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<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">A</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a>  &amp; <font color="Maroon" title="c1">A</font> is  <a href="jordan1.html#V1" title="JORDAN1:attr.1">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/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:116"/><span class="lab"><font color="Green" title="E88">A1</font></span>: 
<font color="Maroon" title="c1">A</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a> 
 <span class="kw">and </span><br/><a NAME="E2:116"/><span class="lab"><font color="Green" title="E89">A2</font></span>: 
<font color="Maroon" title="c1">A</font> is  <a href="jordan1.html#V1" title="JORDAN1:attr.1">Jordan</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">reconsider </span><font color="Maroon" title="c2">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:116"><span class="lab"><font color="Green" title="E89">A2</font></span></a>, <a class="ref" href="jordan1.html#D2" onmouseover="rs('jordan1/D2')" onmouseout="rh()">JORDAN1:def 2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">A1</font>, <font color="Maroon" title="c4">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><span class="kw"> such that </span><br/><a NAME="E5:116"/><span class="lab"><font color="Green" title="E90">A3</font></span>: 
<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="c3">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">A2</font>
 <span class="kw">and </span><br/><a NAME="E6:116"/><span class="lab"><font color="Green" title="E91">A4</font></span>: 
<font color="Maroon" title="c3">A1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">A2</font>
 <span class="kw">and </span><br/><a NAME="E7:116"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">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="c4">A2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">A2</font>
 <span class="kw">and </span><br/><a NAME="E8:116"/><span class="lab"><font color="Green" title="E92">A5</font></span>: 
 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="c3">A1</font> &amp; <font color="Olive" title="b2">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A2</font> holds <br/>( <font color="Olive" title="b1">C1</font> is  <a href="connsp_1.html#V3" title="CONNSP_1:attr.3">a_component</a>  &amp; <font color="Olive" title="b2">C2</font> is  <a href="connsp_1.html#V3" title="CONNSP_1:attr.3">a_component</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:116"><span class="lab"><font color="Green" title="E89">A2</font></span></a>, <a class="ref" href="jordan1.html#D2" onmouseover="rs('jordan1/D2')" onmouseout="rh()">JORDAN1:def 2</a></span>;<br/>
<a NAME="E9:116"/><span class="lab"><font color="Green" title="E93">A6</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:116"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="ref" href="jordan2c.html#T68" target="_self" onmouseover="rs('jordan2c/T68')" onmouseout="rh()">Th53</a></span>;<br/>
<a NAME="E10:116"/><span class="kw">then </span>
 <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">A</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E12:116"/><span class="lab"><font color="Green" title="E94">A7</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E13:116"/><span class="lab"><font color="Green" title="E95">A8</font></span>: 
<font color="Maroon" title="c5">B1</font> is  <a href="connsp_1.html#V3" title="CONNSP_1:attr.3">a_component</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_1.html#D6" onmouseover="rs('connsp_1/D6')" onmouseout="rh()">CONNSP_1:def 6</a></span>;<br/>
<a NAME="E14:116"/><span class="lab"><font color="Green" title="E96">A9</font></span>: 
 <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> (<font color="Maroon" title="c3">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="c3">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:116"><span class="lab"><font color="Green" title="E90">A3</font></span></a>, <a class="ref" href="xboole_1.html#T21" onmouseover="rs('xboole_1/T21')" onmouseout="rh()">XBOOLE_1:21</a></span>;<br/>
<a NAME="E15:116"/><span class="lab"><font color="Green" title="E97">A10</font></span>: 
 <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> (<font color="Maroon" title="c4">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="c4">A2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:116"><span class="lab"><font color="Green" title="E90">A3</font></span></a>, <a class="ref" href="xboole_1.html#T21" onmouseover="rs('xboole_1/T21')" onmouseout="rh()">XBOOLE_1:21</a></span>;<br/>
<a NAME="E16:116"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A11</font></span>: 
 <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> (<font color="Maroon" title="c4">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>) is  <a href="connsp_1.html#V3" title="CONNSP_1:attr.3">a_component</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:116"><span class="lab"><font color="Green" title="E92">A5</font></span></a>, <a class="txt" href="#E14:116"><span class="lab"><font color="Green" title="E96">A9</font></span></a></span>;<br/>
<a NAME="E17:116"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A12</font></span>: 
<font color="Maroon" title="c4">A2</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:116"><span class="lab"><font color="Green" title="E97">A10</font></span></a>, <a class="ref" href="connsp_1.html#D6" onmouseover="rs('connsp_1/D6')" onmouseout="rh()">CONNSP_1:def 6</a></span>;<br/>
<a NAME="E18:116"/><span class="lab"><font color="Green" title="E100">A13</font></span>: 
 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="c2">D</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 
;<br/>
<a NAME="E19:116"/><span class="lab"><font color="Green" title="E101">A14</font></span>: 
 <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> (<font color="Maroon" title="c3">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>) is  <a href="connsp_1.html#V3" title="CONNSP_1:attr.3">a_component</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:116"><span class="lab"><font color="Green" title="E92">A5</font></span></a>, <a class="txt" href="#E14:116"><span class="lab"><font color="Green" title="E96">A9</font></span></a>, <a class="txt" href="#E15:116"><span class="lab"><font color="Green" title="E97">A10</font></span></a></span>;<br/>
<a NAME="E20:116"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A15</font></span>: 
<font color="Maroon" title="c3">A1</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:116"><span class="lab"><font color="Green" title="E96">A9</font></span></a>, <a class="ref" href="connsp_1.html#D6" onmouseover="rs('connsp_1/D6')" onmouseout="rh()">CONNSP_1:def 6</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="116_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:116_1"/>
( <font color="Maroon" title="c5">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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="c3">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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:116"><span class="lab"><font color="Green" title="E96">A9</font></span></a>, <a class="txt" href="#E19:116"><span class="lab"><font color="Green" title="E101">A14</font></span></a>, <a class="txt" href="#E13:116"><span class="lab"><font color="Green" title="E95">A8</font></span></a>, <a class="ref" href="connsp_1.html#T35" onmouseover="rs('connsp_1/T35')" onmouseout="rh()">CONNSP_1:35</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="116_1_1"><span class="kw">suppose </span></a><a NAME="E1:116_1_1"/><span class="lab"><font color="Green" title="E103">A16</font></span>: 
<font color="Maroon" title="c5">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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><span class="lab"><font color="Green" title="E104">A17</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="116_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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="c4">A2</font></span></a><div class="add"><span class="kw">assume </span><a NAME="E1:116_1_1_1"/>
 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="c4">A2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:116_1_1_1"/><span class="lab"><font color="Green" title="E105">A18</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:116_1_1_1"/><span class="lab"><font color="Green" title="E106">A19</font></span>: 
 not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A2</font>
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E6:116_1_1_1"/><span class="lab"><font color="Green" title="E107">A20</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:116_1_1_1"/><span class="lab"><font color="Green" title="E108">A21</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="tarski.html#R2" title="TARSKI: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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:116_1_1_1"><span class="lab"><font color="Green" title="E105">A18</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E9:116_1_1_1"/><span class="lab"><font color="Green" title="E109">A22</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B3</font>
 <span class="kw">and </span><br/><a NAME="E10:116_1_1_1"/><span class="lab"><font color="Green" title="E110">A23</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:116_1_1_1"><span class="lab"><font color="Green" title="E108">A21</font></span></a></span>;<br/><a NAME="E11:116_1_1_1"/><span class="lab"><font color="Green" title="E111">A24</font></span>: 
<font color="Maroon" title="c8">B3</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:116_1_1_1"><span class="lab"><font color="Green" title="E110">A23</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E13:116_1_1_1"/><span class="lab"><font color="Green" title="E112">A25</font></span>: 
<font color="Maroon" title="c9">B4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B3</font>
 <span class="kw">and </span><br/><a NAME="E14:116_1_1_1"/><span class="lab"><font color="Green" title="E113">A26</font></span>: 
<font color="Maroon" title="c9">B4</font> is  <a href="connsp_1.html#V3" title="CONNSP_1:attr.3">a_component</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_1.html#D6" onmouseover="rs('connsp_1/D6')" onmouseout="rh()">CONNSP_1:def 6</a></span>;<br/><a NAME="E15:116_1_1_1"/><span class="lab"><font color="Green" title="E114">A27</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:116"><span class="lab"><font color="Green" title="E100">A13</font></span></a>, <a class="txt" href="#E13:116_1_1_1"><span class="lab"><font color="Green" title="E112">A25</font></span></a>, <a class="txt" href="#E14:116_1_1_1"><span class="lab"><font color="Green" title="E113">A26</font></span></a>, <a class="ref" href="connsp_1.html#T32" onmouseover="rs('connsp_1/T32')" onmouseout="rh()">CONNSP_1:32</a></span>;<br/><a NAME="E16:116_1_1_1"/>
<font color="Maroon" title="c9">B4</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> (<font color="Maroon" title="c3">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E14:116"><span class="lab"><font color="Green" title="E96">A9</font></span></a>, <a class="txt" href="#E12:116"><span class="lab"><font color="Green" title="E94">A7</font></span></a>, <a class="txt" href="#E1:116_1_1"><span class="lab"><font color="Green" title="E103">A16</font></span></a>, <a class="txt" href="#E10:116_1_1_1"><span class="lab"><font color="Green" title="E110">A23</font></span></a>, <a class="txt" href="#E13:116_1_1_1"><span class="lab"><font color="Green" title="E112">A25</font></span></a>, <a class="txt" href="#E9:116"><span class="lab"><font color="Green" title="E93">A6</font></span></a></span>;<br/><a NAME="E17:116_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A28</font></span>: 
<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">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:116"><span class="lab"><font color="Green" title="E96">A9</font></span></a>, <a class="txt" href="#E19:116"><span class="lab"><font color="Green" title="E101">A14</font></span></a>, <a class="txt" href="#E13:116_1_1_1"><span class="lab"><font color="Green" title="E112">A25</font></span></a>, <a class="txt" href="#E14:116_1_1_1"><span class="lab"><font color="Green" title="E113">A26</font></span></a>, <a class="ref" href="connsp_1.html#T35" onmouseover="rs('connsp_1/T35')" onmouseout="rh()">CONNSP_1:35</a></span>;<br/><a NAME="E18:116_1_1_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="c4">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="c4">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>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:116"><span class="lab"><font color="Green" title="E98">A11</font></span></a>, <a class="txt" href="#E14:116_1_1_1"><span class="lab"><font color="Green" title="E113">A26</font></span></a>, <a class="ref" href="connsp_1.html#T35" onmouseover="rs('connsp_1/T35')" onmouseout="rh()">CONNSP_1:35</a></span>;<br/><a NAME="E19:116_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A29</font></span>: 
( <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="c4">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="c4">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> )
 ;<br/><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="c3">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">A2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:116"><span class="lab"><font color="Green" title="E90">A3</font></span></a>, <a class="txt" href="#E11:116_1_1_1"><span class="lab"><font color="Green" title="E111">A24</font></span></a>, <a class="ref" href="sprect_1.html#T5" onmouseover="rs('sprect_1/T5')" onmouseout="rh()">SPRECT_1:5</a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
<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="c3">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="c4">A2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T23" onmouseover="rs('xboole_1/T23')" onmouseout="rh()">XBOOLE_1:23</a></span>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:116"><span class="lab"><font color="Green" title="E97">A10</font></span></a>, <a class="txt" href="#E4:116_1_1_1"><span class="lab"><font color="Green" title="E106">A19</font></span></a>, <a class="txt" href="#E6:116_1_1_1"><span class="lab"><font color="Green" title="E107">A20</font></span></a>, <a class="txt" href="#E9:116_1_1_1"><span class="lab"><font color="Green" title="E109">A22</font></span></a>, <a class="txt" href="#E13:116_1_1_1"><span class="lab"><font color="Green" title="E112">A25</font></span></a>, <a class="txt" href="#E19:116_1_1_1"><span class="lab"><font color="Green" title="E116">A29</font></span></a>, <a class="txt" href="#E17:116_1_1_1"><span class="lab"><font color="Green" title="E115">A28</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E21:116_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:116_1_1_1"><span class="lab"><font color="Green" title="E114">A27</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="116_1_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">A2</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:116_1_1_2"/>
 not <font color="Maroon" title="c4">A2</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:116_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">A2</font> <a href="jordan2c.html#R2" title="JORDAN2C:pred.2">is_outside_component_of</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:116"><span class="lab"><font color="Green" title="E99">A12</font></span></a></span>;<br/><a NAME="E3:116_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T23" target="_self" onmouseover="rs('jordan2c/T23')" onmouseout="rh()">Th14</a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>;<br/><span class="kw">hence </span><a NAME="E4:116_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:116"><span class="lab"><font color="Green" title="E91">A4</font></span></a>, <a class="txt" href="#E12:116"><span class="lab"><font color="Green" title="E94">A7</font></span></a>, <a class="txt" href="#E1:116_1_1"><span class="lab"><font color="Green" title="E103">A16</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:116_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A30</font></span>: 
<font color="Maroon" title="c4">A2</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:116"><span class="lab"><font color="Green" title="E99">A12</font></span></a></span>;<br/><a NAME="E5:116_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T22" target="_self" onmouseover="rs('jordan2c/T22')" onmouseout="rh()">Th13</a></span>;<br/><span class="kw">hence </span><a NAME="E6:116_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116_1_1"><span class="lab"><font color="Green" title="E105">A30</font></span></a>, <a class="txt" href="#E2:116_1_1"><span class="lab"><font color="Green" title="E104">A17</font></span></a>, <a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="116_1_2"><span class="kw">suppose </span></a><a NAME="E1:116_1_2"/><span class="lab"><font color="Green" title="E103">A31</font></span>: 
<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="c3">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"><span class="kw">set </span><font color="Maroon" title="c6">E1</font> =  <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> (<font color="Maroon" title="c3">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/><span class="kw">set </span><font color="Maroon" title="c7">E2</font> =  <a href="connsp_3.html#K4" title="CONNSP_3:func.4">Down</a> (<font color="Maroon" title="c4">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="E2:116_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="c3">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="c4">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:116"><span class="lab"><font color="Green" title="E90">A3</font></span></a>, <a class="txt" href="#E14:116"><span class="lab"><font color="Green" title="E96">A9</font></span></a>, <a class="txt" href="#E15:116"><span class="lab"><font color="Green" title="E97">A10</font></span></a>, <a class="ref" href="pre_topc.html#D5" onmouseover="rs('pre_topc/D5')" onmouseout="rh()">PRE_TOPC:def 5</a></span>;<br/><a NAME="E3:116_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A32</font></span>: 
 <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="c4">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:116"><span class="lab"><font color="Green" title="E97">A10</font></span></a>, <a class="txt" href="#E16:116"><span class="lab"><font color="Green" title="E98">A11</font></span></a>, <a class="txt" href="#E18:116"><span class="lab"><font color="Green" title="E100">A13</font></span></a>, <a class="txt" href="#E12:116"><span class="lab"><font color="Green" title="E94">A7</font></span></a>, <a class="txt" href="#E13:116"><span class="lab"><font color="Green" title="E95">A8</font></span></a>, <a class="txt" href="#E1:116_1_2"><span class="lab"><font color="Green" title="E103">A31</font></span></a>, <a class="ref" href="jordan2c.html#T107" target="_self" onmouseover="rs('jordan2c/T107')" onmouseout="rh()">Th91</a></span>;<br/><a NAME="E4:116_1_2"/><span class="lab"><font color="Green" title="E105">A33</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T27" target="_self" onmouseover="rs('jordan2c/T27')" onmouseout="rh()">Th18</a></span>;<br/><a NAME="E5:116_1_2"/><span class="lab"><font color="Green" title="E106">A34</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T24" target="_self" onmouseover="rs('jordan2c/T24')" onmouseout="rh()">Th15</a></span>;<br/><a NAME="E6:116_1_2"/><span class="lab"><font color="Green" title="E107">A35</font></span>: 
 <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">A1</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="116_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c3">A1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:116_1_2_1"/>
<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c3">A1</font></span><br/>

<a NAME="E2:116_1_2_1"/><span class="kw">then </span>
( <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>  &amp;  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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:116_1_2"><span class="lab"><font color="Green" title="E106">A34</font></span></a>, <a class="txt" href="#E4:116_1_2"><span class="lab"><font color="Green" title="E105">A33</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:116_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="c3">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:116"><span class="lab"><font color="Green" title="E90">A3</font></span></a>, <a class="txt" href="#E3:116_1_2"><span class="lab"><font color="Green" title="E104">A32</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E7:116_1_2"/><span class="lab"><font color="Green" title="E108">A36</font></span>: 
 <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">A</font> is  <a href="rltopsp1.html#V9" title="RLTOPSP1:attr.9">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:116"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="ref" href="jordan2c.html#T106" target="_self" onmouseover="rs('jordan2c/T106')" onmouseout="rh()">Th90</a></span>;<br/><a NAME="E8:116_1_2"/>
<font color="Maroon" title="c3">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="116_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c3">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/>

<span class="kw">assume </span><a NAME="E1:116_1_2_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E2:116_1_2_2"/><span class="kw">then </span>
( <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>  &amp;  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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:116"><span class="lab"><font color="Green" title="E90">A3</font></span></a>, <a class="txt" href="#E6:116"><span class="lab"><font color="Green" title="E91">A4</font></span></a>, <a class="txt" href="#E3:116_1_2"><span class="lab"><font color="Green" title="E104">A32</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:116_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116_1_2"><span class="lab"><font color="Green" title="E105">A33</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E9:116_1_2"/><span class="kw">then </span>
 <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="c3">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:116_1_2"><span class="lab"><font color="Green" title="E107">A35</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:116_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:116"><span class="lab"><font color="Green" title="E102">A15</font></span></a>, <a class="txt" href="#E7:116_1_2"><span class="lab"><font color="Green" title="E108">A36</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
