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<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">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>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>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> holds <br/><font color="Maroon">a<sub>2</sub></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/>
<a NAME="E1:4"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>
 ;<br/>

<b>then 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><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<b>consider </b><font color="Maroon">q</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">g</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:4_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">px</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> )
 <b>by </b><i/>;<br/>
<a NAME="E5:4_1"/>
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>  st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> holds <br/><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</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>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:4_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  &amp; ( 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>  st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> holds <br/><font color="Olive">y</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/>


</div><b>end;</b></div>
<a NAME="E2:4"/>
 ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>, <a href="numbers.html#K1">REAL</a>  st <br/>for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 
<b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i>
<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:4"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> holds <br/>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>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</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/>
<a NAME="E5:4"/>
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> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> as   <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>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E7:4"/>
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">f1</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <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>;<br/>

<a NAME="E1:4_2"/>
<font color="Maroon">p</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> <font color="Maroon">n</font></span>)</span>
 
;<br/>
<a NAME="E2:4_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<b>hence </b><a NAME="E3:4_2"/>
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</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/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:4"/>
 ex <font color="Olive">f</font> being  <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>  st <br/>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="Olive">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/>


</div>
