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<div class="add">

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> non <a href="sppol_1.html#V1">horizontal</a> non <a href="sppol_1.html#V2">vertical</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>set </b><font color="Maroon">f</font> =  <a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font>;<br/>
<b>set </b><font color="Maroon">A</font> =  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span>;<br/>
<a NAME="E1:108"/><i><font color="Green">E47</font></i>: 
 <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="goboard9.html#D2">GOBOARD9:def 2</a></i>;<br/>
<div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:108_1"/><i><font color="Green">E57</font></i>: 
not  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> is <a href="jordan2c.html#V1" target="_self">Bounded</a>
 ;<br/><a NAME="E2:108_1"/>
 <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#R1">misses</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="sprect_1.html#T96">SPRECT_1:96</a></i>;<br/><b>hence </b><a NAME="E3:108_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T21" target="_self">Th21</a>, <a class="ref" href="jordan2c.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:108"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:108"/>
 <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <font color="Olive">B</font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:108_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span>
 ;<br/>

<a NAME="E2:108_2"/>
 <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <font color="Olive">B</font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> </span>}</span> 
 
<b>by </b><i><a class="ref" href="jordan2c.html#T21" target="_self">Th21</a></i>;<br/>
<b>hence </b><a NAME="E3:108_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <font color="Olive">B</font> <a href="jordan2c.html#R1" target="_self">is_inside_component_of</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:108"/>
 <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1" target="_self">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E6:108"/>
 <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1">SpStSeq</a> <font color="Maroon">D</font></span>)</span> is <a href="jordan2c.html#V1" target="_self">Bounded</a>
 <b>by </b><i/>;<br/>


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