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<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">A</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a> &amp; <font color="Maroon" title="c1">A</font> is <a href="jordan1.html#V2" title="JORDAN1:attr.2">Jordan</a> implies  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">A</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c1">A</font> )</span><br/>

<b>assume </b><a NAME="E1:119"/><i><font color="Green" title="E105">A1</font></i>: 
( <font color="Maroon" title="c1">A</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a> &amp; <font color="Maroon" title="c1">A</font> is <a href="jordan1.html#V2" title="JORDAN1:attr.2">Jordan</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">A</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c1">A</font></span><br/>

<b>then consider </b><font color="Maroon" title="c2">A1</font>, <font color="Maroon" title="c3">A2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:119"/><i><font color="Green" title="E106">A2</font></i>: 
( <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font> &amp; <font color="Maroon" title="c2">A1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">A2</font> &amp; <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">A2</font> &amp; ( for <font color="Olive" title="b1">C1</font>, <font color="Olive" title="b2">C2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>  st <font color="Olive" title="b1">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A1</font> &amp; <font color="Olive" title="b2">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A2</font> holds <br/>( <font color="Olive" title="b1">C1</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b2">C2</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> ) ) )
 <b>by </b><i><a class="ref" href="jordan1.html#D2" title="JORDAN1:def.2">JORDAN1:def 2</a></i>;<br/>
<a NAME="E4:119"/><i><font color="Green" title="E107">A3</font></i>: 
 <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> <font color="Maroon" title="c2">A1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A1</font>
 
<b>by </b><i><a class="txt" href="#E3:119"><i><font color="Green" title="E106">A2</font></i></a>, <a class="ref" href="xboole_1.html#T21" title="XBOOLE_1:th.21">XBOOLE_1:21</a></i>;<br/>
<a NAME="E5:119"/><i><font color="Green" title="E108">A4</font></i>: 
 <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> <font color="Maroon" title="c3">A2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A2</font>
 
<b>by </b><i><a class="txt" href="#E3:119"><i><font color="Green" title="E106">A2</font></i></a>, <a class="ref" href="xboole_1.html#T21" title="XBOOLE_1:th.21">XBOOLE_1:21</a></i>;<br/>
<a NAME="E6:119"/><b>then </b><i><font color="Green" title="E109">A5</font></i>: 
(  <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> <font color="Maroon" title="c2">A1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp;  <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> <font color="Maroon" title="c3">A2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="#E3:119"><i><font color="Green" title="E106">A2</font></i></a>, <a class="txt" href="#E4:119"><i><font color="Green" title="E107">A3</font></i></a></i>;<br/>
<a NAME="E7:119"/><b>then </b><i><font color="Green" title="E110">A6</font></i>: 
<font color="Maroon" title="c2">A1</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="txt" href="#E4:119"><i><font color="Green" title="E107">A3</font></i></a>, <a class="ref" href="connsp_1.html#D6" title="CONNSP_1:def.6">CONNSP_1:def 6</a></i>;<br/>
<a NAME="E8:119"/><i><font color="Green" title="E111">A7</font></i>: 
<font color="Maroon" title="c3">A2</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="txt" href="#E5:119"><i><font color="Green" title="E108">A4</font></i></a>, <a class="txt" href="#E6:119"><i><font color="Green" title="E109">A5</font></i></a>, <a class="ref" href="connsp_1.html#D6" title="CONNSP_1:def.6">CONNSP_1:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">D</font> = <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="#E1:119"><i><font color="Green" title="E105">A1</font></i></a>, <a class="ref" href="jordan1.html#D2" title="JORDAN1:def.2">JORDAN1:def 2</a></i>;<br/>
<a NAME="E10:119"/><i><font color="Green" title="E112">A8</font></i>: 
not <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">D</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 
;<br/>
<a NAME="E11:119"/><b>then </b><i><font color="Green" title="E113">A9</font></i>: 
 <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> <font color="Maroon" title="c3">A2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E6:119"><i><font color="Green" title="E109">A5</font></i></a>, <a class="ref" href="connsp_1.html#T34" title="CONNSP_1:th.34">CONNSP_1:34</a></i>;<br/>
<a NAME="E12:119"/><i><font color="Green" title="E114">A10</font></i>: 
 <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">A</font> <a href="jordan2c.html#R2" title="JORDAN2C:pred.2">is_outside_component_of</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="#E1:119"><i><font color="Green" title="E105">A1</font></i></a>, <a class="ref" href="jordan2c.html#T76" target="_self" title="JORDAN2C:th.76">Th76</a></i>;<br/>
<a NAME="E13:119"/><b>then </b>
 <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">A</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="jordan2c.html#D4" target="_self" title="JORDAN2C:def.4">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">B1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E15:119"/><i><font color="Green" title="E115">A11</font></i>: 
( <font color="Maroon" title="c5">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c5">B1</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="connsp_1.html#D6" title="CONNSP_1:def.6">CONNSP_1:def 6</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:119_1"/>
( <font color="Maroon" title="c5">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A1</font> or <font color="Maroon" title="c5">B1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> <font color="Maroon" title="c2">A1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> )
 </span><b>by </b><i><a class="txt" href="#E4:119"><i><font color="Green" title="E107">A3</font></i></a>, <a class="txt" href="#E6:119"><i><font color="Green" title="E109">A5</font></i></a>, <a class="txt" href="#E15:119"><i><font color="Green" title="E115">A11</font></i></a>, <a class="ref" href="connsp_1.html#T37" title="CONNSP_1:th.37">CONNSP_1:37</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_1"><b>suppose </b></a><a NAME="E1:119_1_1"/><i><font color="Green" title="E116">A12</font></i>: 
<font color="Maroon" title="c5">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">A</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c1">A</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:119_1_1_1"/>
not <font color="Maroon" title="c3">A2</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:119_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">A2</font> <a href="jordan2c.html#R2" title="JORDAN2C:pred.2">is_outside_component_of</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="#E8:119"><i><font color="Green" title="E111">A7</font></i></a>, <a class="ref" href="jordan2c.html#D4" target="_self" title="JORDAN2C:def.4">Def4</a></i>;<br/><a NAME="E3:119_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">A2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E5:119"><i><font color="Green" title="E108">A4</font></i></a>, <a class="txt" href="#E11:119"><i><font color="Green" title="E113">A9</font></i></a>, <a class="ref" href="jordan2c.html#T27" target="_self" title="JORDAN2C:th.27">Th27</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/><b>hence </b><a NAME="E4:119_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:119"><i><font color="Green" title="E106">A2</font></i></a>, <a class="txt" href="#E15:119"><i><font color="Green" title="E115">A11</font></i></a>, <a class="txt" href="#E1:119_1_1"><i><font color="Green" title="E116">A12</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E3:119_1_1"/><b>then </b><i><font color="Green" title="E117">A13</font></i>: 
<font color="Maroon" title="c3">A2</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="#E8:119"><i><font color="Green" title="E111">A7</font></i></a>, <a class="ref" href="jordan2c.html#D3" target="_self" title="JORDAN2C:def.3">Def3</a></i>;<br/><a NAME="E4:119_1_1"/><b>then </b><i><font color="Green" title="E118">A14</font></i>: 
<font color="Maroon" title="c3">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T26" target="_self" title="JORDAN2C:th.26">Th26</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:119_1_1_2"/>
not  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:119_1_1_2"/><i><font color="Green" title="E119">A15</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">A</font> &amp; not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A2</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:119_1_1_2"/><i><font color="Green" title="E120">A16</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">y</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B3</font> where <font color="Olive" title="b1">B3</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : <font color="Olive" title="b1">B3</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c1">A</font> </span>}</span>  )
 <b>by </b><i><a class="txt" href="#E3:119_1_1_2"><i><font color="Green" title="E119">A15</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">B3</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E7:119_1_1_2"/><i><font color="Green" title="E121">A17</font></i>: 
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B3</font> &amp; <font color="Maroon" title="c8">B3</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c1">A</font> )
 <b>by </b><i><a class="txt" href="#E5:119_1_1_2"><i><font color="Green" title="E120">A16</font></i></a></i>;<br/><a NAME="E8:119_1_1_2"/><i><font color="Green" title="E122">A18</font></i>: 
<font color="Maroon" title="c8">B3</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="#E7:119_1_1_2"><i><font color="Green" title="E121">A17</font></i></a>, <a class="ref" href="jordan2c.html#D3" target="_self" title="JORDAN2C:def.3">Def3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">B4</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E10:119_1_1_2"/><i><font color="Green" title="E123">A19</font></i>: 
( <font color="Maroon" title="c9">B4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B3</font> &amp; <font color="Maroon" title="c9">B4</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="connsp_1.html#D6" title="CONNSP_1:def.6">CONNSP_1:def 6</a></i>;<br/><a NAME="E11:119_1_1_2"/><i><font color="Green" title="E124">A20</font></i>: 
<font color="Maroon" title="c8">B3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E10:119"><i><font color="Green" title="E112">A8</font></i></a>, <a class="txt" href="#E10:119_1_1_2"><i><font color="Green" title="E123">A19</font></i></a>, <a class="ref" href="connsp_1.html#T34" title="CONNSP_1:th.34">CONNSP_1:34</a></i>;<br/><a NAME="E12:119_1_1_2"/>
( <font color="Maroon" title="c9">B4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> <font color="Maroon" title="c3">A2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> or <font color="Maroon" title="c9">B4</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> <font color="Maroon" title="c3">A2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="#E6:119"><i><font color="Green" title="E109">A5</font></i></a>, <a class="txt" href="#E10:119_1_1_2"><i><font color="Green" title="E123">A19</font></i></a>, <a class="ref" href="connsp_1.html#T37" title="CONNSP_1:th.37">CONNSP_1:37</a></i>;<br/><a NAME="E13:119_1_1_2"/><b>then </b><i><font color="Green" title="E125">A21</font></i>: 
( <font color="Maroon" title="c9">B4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> <font color="Maroon" title="c3">A2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> or <font color="Maroon" title="c9">B4</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> <font color="Maroon" title="c3">A2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:119_1_1_2_1"/>
<font color="Maroon" title="c9">B4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> <font color="Maroon" title="c2">A1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:119_1_1_2_1"/><b>then </b>
 <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">A</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 <b>by </b><i><a class="txt" href="#E4:119"><i><font color="Green" title="E107">A3</font></i></a>, <a class="txt" href="#E15:119"><i><font color="Green" title="E115">A11</font></i></a>, <a class="txt" href="#E1:119_1_1"><i><font color="Green" title="E116">A12</font></i></a>, <a class="txt" href="#E7:119_1_1_2"><i><font color="Green" title="E121">A17</font></i></a>, <a class="txt" href="#E10:119_1_1_2"><i><font color="Green" title="E123">A19</font></i></a>, <a class="ref" href="jordan2c.html#D3" target="_self" title="JORDAN2C:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E3:119_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E12:119"><i><font color="Green" title="E114">A10</font></i></a>, <a class="ref" href="jordan2c.html#D4" target="_self" title="JORDAN2C:def.4">Def4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E15:119_1_1_2"/><b>then </b><i><font color="Green" title="E126">A22</font></i>: 
( <font color="Maroon" title="c8">B3</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">A1</font> &amp; <font color="Maroon" title="c8">B3</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">A2</font> )
 <b>by </b><i><a class="txt" href="#E4:119"><i><font color="Green" title="E107">A3</font></i></a>, <a class="txt" href="#E5:119"><i><font color="Green" title="E108">A4</font></i></a>, <a class="txt" href="#E6:119"><i><font color="Green" title="E109">A5</font></i></a>, <a class="txt" href="#E3:119_1_1_2"><i><font color="Green" title="E119">A15</font></i></a>, <a class="txt" href="#E5:119_1_1_2"><i><font color="Green" title="E120">A16</font></i></a>, <a class="txt" href="#E7:119_1_1_2"><i><font color="Green" title="E121">A17</font></i></a>, <a class="txt" href="#E10:119_1_1_2"><i><font color="Green" title="E123">A19</font></i></a>, <a class="ref" href="connsp_1.html#T37" title="CONNSP_1:th.37">CONNSP_1:37</a></i>;<br/><a NAME="E16:119_1_1_2"/>
<font color="Maroon" title="c8">B3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="#E8:119_1_1_2"><i><font color="Green" title="E122">A18</font></i></a>, <a class="ref" href="sprect_1.html#T7" title="SPRECT_1:th.7">SPRECT_1:7</a></i>;<br/><b>then </b><font color="Maroon" title="c8">B3</font> = 
<font color="Maroon" title="c8">B3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font></span>)</span>
<b>by </b><i><a class="txt" href="#E3:119"><i><font color="Green" title="E106">A2</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">B3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">A1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A2</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="#E5:119"><i><font color="Green" title="E108">A4</font></i></a>, <a class="txt" href="#E5:119_1_1_2"><i><font color="Green" title="E120">A16</font></i></a>, <a class="txt" href="#E7:119_1_1_2"><i><font color="Green" title="E121">A17</font></i></a>, <a class="txt" href="#E10:119_1_1_2"><i><font color="Green" title="E123">A19</font></i></a>, <a class="txt" href="#E13:119_1_1_2"><i><font color="Green" title="E125">A21</font></i></a>, <a class="txt" href="#E15:119_1_1_2"><i><font color="Green" title="E126">A22</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>
;<br/><b>hence </b><a NAME="E18:119_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E11:119_1_1_2"><i><font color="Green" title="E124">A20</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E6:119_1_1"/>
 <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">A</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="#E3:119_1_1"><i><font color="Green" title="E117">A13</font></i></a>, <a class="txt" href="#E4:119_1_1"><i><font color="Green" title="E118">A14</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="119_1_2"><b>suppose </b></a><a NAME="E1:119_1_2"/><i><font color="Green" title="E116">A23</font></i>: 
<font color="Maroon" title="c5">B1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> <font color="Maroon" title="c2">A1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">A</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c1">A</font></span><br/><div class="add"><a NAME="E2:119_1_2"/><i><font color="Green" title="E117">A24</font></i>: 
 <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T28" target="_self" title="JORDAN2C:th.28">Th28</a></i>;<br/><a NAME="E3:119_1_2"/><i><font color="Green" title="E118">A25</font></i>: 
<span class="p1">(<span class="default"><a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="jordan2c.html#T31" target="_self" title="JORDAN2C:th.31">Th31</a></i>;<br/><b>set </b><font color="Maroon" title="c6">E1</font> =  <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> <font color="Maroon" title="c2">A1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>;<br/><b>set </b><font color="Maroon" title="c7">E2</font> =  <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> <font color="Maroon" title="c3">A2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>;<br/><a NAME="E4:119_1_2"/>
<span class="p1">(<span class="default"><a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> <font color="Maroon" title="c2">A1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> <font color="Maroon" title="c3">A2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:119"><i><font color="Green" title="E106">A2</font></i></a>, <a class="txt" href="#E4:119"><i><font color="Green" title="E107">A3</font></i></a>, <a class="txt" href="#E5:119"><i><font color="Green" title="E108">A4</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E5:119_1_2"/><b>then </b><i><font color="Green" title="E119">A26</font></i>: 
 <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A2</font>
 <b>by </b><i><a class="txt" href="#E5:119"><i><font color="Green" title="E108">A4</font></i></a>, <a class="txt" href="#E6:119"><i><font color="Green" title="E109">A5</font></i></a>, <a class="txt" href="#E10:119"><i><font color="Green" title="E112">A8</font></i></a>, <a class="txt" href="#E15:119"><i><font color="Green" title="E115">A11</font></i></a>, <a class="txt" href="#E1:119_1_2"><i><font color="Green" title="E116">A23</font></i></a>, <a class="ref" href="jordan2c.html#T115" target="_self" title="JORDAN2C:th.115">Th115</a></i>;<br/><a NAME="E6:119_1_2"/><i><font color="Green" title="E120">A27</font></i>: 
 <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A1</font> )</span><br/>

<b>assume </b><a NAME="E1:119_1_2_1"/><i><font color="Green" title="E121">A28</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A1</font></span><br/>

<a NAME="E2:119_1_2_1"/><b>then </b><i><font color="Green" title="E122">A29</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="txt" href="#E3:119_1_2"><i><font color="Green" title="E118">A25</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:119_1_2_1"/>
not <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="#E2:119_1_2"><i><font color="Green" title="E117">A24</font></i></a>, <a class="txt" href="#E1:119_1_2_1"><i><font color="Green" title="E121">A28</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E4:119_1_2_1"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A1</font>
 <b>by </b><i><a class="txt" href="#E3:119"><i><font color="Green" title="E106">A2</font></i></a>, <a class="txt" href="#E5:119_1_2"><i><font color="Green" title="E119">A26</font></i></a>, <a class="txt" href="#E2:119_1_2_1"><i><font color="Green" title="E122">A29</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E7:119_1_2"/>
<font color="Maroon" title="c2">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A1</font> or <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">A</font> )</span><br/>

<b>assume </b><a NAME="E1:119_1_2_2"/><i><font color="Green" title="E121">A30</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">A</font></span><br/>

<a NAME="E2:119_1_2_2"/><b>then </b><i><font color="Green" title="E122">A31</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="txt" href="#E3:119"><i><font color="Green" title="E106">A2</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:119_1_2_2"/>
not <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="#E3:119"><i><font color="Green" title="E106">A2</font></i></a>, <a class="txt" href="#E5:119_1_2"><i><font color="Green" title="E119">A26</font></i></a>, <a class="txt" href="#E1:119_1_2_2"><i><font color="Green" title="E121">A30</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E4:119_1_2_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="#E3:119_1_2"><i><font color="Green" title="E118">A25</font></i></a>, <a class="txt" href="#E2:119_1_2_2"><i><font color="Green" title="E122">A31</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E8:119_1_2"/><b>then </b><i><font color="Green" title="E121">A32</font></i>: 
 <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A1</font>
 <b>by </b><i><a class="txt" href="#E6:119_1_2"><i><font color="Green" title="E120">A27</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E9:119_1_2"/>
 <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">A</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 <b>by </b><i><a class="txt" href="#E1:119"><i><font color="Green" title="E105">A1</font></i></a>, <a class="ref" href="jordan2c.html#T114" target="_self" title="JORDAN2C:th.114">Th114</a></i>;<br/><b>hence </b><a NAME="E10:119_1_2"/>
 <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">A</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="#E7:119"><i><font color="Green" title="E110">A6</font></i></a>, <a class="txt" href="#E8:119_1_2"><i><font color="Green" title="E121">A32</font></i></a>, <a class="ref" href="jordan2c.html#D3" target="_self" title="JORDAN2C:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
