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<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>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">T</font>;<br/>





<b>set </b><font color="Maroon">c</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>;<br/>
<b>set </b><font color="Maroon">fc</font> = <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:83"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> holds  <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E2:83"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">t</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st ( for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> holds  <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> ) holds <br/><font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font>
 ;<br/>

<div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><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="E1:83_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 ;<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="E3:83_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</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>thus </b><a NAME="E4:83_1"/>
<font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</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="E1:83_2"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> holds <br/><font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><a NAME="E1:83_2_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/><b>hence </b><a NAME="E2:83_2_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:83_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:83"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K11" target="_self">sup</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="seq_4.html#T63">SEQ_4:63</a></i>;<br/>


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