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<div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b</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">f</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>,<a href="topmetr.html#K3">R^1</a> ;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>







<b>assume </b><a NAME="E1:20"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="jordan2b.html#K1" target="_self">Proj</a> <font color="Olive">p</font>,<font color="Maroon">i</font>
 ;<br/>

<b>thus </b><a NAME="E2:20"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> : ( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="jordan2b.html#K1" target="_self">Proj</a> <font color="Olive">p</font>,<font color="Maroon">i</font> &amp;  <a href="jordan2b.html#K1" target="_self">Proj</a> <font color="Olive">p</font>,<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<a NAME="E1:20_1"/><i><font color="Green">E50</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</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> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:20_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> : ( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="jordan2b.html#K1" target="_self">Proj</a> <font color="Olive">p</font>,<font color="Maroon">i</font> &amp;  <a href="jordan2b.html#K1" target="_self">Proj</a> <font color="Olive">p</font>,<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><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="E1:20_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:20_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> ) </span>}</span>  )
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:20_1_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">px</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E6:20_1_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">px</font> <a href="hidden.html#R1">=</a>  <a href="jordan2b.html#K1" target="_self">Proj</a> <font color="Maroon">px</font>,<font color="Maroon">i</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:20_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> : ( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="jordan2b.html#K1" target="_self">Proj</a> <font color="Olive">p</font>,<font color="Maroon">i</font> &amp;  <a href="jordan2b.html#K1" target="_self">Proj</a> <font color="Olive">p</font>,<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> ) </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:20_1"/>
<span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> : ( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="jordan2b.html#K1" target="_self">Proj</a> <font color="Olive">p</font>,<font color="Maroon">i</font> &amp;  <a href="jordan2b.html#K1" target="_self">Proj</a> <font color="Olive">p</font>,<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_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="E1:20_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> : ( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="jordan2b.html#K1" target="_self">Proj</a> <font color="Olive">p</font>,<font color="Maroon">i</font> &amp;  <a href="jordan2b.html#K1" target="_self">Proj</a> <font color="Olive">p</font>,<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E3:20_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="jordan2b.html#K1" target="_self">Proj</a> <font color="Maroon">p</font>,<font color="Maroon">i</font> &amp;  <a href="jordan2b.html#K1" target="_self">Proj</a> <font color="Maroon">p</font>,<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> )
 ;<br/>
<a NAME="E4:20_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="jordan2b.html#K1" target="_self">Proj</a> <font color="Maroon">p</font>,<font color="Maroon">i</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:20_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:20_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> ) </span>}</span> 
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:20_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> : ( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="jordan2b.html#K1" target="_self">Proj</a> <font color="Olive">p</font>,<font color="Maroon">i</font> &amp;  <a href="jordan2b.html#K1" target="_self">Proj</a> <font color="Olive">p</font>,<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>


</div>
