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<b>let </b><font color="Maroon">a</font> be   <a href="pre_topc.html#NM2">Point</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">X</font> = <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">a</font></span>)</span>;<br/>
<a NAME="E1:8"/><i><font color="Green">E55</font></i>: 
<a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">a</font></span>)</span> is <a href="seq_4.html#V1">bounded_above</a>
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:8_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">a</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:8_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 <b>and </b><br/><a NAME="E4:8_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E5:8_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/><a NAME="E7:8_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, <a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>;<br/><b>hence </b><a NAME="E8:8_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, <a class="ref" href="topreal1.html#D14">TOPREAL1:def 14</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:8_2"/><i><font color="Green">E68</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 ;<br/><b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">a</font> <a href="euclid.html#K22">`2</a>  as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/><b>take </b><font color="Maroon">r</font> = <font color="Maroon">r</font>;<br/><a NAME="E3:8_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">a</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="topreal1.html#T45">TOPREAL1:45</a>, <a class="ref" href="pscomp_1.html#D29">PSCOMP_1:def 29</a></i>;<br/><b>hence </b><a NAME="E4:8_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/><a NAME="E5:8_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> 0
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><b>hence </b><a NAME="E6:8_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:8"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14">proj2</a>  <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="topreal1.html#K8">south_halfline</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>


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