (|[x,q]|,q)) \/ {|[x,0]|}) where q is Element of REAL : q > 0  }  and
A3: for x, y being Element of REAL st y > 0 holds
BB . |[x,y]| =  {  ((Ball (|[x,y]|,q)) /\ y>=0-plane) where q is Element of REAL : q > 0  }  by Def3;
A4: a = |[(b `1),(b `2)]| by EUCLID:53;
 for U being set st U in BB . a holds
U meets A \ {s}
proof
let U be set ; ::_thesis: ( U in BB . a implies U meets A \ {s} )
assume A5: U in BB . a ; ::_thesis: U meets A \ {s}
percases ( b `2 = 0 or b `2 > 0 ) by A4, Th18;
supposeA6: b `2 = 0 ; ::_thesis: U meets A \ {s}
then  BB . a =  {  ((Ball (|[(b `1),q]|,q)) \/ {|[(b `1),0]|}) where q is Element of REAL : q > 0  }  by A2, A4;
then consider q being Real such that
A7: U = (Ball (|[(b `1),q]|,q)) \/ {a} and
A8: q > 0 by A4, A5, A6;
reconsider q = q as positive Real by A8;
consider w1, v1 being rational number  such that
A9: |[w1,v1]| in Ball (|[(b `1),q]|,q) and
A10: |[w1,v1]| <> |[(b `1),q]| by Th31;
A11: |[w1,v1]| in U by A7, A9, XBOOLE_0:def_3;
set q2 = |.(|[w1,v1]| - |[(b `1),q]|).|;
 |[w1,v1]| - |[(b `1),q]| <> 0. (TOP-REAL 2) by A10, EUCLID:43;
then  |.(|[w1,v1]| - |[(b `1),q]|).| <> 0 by EUCLID_2:42;
then reconsider q2 = |.(|[w1,v1]| - |[(b `1),q]|).| as real positive number ;
A12: q2 < q by A9, TOPREAL9:7;
consider w2, v2 being rational number  such that
A13: |[w2,v2]| in Ball (|[(b `1),q]|,q2) and
 |[w2,v2]| <> |[(b `1),q]| by Th31;
 |.(|[w2,v2]| - |[(b `1),q]|).| < q2 by A13, TOPREAL9:7;
then  |.(|[w2,v2]| - |[(b `1),q]|).| < q by A12, XXREAL_0:2;
then A14: |[w2,v2]| in Ball (|[(b `1),q]|,q) by TOPREAL9:7;
then A15: |[w2,v2]| in U by A7, XBOOLE_0:def_3;
A16: Ball (|[(b `1),q]|,q) misses y=0-line by Th21;
 Ball (|[(b `1),q]|,q) c= y>=0-plane by Th20;
then A17: Ball (|[(b `1),q]|,q) c= y>=0-plane \ y=0-line by A16, XBOOLE_1:86;
A18: v1 in RAT by RAT_1:def_2;
 w1 in RAT by RAT_1:def_2;
then  |[w1,v1]| in product <*RAT,RAT*> by A18, FINSEQ_3:124;
then A19: |[w1,v1]| in A by A17, A9, A1, XBOOLE_0:def_4;
A20: ( s = |[w1,v1]| or s <> |[w1,v1]| ) ;
A21: v2 in RAT by RAT_1:def_2;
 w2 in RAT by RAT_1:def_2;
then  |[w2,v2]| in product <*RAT,RAT*> by A21, FINSEQ_3:124;
then A22: |[w2,v2]| in A by A17, A14, A1, XBOOLE_0:def_4;
 |[w2,v2]| <> |[w1,v1]| by A13, TOPREAL9:7;
then  ( |[w1,v1]| in A \ {s} or |[w2,v2]| in A \ {s} ) by A20, A19, A22, ZFMISC_1:56;
hence  U meets A \ {s} by A11, A15, XBOOLE_0:3; ::_thesis: verum
end;
supposeA23: b `2 > 0 ; ::_thesis: U meets A \ {s}
then  BB . a =  {  ((Ball (|[(b `1),(b `2)]|,q)) /\ y>=0-plane) where q is Element of REAL : q > 0  }  by A3, A4;
then consider q being Real such that
A24: U = (Ball (b,q)) /\ y>=0-plane and
A25: q > 0 by A4, A5;
reconsider q = q, b2 = b `2 as positive Real by A23, A25;
reconsider q1 = min (q,b2) as positive Real by XXREAL_0:def_9;
consider w1, v1 being rational number  such that
A26: |[w1,v1]| in Ball (b,q1) and
A27: |[w1,v1]| <> b by A4, Th31;
A28: v1 in RAT by RAT_1:def_2;
set q2 = |.(|[w1,v1]| - b).|;
 |[w1,v1]| - b <> 0. (TOP-REAL 2) by A27, EUCLID:43;
then  |.(|[w1,v1]| - b).| <> 0 by EUCLID_2:42;
then reconsider q2 = |.(|[w1,v1]| - b).| as real positive number ;
A29: q2 < q1 by A26, TOPREAL9:7;
A30: q1 <= b `2 by XXREAL_0:17;
then A31: Ball (b,q1) c= y>=0-plane by A4, Th20;
 Ball (b,q1) misses y=0-line by A30, A4, Th21;
then A32: Ball (b,q1) c= y>=0-plane \ y=0-line by A31, XBOOLE_1:86;
 w1 in RAT by RAT_1:def_2;
then  |[w1,v1]| in product <*RAT,RAT*> by A28, FINSEQ_3:124;
then A33: |[w1,v1]| in A by A32, A26, A1, XBOOLE_0:def_4;
A34: ( s = |[w1,v1]| or s <> |[w1,v1]| ) ;
consider w2, v2 being rational number  such that
A35: |[w2,v2]| in Ball (b,q2) and
 |[w2,v2]| <> b by A4, Th31;
A36: |[w2,v2]| <> |[w1,v1]| by A35, TOPREAL9:7;
 |.(|[w2,v2]| - b).| < q2 by A35, TOPREAL9:7;
then A37: |.(|[w2,v2]| - b).| < q1 by A29, XXREAL_0:2;
then A38: |[w2,v2]| in Ball (b,q1) by TOPREAL9:7;
A39: q1 <= q by XXREAL_0:17;
then  |.(|[w2,v2]| - b).| < q by A37, XXREAL_0:2;
then  |[w2,v2]| in Ball (b,q) by TOPREAL9:7;
then A40: |[w2,v2]| in U by A24, A38, A31, XBOOLE_0:def_4;
A41: v2 in RAT by RAT_1:def_2;
 w2 in RAT by RAT_1:def_2;
then  |[w2,v2]| in product <*RAT,RAT*> by A41, FINSEQ_3:124;
then  |[w2,v2]| in A by A32, A38, A1, XBOOLE_0:def_4;
then A42: ( |[w1,v1]| in A \ {s} or |[w2,v2]| in A \ {s} ) by A34, A36, A33, ZFMISC_1:56;
 q2 < q by A39, A29, XXREAL_0:2;
then  |[w1,v1]| in Ball (b,q) by TOPREAL9:7;
then  |[w1,v1]| in U by A24, A26, A31, XBOOLE_0:def_4;
hence  U meets A \ {s} by A42, A40, XBOOLE_0:3; ::_thesis: verum
end;
end;
end;
hence  x in Cl (A \ {s}) by TOPGEN_2:10; ::_thesis: verum
end;

theorem Th33: :: TOPGEN_5:33
for A being Subset of Niemytzki-plane st A = y>=0-plane \ y=0-line holds
for x being set holds Cl (A \ {x}) = [#] Niemytzki-plane
proof
let A be Subset of Niemytzki-plane; ::_thesis: ( A = y>=0-plane \ y=0-line implies for x being set holds Cl (A \ {x}) = [#] Niemytzki-plane )
assume A1: A = y>=0-plane \ y=0-line ; ::_thesis: for x being set holds Cl (A \ {x}) = [#] Niemytzki-plane
let s be set ; ::_thesis: Cl (A \ {s}) = [#] Niemytzki-plane
reconsider B = A /\ (product <*RAT,RAT*>) as Subset of Niemytzki-plane ;
thus  Cl (A \ {s}) c= [#] Niemytzki-plane ; :: according to XBOOLE_0:def_10 ::_thesis: [#] Niemytzki-plane c= Cl (A \ {s})
 B \ {s} c= A \ {s} by XBOOLE_1:17, XBOOLE_1:33;
then  Cl (B \ {s}) c= Cl (A \ {s}) by PRE_TOPC:19;
hence  [#] Niemytzki-plane c= Cl (A \ {s}) by A1, Th32; ::_thesis: verum
end;

theorem Th34: :: TOPGEN_5:34
for A being Subset of Niemytzki-plane st A = y>=0-plane \ y=0-line holds
Cl A = [#] Niemytzki-plane
proof
let A be Subset of Niemytzki-plane; ::_thesis: ( A = y>=0-plane \ y=0-line implies Cl A = [#] Niemytzki-plane )
 A \ {A} = A
proof
thus  A \ {A} c= A by XBOOLE_1:36; :: according to XBOOLE_0:def_10 ::_thesis: A c= A \ {A}
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in A or x in A \ {A} )
 not A in A ;
hence  ( not x in A or x in A \ {A} ) by ZFMISC_1:56; ::_thesis: verum
end;
hence  ( A = y>=0-plane \ y=0-line implies Cl A = [#] Niemytzki-plane ) by Th33; ::_thesis: verum
end;

theorem Th35: :: TOPGEN_5:35
for A being Subset of Niemytzki-plane st A = y=0-line holds
( Cl A = A & Int A = {} )
proof
let A be Subset of Niemytzki-plane; ::_thesis: ( A = y=0-line implies ( Cl A = A & Int A = {} ) )
assume A1: A = y=0-line ; ::_thesis: ( Cl A = A & Int A = {} )
then A2: A ` = y>=0-plane \ y=0-line by Def3;
thus  Cl A = A by A1, Th26, PRE_TOPC:22; ::_thesis: Int A = {}
thus  Int A = (Cl (A `)) ` by TOPS_1:def_1
.= ([#] Niemytzki-plane) ` by A2, Th34
.= {} by XBOOLE_1:37 ; ::_thesis: verum
end;

theorem Th36: :: TOPGEN_5:36
(y>=0-plane \ y=0-line) /\ (product <*RAT,RAT*>) is dense Subset of Niemytzki-plane
proof
 (y>=0-plane \ y=0-line) /\ (product <*RAT,RAT*>) c= y>=0-plane \ y=0-line by XBOOLE_1:17;
then reconsider A = (y>=0-plane \ y=0-line) /\ (product <*RAT,RAT*>) as Subset of Niemytzki-plane by Th25, XBOOLE_1:1;
 A \ {A} = A
proof
thus  A \ {A} c= A by XBOOLE_1:36; :: according to XBOOLE_0:def_10 ::_thesis: A c= A \ {A}
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in A or x in A \ {A} )
 not A in A ;
hence  ( not x in A or x in A \ {A} ) by ZFMISC_1:56; ::_thesis: verum
end;
then  Cl A = [#] Niemytzki-plane by Th32;
hence  (y>=0-plane \ y=0-line) /\ (product <*RAT,RAT*>) is dense Subset of Niemytzki-plane by TOPS_1:def_3; ::_thesis: verum
end;

theorem :: TOPGEN_5:37
(y>=0-plane \ y=0-line) /\ (product <*RAT,RAT*>) is dense-in-itself Subset of Niemytzki-plane
proof
 (y>=0-plane \ y=0-line) /\ (product <*RAT,RAT*>) c= y>=0-plane \ y=0-line by XBOOLE_1:17;
then reconsider A = (y>=0-plane \ y=0-line) /\ (product <*RAT,RAT*>) as Subset of Niemytzki-plane by Th25, XBOOLE_1:1;
 A is dense-in-itself
proof
let a be set ; :: according to TARSKI:def_3,TOPGEN_1:def_7 ::_thesis: ( not a in A or a in Der A )
assume  a in A ; ::_thesis: a in Der A
then reconsider x = a as Point of Niemytzki-plane ;
 Cl (A \ {x}) = the carrier of Niemytzki-plane by Th32;
then  x is_an_accumulation_point_of A by TOPGEN_1:def_2;
hence  a in Der A by TOPGEN_1:def_3; ::_thesis: verum
end;
hence  (y>=0-plane \ y=0-line) /\ (product <*RAT,RAT*>) is dense-in-itself Subset of Niemytzki-plane ; ::_thesis: verum
end;

theorem :: TOPGEN_5:38
y>=0-plane \ y=0-line is dense Subset of Niemytzki-plane
proof
reconsider A = y>=0-plane \ y=0-line as open Subset of Niemytzki-plane by Th25;
 Cl A = [#] Niemytzki-plane by Th34;
hence  y>=0-plane \ y=0-line is dense Subset of Niemytzki-plane by TOPS_1:def_3; ::_thesis: verum
end;

theorem :: TOPGEN_5:39
y>=0-plane \ y=0-line is dense-in-itself Subset of Niemytzki-plane
proof
 the carrier of Niemytzki-plane = y>=0-plane by Def3;
then reconsider A = y>=0-plane \ y=0-line as Subset of Niemytzki-plane by XBOOLE_1:36;
 A is dense-in-itself
proof
let a be set ; :: according to TARSKI:def_3,TOPGEN_1:def_7 ::_thesis: ( not a in A or a in Der A )
assume  a in A ; ::_thesis: a in Der A
then reconsider x = a as Point of Niemytzki-plane ;
 Cl (A \ {x}) = the carrier of Niemytzki-plane by Th33;
then  x is_an_accumulation_point_of A by TOPGEN_1:def_2;
hence  a in Der A by TOPGEN_1:def_3; ::_thesis: verum
end;
hence  y>=0-plane \ y=0-line is dense-in-itself Subset of Niemytzki-plane ; ::_thesis: verum
end;

theorem :: TOPGEN_5:40
y=0-line is nowhere_dense Subset of Niemytzki-plane
proof
reconsider A = y=0-line as Subset of Niemytzki-plane by Def3, Th19;
Int (Cl A) = Int A by Th26, PRE_TOPC:22
.= {} by Th35 ;
hence  y=0-line is nowhere_dense Subset of Niemytzki-plane by TOPS_3:def_3; ::_thesis: verum
end;

theorem Th41: :: TOPGEN_5:41
for A being Subset of Niemytzki-plane st A = y=0-line holds
Der A is empty
proof
consider BB being Neighborhood_System of Niemytzki-plane  such that
A1: for x being Element of REAL holds BB . |[x,0]| =  {  ((Ball (|[x,r]|,r)) \/ {|[x,0]|}) where r is Element of REAL : r > 0  }  and
 for x, y being Element of REAL st y > 0 holds
BB . |[x,y]| =  {  ((Ball (|[x,y]|,r)) /\ y>=0-plane) where r is Element of REAL : r > 0  }  by Def3;
let A be Subset of Niemytzki-plane; ::_thesis: ( A = y=0-line implies Der A is empty )
assume that
A2: A = y=0-line and
A3: not Der A is empty ; ::_thesis: contradiction
set a = the Element of Der A;
 the Element of Der A in Der A by A3;
then reconsider a = the Element of Der A as Point of Niemytzki-plane ;
A4: a in Der A by A3;
 a is_an_accumulation_point_of A by A3, TOPGEN_1:def_3;
then A5: a in Cl (A \ {a}) by TOPGEN_1:def_2;
 the carrier of Niemytzki-plane = y>=0-plane by Def3;
then  a in y>=0-plane ;
then reconsider b = a as Point of (TOP-REAL 2) ;
A6: a = |[(b `1),(b `2)]| by EUCLID:53;
A7: Der A c= Cl A by TOPGEN_1:28;
 Cl A = A by A2, Th35;
then A8: b `2 = 0 by A4, A7, A2, A6, Th15;
then  BB . a =  {  ((Ball (|[(b `1),r]|,r)) \/ {|[(b `1),0]|}) where r is Element of REAL : r > 0  }  by A1, A6;
then  (Ball (|[(b `1),1]|,1)) \/ {b} in BB . a by A6, A8;
then  (Ball (|[(b `1),1]|,1)) \/ {b} meets A \ {a} by A5, TOPGEN_2:9;
then consider z being set  such that
A9: z in (Ball (|[(b `1),1]|,1)) \/ {b} and
A10: z in A \ {a} by XBOOLE_0:3;
A11: z in A by A10, ZFMISC_1:56;
 z <> a by A10, ZFMISC_1:56;
then A12: z in Ball (|[(b `1),1]|,1) by A9, ZFMISC_1:136;
reconsider z = z as Point of (TOP-REAL 2) by A9;
A13: z = |[(z `1),(z `2)]| by EUCLID:53;
then  z `2 = 0 by A2, A11, Th15;
then A14: z - |[(b `1),1]| = |[((z `1) - (b `1)),(0 - 1)]| by A13, EUCLID:62;
A15: |[((z `1) - (b `1)),(0 - 1)]| `2 = 0 - 1 by EUCLID:52;
 |[((z `1) - (b `1)),(0 - 1)]| `1 = (z `1) - (b `1) by EUCLID:52;
then |.(z - |[(b `1),1]|).| = sqrt ((((z `1) - (b `1)) ^2) + ((- 1) ^2)) by A14, A15, JGRAPH_1:30
.= sqrt ((((z `1) - (b `1)) ^2) + (1 ^2)) ;
then A16: |.(z - |[(b `1),1]|).| >= abs 1 by COMPLEX1:79;
 |.(z - |[(b `1),1]|).| < 1 by A12, TOPREAL9:7;
then  abs 1 < 1 by A16, XXREAL_0:2;
hence  contradiction by ABSVALUE:4; ::_thesis: verum
end;

theorem Th42: :: TOPGEN_5:42
for A being Subset of y=0-line holds A is closed Subset of Niemytzki-plane
proof
reconsider B = y=0-line as closed Subset of Niemytzki-plane by Th26;
let A be Subset of y=0-line; ::_thesis: A is closed Subset of Niemytzki-plane
 A c= B ;
then reconsider A = A as Subset of Niemytzki-plane by XBOOLE_1:1;
 Der A c= Der B by TOPGEN_1:30;
then  Der A c= {} by Th41;
then  Der A = {} ;
then  Cl A = A \/ {} by TOPGEN_1:29;
hence  A is closed Subset of Niemytzki-plane ; ::_thesis: verum
end;

theorem Th43: :: TOPGEN_5:43
RAT is dense Subset of Sorgenfrey-line
proof
reconsider A = RAT as Subset of Sorgenfrey-line by NUMBERS:12, TOPGEN_3:def_2;
consider B being Subset-Family of REAL such that
A1: the topology of Sorgenfrey-line = UniCl B and
A2: B =  {  [.x,q.[ where x, q is Element of REAL : ( x < q & q is rational )  }  by TOPGEN_3:def_2;
 the carrier of Sorgenfrey-line = REAL by TOPGEN_3:def_2;
then A3: B is Basis of Sorgenfrey-line by A1, YELLOW_9:22;
 A is dense
proof
thus  Cl A c= the carrier of Sorgenfrey-line ; :: according to XBOOLE_0:def_10,TOPS_3:def_2 ::_thesis: the carrier of Sorgenfrey-line c= Cl A
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of Sorgenfrey-line or x in Cl A )
assume  x in the carrier of Sorgenfrey-line ; ::_thesis: x in Cl A
then reconsider x = x as Point of Sorgenfrey-line ;
now__::_thesis:_for_C_being_Subset_of_Sorgenfrey-line_st_C_in_B_&_x_in_C_holds_
A_meets_C
let C be Subset of Sorgenfrey-line; ::_thesis: ( C in B & x in C implies A meets C )
assume  C in B ; ::_thesis: ( x in C implies A meets C )
then consider y, q being Real such that
A4: C = [.y,q.[ and
A5: y < q and
 q is rational by A2;
assume  x in C ; ::_thesis: A meets C
consider r being rational set  such that
A6: y < r and
A7: r < q by A5, RAT_1:7;
A8: r in A by RAT_1:def_2;
 r in C by A4, A6, A7, XXREAL_1:3;
hence  A meets C by A8, XBOOLE_0:3; ::_thesis: verum
end;
hence  x in Cl A by A3, YELLOW_9:37; ::_thesis: verum
end;
hence  RAT is dense Subset of Sorgenfrey-line ; ::_thesis: verum
end;

theorem :: TOPGEN_5:44
Sorgenfrey-line is separable
proof
reconsider A = RAT as dense Subset of Sorgenfrey-line by Th43;
 density Sorgenfrey-line c= card A by TOPGEN_1:def_12;
hence  density Sorgenfrey-line c= omega by TOPGEN_3:17; :: according to TOPGEN_1:def_13 ::_thesis: verum
end;

theorem :: TOPGEN_5:45
Niemytzki-plane is separable
proof
reconsider A = (y>=0-plane \ y=0-line) /\ (product <*RAT,RAT*>) as dense Subset of Niemytzki-plane by Th36;
A1: card A c= card (product <*RAT,RAT*>) by CARD_1:11, XBOOLE_1:17;
 density Niemytzki-plane c= card A by TOPGEN_1:def_12;
then  density Niemytzki-plane c= card (product <*RAT,RAT*>) by A1, XBOOLE_1:1;
hence  density Niemytzki-plane c= omega by Th8, CARD_4:6, TOPGEN_3:17; :: according to TOPGEN_1:def_13 ::_thesis: verum
end;

theorem :: TOPGEN_5:46
Niemytzki-plane is T_1
proof
set T = Niemytzki-plane ;
let p, q be Point of Niemytzki-plane; :: according to URYSOHN1:def_7 ::_thesis: ( p = q or ex b1, b2 being Element of bool the carrier of Niemytzki-plane st
( b1 is open & b2 is open & p in b1 & not q in b1 & q in b2 & not p in b2 ) )
assume A1: p <> q ; ::_thesis: ex b1, b2 being Element of bool the carrier of Niemytzki-plane st
( b1 is open & b2 is open & p in b1 & not q in b1 & q in b2 & not p in b2 )
A2: q in the carrier of Niemytzki-plane ;
A3: the carrier of Niemytzki-plane = y>=0-plane by Def3;
 p in the carrier of Niemytzki-plane ;
then reconsider p9 = p, q9 = q as Point of (TOP-REAL 2) by A2, A3;
 p9 - q9 <> 0. (TOP-REAL 2) by A1, EUCLID:43;
then  |.(p9 - q9).| <> 0 by EUCLID_2:42;
then reconsider r = |.(p9 - q9).| as positive Real ;
consider ap being Point of (TOP-REAL 2), Up being open Subset of Niemytzki-plane such that
A4: p in Up and
 ap in Up and
A5: for b being Point of (TOP-REAL 2) st b in Up holds
|.(b - ap).| < r / 2 by Th30;
consider aq being Point of (TOP-REAL 2), Uq being open Subset of Niemytzki-plane such that
A6: q in Uq and
 aq in Uq and
A7: for b being Point of (TOP-REAL 2) st b in Uq holds
|.(b - aq).| < r / 2 by Th30;
take  Up ; ::_thesis: ex b1 being Element of bool the carrier of Niemytzki-plane st
( Up is open & b1 is open & p in Up & not q in Up & q in b1 & not p in b1 )
take  Uq ; ::_thesis: ( Up is open & Uq is open & p in Up & not q in Up & q in Uq & not p in Uq )
thus  ( Up is open & Uq is open & p in Up ) by A4; ::_thesis: ( not q in Up & q in Uq & not p in Uq )
thus  not q in Up ::_thesis: ( q in Uq & not p in Uq )
proof
assume  q in Up ; ::_thesis: contradiction
then A8: |.(q9 - ap).| < r / 2 by A5;
 |.(q9 - ap).| = |.(ap - q9).| by TOPRNS_1:27;
then  |.(p9 - ap).| + |.(ap - q9).| < (r / 2) + (r / 2) by A8, A4, A5, XREAL_1:8;
hence  contradiction by TOPRNS_1:34; ::_thesis: verum
end;
thus  q in Uq by A6; ::_thesis: not p in Uq
assume A9: p in Uq ; ::_thesis: contradiction
A10: |.(q9 - aq).| = |.(aq - q9).| by TOPRNS_1:27;
 |.(q9 - aq).| < r / 2 by A6, A7;
then  |.(p9 - aq).| + |.(aq - q9).| < (r / 2) + (r / 2) by A10, A9, A7, XREAL_1:8;
hence  contradiction by TOPRNS_1:34; ::_thesis: verum
end;

theorem :: TOPGEN_5:47
not Niemytzki-plane is normal
proof
reconsider C = (y>=0-plane \ y=0-line) /\ (product <*RAT,RAT*>) as dense Subset of Niemytzki-plane by Th36;
set T = Niemytzki-plane ;
defpred S1[ set , set ] means  ex U, V being open Subset of Niemytzki-plane st
( $2 = U /\ C & $1 c= U & y=0-line \ $1 c= V & U misses V );
A1: exp (2,omega) in exp (2,(exp (2,omega))) by CARD_5:14;
 card C c= card (product <*RAT,RAT*>) by CARD_1:11, XBOOLE_1:17;
then  card C c= omega by Th8, CARD_4:6, TOPGEN_3:17;
then A2: exp (2,(card C)) c= exp (2,omega) by CARD_2:93;
assume A3: for W, V being Subset of Niemytzki-plane st W <> {} & V <> {} & W is closed & V is closed & W misses V holds
ex P, Q being Subset of Niemytzki-plane st
( P is open & Q is open & W c= P & V c= Q & P misses Q ) ; :: according to COMPTS_1:def_3 ::_thesis: contradiction
A4: for a being set st a in bool y=0-line holds
ex b being set st S1[a,b]
proof
let a be set ; ::_thesis: ( a in bool y=0-line implies ex b being set st S1[a,b] )
assume  a in bool y=0-line ; ::_thesis: ex b being set st S1[a,b]
then reconsider aa = a, a9 = y=0-line \ a as Subset of y=0-line by XBOOLE_1:36;
reconsider A = aa, B = a9 as closed Subset of Niemytzki-plane by Th42;
percases ( a = {} or a = y=0-line or ( a <> {} & a <> y=0-line ) ) ;
supposeA5: a = {} ; ::_thesis: ex b being set st S1[a,b]
take  {} ; ::_thesis: S1[a, {} ]
take  {} Niemytzki-plane ; ::_thesis: ex V being open Subset of Niemytzki-plane st
( {} = ({} Niemytzki-plane) /\ C & a c= {} Niemytzki-plane & y=0-line \ a c= V & {} Niemytzki-plane misses V )
take  [#] Niemytzki-plane ; ::_thesis: ( {} = ({} Niemytzki-plane) /\ C & a c= {} Niemytzki-plane & y=0-line \ a c= [#] Niemytzki-plane & {} Niemytzki-plane misses [#] Niemytzki-plane )
thus  ( {} = ({} Niemytzki-plane) /\ C & a c= {} Niemytzki-plane & y=0-line \ a c= [#] Niemytzki-plane & {} Niemytzki-plane misses [#] Niemytzki-plane ) by A5, Def3, Th19, XBOOLE_1:65; ::_thesis: verum
end;
supposeA6: a = y=0-line ; ::_thesis: ex b being set st S1[a,b]
take  ([#] Niemytzki-plane) /\ C ; ::_thesis: S1[a,([#] Niemytzki-plane) /\ C]
take  [#] Niemytzki-plane ; ::_thesis: ex V being open Subset of Niemytzki-plane st
( ([#] Niemytzki-plane) /\ C = ([#] Niemytzki-plane) /\ C & a c= [#] Niemytzki-plane & y=0-line \ a c= V & [#] Niemytzki-plane misses V )
take  {} Niemytzki-plane ; ::_thesis: ( ([#] Niemytzki-plane) /\ C = ([#] Niemytzki-plane) /\ C & a c= [#] Niemytzki-plane & y=0-line \ a c= {} Niemytzki-plane & [#] Niemytzki-plane misses {} Niemytzki-plane )
thus  ( ([#] Niemytzki-plane) /\ C = ([#] Niemytzki-plane) /\ C & a c= [#] Niemytzki-plane & y=0-line \ a c= {} Niemytzki-plane & [#] Niemytzki-plane misses {} Niemytzki-plane ) by A6, Def3, Th19, XBOOLE_1:37, XBOOLE_1:65; ::_thesis: verum
end;
supposeA7: ( a <> {} & a <> y=0-line ) ; ::_thesis: ex b being set st S1[a,b]
 (aa `) ` = a9 ` ;
then A8: B <> {} y=0-line by A7;
 A misses B by XBOOLE_1:79;
then consider P, Q being Subset of Niemytzki-plane such that
A9: P is open and
A10: Q is open and
A11: A c= P and
A12: B c= Q and
A13: P misses Q by A8, A3, A7;
take  P /\ C ; ::_thesis: S1[a,P /\ C]
thus  S1[a,P /\ C] by A9, A10, A11, A12, A13; ::_thesis: verum
end;
end;
end;
consider G being Function such that
A14: dom G = bool y=0-line and
A15: for a being set st a in bool y=0-line holds
S1[a,G . a] from CLASSES1:sch_1(A4);
 G is one-to-one
proof
let x, y be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not x in proj1 G or not y in proj1 G or not G . x = G . y or x = y )
assume that
A16: x in dom G and
A17: y in dom G ; ::_thesis: ( not G . x = G . y or x = y )
reconsider A = x, B = y as Subset of y=0-line by A16, A17, A14;
assume that
A18: G . x = G . y and
A19: x <> y ; ::_thesis: contradiction
consider z being set  such that
A20: ( ( z in A & not z in B ) or ( z in B & not z in A ) ) by A19, TARSKI:1;
A21: ( z in A \ B or z in B \ A ) by A20, XBOOLE_0:def_5;
consider UB, VB being open Subset of Niemytzki-plane such that
A22: G . B = UB /\ C and
A23: B c= UB and
A24: y=0-line \ B c= VB and
A25: UB misses VB by A15;
consider UA, VA being open Subset of Niemytzki-plane such that
A26: G . A = UA /\ C and
A27: A c= UA and
A28: y=0-line \ A c= VA and
A29: UA misses VA by A15;
 B \ A = B /\ (A `) by SUBSET_1:13;
then A30: B \ A c= UB /\ VA by A28, A23, XBOOLE_1:27;
 A \ B = A /\ (B `) by SUBSET_1:13;
then  A \ B c= UA /\ VB by A27, A24, XBOOLE_1:27;
then  ( C meets UA /\ VB or C meets UB /\ VA ) by A30, A21, TOPS_1:45;
then  ( ex z being set st
( z in C & z in UA /\ VB ) or ex z being set st
( z in C & z in UB /\ VA ) ) by XBOOLE_0:3;
then consider z being set  such that
A31: z in C and
A32: ( z in UA /\ VB or z in UB /\ VA ) ;
 ( ( z in UA & z in VB ) or ( z in UB & z in VA ) ) by A32, XBOOLE_0:def_4;
then  ( ( z in UA & not z in UB ) or ( z in UB & not z in UA ) ) by A29, A25, XBOOLE_0:3;
then  ( ( z in G . A & not z in G . B ) or ( z in G . B & not z in G . A ) ) by A26, A22, A31, XBOOLE_0:def_4;
hence  contradiction by A18; ::_thesis: verum
end;
then A33: card (dom G) c= card (rng G) by CARD_1:10;
 rng G c= bool C
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in rng G or a in bool C )
assume  a in rng G ; ::_thesis: a in bool C
then consider b being set  such that
A34: b in dom G and
A35: a = G . b by FUNCT_1:def_3;
 S1[b,a] by A14, A15, A34, A35;
then  a c= C by XBOOLE_1:17;
hence  a in bool C ; ::_thesis: verum
end;
then  card (rng G) c= card (bool C) by CARD_1:11;
then  card (bool y=0-line) c= card (bool C) by A33, A14, XBOOLE_1:1;
then A36: exp (2,continuum) c= card (bool C) by Th16, CARD_2:31;
 card (bool C) = exp (2,(card C)) by CARD_2:31;
then  exp (2,continuum) c= exp (2,omega) by A36, A2, XBOOLE_1:1;
then  exp (2,omega) in exp (2,omega) by A1, TOPGEN_3:29;
hence  contradiction ; ::_thesis: verum
end;

begin

definition
let T be TopSpace;
attrT is Tychonoff  means :Def4: :: TOPGEN_5:def 4
 for A being closed Subset of T
for a being Point of T st a in A ` holds
ex f being continuous Function of T,I[01] st
( f . a = 0 & f .: A c= {1} );
end;

:: deftheorem Def4 defines Tychonoff TOPGEN_5:def_4_:_
 for T being TopSpace holds
( T is Tychonoff iff for A being closed Subset of T
for a being Point of T st a in A ` holds
ex f being continuous Function of T,I[01] st
( f . a = 0 & f .: A c= {1} ) );

registration
cluster TopSpace-like Tychonoff  ->  regular for TopStruct ;
coherence
 for b1 being TopSpace st b1 is Tychonoff holds
b1 is regular
proof
reconsider z = 0 , j = 1, half = 1 / 2 as Element of I[01] by BORSUK_1:40, XXREAL_1:1;
let T be TopSpace; ::_thesis: ( T is Tychonoff implies T is regular )
assume A1: for A being closed Subset of T
for a being Point of T st a in A ` holds
ex f being continuous Function of T,I[01] st
( f . a = 0 & f .: A c= {1} ) ; :: according to TOPGEN_5:def_4 ::_thesis: T is regular
reconsider A = [.z,half.[, B = ].half,j.] as Subset of I[01] by BORSUK_1:40, XXREAL_1:35, XXREAL_1:36;
reconsider A = A, B = B as open Subset of I[01] by TOPALG_1:4, TOPALG_1:5;
let p be Point of T; :: according to PRE_TOPC:def_11 ::_thesis: for b1 being Element of bool the carrier of T holds
( not b1 is closed or not p in b1 ` or ex b2, b3 being Element of bool the carrier of T st
( b2 is open & b3 is open & p in b2 & b1 c= b3 & b2 misses b3 ) )
let P be Subset of T; ::_thesis: ( not P is closed or not p in P ` or ex b1, b2 being Element of bool the carrier of T st
( b1 is open & b2 is open & p in b1 & P c= b2 & b1 misses b2 ) )
assume that
A2: P is closed and
A3: p in P ` ; ::_thesis: ex b1, b2 being Element of bool the carrier of T st
( b1 is open & b2 is open & p in b1 & P c= b2 & b1 misses b2 )
consider f being continuous Function of T,I[01] such that
A4: f . p = 0 and
A5: f .: P c= {1} by A2, A3, A1;
take W = f " A; ::_thesis: ex b1 being Element of bool the carrier of T st
( W is open & b1 is open & p in W & P c= b1 & W misses b1 )
take V = f " B; ::_thesis: ( W is open & V is open & p in W & P c= V & W misses V )
 [#] I[01] <> {} ;
hence  ( W is open & V is open ) by TOPS_2:43; ::_thesis: ( p in W & P c= V & W misses V )
 0 in A by XXREAL_1:3;
hence  p in W by A3, A4, FUNCT_2:38; ::_thesis: ( P c= V & W misses V )
A6: dom f = the carrier of T by FUNCT_2:def_1;
 1 in B by XXREAL_1:2;
then  {1} c= B by ZFMISC_1:31;
then  f .: P c= B by A5, XBOOLE_1:1;
hence  P c= V by A6, FUNCT_1:93; ::_thesis: W misses V
assume  W meets V ; ::_thesis: contradiction
then consider x being set  such that
A7: x in W and
A8: x in V by XBOOLE_0:3;
A9: f . x in A by A7, FUNCT_1:def_7;
then reconsider fx = f . x as Element of I[01] ;
A10: fx < half by A9, XXREAL_1:3;
 f . x in B by A8, FUNCT_1:def_7;
hence  contradiction by A10, XXREAL_1:2; ::_thesis: verum
end;
cluster non empty TopSpace-like T_4  ->  non empty Tychonoff for TopStruct ;
coherence
 for b1 being non empty TopSpace st b1 is T_4 holds
b1 is Tychonoff
proof
 the carrier of (Closed-Interval-TSpace ((- 1),1)) = [.(- 1),1.] by TOPMETR:18;
then reconsider j = 1, k = - 1 as Point of (Closed-Interval-TSpace ((- 1),1)) by XXREAL_1:1;
reconsider z = 0 , o = 1 as Point of I[01] by BORSUK_1:40, XXREAL_1:1;
let T be non empty TopSpace; ::_thesis: ( T is T_4 implies T is Tychonoff )
assume A11: T is T_4 ; ::_thesis: T is Tychonoff
let A be closed Subset of T; :: according to TOPGEN_5:def_4 ::_thesis: for a being Point of T st a in A ` holds
ex f being continuous Function of T,I[01] st
( f . a = 0 & f .: A c= {1} )
let a be Point of T; ::_thesis: ( a in A ` implies ex f being continuous Function of T,I[01] st
( f . a = 0 & f .: A c= {1} ) )
percases ( A is empty or not A is empty ) ;
supposeA12: A is empty ; ::_thesis: ( a in A ` implies ex f being continuous Function of T,I[01] st
( f . a = 0 & f .: A c= {1} ) )
set f = T --> z;
A13: (T --> z) . a = z by FUNCOP_1:7;
 (T --> z) .: A = {} by A12;
hence  ( a in A ` implies ex f being continuous Function of T,I[01] st
( f . a = 0 & f .: A c= {1} ) ) by A13, XBOOLE_1:2; ::_thesis: verum
end;
suppose not A is empty ; ::_thesis: ( a in A ` implies ex f being continuous Function of T,I[01] st
( f . a = 0 & f .: A c= {1} ) )
then reconsider aa = {a}, A9 = A as non empty closed Subset of T by A11, URYSOHN1:19;
reconsider B = A9 \/ aa as closed Subset of T ;
set h = ((T | A9) --> j) +* ((T | aa) --> k);
A14: (T | aa) --> k = aa --> k by PRE_TOPC:8;
A15: (A9 --> 1) .: A c= {1} by FUNCOP_1:81;
A16: a in aa by TARSKI:def_1;
then A17: a in B by XBOOLE_0:def_3;
assume  a in A ` ; ::_thesis: ex f being continuous Function of T,I[01] st
( f . a = 0 & f .: A c= {1} )
then A18: not a in A by XBOOLE_0:def_5;
then  A9 misses aa by ZFMISC_1:50;
then reconsider h = ((T | A9) --> j) +* ((T | aa) --> k) as continuous Function of (T | B),(Closed-Interval-TSpace ((- 1),1)) by Th11;
consider g being continuous Function of T,(Closed-Interval-TSpace ((- 1),1)) such that
A19: g | B = h by A11, TIETZE:23;
consider p being Function of I[01],(Closed-Interval-TSpace ((- 1),1)) such that
A20: p is being_homeomorphism and
 for r being Real st r in [.0,1.] holds
p . r = (2 * r) - 1 and
A21: p . 0 = - 1 and
A22: p . 1 = 1 by JGRAPH_5:39;
reconsider p9 = p /" as continuous Function of (Closed-Interval-TSpace ((- 1),1)),I[01] by A20, TOPS_2:def_5;
A23: p9 = p " by A20, TOPS_2:def_4;
then A24: p9 . k = z by A20, A21, FUNCT_2:26;
A25: the carrier of (T | aa) = aa by PRE_TOPC:8;
then  dom ((T | aa) --> k) = aa by FUNCOP_1:13;
then A26: h . a = ((T | aa) --> k) . a by A16, FUNCT_4:13
.= - 1 by A25, A16, FUNCOP_1:7 ;
reconsider f = p9 * g as continuous Function of T,I[01] ;
A27: f .: A = p9 .: (g .: A) by RELAT_1:126
.= p9 .: (h .: A) by A19, RELAT_1:129, XBOOLE_1:7 ;
 (T | A9) --> j = A9 --> 1 by PRE_TOPC:8;
then  h .: A c= {1} by A14, A15, A18, BOOLMARK:3, ZFMISC_1:50;
then A28: f .: A c= Im (p9,1) by A27, RELAT_1:123;
take  f ; ::_thesis: ( f . a = 0 & f .: A c= {1} )
thus f . a = p9 . (g . a) by FUNCT_2:15
.= 0 by A26, A17, A19, A24, FUNCT_1:49 ; ::_thesis: f .: A c= {1}
 p9 . j = o by A20, A22, A23, FUNCT_2:26;
hence  f .: A c= {1} by A28, SETWISEO:8; ::_thesis: verum
end;
end;
end;
end;

theorem :: TOPGEN_5:48
for X being T_1 TopSpace st X is Tychonoff holds
for B being prebasis of X
for x being Point of X
for V being Subset of X st x in V & V in B holds
ex f being continuous Function of X,I[01] st
( f . x = 0 & f .: (V `) c= {1} )
proof
let X be T_1 TopSpace; ::_thesis: ( X is Tychonoff implies for B being prebasis of X
for x being Point of X
for V being Subset of X st x in V & V in B holds
ex f being continuous Function of X,I[01] st
( f . x = 0 & f .: (V `) c= {1} ) )
assume A1: X is Tychonoff ; ::_thesis: for B being prebasis of X
for x being Point of X
for V being Subset of X st x in V & V in B holds
ex f being continuous Function of X,I[01] st
( f . x = 0 & f .: (V `) c= {1} )
let B be prebasis of X; ::_thesis: for x being Point of X
for V being Subset of X st x in V & V in B holds
ex f being continuous Function of X,I[01] st
( f . x = 0 & f .: (V `) c= {1} )
let x be Point of X; ::_thesis: for V being Subset of X st x in V & V in B holds
ex f being continuous Function of X,I[01] st
( f . x = 0 & f .: (V `) c= {1} )
let V be Subset of X; ::_thesis: ( x in V & V in B implies ex f being continuous Function of X,I[01] st
( f . x = 0 & f .: (V `) c= {1} ) )
assume that
A2: x in V and
A3: V in B ; ::_thesis: ex f being continuous Function of X,I[01] st
( f . x = 0 & f .: (V `) c= {1} )
A4: (V `) ` = V ;
 V is open by A3, TOPS_2:def_1;
hence  ex f being continuous Function of X,I[01] st
( f . x = 0 & f .: (V `) c= {1} ) by A4, A1, A2, Def4; ::_thesis: verum
end;

theorem Th49: :: TOPGEN_5:49
for X being TopSpace
for R being non empty SubSpace of R^1
for f, g being continuous Function of X,R
for A being Subset of X st ( for x being Point of X holds
( x in A iff f . x <= g . x ) ) holds
A is closed
proof
let X be TopSpace; ::_thesis: for R being non empty SubSpace of R^1
for f, g being continuous Function of X,R
for A being Subset of X st ( for x being Point of X holds
( x in A iff f . x <= g . x ) ) holds
A is closed
let R be non empty SubSpace of R^1 ; ::_thesis: for f, g being continuous Function of X,R
for A being Subset of X st ( for x being Point of X holds
( x in A iff f . x <= g . x ) ) holds
A is closed
let f, g be continuous Function of X,R; ::_thesis: for A being Subset of X st ( for x being Point of X holds
( x in A iff f . x <= g . x ) ) holds
A is closed
let A be Subset of X; ::_thesis: ( ( for x being Point of X holds
( x in A iff f . x <= g . x ) ) implies A is closed )
assume A1: for x being Point of X holds
( x in A iff f . x <= g . x ) ; ::_thesis: A is closed
now__::_thesis:_(_the_topology_of_X_is_Basis_of_X_&_(_for_p_being_Point_of_X_st_p_in_A_`_holds_
ex_B_being_Element_of_bool_the_carrier_of_X_st_
(_B_in_the_topology_of_X_&_p_in_B_&_B_c=_A_`_)_)_)
thus  the topology of X is Basis of X by CANTOR_1:2; ::_thesis: for p being Point of X st p in A ` holds
ex B being Element of bool the carrier of X st
( B in the topology of X & p in B & B c= A ` )
let p be Point of X; ::_thesis: ( p in A ` implies ex B being Element of bool the carrier of X st
( B in the topology of X & p in B & B c= A ` ) )
set r = (f . p) - (g . p);
reconsider U1 = ].((f . p) - (((f . p) - (g . p)) / 2)),((f . p) + (((f . p) - (g . p)) / 2)).[, V1 = ].((g . p) - (((f . p) - (g . p)) / 2)),((g . p) + (((f . p) - (g . p)) / 2)).[ as open Subset of R^1 by JORDAN6:35, TOPMETR:17;
reconsider U = U1 /\ ([#] R), V = V1 /\ ([#] R) as open Subset of R by TOPS_2:24;
A2: g " V is open by TOPS_2:43;
assume A3: p in A ` ; ::_thesis: ex B being Element of bool the carrier of X st
( B in the topology of X & p in B & B c= A ` )
then A4: f . p in [#] R by FUNCT_2:5;
 not p in A by A3, XBOOLE_0:def_5;
then  f . p > g . p by A1;
then reconsider r = (f . p) - (g . p) as real positive number by XREAL_1:50;
A5: f . p < (f . p) + (r / 2) by XREAL_1:29;
take B = (f " U) /\ (g " V); ::_thesis: ( B in the topology of X & p in B & B c= A ` )
A6: g . p < (g . p) + (r / 2) by XREAL_1:29;
A7: g . p in [#] R by A3, FUNCT_2:5;
 (g . p) - (r / 2) < g . p by XREAL_1:44;
then  g . p in V1 by A6, XXREAL_1:4;
then  g . p in V by A7, XBOOLE_0:def_4;
then A8: p in g " V by A3, FUNCT_2:38;
 (f . p) - (r / 2) < f . p by XREAL_1:44;
then  f . p in U1 by A5, XXREAL_1:4;
then  f . p in U by A4, XBOOLE_0:def_4;
then A9: p in f " U by A3, FUNCT_2:38;
 f " U is open by TOPS_2:43;
hence  ( B in the topology of X & p in B ) by A9, A8, A2, PRE_TOPC:def_2, XBOOLE_0:def_4; ::_thesis: B c= A `
thus  B c= A ` ::_thesis: verum
proof
let q be set ; :: according to TARSKI:def_3 ::_thesis: ( not q in B or q in A ` )
assume A10: q in B ; ::_thesis: q in A `
then  q in g " V by XBOOLE_0:def_4;
then  g . q in V by FUNCT_2:38;
then  g . q in V1 by XBOOLE_0:def_4;
then A11: g . q < (g . p) + (r / 2) by XXREAL_1:4;
 q in f " U by A10, XBOOLE_0:def_4;
then  f . q in U by FUNCT_2:38;
then  f . q in U1 by XBOOLE_0:def_4;
then  f . q > (f . p) - (r / 2) by XXREAL_1:4;
then  g . q < f . q by A11, XXREAL_0:2;
then  not q in A by A1;
hence  q in A ` by A10, SUBSET_1:29; ::_thesis: verum
end;
end;
then  A ` is open by YELLOW_9:31;
hence  A is closed by TOPS_1:3; ::_thesis: verum
end;

theorem Th50: :: TOPGEN_5:50
for X being TopSpace
for R being non empty SubSpace of R^1
for f, g being continuous Function of X,R ex h being continuous Function of X,R st
for x being Point of X holds h . x = max ((f . x),(g . x))
proof
let X be TopSpace; ::_thesis: for R being non empty SubSpace of R^1
for f, g being continuous Function of X,R ex h being continuous Function of X,R st
for x being Point of X holds h . x = max ((f . x),(g . x))
let R be non empty SubSpace of R^1 ; ::_thesis: for f, g being continuous Function of X,R ex h being continuous Function of X,R st
for x being Point of X holds h . x = max ((f . x),(g . x))
let f, g be continuous Function of X,R; ::_thesis: ex h being continuous Function of X,R st
for x being Point of X holds h . x = max ((f . x),(g . x))
defpred S1[ set ] means f . $1 >= g . $1;
consider A being Subset of X such that
A1: for a being set holds
( a in A iff ( a in the carrier of X & S1[a] ) ) from SUBSET_1:sch_1();
defpred S2[ set ] means f . $1 <= g . $1;
consider B being Subset of X such that
A2: for a being set holds
( a in B iff ( a in the carrier of X & S2[a] ) ) from SUBSET_1:sch_1();
percases ( X is empty or ( not X is empty & A is empty ) or ( not X is empty & B is empty ) or ( not X is empty & not A is empty & not B is empty ) ) ;
supposeA3: X is empty ; ::_thesis: ex h being continuous Function of X,R st
for x being Point of X holds h . x = max ((f . x),(g . x))
set h = the continuous Function of X,R;
take  the continuous Function of X,R ; ::_thesis: for x being Point of X holds the continuous Function of X,R . x = max ((f . x),(g . x))
let x be Point of X; ::_thesis: the continuous Function of X,R . x = max ((f . x),(g . x))
A4: f . x = 0 by A3;
A5: g . x = 0 by A3;
thus  the continuous Function of X,R . x = max ((f . x),(g . x)) by A3, A4, A5; ::_thesis: verum
end;
supposeA6: ( not X is empty & A is empty ) ; ::_thesis: ex h being continuous Function of X,R st
for x being Point of X holds h . x = max ((f . x),(g . x))
take  g ; ::_thesis: for x being Point of X holds g . x = max ((f . x),(g . x))
let x be Point of X; ::_thesis: g . x = max ((f . x),(g . x))
 f . x < g . x by A6, A1;
hence  g . x = max ((f . x),(g . x)) by XXREAL_0:def_10; ::_thesis: verum
end;
supposeA7: ( not X is empty & B is empty ) ; ::_thesis: ex h being continuous Function of X,R st
for x being Point of X holds h . x = max ((f . x),(g . x))
take  f ; ::_thesis: for x being Point of X holds f . x = max ((f . x),(g . x))
let x be Point of X; ::_thesis: f . x = max ((f . x),(g . x))
 g . x < f . x by A7, A2;
hence  f . x = max ((f . x),(g . x)) by XXREAL_0:def_10; ::_thesis: verum
end;
supposeA8: ( not X is empty & not A is empty & not B is empty ) ; ::_thesis: ex h being continuous Function of X,R st
for x being Point of X holds h . x = max ((f . x),(g . x))
then reconsider X9 = X as non empty TopSpace ;
 for x being Point of X9 holds
( ( x in A implies f . x >= g . x ) & ( f . x >= g . x implies x in A ) & ( x in B implies f . x <= g . x ) & ( f . x <= g . x implies x in B ) ) by A1, A2;
then reconsider A9 = A, B9 = B as non empty closed Subset of X9 by A8, Th49;
reconsider ff = f, gg = g as continuous Function of X9,R ;
A9: the carrier of (X9 | A9) = [#] (X9 | A9)
.= A9 by PRE_TOPC:def_5 ;
A10: dom ff = the carrier of X9 by FUNCT_2:def_1;
then  dom (ff | A9) = A9 by RELAT_1:62;
then reconsider f9 = ff | A9 as continuous Function of (X9 | A9),R by A9, FUNCT_2:def_1, RELSET_1:18, TOPMETR:7;
A11: the carrier of (X9 | B9) = [#] (X9 | B9)
.= B9 by PRE_TOPC:def_5 ;
A12: dom gg = the carrier of X9 by FUNCT_2:def_1;
then  dom (gg | B9) = B9 by RELAT_1:62;
then reconsider g9 = gg | B9 as continuous Function of (X9 | B9),R by A11, FUNCT_2:def_1, RELSET_1:18, TOPMETR:7;
A13: dom g9 = B by A12, RELAT_1:62;
A14: A9 \/ B9 = the carrier of X9
proof
thus  A9 \/ B9 c= the carrier of X9 ; :: according to XBOOLE_0:def_10 ::_thesis: the carrier of X9 c= A9 \/ B9
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in the carrier of X9 or a in A9 \/ B9 )
 ( f . a >= g . a or f . a <= g . a ) ;
then  ( not a in the carrier of X or a in A9 or a in B9 ) by A1, A2;
hence  ( not a in the carrier of X9 or a in A9 \/ B9 ) by XBOOLE_0:def_3; ::_thesis: verum
end;
then A15: X9 | (A9 \/ B9) = X9 | ([#] X9)
.= TopStruct(# the carrier of X, the topology of X #) by TSEP_1:93 ;
A16: TopStruct(# the carrier of R, the topology of R #) = TopStruct(# the carrier of R, the topology of R #) ;
A17: dom f9 = A by A10, RELAT_1:62;
A18: f9 tolerates g9
proof
let a be set ; :: according to PARTFUN1:def_4 ::_thesis: ( not a in (proj1 f9) /\ (proj1 g9) or f9 . a = g9 . a )
assume A19: a in (dom f9) /\ (dom g9) ; ::_thesis: f9 . a = g9 . a
then A20: a in A by A17, XBOOLE_0:def_4;
then A21: f . a >= g . a by A1;
A22: a in B by A19, A13, XBOOLE_0:def_4;
then  f . a <= g . a by A2;
then  f . a = g . a by A21, XXREAL_0:1;
hence f9 . a = g . a by A20, FUNCT_1:49
.= g9 . a by A22, FUNCT_1:49 ;
::_thesis: verum
end;
then  f9 +* g9 is continuous Function of (X9 | (A9 \/ B9)),R by Th10;
then reconsider h = f9 +* g9 as continuous Function of X,R by A16, A15, YELLOW12:36;
take  h ; ::_thesis: for x being Point of X holds h . x = max ((f . x),(g . x))
let x be Point of X; ::_thesis: h . x = max ((f . x),(g . x))
 ( x in A9 or x in B9 ) by A14, XBOOLE_0:def_3;
then  ( ( x in A9 & f . x >= g . x & h . x = f9 . x & f . x = f9 . x ) or ( x in B9 & f . x <= g . x & h . x = g9 . x & g . x = g9 . x ) ) by A1, A17, A13, A18, FUNCT_1:49, FUNCT_4:13, FUNCT_4:15;
hence  h . x = max ((f . x),(g . x)) by XXREAL_0:def_10; ::_thesis: verum
end;
end;
end;

theorem Th51: :: TOPGEN_5:51
for X being non empty TopSpace
for R being non empty SubSpace of R^1
for A being non empty finite set
for F being ManySortedFunction of A st ( for a being set st a in A holds
F . a is continuous Function of X,R ) holds
ex f being continuous Function of X,R st
for x being Point of X
for S being non empty finite Subset of REAL st S = rng ((commute F) . x) holds
f . x = max S
proof
let X be non empty TopSpace; ::_thesis: for R being non empty SubSpace of R^1
for A being non empty finite set
for F being ManySortedFunction of A st ( for a being set st a in A holds
F . a is continuous Function of X,R ) holds
ex f being continuous Function of X,R st
for x being Point of X
for S being non empty finite Subset of REAL st S = rng ((commute F) . x) holds
f . x = max S
let R be non empty SubSpace of R^1 ; ::_thesis: for A being non empty finite set
for F being ManySortedFunction of A st ( for a being set st a in A holds
F . a is continuous Function of X,R ) holds
ex f being continuous Function of X,R st
for x being Point of X
for S being non empty finite Subset of REAL st S = rng ((commute F) . x) holds
f . x = max S
let A be non empty finite set ; ::_thesis: for F being ManySortedFunction of A st ( for a being set st a in A holds
F . a is continuous Function of X,R ) holds
ex f being continuous Function of X,R st
for x being Point of X
for S being non empty finite Subset of REAL st S = rng ((commute F) . x) holds
f . x = max S
let F be ManySortedFunction of A; ::_thesis: ( ( for a being set st a in A holds
F . a is continuous Function of X,R ) implies ex f being continuous Function of X,R st
for x being Point of X
for S being non empty finite Subset of REAL st S = rng ((commute F) . x) holds
f . x = max S )
defpred S1[ set ] means ( $1 is empty or ex f being continuous Function of X,R st
for x being Point of X
for S being non empty finite Subset of REAL st S = rng ((commute (F | $1)) . x) holds
f . x = max S );
A1: S1[ {} ] ;
A2: dom F = A by PARTFUN1:def_2;
assume A3: for a being set st a in A holds
F . a is continuous Function of X,R ; ::_thesis: ex f being continuous Function of X,R st
for x being Point of X
for S being non empty finite Subset of REAL st S = rng ((commute F) . x) holds
f . x = max S
 rng F c= Funcs ( the carrier of X, the carrier of R)
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in rng F or a in Funcs ( the carrier of X, the carrier of R) )
assume  a in rng F ; ::_thesis: a in Funcs ( the carrier of X, the carrier of R)
then  ex b being set st
( b in dom F & a = F . b ) by FUNCT_1:def_3;
then  a is Function of X,R by A3;
hence  a in Funcs ( the carrier of X, the carrier of R) by FUNCT_2:8; ::_thesis: verum
end;
then A4: F in Funcs (A,(Funcs ( the carrier of X, the carrier of R))) by A2, FUNCT_2:def_2;
A5: now__::_thesis:_for_x,_B_being_set_st_x_in_A_&_B_c=_A_&_S1[B]_holds_
S1[B_\/_{x}]
let x, B be set ; ::_thesis: ( x in A & B c= A & S1[B] implies S1[b2 \/ {b1}] )
assume A6: x in A ; ::_thesis: ( B c= A & S1[B] implies S1[b2 \/ {b1}] )
then reconsider fx = F . x as continuous Function of X,R by A3;
assume A7: B c= A ; ::_thesis: ( S1[B] implies S1[b2 \/ {b1}] )
assume A8: S1[B] ; ::_thesis: S1[b2 \/ {b1}]
percases ( B = {} or B <> {} ) ;
supposeA9: B = {} ; ::_thesis: S1[b2 \/ {b1}]
thus  S1[B \/ {x}] ::_thesis: verum
proof
assume  not B \/ {x} is empty ; ::_thesis: ex f being continuous Function of X,R st
for x being Point of X
for S being non empty finite Subset of REAL st S = rng ((commute (F | (B \/ {x}))) . x) holds
f . x = max S
take  fx ; ::_thesis: for x being Point of X
for S being non empty finite Subset of REAL st S = rng ((commute (F | (B \/ {x}))) . x) holds
fx . x = max S
let a be Point of X; ::_thesis: for S being non empty finite Subset of REAL st S = rng ((commute (F | (B \/ {x}))) . a) holds
fx . a = max S
let S be non empty finite Subset of REAL; ::_thesis: ( S = rng ((commute (F | (B \/ {x}))) . a) implies fx . a = max S )
A10: dom fx = the carrier of X by FUNCT_2:def_1;
 F | {x} = x .--> (F . x) by A2, A6, FUNCT_7:6;
then  (commute (F | {x})) . a = x .--> (fx . a) by A10, Th3;
then A11: rng ((commute (F | {x})) . a) = {(fx . a)} by FUNCOP_1:8;
assume  S = rng ((commute (F | (B \/ {x}))) . a) ; ::_thesis: fx . a = max S
hence  fx . a = max S by A11, A9, XXREAL_2:11; ::_thesis: verum
end;
end;
supposeA12: B <> {} ; ::_thesis: S1[b2 \/ {b1}]
then reconsider B9 = B as non empty set ;
consider f being continuous Function of X,R such that
A13: for x being Point of X
for S being non empty finite Subset of REAL st S = rng ((commute (F | B9)) . x) holds
f . x = max S by A8;
consider h being continuous Function of X,R such that
A14: for x being Point of X holds h . x = max ((f . x),(fx . x)) by Th50;
thus  S1[B \/ {x}] ::_thesis: verum
proof
 F is Function of A,(Funcs ( the carrier of X, the carrier of R)) by A4, FUNCT_2:66;
then  F | B is Function of B,(Funcs ( the carrier of X, the carrier of R)) by A7, FUNCT_2:32;
then  F | B in Funcs (B,(Funcs ( the carrier of X, the carrier of R))) by FUNCT_2:8;
then  commute (F | B) in Funcs ( the carrier of X,(Funcs (B, the carrier of R))) by A12, FUNCT_6:55;
then reconsider cFB = commute (F | B) as Function of the carrier of X,(Funcs (B, the carrier of R)) by FUNCT_2:66;
assume  not B \/ {x} is empty ; ::_thesis: ex f being continuous Function of X,R st
for x being Point of X
for S being non empty finite Subset of REAL st S = rng ((commute (F | (B \/ {x}))) . x) holds
f . x = max S
take  h ; ::_thesis: for x being Point of X
for S being non empty finite Subset of REAL st S = rng ((commute (F | (B \/ {x}))) . x) holds
h . x = max S
let a be Point of X; ::_thesis: for S being non empty finite Subset of REAL st S = rng ((commute (F | (B \/ {x}))) . a) holds
h . a = max S
let S be non empty finite Subset of REAL; ::_thesis: ( S = rng ((commute (F | (B \/ {x}))) . a) implies h . a = max S )
reconsider cFBa = cFB . a as Function of B9, the carrier of R ;
A15: dom fx = the carrier of X by FUNCT_2:def_1;
 F | (B \/ {x}) = (F | B) \/ (F | {x}) by RELAT_1:78;
then A16: (commute (F | (B \/ {x}))) . a = (cFB . a) \/ ((commute (F | {x})) . a) by Th7;
assume  S = rng ((commute (F | (B \/ {x}))) . a) ; ::_thesis: h . a = max S
then A17: S = (rng cFBa) \/ (rng ((commute (F | {x})) . a)) by A16, RELAT_1:12;
then  rng cFBa c= S by XBOOLE_1:7;
then reconsider S1 = rng cFBa as non empty finite Subset of REAL by XBOOLE_1:1;
 F | {x} = x .--> (F . x) by A2, A6, FUNCT_7:6;
then  (commute (F | {x})) . a = x .--> (fx . a) by A15, Th3;
then A18: S = S1 \/ {(fx . a)} by A17, FUNCOP_1:8;
 f . a = max S1 by A13;
then  max S = max ((f . a),(max {(fx . a)})) by A18, XXREAL_2:10
.= max ((f . a),(fx . a)) by XXREAL_2:11 ;
hence  h . a = max S by A14; ::_thesis: verum
end;
end;
end;
end;
A19: A is finite ;
 S1[A] from FINSET_1:sch_2(A19, A1, A5);
hence  ex f being continuous Function of X,R st
for x being Point of X
for S being non empty finite Subset of REAL st S = rng ((commute F) . x) holds
f . x = max S ; ::_thesis: verum
end;

theorem Th52: :: TOPGEN_5:52
for X being non empty T_1 TopSpace
for B being prebasis of X st ( for x being Point of X
for V being Subset of X st x in V & V in B holds
ex f being continuous Function of X,I[01] st
( f . x = 0 & f .: (V `) c= {1} ) ) holds
X is Tychonoff
proof
reconsider z = 0 as Point of I[01] by BORSUK_1:40, XXREAL_1:1;
let X be non empty T_1 TopSpace; ::_thesis: for B being prebasis of X st ( for x being Point of X
for V being Subset of X st x in V & V in B holds
ex f being continuous Function of X,I[01] st
( f . x = 0 & f .: (V `) c= {1} ) ) holds
X is Tychonoff
let BB be prebasis of X; ::_thesis: ( ( for x being Point of X
for V being Subset of X st x in V & V in BB holds
ex f being continuous Function of X,I[01] st
( f . x = 0 & f .: (V `) c= {1} ) ) implies X is Tychonoff )
assume A1: for x being Point of X
for V being Subset of X st x in V & V in BB holds
ex f being continuous Function of X,I[01] st
( f . x = 0 & f .: (V `) c= {1} ) ; ::_thesis: X is Tychonoff
let A be closed Subset of X; :: according to TOPGEN_5:def_4 ::_thesis: for a being Point of X st a in A ` holds
ex f being continuous Function of X,I[01] st
( f . a = 0 & f .: A c= {1} )
let a be Point of X; ::_thesis: ( a in A ` implies ex f being continuous Function of X,I[01] st
( f . a = 0 & f .: A c= {1} ) )
A2: FinMeetCl BB is Basis of X by YELLOW_9:23;
assume  a in A ` ; ::_thesis: ex f being continuous Function of X,I[01] st
( f . a = 0 & f .: A c= {1} )
then consider B being Subset of X such that
A3: B in FinMeetCl BB and
A4: a in B and
A5: B c= A ` by A2, YELLOW_9:31;
consider F being Subset-Family of X such that
A6: F c= BB and
A7: F is finite and
A8: B = Intersect F by A3, CANTOR_1:def_3;
percases ( F is empty or not F is empty ) ;
suppose F is empty ; ::_thesis: ex f being continuous Function of X,I[01] st
( f . a = 0 & f .: A c= {1} )
then  B = the carrier of X by A8, SETFAM_1:def_9;
then  (A `) ` = {} X by A5, XBOOLE_1:37;
then A9: (X --> z) .: A = {} ;
 (X --> z) . a = z by FUNCOP_1:7;
hence  ex f being continuous Function of X,I[01] st
( f . a = 0 & f .: A c= {1} ) by A9, XBOOLE_1:2; ::_thesis: verum
end;
suppose not F is empty ; ::_thesis: ex f being continuous Function of X,I[01] st
( f . a = 0 & f .: A c= {1} )
then reconsider F = F as non empty finite Subset-Family of X by A7;
defpred S1[ set , set ] means  ex S being Subset of X ex f being continuous Function of X,I[01] st
( S = $1 & f = $2 & f . a = 0 & f .: (S `) c= {1} );
reconsider Sa = {0} as non empty finite Subset of REAL ;
set z = the Element of F;
set R = I[01] ;
A10: for x being set st x in F holds
ex y being set st S1[x,y]
proof
let x be set ; ::_thesis: ( x in F implies ex y being set st S1[x,y] )
assume A11: x in F ; ::_thesis: ex y being set st S1[x,y]
then reconsider S = x as Subset of X ;
 a in S by A4, A8, A11, SETFAM_1:43;
then consider f being continuous Function of X,I[01] such that
A12: f . a = 0 and
A13: f .: (S `) c= {1} by A6, A11, A1;
take  f ; ::_thesis: S1[x,f]
thus  S1[x,f] by A12, A13; ::_thesis: verum
end;
consider G being Function such that
A14: ( dom G = F & ( for x being set st x in F holds
S1[x,G . x] ) ) from CLASSES1:sch_1(A10);
 G is Function-yielding
proof
let x be set ; :: according to FUNCOP_1:def_6 ::_thesis: ( not x in proj1 G or G . x is set )
assume  x in dom G ; ::_thesis: G . x is set
then  S1[x,G . x] by A14;
hence  G . x is set ; ::_thesis: verum
end;
then reconsider G = G as ManySortedFunction of F by A14, PARTFUN1:def_2, RELAT_1:def_18;
 rng G c= Funcs ( the carrier of X, the carrier of I[01])
proof
let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in rng G or u in Funcs ( the carrier of X, the carrier of I[01]) )
assume  u in rng G ; ::_thesis: u in Funcs ( the carrier of X, the carrier of I[01])
then consider v being set  such that
A15: v in dom G and
A16: u = G . v by FUNCT_1:def_3;
 S1[v,u] by A14, A15, A16;
hence  u in Funcs ( the carrier of X, the carrier of I[01]) by FUNCT_2:8; ::_thesis: verum
end;
then  G is Function of F,(Funcs ( the carrier of X, the carrier of I[01])) by A14, FUNCT_2:2;
then A17: G in Funcs (F,(Funcs ( the carrier of X, the carrier of I[01]))) by FUNCT_2:8;
then  commute G in Funcs ( the carrier of X,(Funcs (F, the carrier of I[01]))) by FUNCT_6:55;
then reconsider cG = commute G as Function of the carrier of X,(Funcs (F, the carrier of I[01])) by FUNCT_2:66;
now__::_thesis:_for_a_being_set_st_a_in_F_holds_
G_._a_is_continuous_Function_of_X,I[01]
let a be set ; ::_thesis: ( a in F implies G . a is continuous Function of X,I[01] )
assume  a in F ; ::_thesis: G . a is continuous Function of X,I[01]
then  S1[a,G . a] by A14;
hence  G . a is continuous Function of X,I[01] ; ::_thesis: verum
end;
then consider f being continuous Function of X,I[01] such that
A18: for x being Point of X
for S being non empty finite Subset of REAL st S = rng ((commute G) . x) holds
f . x = max S by Th51;
take  f ; ::_thesis: ( f . a = 0 & f .: A c= {1} )
reconsider cGa = cG . a as Function of F, the carrier of I[01] ;
A19: dom cGa = F by FUNCT_2:def_1;
 Sa = rng ((commute G) . a)
proof
thus  Sa c= rng ((commute G) . a) :: according to XBOOLE_0:def_10 ::_thesis: rng ((commute G) . a) c= Sa
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Sa or x in rng ((commute G) . a) )
assume  x in Sa ; ::_thesis: x in rng ((commute G) . a)
then A20: x = 0 by TARSKI:def_1;
 S1[ the Element of F,G . the Element of F] by A14;
then  x = ((commute G) . a) . the Element of F by A20, A17, FUNCT_6:56;
hence  x in rng ((commute G) . a) by A19, FUNCT_1:def_3; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng ((commute G) . a) or x in Sa )
assume  x in rng ((commute G) . a) ; ::_thesis: x in Sa
then consider z being set  such that
A21: z in dom cGa and
A22: x = cGa . z by FUNCT_1:def_3;
 S1[z,G . z] by A14, A21;
then  x = 0 by A17, A21, A22, FUNCT_6:56;
hence  x in Sa by TARSKI:def_1; ::_thesis: verum
end;
hence f . a = max Sa by A18
.= 0 by XXREAL_2:11 ;
::_thesis: f .: A c= {1}
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in f .: A or z in {1} )
assume  z in f .: A ; ::_thesis: z in {1}
then consider x being set  such that
A23: x in dom f and
A24: x in A and
A25: z = f . x by FUNCT_1:def_6;
reconsider x = x as Element of X by A23;
 not x in B by A5, A24, XBOOLE_0:def_5;
then consider w being set  such that
A26: w in F and
A27: not x in w by A8, SETFAM_1:43;
reconsider cGx = cG . x as Function of F, the carrier of I[01] ;
reconsider S = rng cGx as non empty finite Subset of REAL by BORSUK_1:40, XBOOLE_1:1;
A28: f . x = max S by A18;
consider T being Subset of X, g being continuous Function of X,I[01] such that
A29: T = w and
A30: g = G . w and
 g . a = 0 and
A31: g .: (T `) c= {1} by A14, A26;
 x in T ` by A27, A29, SUBSET_1:29;
then  g . x in g .: (T `) by FUNCT_2:35;
then  g . x = 1 by A31, TARSKI:def_1;
then A32: cGx . w = 1 by A17, A26, A30, FUNCT_6:56;
 w in dom cGx by A17, A26, A30, FUNCT_6:56;
then A33: 1 in S by A32, FUNCT_1:def_3;
 for r being ext-real number st r in S holds
r <= 1 by BORSUK_1:40, XXREAL_1:1;
then  max S = 1 by A33, XXREAL_2:def_8;
hence  z in {1} by A25, A28, TARSKI:def_1; ::_thesis: verum
end;
end;
end;

theorem Th53: :: TOPGEN_5:53
Sorgenfrey-line is T_1
proof
set T = Sorgenfrey-line ;
consider B being Subset-Family of REAL such that
A1: the topology of Sorgenfrey-line = UniCl B and
A2: B =  {  [.x,q.[ where x, q is Element of REAL : ( x < q & q is rational )  }  by TOPGEN_3:def_2;
let x, y be Point of Sorgenfrey-line; :: according to URYSOHN1:def_7 ::_thesis: ( x = y or ex b1, b2 being Element of bool the carrier of Sorgenfrey-line st
( b1 is open & b2 is open & x in b1 & not y in b1 & y in b2 & not x in b2 ) )
reconsider a = x, b = y as Real by TOPGEN_3:def_2;
A3: B c= the topology of Sorgenfrey-line by A1, CANTOR_1:1;
assume A4: x <> y ; ::_thesis: ex b1, b2 being Element of bool the carrier of Sorgenfrey-line st
( b1 is open & b2 is open & x in b1 & not y in b1 & y in b2 & not x in b2 )
percases ( a < b or a > b ) by A4, XXREAL_0:1;
supposeA5: a < b ; ::_thesis: ex b1, b2 being Element of bool the carrier of Sorgenfrey-line st
( b1 is open & b2 is open & x in b1 & not y in b1 & y in b2 & not x in b2 )
 b < b + 1 by XREAL_1:29;
then consider q being rational number  such that
A6: b < q and
 q < b + 1 by RAT_1:7;
 q is Real by XREAL_0:def_1;
then  [.b,q.[ in B by A2, A6;
then A7: [.b,q.[ in the topology of Sorgenfrey-line by A3;
consider w being rational number  such that
A8: a < w and
A9: w < b by A5, RAT_1:7;
 w is Real by XREAL_0:def_1;
then  [.a,w.[ in B by A2, A8;
then  [.a,w.[ in the topology of Sorgenfrey-line by A3;
then reconsider U = [.a,w.[, V = [.b,q.[ as open Subset of Sorgenfrey-line by A7, PRE_TOPC:def_2;
take  U ; ::_thesis: ex b1 being Element of bool the carrier of Sorgenfrey-line st
( U is open & b1 is open & x in U & not y in U & y in b1 & not x in b1 )
take  V ; ::_thesis: ( U is open & V is open & x in U & not y in U & y in V & not x in V )
thus  ( U is open & V is open ) ; ::_thesis: ( x in U & not y in U & y in V & not x in V )
thus  ( x in U & not y in U & y in V & not x in V ) by A5, A8, A9, A6, XXREAL_1:3; ::_thesis: verum
end;
supposeA10: a > b ; ::_thesis: ex b1, b2 being Element of bool the carrier of Sorgenfrey-line st
( b1 is open & b2 is open & x in b1 & not y in b1 & y in b2 & not x in b2 )
 a < a + 1 by XREAL_1:29;
then consider q being rational number  such that
A11: a < q and
 q < a + 1 by RAT_1:7;
 q is Real by XREAL_0:def_1;
then  [.a,q.[ in B by A2, A11;
then A12: [.a,q.[ in the topology of Sorgenfrey-line by A3;
consider w being rational number  such that
A13: b < w and
A14: w < a by A10, RAT_1:7;
 w is Real by XREAL_0:def_1;
then  [.b,w.[ in B by A2, A13;
then  [.b,w.[ in the topology of Sorgenfrey-line by A3;
then reconsider V = [.b,w.[, U = [.a,q.[ as open Subset of Sorgenfrey-line by A12, PRE_TOPC:def_2;
take  U ; ::_thesis: ex b1 being Element of bool the carrier of Sorgenfrey-line st
( U is open & b1 is open & x in U & not y in U & y in b1 & not x in b1 )
take  V ; ::_thesis: ( U is open & V is open & x in U & not y in U & y in V & not x in V )
thus  ( U is open & V is open ) ; ::_thesis: ( x in U & not y in U & y in V & not x in V )
thus  ( x in U & not y in U & y in V & not x in V ) by A10, A13, A14, A11, XXREAL_1:3; ::_thesis: verum
end;
end;
end;

theorem Th54: :: TOPGEN_5:54
for x being real number holds left_open_halfline x is closed Subset of Sorgenfrey-line
proof
let x be real number ; ::_thesis: left_open_halfline x is closed Subset of Sorgenfrey-line
set T = Sorgenfrey-line ;
reconsider A = right_closed_halfline x as open Subset of Sorgenfrey-line by TOPGEN_3:15;
 the carrier of Sorgenfrey-line = REAL by TOPGEN_3:def_2;
then  left_open_halfline x = A ` by XXREAL_1:224, XXREAL_1:294;
hence  left_open_halfline x is closed Subset of Sorgenfrey-line ; ::_thesis: verum
end;

theorem :: TOPGEN_5:55
for x being real number holds left_closed_halfline x is closed Subset of Sorgenfrey-line
proof
let x be real number ; ::_thesis: left_closed_halfline x is closed Subset of Sorgenfrey-line
set T = Sorgenfrey-line ;
reconsider A = right_open_halfline x as open Subset of Sorgenfrey-line by TOPGEN_3:14;
 the carrier of Sorgenfrey-line = REAL by TOPGEN_3:def_2;
then  ((left_closed_halfline x) `) ` = A ` by XXREAL_1:224, XXREAL_1:288;
hence  left_closed_halfline x is closed Subset of Sorgenfrey-line ; ::_thesis: verum
end;

theorem Th56: :: TOPGEN_5:56
for x being real number holds right_closed_halfline x is closed Subset of Sorgenfrey-line
proof
let x be real number ; ::_thesis: right_closed_halfline x is closed Subset of Sorgenfrey-line
set T = Sorgenfrey-line ;
reconsider A = left_open_halfline x as open Subset of Sorgenfrey-line by TOPGEN_3:13;
 the carrier of Sorgenfrey-line = REAL by TOPGEN_3:def_2;
then  ((right_closed_halfline x) `) ` = A ` by XXREAL_1:224, XXREAL_1:294;
hence  right_closed_halfline x is closed Subset of Sorgenfrey-line ; ::_thesis: verum
end;

theorem Th57: :: TOPGEN_5:57
for x, y being real number holds [.x,y.[ is closed Subset of Sorgenfrey-line
proof
let x, y be real number ; ::_thesis: [.x,y.[ is closed Subset of Sorgenfrey-line
set T = Sorgenfrey-line ;
reconsider A = right_closed_halfline x, B = left_open_halfline y as closed Subset of Sorgenfrey-line by Th54, Th56;
A1: the carrier of Sorgenfrey-line = REAL by TOPGEN_3:def_2;
[.x,y.[ = ([.x,y.[ `) `
.= ((left_open_halfline x) \/ (right_closed_halfline y)) ` by XXREAL_1:382
.= ((left_open_halfline x) `) /\ ((right_closed_halfline y) `) by XBOOLE_1:53
.= ((A `) `) /\ ((right_closed_halfline y) `) by A1, XXREAL_1:224, XXREAL_1:294
.= A /\ B by XXREAL_1:224, XXREAL_1:294 ;
hence  [.x,y.[ is closed Subset of Sorgenfrey-line ; ::_thesis: verum
end;

theorem Th58: :: TOPGEN_5:58
for x being real number
for w being rational number ex f being continuous Function of Sorgenfrey-line,I[01] st
for a being Point of Sorgenfrey-line holds
( ( a in [.x,w.[ implies f . a = 0 ) & ( not a in [.x,w.[ implies f . a = 1 ) )
proof
reconsider 00 = 0 , 01 = 1 as Element of I[01] by BORSUK_1:40, XXREAL_1:1;
let x be real number ; ::_thesis: for w being rational number ex f being continuous Function of Sorgenfrey-line,I[01] st
for a being Point of Sorgenfrey-line holds
( ( a in [.x,w.[ implies f . a = 0 ) & ( not a in [.x,w.[ implies f . a = 1 ) )
set X = Sorgenfrey-line ;
let w be rational number ; ::_thesis: ex f being continuous Function of Sorgenfrey-line,I[01] st
for a being Point of Sorgenfrey-line holds
( ( a in [.x,w.[ implies f . a = 0 ) & ( not a in [.x,w.[ implies f . a = 1 ) )
reconsider V = [.x,w.[ as open closed Subset of Sorgenfrey-line by Th57, TOPGEN_3:11;
defpred S1[ set ] means $1 in [.x,w.[;
deffunc H1( set ) ->  Element of NAT = 0 ;
deffunc H2( set ) ->  Element of NAT = 1;
reconsider f1 = (Sorgenfrey-line | V) --> 00 as continuous Function of (Sorgenfrey-line | V),I[01] ;
reconsider f2 = (Sorgenfrey-line | (V `)) --> 01 as continuous Function of (Sorgenfrey-line | (V `)),I[01] ;
A1: for a being set st a in the carrier of Sorgenfrey-line holds
( ( S1[a] implies H1(a) in the carrier of I[01] ) & ( not S1[a] implies H2(a) in the carrier of I[01] ) ) by BORSUK_1:40, XXREAL_1:1;
consider f being Function of Sorgenfrey-line,I[01] such that
A2: for a being set st a in the carrier of Sorgenfrey-line holds
( ( S1[a] implies f . a = H1(a) ) & ( not S1[a] implies f . a = H2(a) ) ) from FUNCT_2:sch_5(A1);
A3: the carrier of (Sorgenfrey-line | V) = V by PRE_TOPC:8;
then A4: dom f1 = V by FUNCT_2:def_1;
A5: the carrier of (Sorgenfrey-line | (V `)) = V ` by PRE_TOPC:8;
then A6: dom f2 = V ` by FUNCT_2:def_1;
A7: dom f = [#] Sorgenfrey-line by FUNCT_2:def_1;
A8: now__::_thesis:_for_u_being_set_st_u_in_(dom_f1)_\/_(dom_f2)_holds_
(_(_u_in_dom_f2_implies_f_._u_=_f2_._u_)_&_(_not_u_in_dom_f2_implies_f_._u_=_f1_._u_)_)
let u be set ; ::_thesis: ( u in (dom f1) \/ (dom f2) implies ( ( u in dom f2 implies f . u = f2 . u ) & ( not u in dom f2 implies f . u = f1 . u ) ) )
assume  u in (dom f1) \/ (dom f2) ; ::_thesis: ( ( u in dom f2 implies f . u = f2 . u ) & ( not u in dom f2 implies f . u = f1 . u ) )
then reconsider x = u as Point of Sorgenfrey-line by A7, A4, A6, PRE_TOPC:2;
hereby  ::_thesis: ( not u in dom f2 implies f . u = f1 . u )
assume A9: u in dom f2 ; ::_thesis: f . u = f2 . u
then A10: ((V `) --> 1) . u = 1 by A5, FUNCOP_1:7;
 not x in V by A9, A5, XBOOLE_0:def_5;
hence  f . u = f2 . u by A10, A5, A2; ::_thesis: verum
end;
assume  not u in dom f2 ; ::_thesis: f . u = f1 . u
then A11: x in V by A6, SUBSET_1:29;
hence f . u = 0 by A2
.= f1 . u by A3, A11, FUNCOP_1:7 ;
::_thesis: verum
end;
 V \/ (V `) = [#] Sorgenfrey-line by PRE_TOPC:2;
then  f = f1 +* f2 by A8, A7, A4, A6, FUNCT_4:def_1;
then reconsider f = f as continuous Function of Sorgenfrey-line,I[01] by Th12;
take  f ; ::_thesis: for a being Point of Sorgenfrey-line holds
( ( a in [.x,w.[ implies f . a = 0 ) & ( not a in [.x,w.[ implies f . a = 1 ) )
let a be Point of Sorgenfrey-line; ::_thesis: ( ( a in [.x,w.[ implies f . a = 0 ) & ( not a in [.x,w.[ implies f . a = 1 ) )
thus  ( ( a in [.x,w.[ implies f . a = 0 ) & ( not a in [.x,w.[ implies f . a = 1 ) ) by A2; ::_thesis: verum
end;

theorem :: TOPGEN_5:59
Sorgenfrey-line is Tychonoff
proof
set X = Sorgenfrey-line ;
A1: the carrier of Sorgenfrey-line = REAL by TOPGEN_3:def_2;
consider B being Subset-Family of REAL such that
A2: the topology of Sorgenfrey-line = UniCl B and
A3: B =  {  [.x,q.[ where x, q is Element of REAL : ( x < q & q is rational )  }  by TOPGEN_3:def_2;
 B c= UniCl B by CANTOR_1:1;
then  B is Basis of Sorgenfrey-line by A1, A2, CANTOR_1:def_2, TOPS_2:64;
then reconsider B = B as prebasis of Sorgenfrey-line by YELLOW_9:27;
now__::_thesis:_for_x_being_Point_of_Sorgenfrey-line
for_V_being_Subset_of_Sorgenfrey-line_st_x_in_V_&_V_in_B_holds_
ex_f_being_continuous_Function_of_Sorgenfrey-line,I[01]_st_
(_f_._x_=_0_&_f_.:_(V_`)_c=_{1}_)
let x be Point of Sorgenfrey-line; ::_thesis: for V being Subset of Sorgenfrey-line st x in V & V in B holds
ex f being continuous Function of Sorgenfrey-line,I[01] st
( f . x = 0 & f .: (V `) c= {1} )
let V be Subset of Sorgenfrey-line; ::_thesis: ( x in V & V in B implies ex f being continuous Function of Sorgenfrey-line,I[01] st
( f . x = 0 & f .: (V `) c= {1} ) )
assume that
A4: x in V and
A5: V in B ; ::_thesis: ex f being continuous Function of Sorgenfrey-line,I[01] st
( f . x = 0 & f .: (V `) c= {1} )
consider a, q being Real such that
A6: V = [.a,q.[ and
 a < q and
A7: q is rational by A5, A3;
consider f being continuous Function of Sorgenfrey-line,I[01] such that
A8: for b being Point of Sorgenfrey-line holds
( ( b in [.a,q.[ implies f . b = 0 ) & ( not b in [.a,q.[ implies f . b = 1 ) ) by A7, Th58;
take f = f; ::_thesis: ( f . x = 0 & f .: (V `) c= {1} )
thus  f . x = 0 by A4, A6, A8; ::_thesis: f .: (V `) c= {1}
thus  f .: (V `) c= {1} ::_thesis: verum
proof
let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in f .: (V `) or u in {1} )
assume  u in f .: (V `) ; ::_thesis: u in {1}
then consider b being Point of Sorgenfrey-line such that
A9: b in V ` and
A10: u = f . b by FUNCT_2:65;
 not b in [.a,q.[ by A6, A9, XBOOLE_0:def_5;
then  u = 1 by A8, A10;
hence  u in {1} by TARSKI:def_1; ::_thesis: verum
end;
end;
hence  Sorgenfrey-line is Tychonoff by Th52, Th53; ::_thesis: verum
end;

begin

definition
let x be real number ;
let r be real positive number ;
func + (x,r) ->  Function of Niemytzki-plane,I[01] means :Def5: :: TOPGEN_5:def 5
( it . |[x,0]| = 0 & ( for a being real number
for b being real non negative number holds
( ( ( a <> x or b <> 0 ) & not |[a,b]| in Ball (|[x,r]|,r) implies it . |[a,b]| = 1 ) & ( |[a,b]| in Ball (|[x,r]|,r) implies it . |[a,b]| = (|.(|[x,0]| - |[a,b]|).| ^2) / ((2 * r) * b) ) ) ) );
existence
 ex b1 being Function of Niemytzki-plane,I[01] st
( b1 . |[x,0]| = 0 & ( for a being real number
for b being real non negative number holds
( ( ( a <> x or b <> 0 ) & not |[a,b]| in Ball (|[x,r]|,r) implies b1 . |[a,b]| = 1 ) & ( |[a,b]| in Ball (|[x,r]|,r) implies b1 . |[a,b]| = (|.(|[x,0]| - |[a,b]|).| ^2) / ((2 * r) * b) ) ) ) )
proof
deffunc H1( Point of (TOP-REAL 2)) ->  Element of REAL = (|.(|[x,0]| - $1).| ^2) / ((2 * r) * ($1 `2));
deffunc H2( Point of (TOP-REAL 2)) ->  Element of NAT = 1;
deffunc H3( Point of (TOP-REAL 2)) ->  Element of NAT = 0 ;
defpred S1[ set ] means  not $1 in Ball (|[x,r]|,r);
defpred S2[ set ] means $1 = |[x,0]|;
A1: for a being Point of (TOP-REAL 2) st a in the carrier of Niemytzki-plane holds
( ( S2[a] implies H3(a) in the carrier of I[01] ) & ( not S2[a] & S1[a] implies H2(a) in the carrier of I[01] ) & ( not S2[a] & not S1[a] implies H1(a) in the carrier of I[01] ) )
proof
let a be Point of (TOP-REAL 2); ::_thesis: ( a in the carrier of Niemytzki-plane implies ( ( S2[a] implies H3(a) in the carrier of I[01] ) & ( not S2[a] & S1[a] implies H2(a) in the carrier of I[01] ) & ( not S2[a] & not S1[a] implies H1(a) in the carrier of I[01] ) ) )
assume  a in the carrier of Niemytzki-plane ; ::_thesis: ( ( S2[a] implies H3(a) in the carrier of I[01] ) & ( not S2[a] & S1[a] implies H2(a) in the carrier of I[01] ) & ( not S2[a] & not S1[a] implies H1(a) in the carrier of I[01] ) )
thus  ( S2[a] implies H3(a) in the carrier of I[01] ) by BORSUK_1:40, XXREAL_1:1; ::_thesis: ( ( not S2[a] & S1[a] implies H2(a) in the carrier of I[01] ) & ( not S2[a] & not S1[a] implies H1(a) in the carrier of I[01] ) )
thus  ( not S2[a] & S1[a] implies H2(a) in the carrier of I[01] ) by BORSUK_1:40, XXREAL_1:1; ::_thesis: ( not S2[a] & not S1[a] implies H1(a) in the carrier of I[01] )
assume  not S2[a] ; ::_thesis: ( S1[a] or H1(a) in the carrier of I[01] )
A2: a = |[(a `1),(a `2)]| by EUCLID:53;
assume  not S1[a] ; ::_thesis: H1(a) in the carrier of I[01]
then  |.(a - |[x,r]|).| < r by TOPREAL9:7;
then  |.|[((a `1) - x),((a `2) - r)]|.| < r by A2, EUCLID:62;
then A3: |.|[((a `1) - x),((a `2) - r)]|.| ^2 < r ^2 by SQUARE_1:16;
A4: |[((a `1) - x),((a `2) - r)]| `2 = (a `2) - r by EUCLID:52;
 |[((a `1) - x),((a `2) - r)]| `1 = (a `1) - x by EUCLID:52;
then  (((a `1) - x) ^2) + (((a `2) - r) ^2) < r ^2 by A4, A3, JGRAPH_1:29;
then  ((((a `1) - x) ^2) + (((a `2) ^2) - ((2 * (a `2)) * r))) + (r ^2) < 0 + (r ^2) ;
then  ((((a `1) - x) ^2) + ((a `2) ^2)) - ((2 * (a `2)) * r) < 0 by XREAL_1:6;
then A5: (((a `1) - x) ^2) + (((a `2) - 0) ^2) < (2 * r) * (a `2) by XREAL_1:48;
A6: |[((a `1) - x),(a `2)]| `2 = a `2 by EUCLID:52;
 |[((a `1) - x),(a `2)]| `1 = (a `1) - x by EUCLID:52;
then A7: |.|[((a `1) - x),((a `2) - 0)]|.| ^2 < (2 * r) * (a `2) by A6, A5, JGRAPH_1:29;
then  |.(a - |[x,0]|).| ^2 < (2 * r) * (a `2) by A2, EUCLID:62;
then  |.(|[x,0]| - a).| ^2 < (2 * r) * (a `2) by TOPRNS_1:27;
then  H1(a) <= 1 by XREAL_1:183;
hence  H1(a) in the carrier of I[01] by A7, BORSUK_1:40, XXREAL_1:1; ::_thesis: verum
end;
 the carrier of Niemytzki-plane = y>=0-plane by Def3;
then A8: the carrier of Niemytzki-plane c= the carrier of (TOP-REAL 2) ;
consider f being Function of Niemytzki-plane,I[01] such that
A9: for a being Point of (TOP-REAL 2) st a in the carrier of Niemytzki-plane holds
( ( S2[a] implies f . a = H3(a) ) & ( not S2[a] & S1[a] implies f . a = H2(a) ) & ( not S2[a] & not S1[a] implies f . a = H1(a) ) ) from TOPGEN_5:sch_2(A8, A1);
take  f ; ::_thesis: ( f . |[x,0]| = 0 & ( for a being real number
for b being real non negative number holds
( ( ( a <> x or b <> 0 ) & not |[a,b]| in Ball (|[x,r]|,r) implies f . |[a,b]| = 1 ) & ( |[a,b]| in Ball (|[x,r]|,r) implies f . |[a,b]| = (|.(|[x,0]| - |[a,b]|).| ^2) / ((2 * r) * b) ) ) ) )
A10: the carrier of Niemytzki-plane = y>=0-plane by Def3;
then  |[x,0]| in the carrier of Niemytzki-plane by Th18;
hence  f . |[x,0]| = 0 by A9; ::_thesis: for a being real number
for b being real non negative number holds
( ( ( a <> x or b <> 0 ) & not |[a,b]| in Ball (|[x,r]|,r) implies f . |[a,b]| = 1 ) & ( |[a,b]| in Ball (|[x,r]|,r) implies f . |[a,b]| = (|.(|[x,0]| - |[a,b]|).| ^2) / ((2 * r) * b) ) )
let a be real number ; ::_thesis: for b being real non negative number holds
( ( ( a <> x or b <> 0 ) & not |[a,b]| in Ball (|[x,r]|,r) implies f . |[a,b]| = 1 ) & ( |[a,b]| in Ball (|[x,r]|,r) implies f . |[a,b]| = (|.(|[x,0]| - |[a,b]|).| ^2) / ((2 * r) * b) ) )
let b be real non negative number ; ::_thesis: ( ( ( a <> x or b <> 0 ) & not |[a,b]| in Ball (|[x,r]|,r) implies f . |[a,b]| = 1 ) & ( |[a,b]| in Ball (|[x,r]|,r) implies f . |[a,b]| = (|.(|[x,0]| - |[a,b]|).| ^2) / ((2 * r) * b) ) )
A11: |[a,b]| in the carrier of Niemytzki-plane by A10, Th18;
hereby  ::_thesis: ( |[a,b]| in Ball (|[x,r]|,r) implies f . |[a,b]| = (|.(|[x,0]| - |[a,b]|).| ^2) / ((2 * r) * b) )
assume  ( a <> x or b <> 0 ) ; ::_thesis: ( not |[a,b]| in Ball (|[x,r]|,r) implies f . |[a,b]| = 1 )
then  not S2[|[a,b]|] by SPPOL_2:1;
hence  ( not |[a,b]| in Ball (|[x,r]|,r) implies f . |[a,b]| = 1 ) by A9, A11; ::_thesis: verum
end;
assume A12: |[a,b]| in Ball (|[x,r]|,r) ; ::_thesis: f . |[a,b]| = (|.(|[x,0]| - |[a,b]|).| ^2) / ((2 * r) * b)
A13: |[x,0]| in y=0-line by Th15;
A14: |[a,b]| `2 = b by EUCLID:52;
 Ball (|[x,r]|,r) misses y=0-line by Th21;
then  not S2[|[a,b]|] by A13, A12, XBOOLE_0:3;
hence  f . |[a,b]| = (|.(|[x,0]| - |[a,b]|).| ^2) / ((2 * r) * b) by A14, A9, A11, A12; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Function of Niemytzki-plane,I[01] st b1 . |[x,0]| = 0 & ( for a being real number
for b being real non negative number holds
( ( ( a <> x or b <> 0 ) & not |[a,b]| in Ball (|[x,r]|,r) implies b1 . |[a,b]| = 1 ) & ( |[a,b]| in Ball (|[x,r]|,r) implies b1 . |[a,b]| = (|.(|[x,0]| - |[a,b]|).| ^2) / ((2 * r) * b) ) ) ) & b2 . |[x,0]| = 0 & ( for a being real number
for b being real non negative number holds
( ( ( a <> x or b <> 0 ) & not |[a,b]| in Ball (|[x,r]|,r) implies b2 . |[a,b]| = 1 ) & ( |[a,b]| in Ball (|[x,r]|,r) implies b2 . |[a,b]| = (|.(|[x,0]| - |[a,b]|).| ^2) / ((2 * r) * b) ) ) ) holds
b1 = b2
proof
let f, g be Function of Niemytzki-plane,I[01]; ::_thesis: ( f . |[x,0]| = 0 & ( for a being real number
for b being real non negative number holds
( ( ( a <> x or b <> 0 ) & not |[a,b]| in Ball (|[x,r]|,r) implies f . |[a,b]| = 1 ) & ( |[a,b]| in Ball (|[x,r]|,r) implies f . |[a,b]| = (|.(|[x,0]| - |[a,b]|).| ^2) / ((2 * r) * b) ) ) ) & g . |[x,0]| = 0 & ( for a being real number
for b being real non negative number holds
( ( ( a <> x or b <> 0 ) & not |[a,b]| in Ball (|[x,r]|,r) implies g . |[a,b]| = 1 ) & ( |[a,b]| in Ball (|[x,r]|,r) implies g . |[a,b]| = (|.(|[x,0]| - |[a,b]|).| ^2) / ((2 * r) * b) ) ) ) implies f = g )
assume that
A15: f . |[x,0]| = 0 and
A16: for a being real number
for b being real non negative number holds
( ( ( a <> x or b <> 0 ) & not |[a,b]| in Ball (|[x,r]|,r) implies f . |[a,b]| = 1 ) & ( |[a,b]| in Ball (|[x,r]|,r) implies f . |[a,b]| = (|.(|[x,0]| - |[a,b]|).| ^2) / ((2 * r) * b) ) ) and
A17: g . |[x,0]| = 0 and
A18: for a being real number
for b being real non negative number holds
( ( ( a <> x or b <> 0 ) & not |[a,b]| in Ball (|[x,r]|,r) implies g . |[a,b]| = 1 ) & ( |[a,b]| in Ball (|[x,r]|,r) implies g . |[a,b]| = (|.(|[x,0]| - |[a,b]|).| ^2) / ((2 * r) * b) ) ) ; ::_thesis: f = g
A19: the carrier of Niemytzki-plane = y>=0-plane by Def3;
now__::_thesis:_for_p_being_Point_of_Niemytzki-plane_holds_f_._p_=_g_._p
let p be Point of Niemytzki-plane; ::_thesis: f . b1 = g . b1
 p in y>=0-plane by A19;
then reconsider q = p as Point of (TOP-REAL 2) ;
A20: p = |[(q `1),(q `2)]| by EUCLID:53;
then reconsider q2 = q `2 as real non negative number by A19, Th18;
percases ( ( q `1 = x & q2 = 0 ) or ( ( q `1 <> x or q2 <> 0 ) & not |[(q `1),q2]| in Ball (|[x,r]|,r) ) or p in Ball (|[x,r]|,r) ) by EUCLID:53;
suppose ( q `1 = x & q2 = 0 ) ; ::_thesis: f . b1 = g . b1
hence  f . p = g . p by A15, A17, A20; ::_thesis: verum
end;
supposeA21: ( ( q `1 <> x or q2 <> 0 ) & not |[(q `1),q2]| in Ball (|[x,r]|,r) ) ; ::_thesis: f . b1 = g . b1
then  f . p = 1 by A16, A20;
hence  f . p = g . p by A18, A20, A21; ::_thesis: verum
end;
supposeA22: p in Ball (|[x,r]|,r) ; ::_thesis: f . b1 = g . b1
then  f . p = (|.(|[x,0]| - q).| ^2) / ((2 * r) * q2) by A16, A20;
hence  f . p = g . p by A18, A20, A22; ::_thesis: verum
end;
end;
end;
hence  f = g by FUNCT_2:63; ::_thesis: verum
end;
end;

:: deftheorem Def5 defines + TOPGEN_5:def_5_:_
 for x being real number
for r being real positive number
for b3 being Function of Niemytzki-plane,I[01] holds
( b3 = + (x,r) iff ( b3 . |[x,0]| = 0 & ( for a being real number
for b being real non negative number holds
( ( ( a <> x or b <> 0 ) & not |[a,b]| in Ball (|[x,r]|,r) implies b3 . |[a,b]| = 1 ) & ( |[a,b]| in Ball (|[x,r]|,r) implies b3 . |[a,b]| = (|.(|[x,0]| - |[a,b]|).| ^2) / ((2 * r) * b) ) ) ) ) );

theorem Th60: :: TOPGEN_5:60
for p being Point of (TOP-REAL 2) st p `2 >= 0 holds
for x being real number
for r being real positive number st (+ (x,r)) . p = 0 holds
p = |[x,0]|
proof
let p be Point of (TOP-REAL 2); ::_thesis: ( p `2 >= 0 implies for x being real number
for r being real positive number st (+ (x,r)) . p = 0 holds
p = |[x,0]| )
assume A1: p `2 >= 0 ; ::_thesis: for x being real number
for r being real positive number st (+ (x,r)) . p = 0 holds
p = |[x,0]|
set p1 = p `1 ;
set p2 = p `2 ;
let x be real number ; ::_thesis: for r being real positive number st (+ (x,r)) . p = 0 holds
p = |[x,0]|
let r be real positive number ; ::_thesis: ( (+ (x,r)) . p = 0 implies p = |[x,0]| )
assume A2: (+ (x,r)) . p = 0 ; ::_thesis: p = |[x,0]|
A3: p = |[(p `1),(p `2)]| by EUCLID:53;
assume A4: p <> |[x,0]| ; ::_thesis: contradiction
then  ( p `1 <> x or p `2 <> 0 ) by EUCLID:53;
then A5: |[(p `1),(p `2)]| in Ball (|[x,r]|,r) by A1, A2, A3, Def5;
 Ball (|[x,r]|,r) misses y=0-line by Th21;
then  not |[(p `1),(p `2)]| in y=0-line by A5, XBOOLE_0:3;
then  p `2 <> 0 ;
then reconsider p2 = p `2 as real positive number by A1;
 0 / ((2 * r) * p2) = (|.(|[x,0]| - |[(p `1),p2]|).| ^2) / ((2 * r) * p2) by A2, A3, A5, Def5;
then  0 = |.(|[x,0]| - p).| by A3;
hence  contradiction by A4, TOPRNS_1:28; ::_thesis: verum
end;

theorem Th61: :: TOPGEN_5:61
for x, y being real number
for r being real positive number st x <> y holds
(+ (x,r)) . |[y,0]| = 1
proof
let x, y be real number ; ::_thesis: for r being real positive number st x <> y holds
(+ (x,r)) . |[y,0]| = 1
let r be real positive number ; ::_thesis: ( x <> y implies (+ (x,r)) . |[y,0]| = 1 )
A1: ( |.(x - y).| = x - y or |.(x - y).| = - (x - y) ) by ABSVALUE:1;
A2: |[(x - y),r]| `2 = r by EUCLID:52;
 |[(x - y),r]| `1 = x - y by EUCLID:52;
then A3: |.|[(x - y),r]|.| ^2 = (r ^2) + ((x - y) ^2) by A2, JGRAPH_1:29;
assume A4: x <> y ; ::_thesis: (+ (x,r)) . |[y,0]| = 1
then  x - y <> 0 ;
then A5: |.(x - y).| > 0 by COMPLEX1:47;
then  |.(x - y).| ^2 <> 0 ;
then  |.|[(x - y),r]|.| ^2 > r ^2 by A1, A5, A3, XREAL_1:29;
then  |.|[(x - y),(r - 0)]|.| > r by SQUARE_1:15;
then  |.(|[x,r]| - |[y,0]|).| > r by EUCLID:62;
then  |.(|[y,0]| - |[x,r]|).| > r by TOPRNS_1:27;
then  not |[y,0]| in Ball (|[x,r]|,r) by TOPREAL9:7;
hence  (+ (x,r)) . |[y,0]| = 1 by A4, Def5; ::_thesis: verum
end;

theorem Th62: :: TOPGEN_5:62
for p being Point of (TOP-REAL 2)
for x being real number
for a, r being real positive number st a <= 1 & |.(p - |[x,(r * a)]|).| = r * a & p `2 <> 0 holds
(+ (x,r)) . p = a
proof
let p be Point of (TOP-REAL 2); ::_thesis: for x being real number
for a, r being real positive number st a <= 1 & |.(p - |[x,(r * a)]|).| = r * a & p `2 <> 0 holds
(+ (x,r)) . p = a
set p1 = p `1 ;
set p2 = p `2 ;
let x be real number ; ::_thesis: for a, r being real positive number st a <= 1 & |.(p - |[x,(r * a)]|).| = r * a & p `2 <> 0 holds
(+ (x,r)) . p = a
let a, r be real positive number ; ::_thesis: ( a <= 1 & |.(p - |[x,(r * a)]|).| = r * a & p `2 <> 0 implies (+ (x,r)) . p = a )
assume A1: a <= 1 ; ::_thesis: ( not |.(p - |[x,(r * a)]|).| = r * a or not p `2 <> 0 or (+ (x,r)) . p = a )
A2: |[((p `1) - x),((p `2) - (r * a))]| `1 = (p `1) - x by EUCLID:52;
A3: |[((p `1) - x),((p `2) - (r * a))]| `2 = (p `2) - (r * a) by EUCLID:52;
assume that
A4: |.(p - |[x,(r * a)]|).| = r * a and
A5: p `2 <> 0 ; ::_thesis: (+ (x,r)) . p = a
A6: p = |[(p `1),(p `2)]| by EUCLID:53;
then  |.|[((p `1) - x),((p `2) - (r * a))]|.| ^2 = (r * a) ^2 by A4, EUCLID:62;
then A7: (((p `1) - x) ^2) + (((p `2) - (r * a)) ^2) = (r * a) ^2 by A2, A3, JGRAPH_1:29;
then A8: (((p `1) - x) ^2) + ((p `2) ^2) = ((2 * (p `2)) * r) * a ;
 ((p `1) - x) ^2 >= 0 by XREAL_1:63;
then reconsider p2 = p `2 as real positive number by A5, A8;
A9: |[((p `1) - x),(p2 - 0)]| `1 = (p `1) - x by EUCLID:52;
A10: |[((p `1) - x),p2]| `2 = p2 by EUCLID:52;
percases ( a < 1 or a = 1 ) by A1, XXREAL_0:1;
suppose a < 1 ; ::_thesis: (+ (x,r)) . p = a
then  r * a < r by XREAL_1:157;
then reconsider s = r - (r * a) as real positive number by XREAL_1:50;
|.(p - |[x,r]|).| ^2 = |.|[((p `1) - x),(p2 - r)]|.| ^2 by A6, EUCLID:62
.= (((p `1) - x) ^2) + ((p2 - r) ^2) by Th9
.= ((((p `1) - x) ^2) + ((p2 - (a * r)) ^2)) + (((r - (a * r)) ^2) + ((2 * (r - (a * r))) * ((a * r) - p2)))
.= (|.|[((p `1) - x),(p2 - (a * r))]|.| ^2) + (((r - (a * r)) ^2) + ((2 * (r - (a * r))) * ((a * r) - p2))) by Th9
.= ((a * r) ^2) + (((((r * r) - (r * p2)) + ((r * a) * r)) - (r * p2)) - (((((a * r) * r) - ((a * r) * p2)) + ((a * r) ^2)) - ((a * r) * p2))) by A6, A4, EUCLID:62
.= (r ^2) - (((1 + 1) * p2) * s) ;
then  |.(p - |[x,r]|).| ^2 < r ^2 by XREAL_1:44;
then  |.(p - |[x,r]|).| < r by SQUARE_1:15;
then  p in Ball (|[x,r]|,r) by TOPREAL9:7;
then (+ (x,r)) . p = (|.(|[x,0]| - p).| ^2) / ((2 * r) * p2) by A6, Def5
.= (|.(p - |[x,0]|).| ^2) / ((2 * r) * p2) by TOPRNS_1:27
.= (|.|[((p `1) - x),(p2 - 0)]|.| ^2) / ((2 * r) * p2) by A6, EUCLID:62
.= ((((p `1) - x) ^2) + (p2 ^2)) / ((2 * r) * p2) by A9, A10, JGRAPH_1:29 ;
then A11: (+ (x,r)) . p = (((2 * p2) * r) * a) / ((2 * r) * p2) by A7;
 a * (((2 * p2) * r) / ((2 * r) * p2)) = a * 1 by XCMPLX_1:60;
hence  (+ (x,r)) . p = a by A11, XCMPLX_1:74; ::_thesis: verum
end;
supposeA12: a = 1 ; ::_thesis: (+ (x,r)) . p = a
A13: not p2 is negative ;
 not p in Ball (|[x,r]|,r) by A12, A4, TOPREAL9:7;
hence  (+ (x,r)) . p = a by A13, A6, A12, Def5; ::_thesis: verum
end;
end;
end;

theorem Th63: :: TOPGEN_5:63
for p being Point of (TOP-REAL 2)
for x, a being real number
for r being real positive number st a <= 1 & |.(p - |[x,(r * a)]|).| < r * a holds
(+ (x,r)) . p < a
proof
let p be Point of (TOP-REAL 2); ::_thesis: for x, a being real number
for r being real positive number st a <= 1 & |.(p - |[x,(r * a)]|).| < r * a holds
(+ (x,r)) . p < a
set p1 = p `1 ;
set p2 = p `2 ;
let x, a be real number ; ::_thesis: for r being real positive number st a <= 1 & |.(p - |[x,(r * a)]|).| < r * a holds
(+ (x,r)) . p < a
let r be real positive number ; ::_thesis: ( a <= 1 & |.(p - |[x,(r * a)]|).| < r * a implies (+ (x,r)) . p < a )
assume A1: a <= 1 ; ::_thesis: ( not |.(p - |[x,(r * a)]|).| < r * a or (+ (x,r)) . p < a )
A2: |[((p `1) - x),((p `2) - (r * a))]| `2 = (p `2) - (r * a) by EUCLID:52;
A3: p = |[(p `1),(p `2)]| by EUCLID:53;
then A4: ( p `2 = 0 implies p in y=0-line ) ;
set r1 = r * a;
set r2 = r * 1;
A5: |[((p `1) - x),((p `2) - (r * a))]| `1 = (p `1) - x by EUCLID:52;
assume A6: |.(p - |[x,(r * a)]|).| < r * a ; ::_thesis: (+ (x,r)) . p < a
then reconsider r1 = r * a as real positive number ;
A7: p in Ball (|[x,r1]|,r1) by A6, TOPREAL9:7;
 |.(p - |[x,(r * a)]|).| ^2 < (r * a) ^2 by A6, SQUARE_1:16;
then  |.|[((p `1) - x),((p `2) - (r * a))]|.| ^2 < (r * a) ^2 by A3, EUCLID:62;
then  (((p `1) - x) ^2) + (((p `2) - (r * a)) ^2) < (r * a) ^2 by A5, A2, JGRAPH_1:29;
then  (((((p `1) - x) ^2) + ((p `2) ^2)) - ((2 * (p `2)) * (r * a))) + ((r * a) ^2) < (r * a) ^2 ;
then  ((((p `1) - x) ^2) + ((p `2) ^2)) - (((2 * (p `2)) * r) * a) < 0 by XREAL_1:31;
then A8: (((p `1) - x) ^2) + ((p `2) ^2) < ((2 * (p `2)) * r) * a by XREAL_1:48;
A9: Ball (|[x,r1]|,r1) misses y=0-line by Th21;
 Ball (|[x,r1]|,r1) c= y>=0-plane by Th20;
then reconsider p2 = p `2 as real positive number by A3, A7, Th18, A9, A4, XBOOLE_0:3;
A10: |[((p `1) - x),(p2 - 0)]| `1 = (p `1) - x by EUCLID:52;
A11: |[((p `1) - x),p2]| `2 = p2 by EUCLID:52;
 Ball (|[x,r1]|,r1) c= Ball (|[x,(r * 1)]|,(r * 1)) by A1, Th23, XREAL_1:64;
then (+ (x,r)) . p = (|.(|[x,0]| - p).| ^2) / ((2 * r) * p2) by A3, A7, Def5
.= (|.(p - |[x,0]|).| ^2) / ((2 * r) * p2) by TOPRNS_1:27
.= (|.|[((p `1) - x),(p2 - 0)]|.| ^2) / ((2 * r) * p2) by A3, EUCLID:62
.= ((((p `1) - x) ^2) + (p2 ^2)) / ((2 * r) * p2) by A10, A11, JGRAPH_1:29 ;
then A12: (+ (x,r)) . p < (((2 * p2) * r) * a) / ((2 * r) * p2) by A8, XREAL_1:74;
 a * (((2 * p2) * r) / ((2 * r) * p2)) = a * 1 by XCMPLX_1:60;
hence  (+ (x,r)) . p < a by A12, XCMPLX_1:74; ::_thesis: verum
end;

theorem Th64: :: TOPGEN_5:64
for p being Point of (TOP-REAL 2) st p `2 >= 0 holds
for x, a being real number
for r being real positive number st 0 <= a & a < 1 & |.(p - |[x,(r * a)]|).| > r * a holds
(+ (x,r)) . p > a
proof
let p be Point of (TOP-REAL 2); ::_thesis: ( p `2 >= 0 implies for x, a being real number
for r being real positive number st 0 <= a & a < 1 & |.(p - |[x,(r * a)]|).| > r * a holds
(+ (x,r)) . p > a )
assume A1: p `2 >= 0 ; ::_thesis: for x, a being real number
for r being real positive number st 0 <= a & a < 1 & |.(p - |[x,(r * a)]|).| > r * a holds
(+ (x,r)) . p > a
set p1 = p `1 ;
set p2 = p `2 ;
reconsider p2 = p `2 as real non negative number by A1;
let x, a be real number ; ::_thesis: for r being real positive number st 0 <= a & a < 1 & |.(p - |[x,(r * a)]|).| > r * a holds
(+ (x,r)) . p > a
let r be real positive number ; ::_thesis: ( 0 <= a & a < 1 & |.(p - |[x,(r * a)]|).| > r * a implies (+ (x,r)) . p > a )
assume that
A2: 0 <= a and
A3: a < 1 ; ::_thesis: ( not |.(p - |[x,(r * a)]|).| > r * a or (+ (x,r)) . p > a )
reconsider a9 = a as real non negative number by A2;
reconsider ra = r * a as Real by XREAL_0:def_1;
assume A4: |.(p - |[x,(r * a)]|).| > r * a ; ::_thesis: (+ (x,r)) . p > a
|.(|[x,0]| - |[x,(r * a)]|).| = |.(|[x,(r * a)]| - |[x,0]|).| by TOPRNS_1:27
.= |.|[(x - x),(ra - 0)]|.| by EUCLID:62
.= abs ra by TOPREAL6:23
.= r * a9 by ABSVALUE:def_1 ;
then A5: ( p `1 <> x or p2 <> 0 ) by A4, EUCLID:53;
A6: p = |[(p `1),(p `2)]| by EUCLID:53;
then reconsider z = p as Element of Niemytzki-plane by A1, Lm1, Th18;
A7: (+ (x,r)) . z in the carrier of I[01] ;
percases ( a = 0 or (+ (x,r)) . p = 1 or ( a > 0 & (+ (x,r)) . z <> 1 ) ) by A2;
supposeA8: a = 0 ; ::_thesis: (+ (x,r)) . p > a
then  p <> |[x,(r * 0)]| by A4, TOPRNS_1:28;
then  (+ (x,r)) . p <> 0 by A1, Th60;
hence  (+ (x,r)) . p > a by A7, A8, BORSUK_1:40, XXREAL_1:1; ::_thesis: verum
end;
suppose (+ (x,r)) . p = 1 ; ::_thesis: (+ (x,r)) . p > a
hence  (+ (x,r)) . p > a by A3; ::_thesis: verum
end;
supposeA9: ( a > 0 & (+ (x,r)) . z <> 1 ) ; ::_thesis: (+ (x,r)) . p > a
A10: |[((p `1) - x),(p2 - 0)]| `1 = (p `1) - x by EUCLID:52;
A11: |[((p `1) - x),p2]| `2 = p2 by EUCLID:52;
 not p2 is negative ;
then A12: p in Ball (|[x,r]|,r) by A6, A5, A9, Def5;
then A13: (+ (x,r)) . p = (|.(|[x,0]| - p).| ^2) / ((2 * r) * p2) by A6, Def5
.= (|.(p - |[x,0]|).| ^2) / ((2 * r) * p2) by TOPRNS_1:27
.= (|.|[((p `1) - x),(p2 - 0)]|.| ^2) / ((2 * r) * p2) by A6, EUCLID:62
.= ((((p `1) - x) ^2) + (p2 ^2)) / ((2 * r) * p2) by A10, A11, JGRAPH_1:29 ;
 |.(p - |[x,(r * a)]|).| ^2 > (r * a) ^2 by A2, A4, SQUARE_1:16;
then A14: |.|[((p `1) - x),(p2 - (r * a))]|.| ^2 > (r * a) ^2 by A6, EUCLID:62;
A15: |[((p `1) - x),(p2 - (r * a))]| `2 = p2 - (r * a) by EUCLID:52;
 |[((p `1) - x),(p2 - (r * a))]| `1 = (p `1) - x by EUCLID:52;
then  (((p `1) - x) ^2) + ((p2 - (r * a)) ^2) > (r * a) ^2 by A14, A15, JGRAPH_1:29;
then  (((((p `1) - x) ^2) + (p2 ^2)) - ((2 * p2) * (r * a))) + ((r * a) ^2) > (r * a) ^2 ;
then  ((((p `1) - x) ^2) + (p2 ^2)) - (((2 * p2) * r) * a) > 0 by XREAL_1:32;
then A16: (((p `1) - x) ^2) + (p2 ^2) > ((2 * p2) * r) * a by XREAL_1:47;
A17: ( p2 = 0 implies p in y=0-line ) by A6;
 Ball (|[x,r]|,r) misses y=0-line by Th21;
then reconsider p2 = p2 as real positive number by A12, A17, XBOOLE_0:3;
A18: a * (((2 * p2) * r) / ((2 * r) * p2)) = a * 1 by XCMPLX_1:60;
 (+ (x,r)) . p > (((2 * p2) * r) * a) / ((2 * r) * p2) by A13, A16, XREAL_1:74;
hence  (+ (x,r)) . p > a by A18, XCMPLX_1:74; ::_thesis: verum
end;
end;
end;

theorem Th65: :: TOPGEN_5:65
for p being Point of (TOP-REAL 2) st p `2 >= 0 holds
for x, a, b being real number
for r being real positive number st 0 <= a & b <= 1 & (+ (x,r)) . p in ].a,b.[ holds
ex r1 being real positive number st
( r1 <= p `2 & Ball (p,r1) c= (+ (x,r)) " ].a,b.[ )
proof
let p be Point of (TOP-REAL 2); ::_thesis: ( p `2 >= 0 implies for x, a, b being real number
for r being real positive number st 0 <= a & b <= 1 & (+ (x,r)) . p in ].a,b.[ holds
ex r1 being real positive number st
( r1 <= p `2 & Ball (p,r1) c= (+ (x,r)) " ].a,b.[ ) )
assume A1: p `2 >= 0 ; ::_thesis: for x, a, b being real number
for r being real positive number st 0 <= a & b <= 1 & (+ (x,r)) . p in ].a,b.[ holds
ex r1 being real positive number st
( r1 <= p `2 & Ball (p,r1) c= (+ (x,r)) " ].a,b.[ )
let x, a, b be real number ; ::_thesis: for r being real positive number st 0 <= a & b <= 1 & (+ (x,r)) . p in ].a,b.[ holds
ex r1 being real positive number st
( r1 <= p `2 & Ball (p,r1) c= (+ (x,r)) " ].a,b.[ )
let r be real positive number ; ::_thesis: ( 0 <= a & b <= 1 & (+ (x,r)) . p in ].a,b.[ implies ex r1 being real positive number st
( r1 <= p `2 & Ball (p,r1) c= (+ (x,r)) " ].a,b.[ ) )
A2: p = |[(p `1),(p `2)]| by EUCLID:53;
A3: Ball (|[x,r]|,r) misses y=0-line by Th21;
A4: |[((p `1) - x),(p `2)]| `1 = (p `1) - x by EUCLID:52;
assume that
A5: 0 <= a and
A6: b <= 1 and
A7: (+ (x,r)) . p in ].a,b.[ ; ::_thesis: ex r1 being real positive number st
( r1 <= p `2 & Ball (p,r1) c= (+ (x,r)) " ].a,b.[ )
A8: (+ (x,r)) . p > a by A7, XXREAL_1:4;
A9: (+ (x,r)) . p < b by A7, XXREAL_1:4;
then A10: ( p in Ball (|[x,r]|,r) or ( p `1 = x & p `2 = 0 & p <> |[x,0]| ) ) by A8, A1, A2, A5, A6, Def5;
then A11: (+ (x,r)) . p = (|.(|[x,0]| - p).| ^2) / ((2 * r) * (p `2)) by A1, A2, Def5;
 ( p `2 = 0 implies p in y=0-line ) by A2;
then reconsider p2 = p `2 , b9 = b as real positive number by A3, A1, A5, A8, A7, A10, EUCLID:53, XBOOLE_0:3, XXREAL_1:4;
A12: |[((p `1) - x),p2]| `2 = p2 by EUCLID:52;
A13: (2 * r) * p2 > 0 ;
then  |.(|[x,0]| - p).| ^2 > ((2 * r) * (p `2)) * a by A11, A8, XREAL_1:79;
then  |.(p - |[x,0]|).| ^2 > ((2 * r) * (p `2)) * a by TOPRNS_1:27;
then  |.|[((p `1) - x),((p `2) - 0)]|.| ^2 > ((2 * r) * (p `2)) * a by A2, EUCLID:62;
then  (((p `1) - x) ^2) + (p2 ^2) > ((2 * r) * p2) * a by A4, A12, JGRAPH_1:29;
then  ((((p `1) - x) ^2) + (p2 ^2)) - (((2 * r) * p2) * a) > 0 by XREAL_1:50;
then  (((((p `1) - x) ^2) + (p2 ^2)) - (((2 * r) * p2) * a)) + ((r * a) ^2) > (r * a) ^2 by XREAL_1:29;
then A14: (((p `1) - x) ^2) + ((p2 - (r * a)) ^2) > (r * a) ^2 ;
 |.(|[x,0]| - p).| ^2 < ((2 * r) * (p `2)) * b by A13, A11, A9, XREAL_1:77;
then  |.(p - |[x,0]|).| ^2 < ((2 * r) * (p `2)) * b by TOPRNS_1:27;
then  |.|[((p `1) - x),((p `2) - 0)]|.| ^2 < ((2 * r) * (p `2)) * b by A2, EUCLID:62;
then  (((p `1) - x) ^2) + (p2 ^2) < ((2 * r) * p2) * b by A4, A12, JGRAPH_1:29;
then  ((((p `1) - x) ^2) + (p2 ^2)) - (((2 * r) * p2) * b) < 0 by XREAL_1:49;
then  (((((p `1) - x) ^2) + (p2 ^2)) - (((2 * r) * p2) * b)) + ((r * b) ^2) < (r * b) ^2 by XREAL_1:30;
then A15: (((p `1) - x) ^2) + ((p2 - (r * b)) ^2) < (r * b) ^2 ;
A16: |[((p `1) - x),(p2 - (r * b))]| `2 = p2 - (r * b) by EUCLID:52;
A17: |[((p `1) - x),(p2 - (r * a))]| `2 = p2 - (r * a) by EUCLID:52;
 |[((p `1) - x),(p2 - (r * a))]| `1 = (p `1) - x by EUCLID:52;
then  |.|[((p `1) - x),(p2 - (r * a))]|.| ^2 > (r * a) ^2 by A14, A17, JGRAPH_1:29;
then  |.(p - |[x,(r * a)]|).| ^2 > (r * a) ^2 by A2, EUCLID:62;
then A18: |.(p - |[x,(r * a)]|).| > r * a by SQUARE_1:48;
A19: ((r * b) - |.(p - |[x,(r * b)]|).|) + |.(p - |[x,(r * b)]|).| = r * b ;
set r1 = min (((r * b) - |.(p - |[x,(r * b)]|).|),(|.(p - |[x,(r * a)]|).| - (r * a)));
A20: |.(p - |[x,(r * b)]|).| = |.(|[x,(r * b)]| - p).| by TOPRNS_1:27;
 |[((p `1) - x),(p2 - (r * b))]| `1 = (p `1) - x by EUCLID:52;
then  |.|[((p `1) - x),(p2 - (r * b))]|.| ^2 < (r * b) ^2 by A15, A16, JGRAPH_1:29;
then A21: |.(p - |[x,(r * b)]|).| ^2 < (r * b) ^2 by A2, EUCLID:62;
 r * b9 >= 0 ;
then  |.(p - |[x,(r * b)]|).| < r * b by A21, SQUARE_1:48;
then  ( ( (r * b) - |.(p - |[x,(r * b)]|).| > 0 & min (((r * b) - |.(p - |[x,(r * b)]|).|),(|.(p - |[x,(r * a)]|).| - (r * a))) = (r * b) - |.(p - |[x,(r * b)]|).| ) or ( |.(p - |[x,(r * a)]|).| - (r * a) > 0 & min (((r * b) - |.(p - |[x,(r * b)]|).|),(|.(p - |[x,(r * a)]|).| - (r * a))) = |.(p - |[x,(r * a)]|).| - (r * a) ) ) by A18, XREAL_1:50, XXREAL_0:15;
then reconsider r1 = min (((r * b) - |.(p - |[x,(r * b)]|).|),(|.(p - |[x,(r * a)]|).| - (r * a))) as real positive number ;
take  r1 ; ::_thesis: ( r1 <= p `2 & Ball (p,r1) c= (+ (x,r)) " ].a,b.[ )
 r1 <= (r * b) - |.(p - |[x,(r * b)]|).| by XXREAL_0:17;
then A22: |.(|[x,(r * b)]| - p).| + r1 <= r * b by A20, A19, XREAL_1:6;
|.(p - |[(p `1),0]|).| = |.|[((p `1) - (p `1)),(p2 - 0)]|.| by A2, EUCLID:62
.= abs p2 by TOPREAL6:23
.= p2 by ABSVALUE:def_1 ;
then A23: |.(|[x,(r * b)]| - |[(p `1),0]|).| <= |.(|[x,(r * b)]| - p).| + p2 by TOPRNS_1:34;
hereby  ::_thesis: Ball (p,r1) c= (+ (x,r)) " ].a,b.[
assume  r1 > p `2 ; ::_thesis: contradiction
then  |.(|[x,(r * b)]| - p).| + p2 < |.(|[x,(r * b)]| - p).| + r1 by XREAL_1:8;
then  |.(|[x,(r * b)]| - |[(p `1),0]|).| < |.(|[x,(r * b)]| - p).| + r1 by A23, XXREAL_0:2;
then  |.(|[x,(r * b)]| - |[(p `1),0]|).| < r * b by A22, XXREAL_0:2;
then  |.(|[(p `1),0]| - |[x,(r * b)]|).| < r * b by TOPRNS_1:27;
then A24: |[(p `1),0]| in Ball (|[x,(r * b)]|,(r * b)) by TOPREAL9:7;
 |[(p `1),0]| in y=0-line ;
then  Ball (|[x,(r * b)]|,(r * b9)) meets y=0-line by A24, XBOOLE_0:3;
hence  contradiction by Th21; ::_thesis: verum
end;
then A25: Ball (p,r1) c= y>=0-plane by A2, Th20;
let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in Ball (p,r1) or u in (+ (x,r)) " ].a,b.[ )
assume A26: u in Ball (p,r1) ; ::_thesis: u in (+ (x,r)) " ].a,b.[
then reconsider q = u as Point of (TOP-REAL 2) ;
A27: |.(q - p).| < r1 by A26, TOPREAL9:7;
then  |.(q - p).| + |.(p - |[x,(r * b)]|).| < r1 + |.(p - |[x,(r * b)]|).| by XREAL_1:8;
then A28: |.(q - p).| + |.(p - |[x,(r * b)]|).| < r * b by A20, A22, XXREAL_0:2;
 |.(q - |[x,(r * b)]|).| <= |.(q - p).| + |.(p - |[x,(r * b)]|).| by TOPRNS_1:34;
then  |.(q - |[x,(r * b)]|).| < r * b by A28, XXREAL_0:2;
then A29: (+ (x,r)) . q < b by A6, Th63;
A30: |.(p - |[x,(r * a)]|).| <= |.(p - q).| + |.(q - |[x,(r * a)]|).| by TOPRNS_1:34;
 a < b by A8, A9, XXREAL_0:2;
then A31: a < 1 by A6, XXREAL_0:2;
 |.(p - |[x,(r * a)]|).| - (r * a) >= r1 by XXREAL_0:17;
then  |.(p - |[x,(r * a)]|).| - (r * a) > |.(q - p).| by A27, XXREAL_0:2;
then A32: |.(p - |[x,(r * a)]|).| - |.(q - p).| > r * a by XREAL_1:12;
 |.(p - q).| = |.(q - p).| by TOPRNS_1:27;
then  |.(q - |[x,(r * a)]|).| >= |.(p - |[x,(r * a)]|).| - |.(q - p).| by A30, XREAL_1:20;
then A33: |.(q - |[x,(r * a)]|).| > r * a by A32, XXREAL_0:2;
 q = |[(q `1),(q `2)]| by EUCLID:53;
then  q `2 >= 0 by A25, A26, Th18;
then  (+ (x,r)) . q > a by A33, A5, A31, Th64;
then  (+ (x,r)) . q in ].a,b.[ by A29, XXREAL_1:4;
hence  u in (+ (x,r)) " ].a,b.[ by A25, A26, Lm1, FUNCT_2:38; ::_thesis: verum
end;

theorem Th66: :: TOPGEN_5:66
for x being real number
for a, r being real positive number holds Ball (|[x,(r * a)]|,(r * a)) c= (+ (x,r)) " ].0,a.[
proof
let x be real number ; ::_thesis: for a, r being real positive number holds Ball (|[x,(r * a)]|,(r * a)) c= (+ (x,r)) " ].0,a.[
let a, r be real positive number ; ::_thesis: Ball (|[x,(r * a)]|,(r * a)) c= (+ (x,r)) " ].0,a.[
let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in Ball (|[x,(r * a)]|,(r * a)) or u in (+ (x,r)) " ].0,a.[ )
assume A1: u in Ball (|[x,(r * a)]|,(r * a)) ; ::_thesis: u in (+ (x,r)) " ].0,a.[
then reconsider p = u as Point of (TOP-REAL 2) ;
 Ball (|[x,(r * a)]|,(r * a)) c= y>=0-plane by Th20;
then reconsider q = p as Point of Niemytzki-plane by A1, Def3;
 q = |[(p `1),(p `2)]| by EUCLID:53;
then A2: p `2 >= 0 by Lm1, Th18;
A3: now__::_thesis:_not_(+_(x,r))_._p_=_0
assume  (+ (x,r)) . p = 0 ; ::_thesis: contradiction
then  p = |[x,0]| by A2, Th60;
then A4: p in y=0-line by Th15;
 Ball (|[x,(r * a)]|,(r * a)) misses y=0-line by Th21;
hence  contradiction by A4, A1, XBOOLE_0:3; ::_thesis: verum
end;
A5: (+ (x,r)) . q <= 1 by BORSUK_1:40, XXREAL_1:1;
percases ( a > 1 or a <= 1 ) ;
supposeA6: a > 1 ; ::_thesis: u in (+ (x,r)) " ].0,a.[
A7: (+ (x,r)) . q > 0 by A3, BORSUK_1:40, XXREAL_1:1;
 (+ (x,r)) . q < a by A6, A5, XXREAL_0:2;
then  (+ (x,r)) . q in ].0,a.[ by A7, XXREAL_1:4;
hence  u in (+ (x,r)) " ].0,a.[ by FUNCT_2:38; ::_thesis: verum
end;
supposeA8: a <= 1 ; ::_thesis: u in (+ (x,r)) " ].0,a.[
 |.(p - |[x,(r * a)]|).| < r * a by A1, TOPREAL9:7;
then A9: (+ (x,r)) . p < a by A8, Th63;
 (+ (x,r)) . q > 0 by A3, BORSUK_1:40, XXREAL_1:1;
then  (+ (x,r)) . q in ].0,a.[ by A9, XXREAL_1:4;
hence  u in (+ (x,r)) " ].0,a.[ by FUNCT_2:38; ::_thesis: verum
end;
end;
end;

theorem Th67: :: TOPGEN_5:67
for x being real number
for a, r being real positive number holds (Ball (|[x,(r * a)]|,(r * a))) \/ {|[x,0]|} c= (+ (x,r)) " [.0,a.[
proof
let x be real number ; ::_thesis: for a, r being real positive number holds (Ball (|[x,(r * a)]|,(r * a))) \/ {|[x,0]|} c= (+ (x,r)) " [.0,a.[
let a, r be real positive number ; ::_thesis: (Ball (|[x,(r * a)]|,(r * a))) \/ {|[x,0]|} c= (+ (x,r)) " [.0,a.[
let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in (Ball (|[x,(r * a)]|,(r * a))) \/ {|[x,0]|} or u in (+ (x,r)) " [.0,a.[ )
assume A1: u in (Ball (|[x,(r * a)]|,(r * a))) \/ {|[x,0]|} ; ::_thesis: u in (+ (x,r)) " [.0,a.[
then A2: ( u in Ball (|[x,(r * a)]|,(r * a)) or u in {|[x,0]|} ) by XBOOLE_0:def_3;
reconsider p = u as Point of (TOP-REAL 2) by A1;
A3: |[x,0]| in y>=0-plane by Th18;
 Ball (|[x,(r * a)]|,(r * a)) c= y>=0-plane by Th20;
then reconsider q = p as Point of Niemytzki-plane by A3, A2, Lm1, TARSKI:def_1;
A4: (+ (x,r)) . q <= 1 by BORSUK_1:40, XXREAL_1:1;
A5: (+ (x,r)) . q >= 0 by BORSUK_1:40, XXREAL_1:1;
percases ( a > 1 or ( a <= 1 & u in Ball (|[x,(r * a)]|,(r * a)) ) or u = |[x,0]| ) by A2, TARSKI:def_1;
suppose a > 1 ; ::_thesis: u in (+ (x,r)) " [.0,a.[
then  (+ (x,r)) . q < a by A4, XXREAL_0:2;
then  (+ (x,r)) . q in [.0,a.[ by A5, XXREAL_1:3;
hence  u in (+ (x,r)) " [.0,a.[ by FUNCT_2:38; ::_thesis: verum
end;
supposeA6: ( a <= 1 & u in Ball (|[x,(r * a)]|,(r * a)) ) ; ::_thesis: u in (+ (x,r)) " [.0,a.[
then  |.(p - |[x,(r * a)]|).| < r * a by TOPREAL9:7;
then  (+ (x,r)) . p < a by A6, Th63;
then  (+ (x,r)) . q in [.0,a.[ by A5, XXREAL_1:3;
hence  u in (+ (x,r)) " [.0,a.[ by FUNCT_2:38; ::_thesis: verum
end;
suppose u = |[x,0]| ; ::_thesis: u in (+ (x,r)) " [.0,a.[
then  (+ (x,r)) . u = 0 by Def5;
then  (+ (x,r)) . q in [.0,a.[ by XXREAL_1:3;
hence  u in (+ (x,r)) " [.0,a.[ by FUNCT_2:38; ::_thesis: verum
end;
end;
end;

theorem Th68: :: TOPGEN_5:68
for p being Point of (TOP-REAL 2) st p `2 >= 0 holds
for x, a being real number
for r being real positive number st 0 < (+ (x,r)) . p & (+ (x,r)) . p < a & a <= 1 holds
p in Ball (|[x,(r * a)]|,(r * a))
proof
let p be Point of (TOP-REAL 2); ::_thesis: ( p `2 >= 0 implies for x, a being real number
for r being real positive number st 0 < (+ (x,r)) . p & (+ (x,r)) . p < a & a <= 1 holds
p in Ball (|[x,(r * a)]|,(r * a)) )
assume A1: p `2 >= 0 ; ::_thesis: for x, a being real number
for r being real positive number st 0 < (+ (x,r)) . p & (+ (x,r)) . p < a & a <= 1 holds
p in Ball (|[x,(r * a)]|,(r * a))
let x, a be real number ; ::_thesis: for r being real positive number st 0 < (+ (x,r)) . p & (+ (x,r)) . p < a & a <= 1 holds
p in Ball (|[x,(r * a)]|,(r * a))
A2: p = |[(p `1),(p `2)]| by EUCLID:53;
let r be real positive number ; ::_thesis: ( 0 < (+ (x,r)) . p & (+ (x,r)) . p < a & a <= 1 implies p in Ball (|[x,(r * a)]|,(r * a)) )
set r1 = r * a;
assume that
A3: 0 < (+ (x,r)) . p and
A4: (+ (x,r)) . p < a and
A5: a <= 1 ; ::_thesis: p in Ball (|[x,(r * a)]|,(r * a))
A6: ( x <> p `1 implies p <> |[(p `1),0]| ) by A4, A5, Th61;
A7: p <> |[x,0]| by A3, Def5;
assume  not p in Ball (|[x,(r * a)]|,(r * a)) ; ::_thesis: contradiction
then  |.(p - |[x,(r * a)]|).| >= r * a by TOPREAL9:7;
then  ( |.(p - |[x,(r * a)]|).| = r * a or ( |.(p - |[x,(r * a)]|).| > r * a & ( a < 1 or a = 1 ) ) ) by A5, XXREAL_0:1;
then  ( (+ (x,r)) . p = a or ( a < 1 & (+ (x,r)) . p > a ) or ( a = 1 & not p in Ball (|[x,r]|,r) ) ) by A1, A2, A3, A5, A7, A6, Th62, Th64, TOPREAL9:7;
hence  contradiction by A1, A2, A4, A7, A6, Def5; ::_thesis: verum
end;

theorem Th69: :: TOPGEN_5:69
for p being Point of (TOP-REAL 2) st p `2 > 0 holds
for x, a being real number
for r being real positive number st 0 <= a & a < (+ (x,r)) . p holds
ex r1 being real positive number st
( r1 <= p `2 & Ball (p,r1) c= (+ (x,r)) " ].a,1.] )
proof
let p be Point of (TOP-REAL 2); ::_thesis: ( p `2 > 0 implies for x, a being real number
for r being real positive number st 0 <= a & a < (+ (x,r)) . p holds
ex r1 being real positive number st
( r1 <= p `2 & Ball (p,r1) c= (+ (x,r)) " ].a,1.] ) )
assume A1: p `2 > 0 ; ::_thesis: for x, a being real number
for r being real positive number st 0 <= a & a < (+ (x,r)) . p holds
ex r1 being real positive number st
( r1 <= p `2 & Ball (p,r1) c= (+ (x,r)) " ].a,1.] )
let x, a be real number ; ::_thesis: for r being real positive number st 0 <= a & a < (+ (x,r)) . p holds
ex r1 being real positive number st
( r1 <= p `2 & Ball (p,r1) c= (+ (x,r)) " ].a,1.] )
let r be real positive number ; ::_thesis: ( 0 <= a & a < (+ (x,r)) . p implies ex r1 being real positive number st
( r1 <= p `2 & Ball (p,r1) c= (+ (x,r)) " ].a,1.] ) )
set f = + (x,r);
assume that
A2: 0 <= a and
A3: a < (+ (x,r)) . p ; ::_thesis: ex r1 being real positive number st
( r1 <= p `2 & Ball (p,r1) c= (+ (x,r)) " ].a,1.] )
A4: p = |[(p `1),(p `2)]| by EUCLID:53;
then  p in the carrier of Niemytzki-plane by A1, Lm1;
then  (+ (x,r)) . p in the carrier of I[01] by FUNCT_2:5;
then  (+ (x,r)) . p <= 1 by BORSUK_1:40, XXREAL_1:1;
then A5: a < 1 by A3, XXREAL_0:2;
percases ( a = 0 or a > 0 ) by A2;
supposeA6: a = 0 ; ::_thesis: ex r1 being real positive number st
( r1 <= p `2 & Ball (p,r1) c= (+ (x,r)) " ].a,1.] )
reconsider r1 = p `2 as real positive number by A1;
reconsider A = Ball (p,r1) as Subset of Niemytzki-plane by A4, Lm1, Th20;
take  r1 ; ::_thesis: ( r1 <= p `2 & Ball (p,r1) c= (+ (x,r)) " ].a,1.] )
thus  r1 <= p `2 ; ::_thesis: Ball (p,r1) c= (+ (x,r)) " ].a,1.]
let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in Ball (p,r1) or u in (+ (x,r)) " ].a,1.] )
assume A7: u in Ball (p,r1) ; ::_thesis: u in (+ (x,r)) " ].a,1.]
then reconsider q = u as Point of (TOP-REAL 2) ;
A8: q = |[(q `1),(q `2)]| by EUCLID:53;
 q in A by A7;
then A9: q `2 >= 0 by A8, Lm1, Th18;
 q in A by A7;
then reconsider z = q as Element of Niemytzki-plane ;
A10: (+ (x,r)) . z >= 0 by BORSUK_1:40, XXREAL_1:1;
 y=0-line misses Ball (p,r1) by A4, Th21;
then  not q in y=0-line by A7, XBOOLE_0:3;
then A11: q `2 <> 0 by A8;
A12: (+ (x,r)) . z <= 1 by BORSUK_1:40, XXREAL_1:1;
 |[x,0]| `2 = 0 by EUCLID:52;
then  (+ (x,r)) . q <> 0 by A11, A9, Th60;
then  (+ (x,r)) . z in ].a,1.] by A10, A12, A6, XXREAL_1:2;
hence  u in (+ (x,r)) " ].a,1.] by FUNCT_2:38; ::_thesis: verum
end;
suppose a > 0 ; ::_thesis: ex r1 being real positive number st
( r1 <= p `2 & Ball (p,r1) c= (+ (x,r)) " ].a,1.] )
then reconsider b = a as real positive number ;
set r1 = min ((|.(p - |[x,(r * a)]|).| - (r * a)),(p `2));
A13: ( min ((|.(p - |[x,(r * a)]|).| - (r * a)),(p `2)) = |.(p - |[x,(r * a)]|).| - (r * a) or min ((|.(p - |[x,(r * a)]|).| - (r * a)),(p `2)) = p `2 ) by XXREAL_0:def_9;
A14: b <> (+ (x,r)) . p by A3;
 not |.(p - |[x,(r * a)]|).| < r * a by A3, A5, Th63;
then  |.(p - |[x,(r * a)]|).| > r * a by A14, A1, A5, Th62, XXREAL_0:1;
then reconsider r1 = min ((|.(p - |[x,(r * a)]|).| - (r * a)),(p `2)) as real positive number by A13, A1, XREAL_1:50;
take  r1 ; ::_thesis: ( r1 <= p `2 & Ball (p,r1) c= (+ (x,r)) " ].a,1.] )
thus  r1 <= p `2 by XXREAL_0:17; ::_thesis: Ball (p,r1) c= (+ (x,r)) " ].a,1.]
then reconsider A = Ball (p,r1) as Subset of Niemytzki-plane by A4, Lm1, Th20;
let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in Ball (p,r1) or u in (+ (x,r)) " ].a,1.] )
assume A15: u in Ball (p,r1) ; ::_thesis: u in (+ (x,r)) " ].a,1.]
then reconsider q = u as Point of (TOP-REAL 2) ;
 u in A by A15;
then reconsider z = q as Point of Niemytzki-plane ;
A16: q = |[(q `1),(q `2)]| by EUCLID:53;
 q in A by A15;
then A17: q `2 >= 0 by A16, Lm1, Th18;
A18: now__::_thesis:_(+_(x,r))_._z_>_a
 |.(p - |[x,(r * a)]|).| - (r * a) >= r1 by XXREAL_0:17;
then A19: (r * a) + r1 <= (|.(p - |[x,(r * a)]|).| - (r * a)) + (r * a) by XREAL_1:6;
assume  not (+ (x,r)) . z > a ; ::_thesis: contradiction
then  |.(q - |[x,(r * a)]|).| <= r * a by A2, A5, A17, Th64;
then A20: |.(|[x,(r * a)]| - q).| <= r * a by TOPRNS_1:27;
A21: |.(|[x,(r * a)]| - q).| + |.(q - p).| >= |.(|[x,(r * a)]| - p).| by TOPRNS_1:34;
 |.(q - p).| < r1 by A15, TOPREAL9:7;
then  |.(|[x,(r * a)]| - q).| + |.(q - p).| < (r * a) + r1 by A20, XREAL_1:8;
then  |.(|[x,(r * a)]| - p).| < (r * a) + r1 by A21, XXREAL_0:2;
hence  contradiction by A19, TOPRNS_1:27; ::_thesis: verum
end;
 (+ (x,r)) . z <= 1 by BORSUK_1:40, XXREAL_1:1;
then  (+ (x,r)) . z in ].a,1.] by A18, XXREAL_1:2;
hence  u in (+ (x,r)) " ].a,1.] by FUNCT_2:38; ::_thesis: verum
end;
end;
end;

theorem Th70: :: TOPGEN_5:70
for p being Point of (TOP-REAL 2) st p `2 = 0 holds
for x being real number
for r being real positive number st (+ (x,r)) . p = 1 holds
ex r1 being real positive number st (Ball (|[(p `1),r1]|,r1)) \/ {p} c= (+ (x,r)) " {1}
proof
let p be Point of (TOP-REAL 2); ::_thesis: ( p `2 = 0 implies for x being real number
for r being real positive number st (+ (x,r)) . p = 1 holds
ex r1 being real positive number st (Ball (|[(p `1),r1]|,r1)) \/ {p} c= (+ (x,r)) " {1} )
assume A1: p `2 = 0 ; ::_thesis: for x being real number
for r being real positive number st (+ (x,r)) . p = 1 holds
ex r1 being real positive number st (Ball (|[(p `1),r1]|,r1)) \/ {p} c= (+ (x,r)) " {1}
let x be real number ; ::_thesis: for r being real positive number st (+ (x,r)) . p = 1 holds
ex r1 being real positive number st (Ball (|[(p `1),r1]|,r1)) \/ {p} c= (+ (x,r)) " {1}
let r be real positive number ; ::_thesis: ( (+ (x,r)) . p = 1 implies ex r1 being real positive number st (Ball (|[(p `1),r1]|,r1)) \/ {p} c= (+ (x,r)) " {1} )
set r1 = (|.(p - |[x,r]|).| - r) / 2;
set f = + (x,r);
A2: p = |[(p `1),(p `2)]| by EUCLID:53;
assume A3: (+ (x,r)) . p = 1 ; ::_thesis: ex r1 being real positive number st (Ball (|[(p `1),r1]|,r1)) \/ {p} c= (+ (x,r)) " {1}
then  (p `1) - x <> 0 by A2, A1, Def5;
then  ((p `1) - x) ^2 > 0 by SQUARE_1:12;
then  (((p `1) - x) ^2) + ((0 - r) ^2) > 0 + ((0 - r) ^2) by XREAL_1:6;
then  |.|[((p `1) - x),((p `2) - r)]|.| ^2 > r ^2 by A1, Th9;
then  |.(p - |[x,r]|).| ^2 > r ^2 by A2, EUCLID:62;
then  |.(p - |[x,r]|).| > r by SQUARE_1:15;
then  |.(p - |[x,r]|).| - r > 0 by XREAL_1:50;
then reconsider r1 = (|.(p - |[x,r]|).| - r) / 2 as real positive number ;
take  r1 ; ::_thesis: (Ball (|[(p `1),r1]|,r1)) \/ {p} c= (+ (x,r)) " {1}
let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in (Ball (|[(p `1),r1]|,r1)) \/ {p} or u in (+ (x,r)) " {1} )
assume A4: u in (Ball (|[(p `1),r1]|,r1)) \/ {p} ; ::_thesis: u in (+ (x,r)) " {1}
then reconsider q = u as Point of (TOP-REAL 2) ;
A5: Ball (|[(p `1),r1]|,r1) c= y>=0-plane by Th20;
 ( u in Ball (|[(p `1),r1]|,r1) or u = p ) by A4, ZFMISC_1:136;
then reconsider z = q as Point of Niemytzki-plane by A5, A1, A2, Lm1, Th18;
A6: q = |[(q `1),(q `2)]| by EUCLID:53;
A7: now__::_thesis:_(_q_in_Ball_(|[(p_`1),r1]|,r1)_implies_(_not_q_in_Ball_(|[x,r]|,r)_&_q_`2_<>_0_)_)
assume A8: q in Ball (|[(p `1),r1]|,r1) ; ::_thesis: ( not q in Ball (|[x,r]|,r) & q `2 <> 0 )
then  |.(q - |[(p `1),r1]|).| < r1 by TOPREAL9:7;
then A9: |.(q - |[(p `1),r1]|).| + |.(|[(p `1),r1]| - p).| < r1 + |.(|[(p `1),r1]| - p).| by XREAL_1:6;
A10: |.r1.| = r1 by ABSVALUE:def_1;
A11: |.(q - |[(p `1),r1]|).| + |.(|[(p `1),r1]| - p).| >= |.(q - p).| by TOPRNS_1:34;
A12: |.(q - p).| + |.(|[x,r]| - q).| >= |.(|[x,r]| - p).| by TOPRNS_1:34;
A13: |.|[0,r1]|.| = |.r1.| by TOPREAL6:23;
 |[(p `1),r1]| - p = |[((p `1) - (p `1)),(r1 - 0)]| by A1, A2, EUCLID:62;
then  r1 + r1 > |.(q - p).| by A9, A11, A10, A13, XXREAL_0:2;
then  (|.(p - |[x,r]|).| - r) + |.(|[x,r]| - q).| > |.(q - p).| + |.(|[x,r]| - q).| by XREAL_1:6;
then  |.(|[x,r]| - p).| < (|.(p - |[x,r]|).| - r) + |.(|[x,r]| - q).| by A12, XXREAL_0:2;
then A14: |.(|[x,r]| - p).| - (|.(p - |[x,r]|).| - r) < |.(|[x,r]| - q).| by XREAL_1:19;
 |.(p - |[x,r]|).| = |.(|[x,r]| - p).| by TOPRNS_1:27;
then  |.(q - |[x,r]|).| > r by A14, TOPRNS_1:27;
hence  not q in Ball (|[x,r]|,r) by TOPREAL9:7; ::_thesis: q `2 <> 0
 ( q `2 = 0 implies ( q in y=0-line & Ball (|[(p `1),r1]|,r1) misses y=0-line ) ) by A6, Th21;
hence  q `2 <> 0 by A8, XBOOLE_0:3; ::_thesis: verum
end;
 z in y>=0-plane by Lm1;
then  q `2 >= 0 by A6, Th18;
then  (+ (x,r)) . z = 1 by A7, A3, A4, A6, Def5, ZFMISC_1:136;
then  (+ (x,r)) . z in {1} by TARSKI:def_1;
hence  u in (+ (x,r)) " {1} by FUNCT_2:38; ::_thesis: verum
end;

theorem Th71: :: TOPGEN_5:71
for T being non empty TopSpace
for S being SubSpace of T
for B being Basis of T holds  {  (A /\ ([#] S)) where A is Subset of T : ( A in B & A meets [#] S )  }  is Basis of S
proof
let T be non empty TopSpace; ::_thesis: for S being SubSpace of T
for B being Basis of T holds  {  (A /\ ([#] S)) where A is Subset of T : ( A in B & A meets [#] S )  }  is Basis of S
let S be SubSpace of T; ::_thesis: for B being Basis of T holds  {  (A /\ ([#] S)) where A is Subset of T : ( A in B & A meets [#] S )  }  is Basis of S
let B be Basis of T; ::_thesis:  {  (A /\ ([#] S)) where A is Subset of T : ( A in B & A meets [#] S )  }  is Basis of S
set X =  {  (A /\ ([#] S)) where A is Subset of T : ( A in B & A meets [#] S )  }  ;
  {  (A /\ ([#] S)) where A is Subset of T : ( A in B & A meets [#] S )  }  c= bool the carrier of S
proof
let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in  {  (A /\ ([#] S)) where A is Subset of T : ( A in B & A meets [#] S )  }  or u in bool the carrier of S )
assume  u in  {  (A /\ ([#] S)) where A is Subset of T : ( A in B & A meets [#] S )  }  ; ::_thesis: u in bool the carrier of S
then  ex A being Subset of T st
( u = A /\ ([#] S) & A in B & A meets [#] S ) ;
hence  u in bool the carrier of S ; ::_thesis: verum
end;
then reconsider X =  {  (A /\ ([#] S)) where A is Subset of T : ( A in B & A meets [#] S )  }  as Subset-Family of S ;
A1: now__::_thesis:_for_U_being_Subset_of_S_st_U_is_open_holds_
for_x_being_Point_of_S_st_x_in_U_holds_
ex_V_being_Subset_of_S_st_
(_V_in_X_&_x_in_V_&_V_c=_U_)
let U be Subset of S; ::_thesis: ( U is open implies for x being Point of S st x in U holds
ex V being Subset of S st
( V in X & x in V & V c= U ) )
assume  U is open ; ::_thesis: for x being Point of S st x in U holds
ex V being Subset of S st
( V in X & x in V & V c= U )
then consider U0 being Subset of T such that
A2: U0 is open and
A3: U = U0 /\ the carrier of S by TSP_1:def_1;
let x be Point of S; ::_thesis: ( x in U implies ex V being Subset of S st
( V in X & x in V & V c= U ) )
assume A4: x in U ; ::_thesis: ex V being Subset of S st
( V in X & x in V & V c= U )
then  x in U0 by A3, XBOOLE_0:def_4;
then consider V0 being Subset of T such that
A5: V0 in B and
A6: x in V0 and
A7: V0 c= U0 by A2, YELLOW_9:31;
reconsider V = V0 /\ ([#] S) as Subset of S ;
take V = V; ::_thesis: ( V in X & x in V & V c= U )
 V0 meets [#] S by A4, A6, XBOOLE_0:3;
hence  V in X by A5; ::_thesis: ( x in V & V c= U )
thus  x in V by A4, A6, XBOOLE_0:def_4; ::_thesis: V c= U
thus  V c= U by A3, A7, XBOOLE_1:26; ::_thesis: verum
end;
 X c= the topology of S
proof
let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in X or u in the topology of S )
assume  u in X ; ::_thesis: u in the topology of S
then A8: ex A being Subset of T st
( u = A /\ ([#] S) & A in B & A meets [#] S ) ;
 B c= the topology of T by TOPS_2:64;
hence  u in the topology of S by A8, PRE_TOPC:def_4; ::_thesis: verum
end;
hence   {  (A /\ ([#] S)) where A is Subset of T : ( A in B & A meets [#] S )  }  is Basis of S by A1, YELLOW_9:32; ::_thesis: verum
end;

theorem Th72: :: TOPGEN_5:72
 {  ].a,b.[ where a, b is Real : a < b  }  is Basis of R^1
proof
set X =  {  ].a,b.[ where a, b is Real : a < b  }  ;
  {  ].a,b.[ where a, b is Real : a < b  }  c= bool REAL
proof
let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in  {  ].a,b.[ where a, b is Real : a < b  }  or u in bool REAL )
assume  u in  {  ].a,b.[ where a, b is Real : a < b  }  ; ::_thesis: u in bool REAL
then  ex a, b being Real st
( u = ].a,b.[ & a < b ) ;
hence  u in bool REAL ; ::_thesis: verum
end;
then reconsider X =  {  ].a,b.[ where a, b is Real : a < b  }  as Subset-Family of R^1 by TOPMETR:17;
A1: now__::_thesis:_for_U_being_Subset_of_R^1_st_U_is_open_holds_
for_x_being_Point_of_R^1_st_x_in_U_holds_
ex_V_being_Subset_of_R^1_st_
(_V_in_X_&_x_in_V_&_V_c=_U_)
let U be Subset of R^1; ::_thesis: ( U is open implies for x being Point of R^1 st x in U holds
ex V being Subset of R^1 st
( V in X & x in V & V c= U ) )
assume A2: U is open ; ::_thesis: for x being Point of R^1 st x in U holds
ex V being Subset of R^1 st
( V in X & x in V & V c= U )
let x be Point of R^1; ::_thesis: ( x in U implies ex V being Subset of R^1 st
( V in X & x in V & V c= U ) )
assume A3: x in U ; ::_thesis: ex V being Subset of R^1 st
( V in X & x in V & V c= U )
reconsider a = x as Real by TOPMETR:17;
consider r being Real such that
A4: 0 < r and
A5: ].(a - r),(a + r).[ c= U by A2, A3, FRECHET:8;
reconsider V = ].(a - r),(a + r).[ as Subset of R^1 by TOPMETR:17;
take V = V; ::_thesis: ( V in X & x in V & V c= U )
A6: a < a + r by A4, XREAL_1:29;
A7: a - r < a by A4, XREAL_1:44;
then  a - r < a + r by A6, XXREAL_0:2;
hence  V in X ; ::_thesis: ( x in V & V c= U )
thus  x in V by A7, A6, XXREAL_1:4; ::_thesis: V c= U
thus  V c= U by A5; ::_thesis: verum
end;
 X c= the topology of R^1
proof
let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in X or u in the topology of R^1 )
assume  u in X ; ::_thesis: u in the topology of R^1
then  ex a, b being Real st
( u = ].a,b.[ & a < b ) ;
then  u is open Subset of R^1 by JORDAN6:35, TOPMETR:17;
hence  u in the topology of R^1 by PRE_TOPC:def_2; ::_thesis: verum
end;
hence   {  ].a,b.[ where a, b is Real : a < b  }  is Basis of R^1 by A1, YELLOW_9:32; ::_thesis: verum
end;

theorem Th73: :: TOPGEN_5:73
for T being TopSpace
for U, V being Subset of T
for B being set st U in B & V in B & B \/ {(U \/ V)} is Basis of T holds
B is Basis of T
proof
let T be TopSpace; ::_thesis: for U, V being Subset of T
for B being set st U in B & V in B & B \/ {(U \/ V)} is Basis of T holds
B is Basis of T
let U, V be Subset of T; ::_thesis: for B being set st U in B & V in B & B \/ {(U \/ V)} is Basis of T holds
B is Basis of T
let B be set ; ::_thesis: ( U in B & V in B & B \/ {(U \/ V)} is Basis of T implies B is Basis of T )
assume that
A1: U in B and
A2: V in B and
A3: B \/ {(U \/ V)} is Basis of T ; ::_thesis: B is Basis of T
A4: B c= B \/ {(U \/ V)} by XBOOLE_1:7;
then reconsider B9 = B as Subset-Family of T by A3, XBOOLE_1:1;
A5: now__::_thesis:_for_A_being_Subset_of_T_st_A_is_open_holds_
for_p_being_Point_of_T_st_p_in_A_holds_
ex_a_being_Subset_of_T_st_
(_a_in_B9_&_p_in_a_&_a_c=_A_)
A6: V c= U \/ V by XBOOLE_1:7;
A7: U c= U \/ V by XBOOLE_1:7;
let A be Subset of T; ::_thesis: ( A is open implies for p being Point of T st p in A holds
ex a being Subset of T st
( a in B9 & p in a & a c= A ) )
assume A8: A is open ; ::_thesis: for p being Point of T st p in A holds
ex a being Subset of T st
( a in B9 & p in a & a c= A )
let p be Point of T; ::_thesis: ( p in A implies ex a being Subset of T st
( a in B9 & p in a & a c= A ) )
assume  p in A ; ::_thesis: ex a being Subset of T st
( a in B9 & p in a & a c= A )
then consider A9 being Subset of T such that
A9: A9 in B \/ {(U \/ V)} and
A10: p in A9 and
A11: A9 c= A by A3, A8, YELLOW_9:31;
assume A12: for a being Subset of T holds
( not a in B9 or not p in a or not a c= A ) ; ::_thesis: contradiction
 ( A9 in B or A9 = U \/ V ) by A9, ZFMISC_1:136;
then  ( ( p in U & U c= A ) or ( p in V & V c= A ) ) by A10, A11, A12, A7, A6, XBOOLE_0:def_3, XBOOLE_1:1;
hence  contradiction by A1, A2, A12; ::_thesis: verum
end;
 B \/ {(U \/ V)} c= the topology of T by A3, TOPS_2:64;
hence  B is Basis of T by A5, A4, XBOOLE_1:1, YELLOW_9:32; ::_thesis: verum
end;

theorem Th74: :: TOPGEN_5:74
( {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  ) \/  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  is Basis of I[01]
proof
reconsider U = [.0,(2 / 3).[, V = ].(1 / 3),1.] as Subset of I[01] by BORSUK_1:40, XXREAL_1:35, XXREAL_1:36;
reconsider B =  {  ].a,b.[ where a, b is Real : a < b  }  as Basis of R^1 by Th72;
set S = I[01] ;
set T = R^1 ;
set A1 =  {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  ;
set A2 =  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  ;
set A3 =  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  ;
set B9 =  {  (A /\ ([#] I[01])) where A is Subset of R^1 : ( A in B & A meets [#] I[01] )  }  ;
A1:  {  (A /\ ([#] I[01])) where A is Subset of R^1 : ( A in B & A meets [#] I[01] )  }  = (( {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  ) \/  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  ) \/ {([#] I[01])}
proof
reconsider aa = ].(- 1),2.[ as Subset of R^1 by TOPMETR:17;
thus   {  (A /\ ([#] I[01])) where A is Subset of R^1 : ( A in B & A meets [#] I[01] )  }  c= (( {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  ) \/  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  ) \/ {([#] I[01])} :: according to XBOOLE_0:def_10 ::_thesis: (( {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  ) \/  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  ) \/ {([#] I[01])} c=  {  (A /\ ([#] I[01])) where A is Subset of R^1 : ( A in B & A meets [#] I[01] )  }
proof
let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in  {  (A /\ ([#] I[01])) where A is Subset of R^1 : ( A in B & A meets [#] I[01] )  }  or u in (( {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  ) \/  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  ) \/ {([#] I[01])} )
assume  u in  {  (A /\ ([#] I[01])) where A is Subset of R^1 : ( A in B & A meets [#] I[01] )  }  ; ::_thesis: u in (( {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  ) \/  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  ) \/ {([#] I[01])}
then consider A being Subset of R^1 such that
A2: u = A /\ ([#] I[01]) and
A3: A in B and
A4: A meets [#] I[01] ;
consider x being set  such that
A5: x in A and
A6: x in [#] I[01] by A4, XBOOLE_0:3;
consider a, b being Real such that
A7: A = ].a,b.[ and
A8: a < b by A3;
reconsider x = x as Real by A7, A5;
A9: a < x by A7, A5, XXREAL_1:4;
A10: x <= 1 by A6, BORSUK_1:40, XXREAL_1:1;
A11: 0 <= x by A6, BORSUK_1:40, XXREAL_1:1;
percases ( ( a < 0 & b <= 1 ) or ( a < 0 & b > 1 ) or ( a >= 0 & b <= 1 ) or ( a >= 0 & b > 1 ) ) ;
supposeA12: ( a < 0 & b <= 1 ) ; ::_thesis: u in (( {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  ) \/  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  ) \/ {([#] I[01])}
A13: 0 < b by A11, A7, A5, XXREAL_1:4;
 u = [.0,b.[ by A12, A2, A7, BORSUK_1:40, XXREAL_1:148;
then  u in  {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  by A13, A12;
then  u in  {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  by XBOOLE_0:def_3;
then  u in ( {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  ) \/ ( {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  \/ {([#] I[01])}) by XBOOLE_0:def_3;
hence  u in (( {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  ) \/  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  ) \/ {([#] I[01])} by XBOOLE_1:4; ::_thesis: verum
end;
suppose ( a < 0 & b > 1 ) ; ::_thesis: u in (( {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  ) \/  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  ) \/ {([#] I[01])}
then  u = [#] I[01] by A2, A7, BORSUK_1:40, XBOOLE_1:28, XXREAL_1:47;
hence  u in (( {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  ) \/  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  ) \/ {([#] I[01])} by ZFMISC_1:136; ::_thesis: verum
end;
supposeA14: ( a >= 0 & b <= 1 ) ; ::_thesis: u in (( {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  ) \/  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  ) \/ {([#] I[01])}
then  u = A by A2, A7, BORSUK_1:40, XBOOLE_1:28, XXREAL_1:37;
then  u in  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  by A7, A8, A14;
then  u in ( {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  ) \/  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  by XBOOLE_0:def_3;
hence  u in (( {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  ) \/  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  ) \/ {([#] I[01])} by XBOOLE_0:def_3; ::_thesis: verum
end;
supposeA15: ( a >= 0 & b > 1 ) ; ::_thesis: u in (( {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  ) \/  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  ) \/ {([#] I[01])}
A16: a < 1 by A10, A9, XXREAL_0:2;
 u = ].a,1.] by A15, A2, A7, BORSUK_1:40, XXREAL_1:149;
then  u in  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  by A16, A15;
then  u in  {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  by XBOOLE_0:def_3;
then  u in ( {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  ) \/ ( {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  \/ {([#] I[01])}) by XBOOLE_0:def_3;
hence  u in (( {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  ) \/  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  ) \/ {([#] I[01])} by XBOOLE_1:4; ::_thesis: verum
end;
end;
end;
let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in (( {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  ) \/  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  ) \/ {([#] I[01])} or u in  {  (A /\ ([#] I[01])) where A is Subset of R^1 : ( A in B & A meets [#] I[01] )  }  )
assume  u in (( {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  ) \/  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  ) \/ {([#] I[01])} ; ::_thesis: u in  {  (A /\ ([#] I[01])) where A is Subset of R^1 : ( A in B & A meets [#] I[01] )  }
then  ( u in ( {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  ) \/  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  or u in {([#] I[01])} ) by XBOOLE_0:def_3;
then A17: ( u in  {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  or u in  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  or u in {([#] I[01])} ) by XBOOLE_0:def_3;
percases ( u in {([#] I[01])} or u in  {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  or u in  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  or u in  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  ) by A17, XBOOLE_0:def_3;
suppose u in {([#] I[01])} ; ::_thesis: u in  {  (A /\ ([#] I[01])) where A is Subset of R^1 : ( A in B & A meets [#] I[01] )  }
then  u = [#] I[01] by TARSKI:def_1;
then A18: u = aa /\ ([#] I[01]) by BORSUK_1:40, XBOOLE_1:28, XXREAL_1:47;
 [#] I[01] c= aa by BORSUK_1:40, XXREAL_1:47;
then A19: aa meets [#] I[01] by XBOOLE_1:68;
 aa in B ;
hence  u in  {  (A /\ ([#] I[01])) where A is Subset of R^1 : ( A in B & A meets [#] I[01] )  }  by A18, A19; ::_thesis: verum
end;
suppose u in  {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  ; ::_thesis: u in  {  (A /\ ([#] I[01])) where A is Subset of R^1 : ( A in B & A meets [#] I[01] )  }
then consider a being Real such that
A20: u = [.0,a.[ and
A21: 0 < a and
A22: a <= 1 ;
reconsider A = ].(- 1),a.[ as Subset of R^1 by TOPMETR:17;
A23: A in B by A21;
A24: 0 in [.0,1.] by XXREAL_1:1;
 0 in A by A21, XXREAL_1:4;
then A25: A meets [#] I[01] by A24, BORSUK_1:40, XBOOLE_0:3;
 u = A /\ [.0,1.] by A20, A22, XXREAL_1:148;
hence  u in  {  (A /\ ([#] I[01])) where A is Subset of R^1 : ( A in B & A meets [#] I[01] )  }  by A23, A25, BORSUK_1:40; ::_thesis: verum
end;
suppose u in  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  ; ::_thesis: u in  {  (A /\ ([#] I[01])) where A is Subset of R^1 : ( A in B & A meets [#] I[01] )  }
then consider a being Real such that
A26: u = ].a,1.] and
A27: 0 <= a and
A28: a < 1 ;
reconsider A = ].a,2.[ as Subset of R^1 by TOPMETR:17;
 2 > a by A28, XXREAL_0:2;
then A29: A in B ;
A30: 1 in [.0,1.] by XXREAL_1:1;
 1 in A by A28, XXREAL_1:4;
then A31: A meets [#] I[01] by A30, BORSUK_1:40, XBOOLE_0:3;
 u = A /\ [.0,1.] by A26, A27, XXREAL_1:149;
hence  u in  {  (A /\ ([#] I[01])) where A is Subset of R^1 : ( A in B & A meets [#] I[01] )  }  by A29, A31, BORSUK_1:40; ::_thesis: verum
end;
suppose u in  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  ; ::_thesis: u in  {  (A /\ ([#] I[01])) where A is Subset of R^1 : ( A in B & A meets [#] I[01] )  }
then consider a, b being Real such that
A32: u = ].a,b.[ and
A33: 0 <= a and
A34: a < b and
A35: b <= 1 ;
reconsider A = ].a,b.[ as Subset of R^1 by TOPMETR:17;
A36: A c= [.0,1.] by A33, A35, XXREAL_1:37;
 a + b < b + b by A34, XREAL_1:8;
then A37: (a + b) / 2 < (b + b) / 2 by XREAL_1:74;
 a + a < a + b by A34, XREAL_1:8;
then  (a + a) / 2 < (a + b) / 2 by XREAL_1:74;
then  (a + b) / 2 in A by A37, XXREAL_1:4;
then A38: A meets [#] I[01] by A36, BORSUK_1:40, XBOOLE_0:3;
A39: A in B by A34;
 u = A /\ [.0,1.] by A33, A35, A32, XBOOLE_1:28, XXREAL_1:37;
hence  u in  {  (A /\ ([#] I[01])) where A is Subset of R^1 : ( A in B & A meets [#] I[01] )  }  by A39, A38, BORSUK_1:40; ::_thesis: verum
end;
end;
end;
 U in  {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  ;
then  U in  {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  by XBOOLE_0:def_3;
then A40: U in ( {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  ) \/  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  by XBOOLE_0:def_3;
 V in  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  ;
then  V in  {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  by XBOOLE_0:def_3;
then A41: V in ( {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  ) \/  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  by XBOOLE_0:def_3;
 U \/ V = [#] I[01] by BORSUK_1:40, XXREAL_1:175;
hence  ( {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  ) \/  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  is Basis of I[01] by A1, A40, A41, Th71, Th73; ::_thesis: verum
end;

theorem Th75: :: TOPGEN_5:75
for T being non empty TopSpace
for f being Function of T,I[01] holds
( f is continuous iff for a, b being real number st 0 <= a & a < 1 & 0 < b & b <= 1 holds
( f " [.0,b.[ is open & f " ].a,1.] is open ) )
proof
set A3 =  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  ;
set A2 =  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  ;
set A1 =  {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  ;
reconsider B = ( {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  ) \/  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  as Basis of I[01] by Th74;
let T be non empty TopSpace; ::_thesis: for f being Function of T,I[01] holds
( f is continuous iff for a, b being real number st 0 <= a & a < 1 & 0 < b & b <= 1 holds
( f " [.0,b.[ is open & f " ].a,1.] is open ) )
let f be Function of T,I[01]; ::_thesis: ( f is continuous iff for a, b being real number st 0 <= a & a < 1 & 0 < b & b <= 1 holds
( f " [.0,b.[ is open & f " ].a,1.] is open ) )
hereby  ::_thesis: ( ( for a, b being real number st 0 <= a & a < 1 & 0 < b & b <= 1 holds
( f " [.0,b.[ is open & f " ].a,1.] is open ) ) implies f is continuous )
assume A1: f is continuous ; ::_thesis: for a, b being real number st 0 <= a & a < 1 & 0 < b & b <= 1 holds
( f " [.0,b.[ is open & f " ].a,1.] is open )
let a, b be real number ; ::_thesis: ( 0 <= a & a < 1 & 0 < b & b <= 1 implies ( f " [.0,b.[ is open & f " ].a,1.] is open ) )
reconsider x = a, y = b as Real by XREAL_0:def_1;
assume that
A2: 0 <= a and
A3: a < 1 and
A4: 0 < b and
A5: b <= 1 ; ::_thesis: ( f " [.0,b.[ is open & f " ].a,1.] is open )
 ].x,1.] in  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  by A2, A3;
then  ].x,1.] in  {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  by XBOOLE_0:def_3;
then A6: ].x,1.] in B by XBOOLE_0:def_3;
 [.0,y.[ in  {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  by A4, A5;
then  [.0,y.[ in  {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  by XBOOLE_0:def_3;
then  [.0,y.[ in B by XBOOLE_0:def_3;
hence  ( f " [.0,b.[ is open & f " ].a,1.] is open ) by A6, A1, YELLOW_9:34; ::_thesis: verum
end;
assume A7: for a, b being real number st 0 <= a & a < 1 & 0 < b & b <= 1 holds
( f " [.0,b.[ is open & f " ].a,1.] is open ) ; ::_thesis: f is continuous
now__::_thesis:_for_A_being_Subset_of_I[01]_st_A_in_B_holds_
f_"_A_is_open
let A be Subset of I[01]; ::_thesis: ( A in B implies f " b1 is open )
assume  A in B ; ::_thesis: f " b1 is open
then A8: ( A in  {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  \/  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  or A in  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  ) by XBOOLE_0:def_3;
percases ( A in  {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  or A in  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  or A in  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  ) by A8, XBOOLE_0:def_3;
suppose A in  {  [.0,a.[ where a is Real : ( 0 < a & a <= 1 )  }  ; ::_thesis: f " b1 is open
then  ex a being Real st
( A = [.0,a.[ & 0 < a & a <= 1 ) ;
hence  f " A is open by A7; ::_thesis: verum
end;
suppose A in  {  ].a,1.] where a is Real : ( 0 <= a & a < 1 )  }  ; ::_thesis: f " b1 is open
then  ex a being Real st
( A = ].a,1.] & 0 <= a & a < 1 ) ;
hence  f " A is open by A7; ::_thesis: verum
end;
suppose A in  {  ].a,b.[ where a, b is Real : ( 0 <= a & a < b & b <= 1 )  }  ; ::_thesis: f " b1 is open
then consider a, b being Real such that
A9: A = ].a,b.[ and
A10: 0 <= a and
A11: a < b and
A12: b <= 1 ;
 a < 1 by A11, A12, XXREAL_0:2;
then reconsider U = f " [.0,b.[, V = f " ].a,1.] as open Subset of T by A10, A11, A7, A12;
 A = [.0,b.[ /\ ].a,1.] by A9, A10, A12, XXREAL_1:153;
then  f " A = U /\ V by FUNCT_1:68;
hence  f " A is open ; ::_thesis: verum
end;
end;
end;
hence  f is continuous by YELLOW_9:34; ::_thesis: verum
end;

registration
let x be real number ;
let r be real positive number ;
cluster + (x,r) ->  continuous ;
coherence
 + (x,r) is continuous
proof
set f = + (x,r);
consider BB being Neighborhood_System of Niemytzki-plane  such that
A1: for x being Element of REAL holds BB . |[x,0]| =  {  ((Ball (|[x,q]|,q)) \/ {|[x,0]|}) where q is Element of REAL : q > 0  }  and
A2: for x, y being Element of REAL st y > 0 holds
BB . |[x,y]| =  {  ((Ball (|[x,y]|,q)) /\ y>=0-plane) where q is Element of REAL : q > 0  }  by Def3;
A3: dom BB = y>=0-plane by Lm1, PARTFUN1:def_2;
now__::_thesis:_for_a,_b_being_real_number_st_0_<=_a_&_a_<_1_&_0_<_b_&_b_<=_1_holds_
(_(+_(x,r))_"_[.0,b.[_is_open_&_(+_(x,r))_"_].a,1.]_is_open_)
let a, b be real number ; ::_thesis: ( 0 <= a & a < 1 & 0 < b & b <= 1 implies ( (+ (x,r)) " [.0,b.[ is open & (+ (x,r)) " ].a,1.] is open ) )
assume that
A4: 0 <= a and
A5: a < 1 and
A6: 0 < b and
A7: b <= 1 ; ::_thesis: ( (+ (x,r)) " [.0,b.[ is open & (+ (x,r)) " ].a,1.] is open )
now__::_thesis:_for_c_being_Element_of_Niemytzki-plane_st_c_in_(+_(x,r))_"_[.0,b.[_holds_
ex_U_being_Subset_of_Niemytzki-plane_st_
(_U_in_Union_BB_&_c_in_U_&_U_c=_(+_(x,r))_"_[.0,b.[_)
let c be Element of Niemytzki-plane; ::_thesis: ( c in (+ (x,r)) " [.0,b.[ implies ex U being Subset of Niemytzki-plane st
( b2 in Union BB & U in b2 & b2 c= (+ (x,r)) " [.0,b.[ ) )
assume  c in (+ (x,r)) " [.0,b.[ ; ::_thesis: ex U being Subset of Niemytzki-plane st
( b2 in Union BB & U in b2 & b2 c= (+ (x,r)) " [.0,b.[ )
then A8: (+ (x,r)) . c in [.0,b.[ by FUNCT_1:def_7;
 c in y>=0-plane by Lm1;
then reconsider z = c as Point of (TOP-REAL 2) ;
 z = |[(z `1),(z `2)]| by EUCLID:53;
then A9: z `2 >= 0 by Lm1, Th18;
percases ( (+ (x,r)) . c = 0 or ( 0 < (+ (x,r)) . c & (+ (x,r)) . c < b ) ) by A8, XXREAL_1:3;
supposeA10: (+ (x,r)) . c = 0 ; ::_thesis: ex U being Subset of Niemytzki-plane st
( b2 in Union BB & U in b2 & b2 c= (+ (x,r)) " [.0,b.[ )
reconsider r1 = r * b as real positive number by A6;
reconsider U = (Ball (|[x,r1]|,r1)) \/ {|[x,0]|} as Subset of Niemytzki-plane by Th27;
take U = U; ::_thesis: ( U in Union BB & c in U & U c= (+ (x,r)) " [.0,b.[ )
A11: x is Real by XREAL_0:def_1;
then A12: |[x,0]| in y>=0-plane ;
 r1 is Real by XREAL_0:def_1;
then  U in  {  ((Ball (|[x,q]|,q)) \/ {|[x,0]|}) where q is Element of REAL : q > 0  }  ;
then A13: U in BB . |[x,0]| by A1, A11;
 z = |[x,0]| by A10, A9, Th60;
hence  ( U in Union BB & c in U & U c= (+ (x,r)) " [.0,b.[ ) by A13, A12, A3, A6, Th67, CARD_5:2, ZFMISC_1:136; ::_thesis: verum
end;
supposeA14: ( 0 < (+ (x,r)) . c & (+ (x,r)) . c < b ) ; ::_thesis: ex U being Subset of Niemytzki-plane st
( b2 in Union BB & U in b2 & b2 c= (+ (x,r)) " [.0,b.[ )
reconsider r1 = r * b as real positive number by A6;
A15: r1 is Real by XREAL_0:def_1;
reconsider U = Ball (|[x,r1]|,r1) as Subset of Niemytzki-plane by Th29;
take U = U; ::_thesis: ( U in Union BB & c in U & U c= (+ (x,r)) " [.0,b.[ )
A16: x is Real by XREAL_0:def_1;
then A17: |[x,r1]| in y>=0-plane by A15;
 U c= y>=0-plane by Th20;
then  U /\ y>=0-plane = U by XBOOLE_1:28;
then  U in  {  ((Ball (|[x,r1]|,q)) /\ y>=0-plane) where q is Element of REAL : q > 0  }  by A15;
then  U in BB . |[x,r1]| by A2, A15, A16;
hence  U in Union BB by A17, A3, CARD_5:2; ::_thesis: ( c in U & U c= (+ (x,r)) " [.0,b.[ )
A18: (+ (x,r)) " ].0,b.[ c= (+ (x,r)) " [.0,b.[ by RELAT_1:143, XXREAL_1:45;
 Ball (|[x,(r * b)]|,(r * b)) c= (+ (x,r)) " ].0,b.[ by A14, Th66;
hence  ( c in U & U c= (+ (x,r)) " [.0,b.[ ) by A18, A14, A7, A9, Th68, XBOOLE_1:1; ::_thesis: verum
end;
end;
end;
hence  (+ (x,r)) " [.0,b.[ is open by YELLOW_9:31; ::_thesis: (+ (x,r)) " ].a,1.] is open
now__::_thesis:_for_c_being_Element_of_Niemytzki-plane_st_c_in_(+_(x,r))_"_].a,1.]_holds_
ex_U_being_Subset_of_Niemytzki-plane_st_
(_U_in_Union_BB_&_c_in_U_&_U_c=_(+_(x,r))_"_].a,1.]_)
let c be Element of Niemytzki-plane; ::_thesis: ( c in (+ (x,r)) " ].a,1.] implies ex U being Subset of Niemytzki-plane st
( b2 in Union BB & U in b2 & b2 c= (+ (x,r)) " ].a,1.] ) )
 c in y>=0-plane by Lm1;
then reconsider z = c as Point of (TOP-REAL 2) ;
assume  c in (+ (x,r)) " ].a,1.] ; ::_thesis: ex U being Subset of Niemytzki-plane st
( b2 in Union BB & U in b2 & b2 c= (+ (x,r)) " ].a,1.] )
then A19: (+ (x,r)) . c in ].a,1.] by FUNCT_1:def_7;
then A20: (+ (x,r)) . c > a by XXREAL_1:2;
A21: (+ (x,r)) . c <= 1 by A19, XXREAL_1:2;
A22: z = |[(z `1),(z `2)]| by EUCLID:53;
then A23: z `2 >= 0 by Lm1, Th18;
percases ( z `2 > 0 or ( (+ (x,r)) . c = 1 & z `2 = 0 ) or ( a < (+ (x,r)) . c & (+ (x,r)) . c < 1 ) ) by A22, A19, A21, Lm1, Th18, XXREAL_0:1, XXREAL_1:2;
suppose z `2 > 0 ; ::_thesis: ex U being Subset of Niemytzki-plane st
( b2 in Union BB & U in b2 & b2 c= (+ (x,r)) " ].a,1.] )
then consider r1 being real positive number  such that
A24: r1 <= z `2 and
A25: Ball (z,r1) c= (+ (x,r)) " ].a,1.] by A4, A20, Th69;
reconsider U = Ball (z,r1) as Subset of Niemytzki-plane by A22, A24, Th29;
 U c= y>=0-plane by A22, A24, Th20;
then A26: U /\ y>=0-plane = U by XBOOLE_1:28;
 r1 is Real by XREAL_0:def_1;
then  U in  {  ((Ball (|[(z `1),(z `2)]|,q)) /\ y>=0-plane) where q is Element of REAL : q > 0  }  by A26, A22;
then A27: U in BB . |[(z `1),(z `2)]| by A24, A2;
take U = U; ::_thesis: ( U in Union BB & c in U & U c= (+ (x,r)) " ].a,1.] )
 |[(z `1),(z `2)]| in y>=0-plane by A24;
hence  U in Union BB by A27, A3, CARD_5:2; ::_thesis: ( c in U & U c= (+ (x,r)) " ].a,1.] )
thus  ( c in U & U c= (+ (x,r)) " ].a,1.] ) by A25, Th13; ::_thesis: verum
end;
supposeA28: ( (+ (x,r)) . c = 1 & z `2 = 0 ) ; ::_thesis: ex U being Subset of Niemytzki-plane st
( b2 in Union BB & U in b2 & b2 c= (+ (x,r)) " ].a,1.] )
then consider r1 being real positive number  such that
A29: (Ball (|[(z `1),r1]|,r1)) \/ {z} c= (+ (x,r)) " {1} by Th70;
reconsider U = (Ball (|[(z `1),r1]|,r1)) \/ {z} as Subset of Niemytzki-plane by A22, A28, Th27;
 r1 is Real by XREAL_0:def_1;
then  U in  {  ((Ball (|[(z `1),q]|,q)) \/ {|[(z `1),0]|}) where q is Element of REAL : q > 0  }  by A22, A28;
then A30: U in BB . |[(z `1),(z `2)]| by A1, A28;
take U = U; ::_thesis: ( U in Union BB & c in U & U c= (+ (x,r)) " ].a,1.] )
 |[(z `1),(z `2)]| in y>=0-plane by A28;
hence  U in Union BB by A30, A3, CARD_5:2; ::_thesis: ( c in U & U c= (+ (x,r)) " ].a,1.] )
thus  c in U by ZFMISC_1:136; ::_thesis: U c= (+ (x,r)) " ].a,1.]
 1 in ].a,1.] by A5, XXREAL_1:2;
then  {1} c= ].a,1.] by ZFMISC_1:31;
then  (+ (x,r)) " {1} c= (+ (x,r)) " ].a,1.] by RELAT_1:143;
hence  U c= (+ (x,r)) " ].a,1.] by A29, XBOOLE_1:1; ::_thesis: verum
end;
suppose ( a < (+ (x,r)) . c & (+ (x,r)) . c < 1 ) ; ::_thesis: ex U being Subset of Niemytzki-plane st
( b2 in Union BB & U in b2 & b2 c= (+ (x,r)) " ].a,1.] )
then  (+ (x,r)) . c in ].a,1.[ by XXREAL_1:4;
then consider r1 being real positive number  such that
A31: r1 <= z `2 and
A32: Ball (z,r1) c= (+ (x,r)) " ].a,1.[ by A4, A23, Th65;
reconsider U = Ball (z,r1) as Subset of Niemytzki-plane by A22, A31, Th29;
 U c= y>=0-plane by A22, A31, Th20;
then A33: U /\ y>=0-plane = U by XBOOLE_1:28;
 r1 is Real by XREAL_0:def_1;
then  U in  {  ((Ball (|[(z `1),(z `2)]|,q)) /\ y>=0-plane) where q is Element of REAL : q > 0  }  by A33, A22;
then A34: U in BB . |[(z `1),(z `2)]| by A31, A2;
take U = U; ::_thesis: ( U in Union BB & c in U & U c= (+ (x,r)) " ].a,1.] )
 |[(z `1),(z `2)]| in y>=0-plane by A31;
hence  U in Union BB by A34, A3, CARD_5:2; ::_thesis: ( c in U & U c= (+ (x,r)) " ].a,1.] )
 (+ (x,r)) " ].a,1.[ c= (+ (x,r)) " ].a,1.] by RELAT_1:143, XXREAL_1:41;
hence  ( c in U & U c= (+ (x,r)) " ].a,1.] ) by A32, Th13, XBOOLE_1:1; ::_thesis: verum
end;
end;
end;
hence  (+ (x,r)) " ].a,1.] is open by YELLOW_9:31; ::_thesis: verum
end;
hence  + (x,r) is continuous by Th75; ::_thesis: verum
end;
end;

theorem Th76: :: TOPGEN_5:76
for U being Subset of Niemytzki-plane
for x being Element of REAL
for r being real positive number st U = (Ball (|[x,r]|,r)) \/ {|[x,0]|} holds
ex f being continuous Function of Niemytzki-plane,I[01] st
( f . |[x,0]| = 0 & ( for a, b being real number holds
( ( |[a,b]| in U ` implies f . |[a,b]| = 1 ) & ( |[a,b]| in U \ {|[x,0]|} implies f . |[a,b]| = (|.(|[x,0]| - |[a,b]|).| ^2) / ((2 * r) * b) ) ) ) )
proof
let U be Subset of Niemytzki-plane; ::_thesis: for x being Element of REAL
for r being real positive number st U = (Ball (|[x,r]|,r)) \/ {|[x,0]|} holds
ex f being continuous Function of Niemytzki-plane,I[01] st
( f . |[x,0]| = 0 & ( for a, b being real number holds
( ( |[a,b]| in U ` implies f . |[a,b]| = 1 ) & ( |[a,b]| in U \ {|[x,0]|} implies f . |[a,b]| = (|.(|[x,0]| - |[a,b]|).| ^2) / ((2 * r) * b) ) ) ) )
let x be Element of REAL ; ::_thesis: for r being real positive number st U = (Ball (|[x,r]|,r)) \/ {|[x,0]|} holds
ex f being continuous Function of Niemytzki-plane,I[01] st
( f . |[x,0]| = 0 & ( for a, b being real number holds
( ( |[a,b]| in U ` implies f . |[a,b]| = 1 ) & ( |[a,b]| in U \ {|[x,0]|} implies f . |[a,b]| = (|.(|[x,0]| - |[a,b]|).| ^2) / ((2 * r) * b) ) ) ) )
let r be real positive number ; ::_thesis: ( U = (Ball (|[x,r]|,r)) \/ {|[x,0]|} implies ex f being continuous Function of Niemytzki-plane,I[01] st
( f . |[x,0]| = 0 & ( for a, b being real number holds
( ( |[a,b]| in U ` implies f . |[a,b]| = 1 ) & ( |[a,b]| in U \ {|[x,0]|} implies f . |[a,b]| = (|.(|[x,0]| - |[a,b]|).| ^2) / ((2 * r) * b) ) ) ) ) )
assume A1: U = (Ball (|[x,r]|,r)) \/ {|[x,0]|} ; ::_thesis: ex f being continuous Function of Niemytzki-plane,I[01] st
( f . |[x,0]| = 0 & ( for a, b being real number holds
( ( |[a,b]| in U ` implies f . |[a,b]| = 1 ) & ( |[a,b]| in U \ {|[x,0]|} implies f . |[a,b]| = (|.(|[x,0]| - |[a,b]|).| ^2) / ((2 * r) * b) ) ) ) )
take f = + (x,r); ::_thesis: ( f . |[x,0]| = 0 & ( for a, b being real number holds
( ( |[a,b]| in U ` implies f . |[a,b]| = 1 ) & ( |[a,b]| in U \ {|[x,0]|} implies f . |[a,b]| = (|.(|[x,0]| - |[a,b]|).| ^2) / ((2 * r) * b) ) ) ) )
thus  f . |[x,0]| = 0 by Def5; ::_thesis: for a, b being real number holds
( ( |[a,b]| in U ` implies f . |[a,b]| = 1 ) & ( |[a,b]| in U \ {|[x,0]|} implies f . |[a,b]| = (|.(|[x,0]| - |[a,b]|).| ^2) / ((2 * r) * b) ) )
let a, b be real number ; ::_thesis: ( ( |[a,b]| in U ` implies f . |[a,b]| = 1 ) & ( |[a,b]| in U \ {|[x,0]|} implies f . |[a,b]| = (|.(|[x,0]| - |[a,b]|).| ^2) / ((2 * r) * b) ) )
thus  ( |[a,b]| in U ` implies f . |[a,b]| = 1 ) ::_thesis: ( |[a,b]| in U \ {|[x,0]|} implies f . |[a,b]| = (|.(|[x,0]| - |[a,b]|).| ^2) / ((2 * r) * b) )
proof
assume A2: |[a,b]| in U ` ; ::_thesis: f . |[a,b]| = 1
then A3: not |[a,b]| in U by XBOOLE_0:def_5;
then A4: not |[a,b]| in Ball (|[x,r]|,r) by A1, ZFMISC_1:136;
A5: ( a <> x or b <> 0 ) by A3, A1, ZFMISC_1:136;
 b >= 0 by A2, Lm1, Th18;
hence  f . |[a,b]| = 1 by A4, A5, Def5; ::_thesis: verum
end;
assume A6: |[a,b]| in U \ {|[x,0]|} ; ::_thesis: f . |[a,b]| = (|.(|[x,0]| - |[a,b]|).| ^2) / ((2 * r) * b)
then A7: not |[a,b]| in {|[x,0]|} by XBOOLE_0:def_5;
 |[a,b]| in U by A6, XBOOLE_0:def_5;
then A8: |[a,b]| in Ball (|[x,r]|,r) by A7, A1, XBOOLE_0:def_3;
 b >= 0 by A6, Lm1, Th18;
hence  f . |[a,b]| = (|.(|[x,0]| - |[a,b]|).| ^2) / ((2 * r) * b) by A8, Def5; ::_thesis: verum
end;

definition
let x, y be real number ;
let r be real positive number ;
func + (x,y,r) ->  Function of Niemytzki-plane,I[01] means :Def6: :: TOPGEN_5:def 6
 for a being real number
for b being real non negative number holds
( ( not |[a,b]| in Ball (|[x,y]|,r) implies it . |[a,b]| = 1 ) & ( |[a,b]| in Ball (|[x,y]|,r) implies it . |[a,b]| = |.(|[x,y]| - |[a,b]|).| / r ) );
existence
 ex b1 being Function of Niemytzki-plane,I[01] st
for a being real number
for b being real non negative number holds
( ( not |[a,b]| in Ball (|[x,y]|,r) implies b1 . |[a,b]| = 1 ) & ( |[a,b]| in Ball (|[x,y]|,r) implies b1 . |[a,b]| = |.(|[x,y]| - |[a,b]|).| / r ) )
proof
deffunc H1( Point of (TOP-REAL 2)) ->  Element of NAT = 1;
deffunc H2( Point of (TOP-REAL 2)) ->  Element of REAL = |.(|[x,y]| - $1).| / r;
defpred S1[ set ] means  not $1 in Ball (|[x,y]|,r);
A1: for a being Point of (TOP-REAL 2) st a in the carrier of Niemytzki-plane holds
( ( S1[a] implies H1(a) in the carrier of I[01] ) & ( not S1[a] implies H2(a) in the carrier of I[01] ) )
proof
let a be Point of (TOP-REAL 2); ::_thesis: ( a in the carrier of Niemytzki-plane implies ( ( S1[a] implies H1(a) in the carrier of I[01] ) & ( not S1[a] implies H2(a) in the carrier of I[01] ) ) )
assume  a in the carrier of Niemytzki-plane ; ::_thesis: ( ( S1[a] implies H1(a) in the carrier of I[01] ) & ( not S1[a] implies H2(a) in the carrier of I[01] ) )
thus  ( S1[a] implies H1(a) in the carrier of I[01] ) by BORSUK_1:40, XXREAL_1:1; ::_thesis: ( not S1[a] implies H2(a) in the carrier of I[01] )
assume  not S1[a] ; ::_thesis: H2(a) in the carrier of I[01]
then  |.(a - |[x,y]|).| < r by TOPREAL9:7;
then  |.(|[x,y]| - a).| < r by TOPRNS_1:27;
then A2: H2(a) < r / r by XREAL_1:74;
 r / r = 1 by XCMPLX_1:60;
hence  H2(a) in the carrier of I[01] by A2, BORSUK_1:40, XXREAL_1:1; ::_thesis: verum
end;
 the carrier of Niemytzki-plane = y>=0-plane by Def3;
then A3: the carrier of Niemytzki-plane c= the carrier of (TOP-REAL 2) ;
consider f being Function of Niemytzki-plane,I[01] such that
A4: for a being Point of (TOP-REAL 2) st a in the carrier of Niemytzki-plane holds
( ( S1[a] implies f . a = H1(a) ) & ( not S1[a] implies f . a = H2(a) ) ) from TOPGEN_5:sch_1(A3, A1);
take  f ; ::_thesis: for a being real number
for b being real non negative number holds
( ( not |[a,b]| in Ball (|[x,y]|,r) implies f . |[a,b]| = 1 ) & ( |[a,b]| in Ball (|[x,y]|,r) implies f . |[a,b]| = |.(|[x,y]| - |[a,b]|).| / r ) )
let a be real number ; ::_thesis: for b being real non negative number holds
( ( not |[a,b]| in Ball (|[x,y]|,r) implies f . |[a,b]| = 1 ) & ( |[a,b]| in Ball (|[x,y]|,r) implies f . |[a,b]| = |.(|[x,y]| - |[a,b]|).| / r ) )
let b be real non negative number ; ::_thesis: ( ( not |[a,b]| in Ball (|[x,y]|,r) implies f . |[a,b]| = 1 ) & ( |[a,b]| in Ball (|[x,y]|,r) implies f . |[a,b]| = |.(|[x,y]| - |[a,b]|).| / r ) )
 |[a,b]| in the carrier of Niemytzki-plane by Lm1, Th18;
hence  ( ( not |[a,b]| in Ball (|[x,y]|,r) implies f . |[a,b]| = 1 ) & ( |[a,b]| in Ball (|[x,y]|,r) implies f . |[a,b]| = |.(|[x,y]| - |[a,b]|).| / r ) ) by A4; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Function of Niemytzki-plane,I[01] st ( for a being real number
for b being real non negative number holds
( ( not |[a,b]| in Ball (|[x,y]|,r) implies b1 . |[a,b]| = 1 ) & ( |[a,b]| in Ball (|[x,y]|,r) implies b1 . |[a,b]| = |.(|[x,y]| - |[a,b]|).| / r ) ) ) & ( for a being real number
for b being real non negative number holds
( ( not |[a,b]| in Ball (|[x,y]|,r) implies b2 . |[a,b]| = 1 ) & ( |[a,b]| in Ball (|[x,y]|,r) implies b2 . |[a,b]| = |.(|[x,y]| - |[a,b]|).| / r ) ) ) holds
b1 = b2
proof
let f, g be Function of Niemytzki-plane,I[01]; ::_thesis: ( ( for a being real number
for b being real non negative number holds
( ( not |[a,b]| in Ball (|[x,y]|,r) implies f . |[a,b]| = 1 ) & ( |[a,b]| in Ball (|[x,y]|,r) implies f . |[a,b]| = |.(|[x,y]| - |[a,b]|).| / r ) ) ) & ( for a being real number
for b being real non negative number holds
( ( not |[a,b]| in Ball (|[x,y]|,r) implies g . |[a,b]| = 1 ) & ( |[a,b]| in Ball (|[x,y]|,r) implies g . |[a,b]| = |.(|[x,y]| - |[a,b]|).| / r ) ) ) implies f = g )
assume that
A5: for a being real number
for b being real non negative number holds
( ( not |[a,b]| in Ball (|[x,y]|,r) implies f . |[a,b]| = 1 ) & ( |[a,b]| in Ball (|[x,y]|,r) implies f . |[a,b]| = |.(|[x,y]| - |[a,b]|).| / r ) ) and
A6: for a being real number
for b being real non negative number holds
( ( not |[a,b]| in Ball (|[x,y]|,r) implies g . |[a,b]| = 1 ) & ( |[a,b]| in Ball (|[x,y]|,r) implies g . |[a,b]| = |.(|[x,y]| - |[a,b]|).| / r ) ) ; ::_thesis: f = g
A7: the carrier of Niemytzki-plane = y>=0-plane by Def3;
now__::_thesis:_for_p_being_Point_of_Niemytzki-plane_holds_f_._p_=_g_._p
let p be Point of Niemytzki-plane; ::_thesis: f . b1 = g . b1
 p in y>=0-plane by A7;
then reconsider q = p as Point of (TOP-REAL 2) ;
A8: p = |[(q `1),(q `2)]| by EUCLID:53;
then reconsider q2 = q `2 as real non negative number by A7, Th18;
percases ( not |[(q `1),q2]| in Ball (|[x,y]|,r) or |[(q `1),q2]| in Ball (|[x,y]|,r) ) ;
supposeA9: not |[(q `1),q2]| in Ball (|[x,y]|,r) ; ::_thesis: f . b1 = g . b1
then  f . p = 1 by A5, A8;
hence  f . p = g . p by A9, A6, A8; ::_thesis: verum
end;
supposeA10: |[(q `1),q2]| in Ball (|[x,y]|,r) ; ::_thesis: f . b1 = g . b1
then  f . p = |.(|[x,y]| - q).| / r by A5, A8;
hence  f . p = g . p by A10, A6, A8; ::_thesis: verum
end;
end;
end;
hence  f = g by FUNCT_2:63; ::_thesis: verum
end;
end;

:: deftheorem Def6 defines + TOPGEN_5:def_6_:_
 for x, y being real number
for r being real positive number
for b4 being Function of Niemytzki-plane,I[01] holds
( b4 = + (x,y,r) iff for a being real number
for b being real non negative number holds
( ( not |[a,b]| in Ball (|[x,y]|,r) implies b4 . |[a,b]| = 1 ) & ( |[a,b]| in Ball (|[x,y]|,r) implies b4 . |[a,b]| = |.(|[x,y]| - |[a,b]|).| / r ) ) );

theorem Th77: :: TOPGEN_5:77
for p being Point of (TOP-REAL 2) st p `2 >= 0 holds
for x being real number
for y being real non negative number
for r being real positive number holds
( (+ (x,y,r)) . p = 0 iff p = |[x,y]| )
proof
let p be Point of (TOP-REAL 2); ::_thesis: ( p `2 >= 0 implies for x being real number
for y being real non negative number
for r being real positive number holds
( (+ (x,y,r)) . p = 0 iff p = |[x,y]| ) )
assume A1: p `2 >= 0 ; ::_thesis: for x being real number
for y being real non negative number
for r being real positive number holds
( (+ (x,y,r)) . p = 0 iff p = |[x,y]| )
let x be real number ; ::_thesis: for y being real non negative number
for r being real positive number holds
( (+ (x,y,r)) . p = 0 iff p = |[x,y]| )
let y be real non negative number ; ::_thesis: for r being real positive number holds
( (+ (x,y,r)) . p = 0 iff p = |[x,y]| )
let r be real positive number ; ::_thesis: ( (+ (x,y,r)) . p = 0 iff p = |[x,y]| )
A2: p = |[(p `1),(p `2)]| by EUCLID:53;
hereby  ::_thesis: ( p = |[x,y]| implies (+ (x,y,r)) . p = 0 )
assume A3: (+ (x,y,r)) . p = 0 ; ::_thesis: p = |[x,y]|
then  p in Ball (|[x,y]|,r) by A1, A2, Def6;
then  0 = |.(|[x,y]| - p).| / r by A1, A2, A3, Def6;
then  0 * r = |.(|[x,y]| - p).| ;
hence  p = |[x,y]| by TOPRNS_1:28; ::_thesis: verum
end;
assume A4: p = |[x,y]| ; ::_thesis: (+ (x,y,r)) . p = 0
then  p in Ball (|[x,y]|,r) by Th13;
hence (+ (x,y,r)) . p = |.(|[x,y]| - p).| / r by A4, Def6
.= 0 / r by A4, TOPRNS_1:28
.= 0 ;
::_thesis: verum
end;

theorem Th78: :: TOPGEN_5:78
for x being real number
for y being real non negative number
for r, a being real positive number st a <= 1 holds
(+ (x,y,r)) " [.0,a.[ = (Ball (|[x,y]|,(r * a))) /\ y>=0-plane
proof
let x be real number ; ::_thesis: for y being real non negative number
for r, a being real positive number st a <= 1 holds
(+ (x,y,r)) " [.0,a.[ = (Ball (|[x,y]|,(r * a))) /\ y>=0-plane
let y be real non negative number ; ::_thesis: for r, a being real positive number st a <= 1 holds
(+ (x,y,r)) " [.0,a.[ = (Ball (|[x,y]|,(r * a))) /\ y>=0-plane
let r, a be real positive number ; ::_thesis: ( a <= 1 implies (+ (x,y,r)) " [.0,a.[ = (Ball (|[x,y]|,(r * a))) /\ y>=0-plane )
set f = + (x,y,r);
assume A1: a <= 1 ; ::_thesis: (+ (x,y,r)) " [.0,a.[ = (Ball (|[x,y]|,(r * a))) /\ y>=0-plane
then  r * a <= r * 1 by XREAL_1:64;
then A2: Ball (|[x,y]|,(r * a)) c= Ball (|[x,y]|,r) by JORDAN:18;
thus  (+ (x,y,r)) " [.0,a.[ c= (Ball (|[x,y]|,(r * a))) /\ y>=0-plane :: according to XBOOLE_0:def_10 ::_thesis: (Ball (|[x,y]|,(r * a))) /\ y>=0-plane c= (+ (x,y,r)) " [.0,a.[
proof
let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in (+ (x,y,r)) " [.0,a.[ or u in (Ball (|[x,y]|,(r * a))) /\ y>=0-plane )
assume A3: u in (+ (x,y,r)) " [.0,a.[ ; ::_thesis: u in (Ball (|[x,y]|,(r * a))) /\ y>=0-plane
then reconsider p = u as Point of Niemytzki-plane ;
 p in y>=0-plane by Lm1;
then reconsider q = p as Element of (TOP-REAL 2) ;
 (+ (x,y,r)) . p in [.0,a.[ by A3, FUNCT_2:38;
then A4: (+ (x,y,r)) . p < a by XXREAL_1:3;
A5: p = |[(q `1),(q `2)]| by EUCLID:53;
then A6: q `2 >= 0 by Lm1, Th18;
then  p in Ball (|[x,y]|,r) by A4, A1, A5, Def6;
then  (+ (x,y,r)) . p = |.(|[x,y]| - q).| / r by A5, A6, Def6;
then A7: |.(|[x,y]| - q).| < r * a by A4, XREAL_1:77;
 |.(|[x,y]| - q).| = |.(q - |[x,y]|).| by TOPRNS_1:27;
then  p in Ball (|[x,y]|,(r * a)) by A7, TOPREAL9:7;
hence  u in (Ball (|[x,y]|,(r * a))) /\ y>=0-plane by Lm1, XBOOLE_0:def_4; ::_thesis: verum
end;
let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in (Ball (|[x,y]|,(r * a))) /\ y>=0-plane or u in (+ (x,y,r)) " [.0,a.[ )
assume A8: u in (Ball (|[x,y]|,(r * a))) /\ y>=0-plane ; ::_thesis: u in (+ (x,y,r)) " [.0,a.[
then reconsider p = u as Point of Niemytzki-plane by Lm1, XBOOLE_0:def_4;
reconsider q = p as Element of (TOP-REAL 2) by A8;
A9: u in Ball (|[x,y]|,(r * a)) by A8, XBOOLE_0:def_4;
then A10: |.(q - |[x,y]|).| < r * a by TOPREAL9:7;
A11: |.(|[x,y]| - q).| = |.(q - |[x,y]|).| by TOPRNS_1:27;
A12: p = |[(q `1),(q `2)]| by EUCLID:53;
 u in y>=0-plane by A8, XBOOLE_0:def_4;
then  q `2 >= 0 by A12, Th18;
then A13: (+ (x,y,r)) . p = |.(|[x,y]| - q).| / r by A2, A9, A12, Def6;
then  r * ((+ (x,y,r)) . p) = |.(|[x,y]| - q).| by XCMPLX_1:87;
then  (+ (x,y,r)) . p < a by A10, A11, XREAL_1:64;
then  (+ (x,y,r)) . p in [.0,a.[ by A13, XXREAL_1:3;
hence  u in (+ (x,y,r)) " [.0,a.[ by FUNCT_2:38; ::_thesis: verum
end;

theorem Th79: :: TOPGEN_5:79
for p being Point of (TOP-REAL 2) st p `2 > 0 holds
for x being real number
for a being real non negative number
for y, r being real positive number st (+ (x,y,r)) . p > a holds
( |.(|[x,y]| - p).| > r * a & (Ball (p,(|.(|[x,y]| - p).| - (r * a)))) /\ y>=0-plane c= (+ (x,y,r)) " ].a,1.] )
proof
let p be Point of (TOP-REAL 2); ::_thesis: ( p `2 > 0 implies for x being real number
for a being real non negative number
for y, r being real positive number st (+ (x,y,r)) . p > a holds
( |.(|[x,y]| - p).| > r * a & (Ball (p,(|.(|[x,y]| - p).| - (r * a)))) /\ y>=0-plane c= (+ (x,y,r)) " ].a,1.] ) )
assume A1: p `2 > 0 ; ::_thesis: for x being real number
for a being real non negative number
for y, r being real positive number st (+ (x,y,r)) . p > a holds
( |.(|[x,y]| - p).| > r * a & (Ball (p,(|.(|[x,y]| - p).| - (r * a)))) /\ y>=0-plane c= (+ (x,y,r)) " ].a,1.] )
let x be real number ; ::_thesis: for a being real non negative number
for y, r being real positive number st (+ (x,y,r)) . p > a holds
( |.(|[x,y]| - p).| > r * a & (Ball (p,(|.(|[x,y]| - p).| - (r * a)))) /\ y>=0-plane c= (+ (x,y,r)) " ].a,1.] )
let a be real non negative number ; ::_thesis: for y, r being real positive number st (+ (x,y,r)) . p > a holds
( |.(|[x,y]| - p).| > r * a & (Ball (p,(|.(|[x,y]| - p).| - (r * a)))) /\ y>=0-plane c= (+ (x,y,r)) " ].a,1.] )
let y, r be real positive number ; ::_thesis: ( (+ (x,y,r)) . p > a implies ( |.(|[x,y]| - p).| > r * a & (Ball (p,(|.(|[x,y]| - p).| - (r * a)))) /\ y>=0-plane c= (+ (x,y,r)) " ].a,1.] ) )
set f = + (x,y,r);
A2: p = |[(p `1),(p `2)]| by EUCLID:53;
then  p in y>=0-plane by A1;
then  (+ (x,y,r)) . p in [.0,1.] by Lm1, BORSUK_1:40, FUNCT_2:5;
then A3: (+ (x,y,r)) . p <= 1 by XXREAL_1:1;
assume A4: (+ (x,y,r)) . p > a ; ::_thesis: ( |.(|[x,y]| - p).| > r * a & (Ball (p,(|.(|[x,y]| - p).| - (r * a)))) /\ y>=0-plane c= (+ (x,y,r)) " ].a,1.] )
then A5: a < 1 by A3, XXREAL_0:2;
A6: |.(|[x,y]| - p).| = |.(p - |[x,y]|).| by TOPRNS_1:27;
thus  |.(|[x,y]| - p).| > r * a ::_thesis: (Ball (p,(|.(|[x,y]| - p).| - (r * a)))) /\ y>=0-plane c= (+ (x,y,r)) " ].a,1.]
proof
percases ( (+ (x,y,r)) . p < 1 or (+ (x,y,r)) . p = 1 ) by A3, XXREAL_0:1;
suppose (+ (x,y,r)) . p < 1 ; ::_thesis: |.(|[x,y]| - p).| > r * a
then  p in Ball (|[x,y]|,r) by A1, A2, Def6;
then  (+ (x,y,r)) . p = |.(|[x,y]| - p).| / r by A1, A2, Def6;
hence  |.(|[x,y]| - p).| > r * a by A4, XREAL_1:79; ::_thesis: verum
end;
supposeA7: (+ (x,y,r)) . p = 1 ; ::_thesis: |.(|[x,y]| - p).| > r * a
now__::_thesis:_not_p_in_Ball_(|[x,y]|,r)
A8: r / r = 1 by XCMPLX_1:60;
assume A9: p in Ball (|[x,y]|,r) ; ::_thesis: contradiction
then A10: |.(p - |[x,y]|).| < r by TOPREAL9:7;
 1 = |.(|[x,y]| - p).| / r by A9, A1, A2, A7, Def6;
hence  contradiction by A10, A8, A6, XREAL_1:74; ::_thesis: verum
end;
then A11: |.(p - |[x,y]|).| >= r by TOPREAL9:7;
 r * 1 > r * a by A5, XREAL_1:68;
hence  |.(|[x,y]| - p).| > r * a by A11, A6, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
then reconsider r1 = |.(|[x,y]| - p).| - (r * a) as real positive number by XREAL_1:50;
let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in (Ball (p,(|.(|[x,y]| - p).| - (r * a)))) /\ y>=0-plane or u in (+ (x,y,r)) " ].a,1.] )
assume A12: u in (Ball (p,(|.(|[x,y]| - p).| - (r * a)))) /\ y>=0-plane ; ::_thesis: u in (+ (x,y,r)) " ].a,1.]
then reconsider z = u as Point of Niemytzki-plane by Lm1, XBOOLE_0:def_4;
reconsider q = z as Element of (TOP-REAL 2) by A12;
 u in Ball (p,(|.(|[x,y]| - p).| - (r * a))) by A12, XBOOLE_0:def_4;
then  |.(q - p).| < r1 by TOPREAL9:7;
then A13: |.(|[x,y]| - q).| + |.(q - p).| < |.(|[x,y]| - q).| + r1 by XREAL_1:6;
A14: q = |[(q `1),(q `2)]| by EUCLID:53;
then A15: q `2 >= 0 by Lm1, Th18;
 |.(|[x,y]| - p).| <= |.(|[x,y]| - q).| + |.(q - p).| by TOPRNS_1:34;
then  |.(|[x,y]| - p).| < |.(|[x,y]| - q).| + r1 by A13, XXREAL_0:2;
then A16: |.(|[x,y]| - p).| - r1 < (|.(|[x,y]| - q).| + r1) - r1 by XREAL_1:9;
A17: now__::_thesis:_(_q_in_Ball_(|[x,y]|,r)_implies_(+_(x,y,r))_._z_>_a_)
assume  q in Ball (|[x,y]|,r) ; ::_thesis: (+ (x,y,r)) . z > a
then  (+ (x,y,r)) . q = |.(|[x,y]| - q).| / r by A14, A15, Def6;
hence  (+ (x,y,r)) . z > a by A16, XREAL_1:81; ::_thesis: verum
end;
 (+ (x,y,r)) . z in [.0,1.] by BORSUK_1:40;
then A18: (+ (x,y,r)) . z <= 1 by XXREAL_1:1;
 ( not q in Ball (|[x,y]|,r) implies (+ (x,y,r)) . q = 1 ) by A15, A14, Def6;
then  (+ (x,y,r)) . z in ].a,1.] by A18, A5, A17, XXREAL_1:2;
hence  u in (+ (x,y,r)) " ].a,1.] by FUNCT_2:38; ::_thesis: verum
end;

theorem Th80: :: TOPGEN_5:80
for p being Point of (TOP-REAL 2) st p `2 = 0 holds
for x being real number
for a being real non negative number
for y, r being real positive number st (+ (x,y,r)) . p > a holds
( |.(|[x,y]| - p).| > r * a & ex r1 being real positive number st
( r1 = (|.(|[x,y]| - p).| - (r * a)) / 2 & (Ball (|[(p `1),r1]|,r1)) \/ {p} c= (+ (x,y,r)) " ].a,1.] ) )
proof
let p be Point of (TOP-REAL 2); ::_thesis: ( p `2 = 0 implies for x being real number
for a being real non negative number
for y, r being real positive number st (+ (x,y,r)) . p > a holds
( |.(|[x,y]| - p).| > r * a & ex r1 being real positive number st
( r1 = (|.(|[x,y]| - p).| - (r * a)) / 2 & (Ball (|[(p `1),r1]|,r1)) \/ {p} c= (+ (x,y,r)) " ].a,1.] ) ) )
A1: p = |[(p `1),(p `2)]| by EUCLID:53;
assume A2: p `2 = 0 ; ::_thesis: for x being real number
for a being real non negative number
for y, r being real positive number st (+ (x,y,r)) . p > a holds
( |.(|[x,y]| - p).| > r * a & ex r1 being real positive number st
( r1 = (|.(|[x,y]| - p).| - (r * a)) / 2 & (Ball (|[(p `1),r1]|,r1)) \/ {p} c= (+ (x,y,r)) " ].a,1.] ) )
then reconsider p9 = p as Point of Niemytzki-plane by A1, Lm1, Th18;
let x be real number ; ::_thesis: for a being real non negative number
for y, r being real positive number st (+ (x,y,r)) . p > a holds
( |.(|[x,y]| - p).| > r * a & ex r1 being real positive number st
( r1 = (|.(|[x,y]| - p).| - (r * a)) / 2 & (Ball (|[(p `1),r1]|,r1)) \/ {p} c= (+ (x,y,r)) " ].a,1.] ) )
let a be real non negative number ; ::_thesis: for y, r being real positive number st (+ (x,y,r)) . p > a holds
( |.(|[x,y]| - p).| > r * a & ex r1 being real positive number st
( r1 = (|.(|[x,y]| - p).| - (r * a)) / 2 & (Ball (|[(p `1),r1]|,r1)) \/ {p} c= (+ (x,y,r)) " ].a,1.] ) )
let y, r be real positive number ; ::_thesis: ( (+ (x,y,r)) . p > a implies ( |.(|[x,y]| - p).| > r * a & ex r1 being real positive number st
( r1 = (|.(|[x,y]| - p).| - (r * a)) / 2 & (Ball (|[(p `1),r1]|,r1)) \/ {p} c= (+ (x,y,r)) " ].a,1.] ) ) )
set f = + (x,y,r);
 p in y>=0-plane by A2, A1;
then  (+ (x,y,r)) . p in [.0,1.] by Lm1, BORSUK_1:40, FUNCT_2:5;
then A3: (+ (x,y,r)) . p <= 1 by XXREAL_1:1;
assume A4: (+ (x,y,r)) . p > a ; ::_thesis: ( |.(|[x,y]| - p).| > r * a & ex r1 being real positive number st
( r1 = (|.(|[x,y]| - p).| - (r * a)) / 2 & (Ball (|[(p `1),r1]|,r1)) \/ {p} c= (+ (x,y,r)) " ].a,1.] ) )
then A5: a < 1 by A3, XXREAL_0:2;
A6: |.(|[x,y]| - p).| = |.(p - |[x,y]|).| by TOPRNS_1:27;
thus  |.(|[x,y]| - p).| > r * a ::_thesis: ex r1 being real positive number st
( r1 = (|.(|[x,y]| - p).| - (r * a)) / 2 & (Ball (|[(p `1),r1]|,r1)) \/ {p} c= (+ (x,y,r)) " ].a,1.] )
proof
percases ( (+ (x,y,r)) . p < 1 or (+ (x,y,r)) . p = 1 ) by A3, XXREAL_0:1;
suppose (+ (x,y,r)) . p < 1 ; ::_thesis: |.(|[x,y]| - p).| > r * a
then  p in Ball (|[x,y]|,r) by A2, A1, Def6;
then  (+ (x,y,r)) . p = |.(|[x,y]| - p).| / r by A2, A1, Def6;
hence  |.(|[x,y]| - p).| > r * a by A4, XREAL_1:79; ::_thesis: verum
end;
supposeA7: (+ (x,y,r)) . p = 1 ; ::_thesis: |.(|[x,y]| - p).| > r * a
now__::_thesis:_not_p_in_Ball_(|[x,y]|,r)
A8: r / r = 1 by XCMPLX_1:60;
assume A9: p in Ball (|[x,y]|,r) ; ::_thesis: contradiction
then A10: |.(p - |[x,y]|).| < r by TOPREAL9:7;
 1 = |.(|[x,y]| - p).| / r by A9, A2, A1, A7, Def6;
hence  contradiction by A10, A8, A6, XREAL_1:74; ::_thesis: verum
end;
then A11: |.(p - |[x,y]|).| >= r by TOPREAL9:7;
 r * 1 > r * a by A5, XREAL_1:68;
hence  |.(|[x,y]| - p).| > r * a by A11, A6, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
then reconsider r9 = |.(|[x,y]| - p).| - (r * a) as real positive number by XREAL_1:50;
take r1 = r9 / 2; ::_thesis: ( r1 = (|.(|[x,y]| - p).| - (r * a)) / 2 & (Ball (|[(p `1),r1]|,r1)) \/ {p} c= (+ (x,y,r)) " ].a,1.] )
thus  r1 = (|.(|[x,y]| - p).| - (r * a)) / 2 ; ::_thesis: (Ball (|[(p `1),r1]|,r1)) \/ {p} c= (+ (x,y,r)) " ].a,1.]
let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in (Ball (|[(p `1),r1]|,r1)) \/ {p} or u in (+ (x,y,r)) " ].a,1.] )
A12: Ball (|[(p `1),r1]|,r1) c= y>=0-plane by Th20;
assume A13: u in (Ball (|[(p `1),r1]|,r1)) \/ {p} ; ::_thesis: u in (+ (x,y,r)) " ].a,1.]
then  ( u in Ball (|[(p `1),r1]|,r1) or u = p9 ) by ZFMISC_1:136;
then reconsider z = u as Point of Niemytzki-plane by A12, Def3;
reconsider q = z as Element of (TOP-REAL 2) by A13;
A14: q = |[(q `1),(q `2)]| by EUCLID:53;
then A15: q `2 >= 0 by Lm1, Th18;
then A16: ( not q in Ball (|[x,y]|,r) implies (+ (x,y,r)) . q = 1 ) by A14, Def6;
percases ( u = p9 or u in Ball (|[(p `1),r1]|,r1) ) by A13, ZFMISC_1:136;
suppose u = p9 ; ::_thesis: u in (+ (x,y,r)) " ].a,1.]
then  (+ (x,y,r)) . z in ].a,1.] by A4, A3, XXREAL_1:2;
hence  u in (+ (x,y,r)) " ].a,1.] by FUNCT_2:38; ::_thesis: verum
end;
suppose u in Ball (|[(p `1),r1]|,r1) ; ::_thesis: u in (+ (x,y,r)) " ].a,1.]
then  |.(q - |[(p `1),r1]|).| < r1 by TOPREAL9:7;
then A17: |.(q - |[(p `1),r1]|).| + |.(|[(p `1),r1]| - p).| < r1 + |.(|[(p `1),r1]| - p).| by XREAL_1:6;
 |.(q - p).| <= |.(q - |[(p `1),r1]|).| + |.(|[(p `1),r1]| - p).| by TOPRNS_1:34;
then A18: |.(q - p).| < r1 + |.(|[(p `1),r1]| - p).| by A17, XXREAL_0:2;
A19: |.|[0,r1]|.| = abs r1 by TOPREAL6:23;
A20: |.(|[x,y]| - p).| <= |.(|[x,y]| - q).| + |.(q - p).| by TOPRNS_1:34;
A21: abs r1 = r1 by ABSVALUE:def_1;
 |.(|[(p `1),r1]| - p).| = |.|[((p `1) - (p `1)),(r1 - 0)]|.| by A2, A1, EUCLID:62;
then  |.(|[x,y]| - q).| + |.(q - p).| < |.(|[x,y]| - q).| + (r1 + r1) by A18, A19, A21, XREAL_1:6;
then  |.(|[x,y]| - p).| < |.(|[x,y]| - q).| + (r1 + r1) by A20, XXREAL_0:2;
then A22: |.(|[x,y]| - p).| - (r1 + r1) < (|.(|[x,y]| - q).| + (r1 + r1)) - (r1 + r1) by XREAL_1:9;
A23: now__::_thesis:_(_q_in_Ball_(|[x,y]|,r)_implies_(+_(x,y,r))_._z_>_a_)
assume  q in Ball (|[x,y]|,r) ; ::_thesis: (+ (x,y,r)) . z > a
then  (+ (x,y,r)) . q = |.(|[x,y]| - q).| / r by A14, A15, Def6;
hence  (+ (x,y,r)) . z > a by A22, XREAL_1:81; ::_thesis: verum
end;
 (+ (x,y,r)) . z in [.0,1.] by BORSUK_1:40;
then  (+ (x,y,r)) . z <= 1 by XXREAL_1:1;
then  (+ (x,y,r)) . z in ].a,1.] by A5, A16, A23, XXREAL_1:2;
hence  u in (+ (x,y,r)) " ].a,1.] by FUNCT_2:38; ::_thesis: verum
end;
end;
end;

registration
let x be real number ;
let y, r be real positive number ;
cluster + (x,y,r) ->  continuous ;
coherence
 + (x,y,r) is continuous
proof
set f = + (x,y,r);
consider BB being Neighborhood_System of Niemytzki-plane  such that
A1: for x being Element of REAL holds BB . |[x,0]| =  {  ((Ball (|[x,q]|,q)) \/ {|[x,0]|}) where q is Element of REAL : q > 0  }  and
A2: for x, y being Element of REAL st y > 0 holds
BB . |[x,y]| =  {  ((Ball (|[x,y]|,q)) /\ y>=0-plane) where q is Element of REAL : q > 0  }  by Def3;
A3: dom BB = y>=0-plane by Lm1, PARTFUN1:def_2;
now__::_thesis:_for_a,_b_being_real_number_st_0_<=_a_&_a_<_1_&_0_<_b_&_b_<=_1_holds_
(_(+_(x,y,r))_"_[.0,b.[_is_open_&_(+_(x,y,r))_"_].a,1.]_is_open_)
let a, b be real number ; ::_thesis: ( 0 <= a & a < 1 & 0 < b & b <= 1 implies ( (+ (x,y,r)) " [.0,b.[ is open & (+ (x,y,r)) " ].a,1.] is open ) )
assume that
A4: 0 <= a and
 a < 1 and
A5: 0 < b and
A6: b <= 1 ; ::_thesis: ( (+ (x,y,r)) " [.0,b.[ is open & (+ (x,y,r)) " ].a,1.] is open )
 (+ (x,y,r)) " [.0,b.[ = (Ball (|[x,y]|,(r * b))) /\ y>=0-plane by A5, A6, Th78;
hence  (+ (x,y,r)) " [.0,b.[ is open by A5, Th28; ::_thesis: (+ (x,y,r)) " ].a,1.] is open
now__::_thesis:_for_c_being_Element_of_Niemytzki-plane_st_c_in_(+_(x,y,r))_"_].a,1.]_holds_
ex_U_being_Subset_of_Niemytzki-plane_st_
(_U_in_Union_BB_&_c_in_U_&_U_c=_(+_(x,y,r))_"_].a,1.]_)
let c be Element of Niemytzki-plane; ::_thesis: ( c in (+ (x,y,r)) " ].a,1.] implies ex U being Subset of Niemytzki-plane st
( b2 in Union BB & U in b2 & b2 c= (+ (x,y,r)) " ].a,1.] ) )
 c in y>=0-plane by Lm1;
then reconsider z = c as Point of (TOP-REAL 2) ;
A7: z = |[(z `1),(z `2)]| by EUCLID:53;
assume  c in (+ (x,y,r)) " ].a,1.] ; ::_thesis: ex U being Subset of Niemytzki-plane st
( b2 in Union BB & U in b2 & b2 c= (+ (x,y,r)) " ].a,1.] )
then  (+ (x,y,r)) . c in ].a,1.] by FUNCT_1:def_7;
then A8: (+ (x,y,r)) . c > a by XXREAL_1:2;
percases ( z `2 > 0 or z `2 = 0 ) by A7, Lm1, Th18;
supposeA9: z `2 > 0 ; ::_thesis: ex U being Subset of Niemytzki-plane st
( b2 in Union BB & U in b2 & b2 c= (+ (x,y,r)) " ].a,1.] )
then reconsider r1 = |.(|[x,y]| - z).| - (r * a) as real positive number by A4, A8, Th79, XREAL_1:50;
reconsider U = (Ball (z,r1)) /\ y>=0-plane as Subset of Niemytzki-plane by A7, A9, Th28;
 U in  {  ((Ball (|[(z `1),(z `2)]|,q)) /\ y>=0-plane) where q is Element of REAL : q > 0  }  by A7;
then A10: U in BB . |[(z `1),(z `2)]| by A2, A9;
take U = U; ::_thesis: ( U in Union BB & c in U & U c= (+ (x,y,r)) " ].a,1.] )
 |[(z `1),(z `2)]| in y>=0-plane by A9;
hence  U in Union BB by A10, A3, CARD_5:2; ::_thesis: ( c in U & U c= (+ (x,y,r)) " ].a,1.] )
 c in Ball (z,r1) by Th13;
hence  ( c in U & U c= (+ (x,y,r)) " ].a,1.] ) by A4, A8, A9, Lm1, Th79, XBOOLE_0:def_4; ::_thesis: verum
end;
supposeA11: z `2 = 0 ; ::_thesis: ex U being Subset of Niemytzki-plane st
( b2 in Union BB & U in b2 & b2 c= (+ (x,y,r)) " ].a,1.] )
then consider r1 being real positive number  such that
 r1 = (|.(|[x,y]| - z).| - (r * a)) / 2 and
A12: (Ball (|[(z `1),r1]|,r1)) \/ {z} c= (+ (x,y,r)) " ].a,1.] by A4, A8, Th80;
reconsider U = (Ball (|[(z `1),r1]|,r1)) \/ {z} as Subset of Niemytzki-plane by A7, A11, Th27;
 r1 is Real by XREAL_0:def_1;
then  U in  {  ((Ball (|[(z `1),q]|,q)) \/ {|[(z `1),0]|}) where q is Element of REAL : q > 0  }  by A7, A11;
then A13: U in BB . |[(z `1),(z `2)]| by A1, A11;
take U = U; ::_thesis: ( U in Union BB & c in U & U c= (+ (x,y,r)) " ].a,1.] )
 |[(z `1),(z `2)]| in y>=0-plane by A11;
hence  U in Union BB by A13, A3, CARD_5:2; ::_thesis: ( c in U & U c= (+ (x,y,r)) " ].a,1.] )
thus  c in U by ZFMISC_1:136; ::_thesis: U c= (+ (x,y,r)) " ].a,1.]
thus  U c= (+ (x,y,r)) " ].a,1.] by A12; ::_thesis: verum
end;
end;
end;
hence  (+ (x,y,r)) " ].a,1.] is open by YELLOW_9:31; ::_thesis: verum
end;
hence  + (x,y,r) is continuous by Th75; ::_thesis: verum
end;
end;

theorem Th81: :: TOPGEN_5:81
for U being Subset of Niemytzki-plane
for x, y being Element of REAL
for r being real positive number st y > 0 & U = (Ball (|[x,y]|,r)) /\ y>=0-plane holds
ex f being continuous Function of Niemytzki-plane,I[01] st
( f . |[x,y]| = 0 & ( for a, b being real number holds
( ( |[a,b]| in U ` implies f . |[a,b]| = 1 ) & ( |[a,b]| in U implies f . |[a,b]| = |.(|[x,y]| - |[a,b]|).| / r ) ) ) )
proof
let U be Subset of Niemytzki-plane; ::_thesis: for x, y being Element of REAL
for r being real positive number st y > 0 & U = (Ball (|[x,y]|,r)) /\ y>=0-plane holds
ex f being continuous Function of Niemytzki-plane,I[01] st
( f . |[x,y]| = 0 & ( for a, b being real number holds
( ( |[a,b]| in U ` implies f . |[a,b]| = 1 ) & ( |[a,b]| in U implies f . |[a,b]| = |.(|[x,y]| - |[a,b]|).| / r ) ) ) )
let x, y be Element of REAL ; ::_thesis: for r being real positive number st y > 0 & U = (Ball (|[x,y]|,r)) /\ y>=0-plane holds
ex f being continuous Function of Niemytzki-plane,I[01] st
( f . |[x,y]| = 0 & ( for a, b being real number holds
( ( |[a,b]| in U ` implies f . |[a,b]| = 1 ) & ( |[a,b]| in U implies f . |[a,b]| = |.(|[x,y]| - |[a,b]|).| / r ) ) ) )
let r be real positive number ; ::_thesis: ( y > 0 & U = (Ball (|[x,y]|,r)) /\ y>=0-plane implies ex f being continuous Function of Niemytzki-plane,I[01] st
( f . |[x,y]| = 0 & ( for a, b being real number holds
( ( |[a,b]| in U ` implies f . |[a,b]| = 1 ) & ( |[a,b]| in U implies f . |[a,b]| = |.(|[x,y]| - |[a,b]|).| / r ) ) ) ) )
assume that
A1: y > 0 and
A2: U = (Ball (|[x,y]|,r)) /\ y>=0-plane ; ::_thesis: ex f being continuous Function of Niemytzki-plane,I[01] st
( f . |[x,y]| = 0 & ( for a, b being real number holds
( ( |[a,b]| in U ` implies f . |[a,b]| = 1 ) & ( |[a,b]| in U implies f . |[a,b]| = |.(|[x,y]| - |[a,b]|).| / r ) ) ) )
reconsider y9 = y as real positive number by A1;
take f = + (x,y9,r); ::_thesis: ( f . |[x,y]| = 0 & ( for a, b being real number holds
( ( |[a,b]| in U ` implies f . |[a,b]| = 1 ) & ( |[a,b]| in U implies f . |[a,b]| = |.(|[x,y]| - |[a,b]|).| / r ) ) ) )
 |[x,y9]| `2 = y by EUCLID:52;
hence  f . |[x,y]| = 0 by Th77; ::_thesis: for a, b being real number holds
( ( |[a,b]| in U ` implies f . |[a,b]| = 1 ) & ( |[a,b]| in U implies f . |[a,b]| = |.(|[x,y]| - |[a,b]|).| / r ) )
let a, b be real number ; ::_thesis: ( ( |[a,b]| in U ` implies f . |[a,b]| = 1 ) & ( |[a,b]| in U implies f . |[a,b]| = |.(|[x,y]| - |[a,b]|).| / r ) )
thus  ( |[a,b]| in U ` implies f . |[a,b]| = 1 ) ::_thesis: ( |[a,b]| in U implies f . |[a,b]| = |.(|[x,y]| - |[a,b]|).| / r )
proof
assume A3: |[a,b]| in U ` ; ::_thesis: f . |[a,b]| = 1
then  not |[a,b]| in U by XBOOLE_0:def_5;
then A4: not |[a,b]| in Ball (|[x,y]|,r) by A2, A3, Lm1, XBOOLE_0:def_4;
 b >= 0 by A3, Lm1, Th18;
hence  f . |[a,b]| = 1 by A4, Def6; ::_thesis: verum
end;
assume A5: |[a,b]| in U ; ::_thesis: f . |[a,b]| = |.(|[x,y]| - |[a,b]|).| / r
then A6: |[a,b]| in Ball (|[x,y]|,r) by A2, XBOOLE_0:def_4;
 b >= 0 by A5, Lm1, Th18;
hence  f . |[a,b]| = |.(|[x,y]| - |[a,b]|).| / r by A6, Def6; ::_thesis: verum
end;

theorem Th82: :: TOPGEN_5:82
Niemytzki-plane is T_1
proof
set X = Niemytzki-plane ;
let x, y be Point of Niemytzki-plane; :: according to URYSOHN1:def_7 ::_thesis: ( x = y or ex b1, b2 being Element of bool the carrier of Niemytzki-plane st
( b1 is open & b2 is open & x in b1 & not y in b1 & y in b2 & not x in b2 ) )
A1: the carrier of Niemytzki-plane = y>=0-plane by Def3;
then A2: y in y>=0-plane ;
 x in y>=0-plane by A1;
then reconsider a = x, b = y as Point of (TOP-REAL 2) by A2;
assume  x <> y ; ::_thesis: ex b1, b2 being Element of bool the carrier of Niemytzki-plane st
( b1 is open & b2 is open & x in b1 & not y in b1 & y in b2 & not x in b2 )
then  |.(a - b).| <> 0 by TOPRNS_1:28;
then reconsider r = |.(a - b).| as real positive number ;
consider q being Point of (TOP-REAL 2), U being open Subset of Niemytzki-plane such that
A3: x in U and
 q in U and
A4: for c being Point of (TOP-REAL 2) st c in U holds
|.(c - q).| < r / 2 by Th30;
consider p being Point of (TOP-REAL 2), V being open Subset of Niemytzki-plane such that
A5: y in V and
 p in V and
A6: for c being Point of (TOP-REAL 2) st c in V holds
|.(c - p).| < r / 2 by Th30;
take  U ; ::_thesis: ex b1 being Element of bool the carrier of Niemytzki-plane st
( U is open & b1 is open & x in U & not y in U & y in b1 & not x in b1 )
take  V ; ::_thesis: ( U is open & V is open & x in U & not y in U & y in V & not x in V )
thus  ( U is open & V is open & x in U ) by A3; ::_thesis: ( not y in U & y in V & not x in V )
hereby  ::_thesis: ( y in V & not x in V )
assume  y in U ; ::_thesis: contradiction
then  |.(b - q).| < r / 2 by A4;
then A7: |.(a - q).| + |.(b - q).| < (r / 2) + (r / 2) by A3, A4, XREAL_1:8;
 |.(a - b).| <= |.(a - q).| + |.(q - b).| by TOPRNS_1:34;
hence  contradiction by A7, TOPRNS_1:27; ::_thesis: verum
end;
thus  y in V by A5; ::_thesis: not x in V
assume A8: x in V ; ::_thesis: contradiction
A9: |.(a - b).| <= |.(a - p).| + |.(p - b).| by TOPRNS_1:34;
 |.(b - p).| < r / 2 by A5, A6;
then  |.(a - p).| + |.(b - p).| < (r / 2) + (r / 2) by A8, A6, XREAL_1:8;
hence  contradiction by A9, TOPRNS_1:27; ::_thesis: verum
end;

theorem :: TOPGEN_5:83
Niemytzki-plane is Tychonoff
proof
set X = Niemytzki-plane ;
consider B being Neighborhood_System of Niemytzki-plane  such that
A1: for x being Element of REAL holds B . |[x,0]| =  {  ((Ball (|[x,r]|,r)) \/ {|[x,0]|}) where r is Element of REAL : r > 0  }  and
A2: for x, y being Element of REAL st y > 0 holds
B . |[x,y]| =  {  ((Ball (|[x,y]|,r)) /\ y>=0-plane) where r is Element of REAL : r > 0  }  by Def3;
reconsider uB = Union B as prebasis of Niemytzki-plane by YELLOW_9:27;
A3: the carrier of Niemytzki-plane = y>=0-plane by Def3;
now__::_thesis:_for_x_being_Point_of_Niemytzki-plane
for_V_being_Subset_of_Niemytzki-plane_st_x_in_V_&_V_in_uB_holds_
ex_f_being_continuous_Function_of_Niemytzki-plane,I[01]_st_
(_f_._x_=_0_&_f_.:_(V_`)_c=_{1}_)
let x be Point of Niemytzki-plane; ::_thesis: for V being Subset of Niemytzki-plane st x in V & V in uB holds
ex f being continuous Function of Niemytzki-plane,I[01] st
( b4 . f = 0 & b4 .: (b3 `) c= {1} )
let V be Subset of Niemytzki-plane; ::_thesis: ( x in V & V in uB implies ex f being continuous Function of Niemytzki-plane,I[01] st
( b3 . f = 0 & b3 .: (b2 `) c= {1} ) )
assume that
A4: x in V and
A5: V in uB ; ::_thesis: ex f being continuous Function of Niemytzki-plane,I[01] st
( b3 . f = 0 & b3 .: (b2 `) c= {1} )
 V is open by A5, TOPS_2:def_1;
then consider V9 being Subset of Niemytzki-plane such that
A6: V9 in B . x and
A7: V9 c= V by A4, YELLOW_8:def_1;
 x in y>=0-plane by A3;
then reconsider x9 = x as Point of (TOP-REAL 2) ;
A8: x = |[(x9 `1),(x9 `2)]| by EUCLID:53;
percases ( x9 `2 = 0 or x9 `2 > 0 ) by A3, A8, Th18;
supposeA9: x9 `2 = 0 ; ::_thesis: ex f being continuous Function of Niemytzki-plane,I[01] st
( b3 . f = 0 & b3 .: (b2 `) c= {1} )
then  B . x =  {  ((Ball (|[(x9 `1),r]|,r)) \/ {|[(x9 `1),0]|}) where r is Element of REAL : r > 0  }  by A1, A8;
then consider r being Real such that
A10: V9 = (Ball (|[(x9 `1),r]|,r)) \/ {|[(x9 `1),0]|} and
A11: r > 0 by A6;
consider f being continuous Function of Niemytzki-plane,I[01] such that
A12: f . |[(x9 `1),0]| = 0 and
A13: for a, b being real number holds
( ( |[a,b]| in V9 ` implies f . |[a,b]| = 1 ) & ( |[a,b]| in V9 \ {|[(x9 `1),0]|} implies f . |[a,b]| = (|.(|[(x9 `1),0]| - |[a,b]|).| ^2) / ((2 * r) * b) ) ) by A10, A11, Th76;
take f = f; ::_thesis: ( f . x = 0 & f .: (V `) c= {1} )
thus  f . x = 0 by A9, A12, EUCLID:53; ::_thesis: f .: (V `) c= {1}
thus  f .: (V `) c= {1} ::_thesis: verum
proof
let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in f .: (V `) or u in {1} )
assume  u in f .: (V `) ; ::_thesis: u in {1}
then consider b being Point of Niemytzki-plane such that
A14: b in V ` and
A15: u = f . b by FUNCT_2:65;
 b in y>=0-plane by A3;
then reconsider c = b as Element of (TOP-REAL 2) ;
A16: V ` c= V9 ` by A7, SUBSET_1:12;
 b = |[(c `1),(c `2)]| by EUCLID:53;
then  u = 1 by A16, A14, A13, A15;
hence  u in {1} by TARSKI:def_1; ::_thesis: verum
end;
end;
supposeA17: x9 `2 > 0 ; ::_thesis: ex f being continuous Function of Niemytzki-plane,I[01] st
( b3 . f = 0 & b3 .: (b2 `) c= {1} )
then  B . x =  {  ((Ball (|[(x9 `1),(x9 `2)]|,r)) /\ y>=0-plane) where r is Element of REAL : r > 0  }  by A2, A8;
then consider r being Real such that
A18: V9 = (Ball (|[(x9 `1),(x9 `2)]|,r)) /\ y>=0-plane and
A19: r > 0 by A6;
consider f being continuous Function of Niemytzki-plane,I[01] such that
A20: f . |[(x9 `1),(x9 `2)]| = 0 and
A21: for a, b being real number holds
( ( |[a,b]| in V9 ` implies f . |[a,b]| = 1 ) & ( |[a,b]| in V9 implies f . |[a,b]| = |.(|[(x9 `1),(x9 `2)]| - |[a,b]|).| / r ) ) by A17, A18, A19, Th81;
take f = f; ::_thesis: ( f . x = 0 & f .: (V `) c= {1} )
thus  f . x = 0 by A20, EUCLID:53; ::_thesis: f .: (V `) c= {1}
thus  f .: (V `) c= {1} ::_thesis: verum
proof
let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in f .: (V `) or u in {1} )
assume  u in f .: (V `) ; ::_thesis: u in {1}
then consider b being Point of Niemytzki-plane such that
A22: b in V ` and
A23: u = f . b by FUNCT_2:65;
 b in y>=0-plane by A3;
then reconsider c = b as Element of (TOP-REAL 2) ;
A24: V ` c= V9 ` by A7, SUBSET_1:12;
 b = |[(c `1),(c `2)]| by EUCLID:53;
then  u = 1 by A24, A22, A21, A23;
hence  u in {1} by TARSKI:def_1; ::_thesis: verum
end;
end;
end;
end;
hence  Niemytzki-plane is Tychonoff by Th52, Th82; ::_thesis: verum
end;