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<div class="add">

<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> non <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a> non <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</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>;<br/>

<b>set </b><font color="Maroon" title="c2">f</font> =  <a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font>;<br/>
<b>set </b><font color="Maroon" title="c3">A</font> =  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span>;<br/>
<a NAME="E1:105"/><i><font color="Green" title="E94">A1</font></i>: 
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="goboard9.html#D1" title="GOBOARD9:def.1">GOBOARD9:def 1</a></i>;<br/>
<a NAME="E2:105"/>
not  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T98" target="_self" title="JORDAN2C:th.98">Th98</a></i>;<br/>
<a NAME="E3:105"/><b>then </b><i><font color="Green" title="E95">A2</font></i>: 
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="jordan2c.html#R2" title="JORDAN2C:pred.2">is_outside_component_of</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E1:105"><i><font color="Green" title="E94">A1</font></i></a>, <a class="ref" href="jordan2c.html#D4" target="_self" title="JORDAN2C:def.4">Def4</a></i>;<br/>
<a NAME="E4:105"/>
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</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">B</font> <a href="jordan2c.html#R2" title="JORDAN2C:pred.2">is_outside_component_of</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</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><br/>

<b>assume </b><a NAME="E1:105_1"/><i><font color="Green" title="E96">A3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span>
 ;<br/>

<a NAME="E2:105_1"/>
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</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">B</font> <a href="jordan2c.html#R2" title="JORDAN2C:pred.2">is_outside_component_of</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan2c.html#E3:105"><i><font color="Green" title="E95">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:105_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</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">B</font> <a href="jordan2c.html#R2" title="JORDAN2C:pred.2">is_outside_component_of</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:105_1"><i><font color="Green" title="E96">A3</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:105"/>
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" title="SPRECT_1:func.1">SpStSeq</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
 ;<br/>


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