<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>thus </b><a NAME="E1:25"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="compts_1.html#NR3" target="_self">is_T3</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="urysohn1.html#NR1" target="_self">is_T1</a>  &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="nagata_1.html#NM1">FamilySequence</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="nagata_1.html#V6">Basis_sigma_locally_finite</a> implies <font color="Maroon">c<sub>1</sub></font> is <a href="pcomps_1.html#V3">metrizable</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:25_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="compts_1.html#NR3" target="_self">is_T3</a> 
 <b>and </b><br/><a NAME="E2:25_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="urysohn1.html#NR1" target="_self">is_T1</a> 
 <b>and </b><br/><a NAME="E3:25_1"/><i><font color="Green">E26</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nagata_1.html#NM1">FamilySequence</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="nagata_1.html#V6">Basis_sigma_locally_finite</a>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="nagata_1.html#NM1">FamilySequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:25_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="nagata_1.html#V6">Basis_sigma_locally_finite</a>
 <b>by </b><i><a class="txt" href="nagata_2.html#E3:25_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E6:25_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="nagata_1.html#V3">sigma_locally_finite</a> &amp;  <a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E5:25_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="nagata_1.html#D6">NAGATA_1:def 6</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</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> ,  <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Maroon">a<sub>3</sub></font> <a href="funct_2.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> is <a href="pscomp_1.html#V9">continuous</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="nagata_1.html#R1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; ( for <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st ex <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>7</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">b<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>6</sub></font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">b<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>7</sub></font> &amp; not <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>7</sub></font> ) holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> 1 ) ) ) );<br/>
<a NAME="E7:25_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>3</sub></font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<a NAME="E1:25_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> is <a href="pcomps_1.html#V1">locally_finite</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> is <a href="pcomps_1.html#V1">locally_finite</a> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E6:25_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="nagata_1.html#D3">NAGATA_1:def 3</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a<sub>1</sub></font> ) </span>}</span> ;<br/>
<a NAME="E2:25_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:25_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> ) </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:25_1_1_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>9</sub></font>)
 ;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:25_1_1_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:25_1_1_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E3:25_1_1_1_1"><i><font color="Green">E33</font></i></a></i>;<br/><b>thus </b><a NAME="E6:25_1_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="nagata_2.html#E3:25_1_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="nagata_2.html#E5:25_1_1_1_1"><i><font color="Green">E34</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:25_1_1_1"/><b>then </b>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>9</sub></font>) <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:25_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>9</sub></font>) <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:25_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2">FUNCT_2:sch 2</a>(<a class="txt" href="nagata_2.html#E2:25_1_1"><i><font color="Green">E31</font></i></a>);<br/></i>
<a NAME="E5:25_1_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>set </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> ) </span>}</span> ;<br/>
<a NAME="E1:25_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></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="E1:25_1_1_2_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:25_1_1_2_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:25_1_1_2_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:25_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></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="E1:25_1_1_2_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>

<a NAME="E2:25_1_1_2_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:25_1_1_2_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:25_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> is <a href="pcomps_1.html#V1">locally_finite</a>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="pcomps_1.html#T12">PCOMPS_1:12</a></i>;<br/>
<a NAME="E5:25_1_1_2"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2">clf</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T23">PCOMPS_1:23</a></i>;<br/>
<a NAME="E6:25_1_1_2"/>
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2">clf</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></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="E1:25_1_1_2_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2">clf</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:25_1_1_2_3"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K2">clf</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="nagata_2.html#E3:25_1_1_2_3"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:25_1_1_2_3"/><i><font color="Green">E36</font></i>: 
(  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E3:25_1_1_2_3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pcomps_1.html#D3">PCOMPS_1:def 3</a></i>;<br/>
<a NAME="E7:25_1_1_2_3"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E6:25_1_1_2_3"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:25_1_1_2_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="nagata_2.html#E3:25_1_1_2_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="nagata_2.html#E6:25_1_1_2_3"><i><font color="Green">E36</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:25_1_1_2"/>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="nagata_2.html#E4:25_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="nagata_2.html#E5:25_1_1_2"><i><font color="Green">E34</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pscomp_1.html#V9">continuous</a> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; ( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  implies <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> implies <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 ) ) ) );<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>;<br/>
<a NAME="E6:25_1_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:25_1_1_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E2:25_1_1_3"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E6:25_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="cantor_1.html#D2">CANTOR_1:def 2</a>, <a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E4:25_1_1_3"/>
(  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E5:25_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E5:25_1_1_3"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#R1">misses</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="compts_1.html#NR4" target="_self">is_T4</a>  )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="nagata_2.html#E2:25_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="nagata_2.html#E5:25_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="nagata_1.html#T20">NAGATA_1:20</a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E7:25_1_1_3"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K3">`</a>  implies <font color="Maroon">c<sub>13</sub></font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> implies <font color="Maroon">c<sub>13</sub></font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) ) ) )
 <b>by </b><i><a class="ref" href="urysohn3.html#T23">URYSOHN3:23</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E9:25_1_1_3"/><i><font color="Green">E36</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pscomp_1.html#V9">continuous</a> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; ( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  implies <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> implies <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 ) ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_3_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>14</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/>

<b>thus </b><a NAME="E1:25_1_1_3_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> is <a href="pscomp_1.html#V9">continuous</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K3">`</a>  implies <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> implies <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) ) ) )
 <b>by </b><i><a class="txt" href="nagata_2.html#E7:25_1_1_3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="topreal6.html#T83">TOPREAL6:83</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:25_1_1_3"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>hence </b><a NAME="E11:25_1_1_3"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>11</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 <b>by </b><i><a class="txt" href="nagata_2.html#E9:25_1_1_3"><i><font color="Green">E36</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>, <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E8:25_1_1"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="nagata_2.html#E6:25_1_1"><i><font color="Green">E34</font></i></a>);<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">a<sub>1</sub></font> ) </span>}</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ,  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">a<sub>2</sub></font>) is <a href="finset_1.html#V1">finite</a> );<br/>
<a NAME="E9:25_1_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="pcomps_1.html#D2">PCOMPS_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E11:25_1_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="nagata_2.html#E9:25_1_1"><i><font color="Green">E36</font></i></a>);<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font>) ) </span>}</span> ;<br/>
<a NAME="E12:25_1_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) is <a href="finset_1.html#V1">finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>) </span>}</span> ;<br/>
<a NAME="E1:25_1_1_4"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>) is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E11:25_1_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E2:25_1_1_4"/><i><font color="Green">E40</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>(<a class="txt" href="nagata_2.html#E1:25_1_1_4"><i><font color="Green">E39</font></i></a>);<br/></i>
<a NAME="E3:25_1_1_4"/>
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>13</sub></font>) <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></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="E1:25_1_1_4_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>13</sub></font>)
 ;<br/>

<a NAME="E2:25_1_1_4_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>) )
 
;<br/>
<b>hence </b><a NAME="E3:25_1_1_4_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>) </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:25_1_1_4"/>
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>13</sub></font>) is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="nagata_2.html#E2:25_1_1_4"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font>,  <a href="funct_1.html#NM1">Function</a>] means for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">a<sub>2</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E13:25_1_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> ex <font color="Olive">b<sub>2</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>,  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>,   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>2</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E1:25_1_1_5"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>]
 
;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:25_1_1_5"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="metric_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>(<a class="txt" href="nagata_2.html#E1:25_1_1_5"><i><font color="Green">E40</font></i></a>);<br/></i>
<a NAME="E4:25_1_1_5"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E5:25_1_1_5"/><b>then </b>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>] )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E3:25_1_1_5"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>hence </b><a NAME="E6:25_1_1_5"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>13</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> , <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E15:25_1_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a>  holds <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="nagata_2.html#E13:25_1_1"><i><font color="Green">E39</font></i></a>);<br/></i>
<a NAME="E16:25_1_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) holds <br/>( <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> is  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:25_1_1_6"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>14</sub></font>)
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:25_1_1_6"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font>) )
 ;<br/>
<a NAME="E4:25_1_1_6"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font>]
 
<b>by </b><i><a class="txt" href="nagata_2.html#E8:25_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="nagata_2.html#E3:25_1_1_6"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E5:25_1_1_6"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font>] &amp; <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E15:25_1_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> as   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<a NAME="E7:25_1_1_6"/>
<font color="Maroon">c<sub>17</sub></font> is <a href="pscomp_1.html#V9">continuous</a>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E3:25_1_1_6"><i><font color="Green">E42</font></i></a>, <a class="txt" href="nagata_2.html#E4:25_1_1_6"><i><font color="Green">E43</font></i></a>, <a class="txt" href="nagata_2.html#E5:25_1_1_6"><i><font color="Green">E44</font></i></a>, <a class="ref" href="nagata_1.html#T21">NAGATA_1:21</a></i>;<br/>
<b>hence </b><a NAME="E8:25_1_1_6"/>
( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font>] &amp; <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> is  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E15:25_1_1"><i><font color="Green">E40</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> </span>}</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="hidden.html#M1">set</a> ,  <a href="funct_1.html#NM1">Function</a>] means ( <font color="Maroon">a<sub>2</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>2</sub></font>)) ) );<br/>
<a NAME="E17:25_1_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>2</sub></font>)) is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>2</sub></font>)) <a href="tarski.html#R1">c=</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:25_1_1_7"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>14</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>15</sub></font>)) <a href="tarski.html#R1">c=</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></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="E1:25_1_1_7_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>14</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>15</sub></font>))
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:25_1_1_7_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>14</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>15</sub></font>) )
 ;<br/>
<a NAME="E4:25_1_1_7_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>hence </b><a NAME="E5:25_1_1_7_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="txt" href="nagata_2.html#E3:25_1_1_7_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(  <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a>  : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>14</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>15</sub></font>) </span>}</span> ;<br/>
<a NAME="E2:25_1_1_7"/>
( <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>14</sub></font>) is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>15</sub></font>) is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E12:25_1_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E3:25_1_1_7"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>14</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>15</sub></font>) is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T14">FINSET_1:14</a></i>;<br/>
<a NAME="E4:25_1_1_7"/><i><font color="Green">E45</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a>  : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>14</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>15</sub></font>) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>(<a class="txt" href="nagata_2.html#E3:25_1_1_7"><i><font color="Green">E44</font></i></a>);<br/></i>
<a NAME="E5:25_1_1_7"/>
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>14</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>15</sub></font>)) <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a>  : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>14</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>15</sub></font>) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_7_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></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="E1:25_1_1_7_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>14</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>15</sub></font>))
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:25_1_1_7_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>14</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>15</sub></font>) )
 ;<br/>
<a NAME="E4:25_1_1_7_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>hence </b><a NAME="E5:25_1_1_7_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a>  : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>14</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>15</sub></font>) </span>}</span> 
 <b>by </b><i><a class="txt" href="nagata_2.html#E3:25_1_1_7_2"><i><font color="Green">E46</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:25_1_1_7"/>
( <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>14</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>15</sub></font>)) is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>14</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>15</sub></font>)) <a href="tarski.html#R1">c=</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  )
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1_1_7"><i><font color="Green">E43</font></i></a>, <a class="txt" href="nagata_2.html#E4:25_1_1_7"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:25_1_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> </span>)</span> <a href="finseq_1.html#K13">*</a>  st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>

<a NAME="E1:25_1_1_8"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
 
;<br/>
<a NAME="E2:25_1_1_8"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:25_1_1_8"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="nagata_2.html#E4:25_1_1_8"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E6:25_1_1_8"/>
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>17</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>18</sub></font>)) is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E17:25_1_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E8:25_1_1_8"/><i><font color="Green">E45</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>17</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>18</sub></font>)) &amp; <font color="Maroon">c<sub>19</sub></font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>by </b><i><a class="ref" href="finseq_4.html#T73">FINSEQ_4:73</a></i>;<br/>
<a NAME="E9:25_1_1_8"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="txt" href="nagata_2.html#E17:25_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="nagata_2.html#E8:25_1_1_8"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E11:25_1_1_8"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> </span>)</span> <a href="finseq_1.html#K13">*</a>  &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>20</sub></font>] )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E4:25_1_1_8"><i><font color="Green">E44</font></i></a>, <a class="txt" href="nagata_2.html#E8:25_1_1_8"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>hence </b><a NAME="E12:25_1_1_8"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> </span>)</span> <a href="finseq_1.html#K13">*</a>  st <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">c<sub>14</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> </span>)</span> <a href="finseq_1.html#K13">*</a> <b> such that </b><br/><a NAME="E20:25_1_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> holds <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="nagata_2.html#E18:25_1_1"><i><font color="Green">E43</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>6</sub></font>[   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> ,   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> ,   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>1</sub></font> <a href="nagata_1.html#K3">+</a> <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>3</sub></font>;<br/>
<a NAME="E21:25_1_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> ex <font color="Olive">b<sub>3</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  st <font color="Maroon">S<sub>6</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font>;<br/>
<a NAME="E1:25_1_1_9"/>
<font color="Maroon">c<sub>15</sub></font> <a href="nagata_1.html#K3">+</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>hence </b><a NAME="E2:25_1_1_9"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  st <font color="Maroon">S<sub>6</sub></font>[<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="binop_1.html#NM2">BinOp</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E23:25_1_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  holds <font color="Maroon">S<sub>6</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>(<a class="txt" href="nagata_2.html#E21:25_1_1"><i><font color="Green">E45</font></i></a>);<br/></i>
<a NAME="E24:25_1_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> holds <font color="Maroon">c<sub>15</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="nagata_1.html#K3">+</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> be   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<a NAME="E2:25_1_1_10"/>
<font color="Maroon">c<sub>15</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nagata_1.html#K3">+</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E23:25_1_1"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:25_1_1_10"/>
<font color="Maroon">c<sub>15</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nagata_1.html#K3">+</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>7</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Olive">b<sub>1</sub></font>;<br/>
<a NAME="E25:25_1_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">S<sub>7</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:25_1_1_11"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
 ;<br/>

<a NAME="E2:25_1_1_11"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> </span>)</span> <a href="finseq_1.html#K13">*</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E4:25_1_1_11"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> holds <br/><font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>17</sub></font>
 
;<br/>
<b>hence </b><a NAME="E5:25_1_1_11"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon">c<sub>16</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>, <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E27:25_1_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> holds <br/><font color="Maroon">S<sub>7</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="nagata_2.html#E25:25_1_1"><i><font color="Green">E48</font></i></a>);<br/></i>
<a NAME="E28:25_1_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> holds <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> is  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>20</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:25_1_1_12"/><i><font color="Green">E51</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>19</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>20</sub></font>)) )
 <b>by </b><i><a class="txt" href="nagata_2.html#E20:25_1_1"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E4:25_1_1_12"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font> holds <br/><font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>21</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:25_1_1_12_1"/><i><font color="Green">E52</font></i>: 
( 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>18</sub></font> )
 ;<br/>

<a NAME="E2:25_1_1_12_1"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E3:25_1_1_12_1"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1_1_12_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E4:25_1_1_12_1"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>19</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>20</sub></font>))
 
<b>by </b><i><a class="txt" href="nagata_2.html#E3:25_1_1_12"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:25_1_1_12_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>19</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>20</sub></font>) )
 ;<br/>
<a NAME="E7:25_1_1_12_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<a NAME="E8:25_1_1_12_1"/>
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>19</sub></font>) or <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>20</sub></font>) )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E6:25_1_1_12_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E9:25_1_1_12_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> is  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i><a class="txt" href="nagata_2.html#E16:25_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="nagata_2.html#E6:25_1_1_12_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="nagata_2.html#E7:25_1_1_12_1"><i><font color="Green">E54</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:25_1_1_12"/><b>then </b>
( <font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>18</sub></font> is  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E24:25_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="nagata_1.html#T25">NAGATA_1:25</a></i>;<br/>
<b>hence </b><a NAME="E6:25_1_1_12"/>
<font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font> is  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i><a class="txt" href="nagata_2.html#E27:25_1_1"><i><font color="Green">E49</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>, <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <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="E30:25_1_1"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> holds <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topmetr.html#K3">R^1</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="nagata_2.html#E28:25_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E2:25_1_1_13"/>
<font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> is  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E28:25_1_1"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E3:25_1_1_13"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="topreal6.html#T83">TOPREAL6:83</a></i>;<br/>
<b>hence </b><a NAME="E4:25_1_1_13"/>
<font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topmetr.html#K3">R^1</a> 
 ;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>8</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>;<br/>
<a NAME="E31:25_1_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> &amp; <font color="Maroon">S<sub>8</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_14"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:25_1_1_14"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
 ;<br/>

<a NAME="E2:25_1_1_14"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>20</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:25_1_1_14"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>20</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="nagata_2.html#E4:25_1_1_14"><i><font color="Green">E53</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_14_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>23</sub></font>, <font color="Maroon">c<sub>24</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:25_1_1_14_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font></span><a href="domain_1.html#K1">]</a></span>
 ;<br/><a NAME="E2:25_1_1_14_1"/><b>then </b>
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E4:25_1_1_14"><i><font color="Green">E53</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E3:25_1_1_14_1"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:25_1_1_14"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> &amp; <font color="Maroon">S<sub>8</sub></font>[<font color="Maroon">c<sub>18</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>, <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E33:25_1_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> holds <br/><font color="Maroon">S<sub>8</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="nagata_2.html#E31:25_1_1"><i><font color="Green">E52</font></i></a>);<br/></i>
<a NAME="E34:25_1_1"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_15"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>

<a NAME="E1:25_1_1_15"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
 
;<br/>
<a NAME="E2:25_1_1_15"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>20</sub></font>, <font color="Maroon">c<sub>21</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:25_1_1_15"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>20</sub></font>, <font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>21</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="nagata_2.html#E4:25_1_1_15"><i><font color="Green">E55</font></i></a></i>;<br/>
<a NAME="E6:25_1_1_15"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E11:25_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="nagata_2.html#E33:25_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="nagata_2.html#E4:25_1_1_15"><i><font color="Green">E55</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:25_1_1_15"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E4:25_1_1_15"><i><font color="Green">E55</font></i></a>, <a class="ref" href="borsuk_1.html#T46">BORSUK_1:46</a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E35:25_1_1"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  st <font color="Olive">b<sub>2</sub></font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">c<sub>15</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_16"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/><b>let </b><font color="Maroon">c<sub>20</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:25_1_1_16"/>
<font color="Maroon">c<sub>20</sub></font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<a NAME="E2:25_1_1_16"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="ref" href="binop_1.html#T11">BINOP_1:11</a></i>;<br/>
<a NAME="E3:25_1_1_16"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nagata_1.html#K3">+</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E23:25_1_1"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E4:25_1_1_16"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="ref" href="nagata_1.html#D7">NAGATA_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E5:25_1_1_16"/>
<font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>


</div><b>end;</b></div>
<a NAME="E36:25_1_1"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> implies <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_17"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>20</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>

<b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>,  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>(  not <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) ) ) </span>}</span> ;<br/>
<a NAME="E1:25_1_1_17"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span> holds <br/><font color="Maroon">H<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>3</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>4</sub></font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>21</sub></font>, <font color="Maroon">c<sub>22</sub></font>, <font color="Maroon">c<sub>23</sub></font>, <font color="Maroon">c<sub>24</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:25_1_1_17_1"/><i><font color="Green">E58</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font></span><a href="domain_1.html#K1">]</a></span>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>25</sub></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="E2:25_1_1_17_1"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font>)
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>26</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:25_1_1_17_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>(  not <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) ) )
 ;<br/>
<a NAME="E5:25_1_1_17_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="txt" href="nagata_2.html#E4:25_1_1_17_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> as   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>28</sub></font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font></span><a href="domain_1.html#K1">]</a></span> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E8:25_1_1_17_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>27</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">c<sub>27</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E4:25_1_1_17_1"><i><font color="Green">E59</font></i></a></i>;<br/>
<a NAME="E9:25_1_1_17_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>]
 
<b>by </b><i><a class="txt" href="nagata_2.html#E8:25_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="nagata_2.html#E4:25_1_1_17_1"><i><font color="Green">E59</font></i></a></i>;<br/>
<a NAME="E10:25_1_1_17_1"/><b>then </b>
(  not <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>26</sub></font> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>26</sub></font> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E8:25_1_1_17_1"><i><font color="Green">E60</font></i></a></i>;<br/>
<a NAME="E11:25_1_1_17_1"/><b>then </b>
( (  not <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>26</sub></font> or  not <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>26</sub></font> ) &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E33:25_1_1"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E12:25_1_1_17_1"/><b>then </b>
( ( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>26</sub></font> or <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>26</sub></font> ) &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1_1_17_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E13:25_1_1_17_1"/><b>then </b>
( <font color="Maroon">c<sub>26</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> or <font color="Maroon">c<sub>26</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E14:25_1_1_17_1"/><b>then </b>
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font>) or <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font>) )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E4:25_1_1_17_1"><i><font color="Green">E59</font></i></a></i>;<br/>
<a NAME="E15:25_1_1_17_1"/><b>then </b>
( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>23</sub></font>) or <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>24</sub></font>) )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E4:25_1_1_17_1"><i><font color="Green">E59</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:25_1_1_17_1"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>23</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>24</sub></font>)
 <b>by </b><i><a class="txt" href="nagata_2.html#E4:25_1_1_17_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:25_1_1_17"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>4</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>5</sub></font>)) <a href="subset_1.html#K6">\</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>5</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>)) holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_17_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>21</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>22</sub></font>, <font color="Maroon">c<sub>23</sub></font>, <font color="Maroon">c<sub>24</sub></font>, <font color="Maroon">c<sub>25</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:25_1_1_17_2"/><i><font color="Green">E59</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>24</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>25</sub></font>)) <a href="subset_1.html#K6">\</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font>))
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>26</sub></font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="domain_1.html#K1">]</a></span> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_17_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_17_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1_17_2_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_17_2_1_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_17_2_1_1_1"/><i><font color="Green">E60</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:25_1_1_17_2_1_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a> 
 <b>by </b><i><a class="txt" href="nagata_2.html#E24:25_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="nagata_1.html#T23">NAGATA_1:23</a></i>;<br/><a NAME="E3:25_1_1_17_2_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">c<sub>15</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  st <font color="Olive">b<sub>1</sub></font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1_1_17_2_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finsop_1.html#D1">FINSOP_1:def 1</a>, <a class="ref" href="setwiseo.html#D2">SETWISEO:def 2</a></i>;<br/><a NAME="E4:25_1_1_17_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>21</sub></font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="binop_1.html#D8">BINOP_1:def 8</a></i>;<br/><b>hence </b><a NAME="E5:25_1_1_17_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="nagata_2.html#E35:25_1_1"><i><font color="Green">E55</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_17_2_1_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_17_2_1_1_2"/><i><font color="Green">E60</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:25_1_1_17_2_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>27</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:25_1_1_17_2_1_1_2"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>27</sub></font> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> 1 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> holds <br/><font color="Maroon">c<sub>27</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) &amp; <font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="finsop_1.html#D1">FINSOP_1:def 1</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>9</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> implies <span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> 0 );<br/><a NAME="E5:25_1_1_17_2_1_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">S<sub>9</sub></font>[0]
 ;<br/><a NAME="E6:25_1_1_17_2_1_1_2"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>9</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>9</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_17_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>28</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:25_1_1_17_2_1_1_2_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">S<sub>9</sub></font>[<font color="Maroon">c<sub>28</sub></font>]
 ;<br/>

<b>assume </b><a NAME="E2:25_1_1_17_2_1_1_2_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/>

<a NAME="E3:25_1_1_17_2_1_1_2_1"/><b>then </b><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E4:25_1_1_17_2_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:25_1_1_17_2_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>24</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>25</sub></font>)) <a href="subset_1.html#K6">\</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font>))
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1_1_17_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:25_1_1_17_2_1_1_2_1"/><b>then </b><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>24</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>25</sub></font>)) &amp; not <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font>)) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>29</sub></font> being   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:25_1_1_17_2_1_1_2_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>24</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>25</sub></font>) )
 ;<br/>
<a NAME="E9:25_1_1_17_2_1_1_2_1"/>
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<a NAME="E10:25_1_1_17_2_1_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">c<sub>27</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">c<sub>27</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E8:25_1_1_17_2_1_1_2_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font>, <font color="Maroon">c<sub>31</sub></font> = <font color="Maroon">c<sub>27</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <font color="Maroon">c<sub>32</sub></font> = <font color="Maroon">c<sub>27</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font>, <font color="Maroon">c<sub>33</sub></font> = <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<a NAME="E12:25_1_1_17_2_1_1_2_1"/>
( <font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_17_2_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1_1_17_2_1_1_2_1_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_17_2_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1_17_2_1_1_2_1_1_1"/>
( <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>24</sub></font>) or <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>25</sub></font>) )
 </span><b>by </b><i><a class="txt" href="nagata_2.html#E8:25_1_1_17_2_1_1_2_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_17_2_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_17_2_1_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>24</sub></font>)
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>34</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:25_1_1_17_2_1_1_2_1_1_1_1"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font>) )
 ;<br/><a NAME="E4:25_1_1_17_2_1_1_2_1_1_1_1"/>
not <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font>)
 <b>by </b><i><a class="txt" href="nagata_2.html#E6:25_1_1_17_2_1_1_2_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="nagata_2.html#E8:25_1_1_17_2_1_1_2_1"><i><font color="Green">E68</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>35</sub></font> being   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:25_1_1_17_2_1_1_2_1_1_1_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>35</sub></font> &amp; not <font color="Maroon">c<sub>35</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; not <font color="Maroon">c<sub>35</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="nagata_2.html#E3:25_1_1_17_2_1_1_2_1_1_1_1"><i><font color="Green">E70</font></i></a></i>;<br/><a NAME="E7:25_1_1_17_2_1_1_2_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>34</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font>]
 <b>by </b><i><a class="txt" href="nagata_2.html#E8:25_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="nagata_2.html#E3:25_1_1_17_2_1_1_2_1_1_1_1"><i><font color="Green">E70</font></i></a></i>;<br/><b>hence </b><a NAME="E8:25_1_1_17_2_1_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1_1_17_2_1_1_2_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="nagata_2.html#E3:25_1_1_17_2_1_1_2_1_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="nagata_2.html#E6:25_1_1_17_2_1_1_2_1_1_1_1"><i><font color="Green">E71</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_17_2_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_17_2_1_1_2_1_1_1_2"/>
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>25</sub></font>)
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>34</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:25_1_1_17_2_1_1_2_1_1_1_2"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font>) )
 ;<br/><a NAME="E4:25_1_1_17_2_1_1_2_1_1_1_2"/>
not <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font>)
 <b>by </b><i><a class="txt" href="nagata_2.html#E6:25_1_1_17_2_1_1_2_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="nagata_2.html#E8:25_1_1_17_2_1_1_2_1"><i><font color="Green">E68</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>35</sub></font> being   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:25_1_1_17_2_1_1_2_1_1_1_2"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>35</sub></font> &amp; not <font color="Maroon">c<sub>35</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; not <font color="Maroon">c<sub>35</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="nagata_2.html#E3:25_1_1_17_2_1_1_2_1_1_1_2"><i><font color="Green">E70</font></i></a></i>;<br/><a NAME="E7:25_1_1_17_2_1_1_2_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>34</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font>]
 <b>by </b><i><a class="txt" href="nagata_2.html#E8:25_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="nagata_2.html#E3:25_1_1_17_2_1_1_2_1_1_1_2"><i><font color="Green">E70</font></i></a></i>;<br/><b>hence </b><a NAME="E8:25_1_1_17_2_1_1_2_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1_1_17_2_1_1_2_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="nagata_2.html#E3:25_1_1_17_2_1_1_2_1_1_1_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="nagata_2.html#E6:25_1_1_17_2_1_1_2_1_1_1_2"><i><font color="Green">E71</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E13:25_1_1_17_2_1_1_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<a NAME="E14:25_1_1_17_2_1_1_2_1"/><b>then </b><i><font color="Green">E69</font></i>: 
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>30</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E15:25_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_17_2_1_1_2_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1_17_2_1_1_2_1_2"/>
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>28</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_17_2_1_1_2_1_2_1"><b>suppose </b></a><a NAME="E1:25_1_1_17_2_1_1_2_1_2_1"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:25_1_1_17_2_1_1_2_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="nagata_2.html#E4:25_1_1_17_2_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="nagata_2.html#E8:25_1_1_17_2_1_1_2_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="nagata_2.html#E14:25_1_1_17_2_1_1_2_1"><i><font color="Green">E69</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_17_2_1_1_2_1_2_2"><b>suppose </b></a><a NAME="E1:25_1_1_17_2_1_1_2_1_2_2"/>
<font color="Maroon">c<sub>28</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:25_1_1_17_2_1_1_2_1_2_2"/><b>then </b>
( <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="nagata_2.html#E4:25_1_1_17_2_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="nagata_2.html#E3:25_1_1_17_2_1_1_2_1"><i><font color="Green">E66</font></i></a></i>;<br/><a NAME="E3:25_1_1_17_2_1_1_2_1_2_2"/><b>then </b>
( <font color="Maroon">c<sub>31</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>32</sub></font> <a href="nagata_1.html#K3">+</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="nagata_2.html#E24:25_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="nagata_2.html#E1:25_1_1_17_2_1_1_2_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="nagata_2.html#E2:25_1_1_17_2_1_1_2_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:25_1_1_17_2_1_1_2_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>31</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nagata_1.html#D7">NAGATA_1:def 7</a></i>;<br/><b>hence </b><a NAME="E5:25_1_1_17_2_1_1_2_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="nagata_2.html#E8:25_1_1_17_2_1_1_2_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="nagata_2.html#E14:25_1_1_17_2_1_1_2_1"><i><font color="Green">E69</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E7:25_1_1_17_2_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>9</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="nagata_2.html#E5:25_1_1_17_2_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="nagata_2.html#E6:25_1_1_17_2_1_1_2"><i><font color="Green">E63</font></i></a>);<br/></i><b>hence </b><a NAME="E8:25_1_1_17_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1_1_17_2_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="nagata_2.html#E4:25_1_1_17_2_1_1_2"><i><font color="Green">E61</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:25_1_1_17_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> 0
 ;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>23</sub></font>, <font color="Maroon">c<sub>24</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:25_1_1_17"/><i><font color="Green">E59</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>23</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>24</sub></font>)) )
 <b>by </b><i><a class="txt" href="nagata_2.html#E20:25_1_1"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>25</sub></font>, <font color="Maroon">c<sub>26</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:25_1_1_17"/><i><font color="Green">E60</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>25</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>26</sub></font>)) )
 <b>by </b><i><a class="txt" href="nagata_2.html#E20:25_1_1"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E8:25_1_1_17"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>)) <a href="tarski.html#R1">c=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>23</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>24</sub></font>))
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_17_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>27</sub></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="E1:25_1_1_17_3"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>))
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>28</sub></font> being   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:25_1_1_17_3"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>) )
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1_1_17_3"><i><font color="Green">E62</font></i></a></i>;<br/>
<a NAME="E4:25_1_1_17_3"/>
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E11:25_1_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E5:25_1_1_17_3"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E6:25_1_1_17_3"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E33:25_1_1"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E7:25_1_1_17_3"/><b>then </b>
<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>23</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>24</sub></font>)
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1_1_17"><i><font color="Green">E57</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:25_1_1_17_3"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>23</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>24</sub></font>))
 <b>by </b><i><a class="txt" href="nagata_2.html#E3:25_1_1_17_3"><i><font color="Green">E63</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E10:25_1_1_17"/>
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>23</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>24</sub></font>)) is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E17:25_1_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E11:25_1_1_17"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>)) is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E8:25_1_1_17"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>29</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E13:25_1_1_17"/><i><font color="Green">E63</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>)) &amp; <font color="Maroon">c<sub>29</sub></font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>by </b><i><a class="ref" href="finseq_4.html#T73">FINSEQ_4:73</a></i>;<br/>
<a NAME="E14:25_1_1_17"/>
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>23</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>24</sub></font>)) <a href="tarski.html#R1">c=</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="txt" href="nagata_2.html#E17:25_1_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E15:25_1_1_17"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>)) <a href="tarski.html#R1">c=</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="txt" href="nagata_2.html#E8:25_1_1_17"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>29</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="txt" href="nagata_2.html#E13:25_1_1_17"><i><font color="Green">E63</font></i></a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E17:25_1_1_17"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  &amp; <font color="Maroon">c<sub>15</sub></font> is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">c<sub>15</sub></font> is <a href="binop_1.html#V2">associative</a> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E24:25_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="nagata_1.html#T23">NAGATA_1:23</a></i>;<br/>
<a NAME="E18:25_1_1_17"/>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>30</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E20:25_1_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="nagata_2.html#E5:25_1_1_17"><i><font color="Green">E59</font></i></a>, <a class="txt" href="nagata_2.html#E8:25_1_1_17"><i><font color="Green">E61</font></i></a>, <a class="txt" href="nagata_2.html#E13:25_1_1_17"><i><font color="Green">E63</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>31</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E20:25_1_1_17"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>31</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>30</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>30</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>31</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E13:25_1_1_17"><i><font color="Green">E63</font></i></a>, <a class="txt" href="nagata_2.html#E17:25_1_1_17"><i><font color="Green">E65</font></i></a>, <a class="ref" href="nagata_2.html#T18" target="_self">Th18</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>32</sub></font> = <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>)) <a href="xboole_0.html#K3">/\</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>25</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>26</sub></font>));<br/>
<a NAME="E21:25_1_1_17"/>
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>)) <a href="xboole_0.html#K3">/\</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>25</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>26</sub></font>)) is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E11:25_1_1_17"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finset_1.html#T15">FINSET_1:15</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>33</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E23:25_1_1_17"/><i><font color="Green">E67</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>)) <a href="xboole_0.html#K3">/\</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>25</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>26</sub></font>)) &amp; <font color="Maroon">c<sub>33</sub></font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>by </b><i><a class="ref" href="finseq_4.html#T73">FINSEQ_4:73</a></i>;<br/>
<a NAME="E24:25_1_1_17"/>
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>)) <a href="xboole_0.html#K3">/\</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>25</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>26</sub></font>)) <a href="tarski.html#R1">c=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>))
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E25:25_1_1_17"/><b>then </b>
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>)) <a href="xboole_0.html#K3">/\</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>25</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>26</sub></font>)) <a href="tarski.html#R1">c=</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="txt" href="nagata_2.html#E15:25_1_1_17"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>34</sub></font> = <font color="Maroon">c<sub>33</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="txt" href="nagata_2.html#E23:25_1_1_17"><i><font color="Green">E67</font></i></a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E27:25_1_1_17"/>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>34</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>30</sub></font> &amp; <font color="Maroon">c<sub>30</sub></font> is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E13:25_1_1_17"><i><font color="Green">E63</font></i></a>, <a class="txt" href="nagata_2.html#E23:25_1_1_17"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>35</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E29:25_1_1_17"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">c<sub>35</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>30</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E17:25_1_1_17"><i><font color="Green">E65</font></i></a>, <a class="txt" href="nagata_2.html#E23:25_1_1_17"><i><font color="Green">E67</font></i></a>, <a class="ref" href="nagata_2.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E30:25_1_1_17"/>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>34</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E20:25_1_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="nagata_2.html#E7:25_1_1_17"><i><font color="Green">E60</font></i></a>, <a class="txt" href="nagata_2.html#E23:25_1_1_17"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>36</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E32:25_1_1_17"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">c<sub>36</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>36</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E17:25_1_1_17"><i><font color="Green">E65</font></i></a>, <a class="txt" href="nagata_2.html#E23:25_1_1_17"><i><font color="Green">E67</font></i></a>, <a class="ref" href="nagata_2.html#T18" target="_self">Th18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>37</sub></font> = <font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>35</sub></font>, <font color="Maroon">c<sub>38</sub></font> = <font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>30</sub></font>, <font color="Maroon">c<sub>39</sub></font> = <font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>31</sub></font>, <font color="Maroon">c<sub>40</sub></font> = <font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>36</sub></font>, <font color="Maroon">c<sub>41</sub></font> = <font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>34</sub></font>, <font color="Maroon">c<sub>42</sub></font> = <font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>27</sub></font>, <font color="Maroon">c<sub>43</sub></font> = <font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>28</sub></font> as   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<a NAME="E34:25_1_1_17"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>37</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_17_4"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>44</sub></font> = <font color="Maroon">c<sub>35</sub></font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_17_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1_17_4_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_17_4_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_17_4_1_1"/><i><font color="Green">E71</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:25_1_1_17_4_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a> 
 <b>by </b><i><a class="txt" href="nagata_2.html#E24:25_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="nagata_1.html#T23">NAGATA_1:23</a></i>;<br/><a NAME="E3:25_1_1_17_4_1_1"/><b>then </b>
( <font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">c<sub>15</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  st <font color="Olive">b<sub>1</sub></font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1_1_17_4_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="finsop_1.html#D1">FINSOP_1:def 1</a>, <a class="ref" href="setwiseo.html#D2">SETWISEO:def 2</a></i>;<br/><a NAME="E4:25_1_1_17_4_1_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>35</sub></font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="binop_1.html#D8">BINOP_1:def 8</a></i>;<br/><b>hence </b><a NAME="E5:25_1_1_17_4_1_1"/>
<font color="Maroon">c<sub>37</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="nagata_2.html#E5:25_1_1_17"><i><font color="Green">E59</font></i></a>, <a class="txt" href="nagata_2.html#E35:25_1_1"><i><font color="Green">E55</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_17_4_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_17_4_1_2"/><i><font color="Green">E71</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:25_1_1_17_4_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>35</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>45</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:25_1_1_17_4_1_2"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c<sub>45</sub></font> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>35</sub></font> <a href="funct_1.html#K1">.</a> 1 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>35</sub></font> holds <br/><font color="Maroon">c<sub>45</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>35</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) &amp; <font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>45</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="finsop_1.html#D1">FINSOP_1:def 1</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>9</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>35</sub></font> implies for <font color="Olive">b<sub>1</sub></font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>45</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 );<br/><a NAME="E5:25_1_1_17_4_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">S<sub>9</sub></font>[0]
 ;<br/><a NAME="E6:25_1_1_17_4_1_2"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>9</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>9</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_17_4_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>46</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:25_1_1_17_4_1_2_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">S<sub>9</sub></font>[<font color="Maroon">c<sub>46</sub></font>]
 ;<br/>

<b>assume </b><a NAME="E2:25_1_1_17_4_1_2_1"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c<sub>46</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>46</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>35</sub></font> )
 ;<br/>

<a NAME="E3:25_1_1_17_4_1_2_1"/><b>then </b><i><font color="Green">E77</font></i>: 
( <font color="Maroon">c<sub>46</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>46</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>46</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E4:25_1_1_17_4_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>46</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>35</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:25_1_1_17_4_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>35</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>46</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>30</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E29:25_1_1_17"><i><font color="Green">E68</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:25_1_1_17_4_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>35</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>46</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>))
 
<b>by </b><i><a class="txt" href="nagata_2.html#E13:25_1_1_17"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>47</sub></font> being   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:25_1_1_17_4_1_2_1"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>35</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>46</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>) )
 ;<br/>
<a NAME="E9:25_1_1_17_4_1_2_1"/>
<font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<a NAME="E10:25_1_1_17_4_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">c<sub>45</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">c<sub>45</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>46</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">c<sub>35</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>46</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E8:25_1_1_17_4_1_2_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>48</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>47</sub></font>, <font color="Maroon">c<sub>49</sub></font> = <font color="Maroon">c<sub>45</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>46</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <font color="Maroon">c<sub>50</sub></font> = <font color="Maroon">c<sub>45</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>46</sub></font>, <font color="Maroon">c<sub>51</sub></font> = <font color="Maroon">c<sub>35</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>46</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<a NAME="E12:25_1_1_17_4_1_2_1"/>
<font color="Maroon">c<sub>47</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<a NAME="E13:25_1_1_17_4_1_2_1"/><b>then </b><i><font color="Green">E79</font></i>: 
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">c<sub>48</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E15:25_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<div><i><font color="Green">E80</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_17_4_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_17_4_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1_17_4_1_2_1_1_1"/>
( <font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>46</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_17_4_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_17_4_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:25_1_1_17_4_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>49</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="nagata_2.html#E4:25_1_1_17_4_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="nagata_2.html#E8:25_1_1_17_4_1_2_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="nagata_2.html#E13:25_1_1_17_4_1_2_1"><i><font color="Green">E79</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_17_4_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_17_4_1_2_1_1_1_2"/>
<font color="Maroon">c<sub>46</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:25_1_1_17_4_1_2_1_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>49</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>50</sub></font>,<font color="Maroon">c<sub>51</sub></font> &amp; <font color="Maroon">c<sub>46</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="nagata_2.html#E4:25_1_1_17_4_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="nagata_2.html#E3:25_1_1_17_4_1_2_1"><i><font color="Green">E77</font></i></a></i>;<br/><a NAME="E3:25_1_1_17_4_1_2_1_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>49</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>50</sub></font> <a href="nagata_1.html#K3">+</a> <font color="Maroon">c<sub>51</sub></font> &amp; <font color="Maroon">c<sub>50</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">c<sub>51</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="nagata_2.html#E24:25_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="nagata_2.html#E1:25_1_1_17_4_1_2_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="nagata_2.html#E2:25_1_1_17_4_1_2_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="nagata_2.html#E8:25_1_1_17_4_1_2_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="nagata_2.html#E13:25_1_1_17_4_1_2_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:25_1_1_17_4_1_2_1_1_1_2"/><b>then </b>
( 0 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>50</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>51</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>50</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>51</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>49</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font> )
 <b>by </b><i><a class="ref" href="nagata_1.html#D7">NAGATA_1:def 7</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a>, <a class="txt" href="nagata_2.html#E5:25_1_1_17"><i><font color="Green">E59</font></i></a></i>;<br/><b>hence </b><a NAME="E5:25_1_1_17_4_1_2_1_1_1_2"/>
<font color="Maroon">c<sub>49</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>let </b><font color="Maroon">c<sub>52</sub></font> be   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>

<b>assume </b><a NAME="E15:25_1_1_17_4_1_2_1"/>
<font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>45</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>46</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<b>hence </b><a NAME="E16:25_1_1_17_4_1_2_1"/>
<font color="Maroon">c<sub>52</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="nagata_2.html#E14:25_1_1_17_4_1_2_1"><i><font color="Green">E80</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:25_1_1_17_4_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>9</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="nagata_2.html#E5:25_1_1_17_4_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="nagata_2.html#E6:25_1_1_17_4_1_2"><i><font color="Green">E74</font></i></a>);<br/></i><b>hence </b><a NAME="E8:25_1_1_17_4_1_2"/>
<font color="Maroon">c<sub>37</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1_1_17_4_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="nagata_2.html#E4:25_1_1_17_4_1_2"><i><font color="Green">E72</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E35:25_1_1_17"/>
( <font color="Maroon">c<sub>42</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>38</sub></font> <a href="nagata_1.html#K3">+</a> <font color="Maroon">c<sub>39</sub></font> &amp; <font color="Maroon">c<sub>38</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>41</sub></font> <a href="nagata_1.html#K3">+</a> <font color="Maroon">c<sub>37</sub></font> &amp; <font color="Maroon">c<sub>43</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>41</sub></font> <a href="nagata_1.html#K3">+</a> <font color="Maroon">c<sub>40</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>25</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>26</sub></font>)) <a href="subset_1.html#K6">\</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>)) )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E24:25_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="nagata_2.html#E7:25_1_1_17"><i><font color="Green">E60</font></i></a>, <a class="txt" href="nagata_2.html#E20:25_1_1_17"><i><font color="Green">E66</font></i></a>, <a class="txt" href="nagata_2.html#E23:25_1_1_17"><i><font color="Green">E67</font></i></a>, <a class="txt" href="nagata_2.html#E29:25_1_1_17"><i><font color="Green">E68</font></i></a>, <a class="txt" href="nagata_2.html#E32:25_1_1_17"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xboole_1.html#T47">XBOOLE_1:47</a></i>;<br/>
<a NAME="E36:25_1_1_17"/><b>then </b><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>42</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>38</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>39</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>38</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>37</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>43</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>40</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>39</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>40</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E2:25_1_1_17"><i><font color="Green">E58</font></i></a>, <a class="txt" href="nagata_2.html#E5:25_1_1_17"><i><font color="Green">E59</font></i></a>, <a class="txt" href="nagata_2.html#E13:25_1_1_17"><i><font color="Green">E63</font></i></a>, <a class="txt" href="nagata_2.html#E20:25_1_1_17"><i><font color="Green">E66</font></i></a>, <a class="ref" href="nagata_1.html#D7">NAGATA_1:def 7</a></i>;<br/>
<b>thus </b><a NAME="E37:25_1_1_17"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font> implies <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_17_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1_1_17_5"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_17_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>44</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:25_1_1_17_5_1"/>
<font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>))
 ;<br/><b>then consider </b><font color="Maroon">c<sub>45</sub></font> being   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:25_1_1_17_5_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>45</sub></font> &amp; <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>) )
 ;<br/><a NAME="E4:25_1_1_17_5_1"/>
<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>25</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>26</sub></font>)
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1_1_17"><i><font color="Green">E57</font></i></a>, <a class="txt" href="nagata_2.html#E5:25_1_1_17"><i><font color="Green">E59</font></i></a>, <a class="txt" href="nagata_2.html#E7:25_1_1_17"><i><font color="Green">E60</font></i></a>, <a class="txt" href="nagata_2.html#E1:25_1_1_17_5"><i><font color="Green">E72</font></i></a></i>;<br/><b>hence </b><a NAME="E5:25_1_1_17_5_1"/>
<font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>25</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>26</sub></font>))
 <b>by </b><i><a class="txt" href="nagata_2.html#E3:25_1_1_17_5_1"><i><font color="Green">E73</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:25_1_1_17_5"/><b>then </b>
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>)) <a href="tarski.html#R1">c=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>25</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>26</sub></font>))
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:25_1_1_17_5"/><b>then </b>
( <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>)) <a href="xboole_0.html#K3">/\</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>25</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>26</sub></font>)) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>)) &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>)) <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>)) <a href="xboole_0.html#K3">/\</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>25</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>26</sub></font>))</span>)</span> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E13:25_1_1_17"><i><font color="Green">E63</font></i></a>, <a class="txt" href="nagata_2.html#E23:25_1_1_17"><i><font color="Green">E67</font></i></a>, <a class="txt" href="nagata_2.html#E29:25_1_1_17"><i><font color="Green">E68</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E5:25_1_1_17_5"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<a NAME="E6:25_1_1_17_5"/><b>then </b>
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T27">FINSEQ_1:27</a></i>;<br/>
<a NAME="E7:25_1_1_17_5"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E24:25_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a>, <a class="ref" href="nagata_1.html#T23">NAGATA_1:23</a></i>;<br/>
<a NAME="E8:25_1_1_17_5"/><b>then </b>
( <font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">c<sub>15</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  st <font color="Olive">b<sub>1</sub></font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">c<sub>15</sub></font> )
 
<b>by </b><i><a class="ref" href="finsop_1.html#D1">FINSOP_1:def 1</a>, <a class="ref" href="setwiseo.html#D2">SETWISEO:def 2</a></i>;<br/>
<a NAME="E9:25_1_1_17_5"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>35</sub></font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="ref" href="binop_1.html#D8">BINOP_1:def 8</a></i>;<br/>
<a NAME="E10:25_1_1_17_5"/><b>then </b>
( <font color="Maroon">c<sub>42</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>43</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>42</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E27:25_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="nagata_2.html#E35:25_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="nagata_2.html#E36:25_1_1_17"><i><font color="Green">E71</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:25_1_1_17_5"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="nagata_2.html#E27:25_1_1"><i><font color="Green">E49</font></i></a></i>;<br/>


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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_17_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>44</sub></font>, <font color="Maroon">c<sub>45</sub></font> be   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/><b>assume </b><a NAME="E1:25_1_1_17_6"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font> )
 ;<br/><a NAME="E2:25_1_1_17_6"/><b>then </b>
( <font color="Maroon">c<sub>43</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>42</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>45</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>43</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E27:25_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="nagata_2.html#E34:25_1_1_17"><i><font color="Green">E70</font></i></a>, <a class="txt" href="nagata_2.html#E36:25_1_1_17"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a>, <a class="txt" href="nagata_2.html#E5:25_1_1_17"><i><font color="Green">E59</font></i></a></i>;<br/><b>hence </b><a NAME="E3:25_1_1_17_6"/>
<font color="Maroon">c<sub>44</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>45</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="nagata_2.html#E5:25_1_1_17"><i><font color="Green">E59</font></i></a>, <a class="txt" href="nagata_2.html#E27:25_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="nagata_2.html#E1:25_1_1_17_6"><i><font color="Green">E72</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E39:25_1_1_17"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E37:25_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="nagata_1.html#K4">Toler</a>  as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="nagata_1.html#K4">Toler</a>  as   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E40:25_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E36:25_1_1"><i><font color="Green">E56</font></i></a></i>;<br/>
<a NAME="E41:25_1_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="nagata_1.html#K4">Toler</a>  is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E30:25_1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="nagata_2.html#E34:25_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="nagata_1.html#T26">NAGATA_1:26</a></i>;<br/>
<a NAME="E42:25_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>20</sub></font> is <a href="pscomp_1.html#V9">continuous</a>
 
<b>by </b><i><a class="ref" href="topreal6.html#T83">TOPREAL6:83</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_18"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>21</sub></font>, <font color="Maroon">c<sub>22</sub></font>, <font color="Maroon">c<sub>23</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>thus </b><a NAME="E1:25_1_1_18"/>
<font color="Maroon">c<sub>19</sub></font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_18_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K1">]</a></span> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_18_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_18_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1_18_1_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_18_1_1_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_18_1_1_1_1"/><i><font color="Green">E58</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:25_1_1_18_1_1_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a> 
 <b>by </b><i><a class="txt" href="nagata_2.html#E24:25_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="nagata_1.html#T23">NAGATA_1:23</a></i>;<br/><a NAME="E3:25_1_1_18_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">c<sub>15</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  st <font color="Olive">b<sub>1</sub></font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1_1_18_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finsop_1.html#D1">FINSOP_1:def 1</a>, <a class="ref" href="setwiseo.html#D2">SETWISEO:def 2</a></i>;<br/><a NAME="E4:25_1_1_18_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>25</sub></font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="binop_1.html#D8">BINOP_1:def 8</a></i>;<br/><b>hence </b><a NAME="E5:25_1_1_18_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="nagata_2.html#E35:25_1_1"><i><font color="Green">E55</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_18_1_1_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_18_1_1_1_2"/><i><font color="Green">E58</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:25_1_1_18_1_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>26</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:25_1_1_18_1_1_1_2"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> 1 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font> holds <br/><font color="Maroon">c<sub>26</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) &amp; <font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="finsop_1.html#D1">FINSOP_1:def 1</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>9</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font> implies <span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> 0 );<br/><a NAME="E5:25_1_1_18_1_1_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">S<sub>9</sub></font>[0]
 ;<br/><a NAME="E6:25_1_1_18_1_1_1_2"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>9</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>9</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_18_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>27</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:25_1_1_18_1_1_1_2_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">S<sub>9</sub></font>[<font color="Maroon">c<sub>27</sub></font>]
 ;<br/>

<b>assume </b><a NAME="E2:25_1_1_18_1_1_1_2_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font> )
 ;<br/>

<a NAME="E3:25_1_1_18_1_1_1_2_1"/><b>then </b><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E4:25_1_1_18_1_1_1_2_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>28</sub></font>, <font color="Maroon">c<sub>29</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:25_1_1_18_1_1_1_2_1"/><i><font color="Green">E66</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>28</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>29</sub></font>)) )
 <b>by </b><i><a class="txt" href="nagata_2.html#E20:25_1_1"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E7:25_1_1_18_1_1_1_2_1"/>
<font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>28</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>29</sub></font>))
 
<b>by </b><i><a class="txt" href="nagata_2.html#E4:25_1_1_18_1_1_1_2_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="nagata_2.html#E6:25_1_1_18_1_1_1_2_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>30</sub></font> being   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:25_1_1_18_1_1_1_2_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>28</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>29</sub></font>) )
 ;<br/>
<a NAME="E10:25_1_1_18_1_1_1_2_1"/>
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<a NAME="E11:25_1_1_18_1_1_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">c<sub>26</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">c<sub>26</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E9:25_1_1_18_1_1_1_2_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>31</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font>, <font color="Maroon">c<sub>32</sub></font> = <font color="Maroon">c<sub>26</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <font color="Maroon">c<sub>33</sub></font> = <font color="Maroon">c<sub>26</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>27</sub></font>, <font color="Maroon">c<sub>34</sub></font> = <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<a NAME="E13:25_1_1_18_1_1_1_2_1"/>
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<a NAME="E14:25_1_1_18_1_1_1_2_1"/><b>then </b><i><font color="Green">E68</font></i>: 
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>31</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E15:25_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_18_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1_18_1_1_1_2_1_1"/>
( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>27</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_18_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_18_1_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:25_1_1_18_1_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="nagata_2.html#E4:25_1_1_18_1_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="nagata_2.html#E9:25_1_1_18_1_1_1_2_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="nagata_2.html#E14:25_1_1_18_1_1_1_2_1"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_18_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_18_1_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>27</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:25_1_1_18_1_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font>,<font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="nagata_2.html#E4:25_1_1_18_1_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="nagata_2.html#E3:25_1_1_18_1_1_1_2_1"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E3:25_1_1_18_1_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>32</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>33</sub></font> <a href="nagata_1.html#K3">+</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="nagata_2.html#E24:25_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="nagata_2.html#E1:25_1_1_18_1_1_1_2_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="nagata_2.html#E2:25_1_1_18_1_1_1_2_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:25_1_1_18_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>32</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>34</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nagata_1.html#D7">NAGATA_1:def 7</a></i>;<br/><b>hence </b><a NAME="E5:25_1_1_18_1_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="nagata_2.html#E9:25_1_1_18_1_1_1_2_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="nagata_2.html#E14:25_1_1_18_1_1_1_2_1"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E7:25_1_1_18_1_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>9</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="nagata_2.html#E5:25_1_1_18_1_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="nagata_2.html#E6:25_1_1_18_1_1_1_2"><i><font color="Green">E61</font></i></a>);<br/></i><b>hence </b><a NAME="E8:25_1_1_18_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1_1_18_1_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="nagata_2.html#E4:25_1_1_18_1_1_1_2"><i><font color="Green">E59</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E4:25_1_1_18_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>25</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E27:25_1_1"><i><font color="Green">E49</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:25_1_1_18_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nagata_1.html#D8">NAGATA_1:def 8</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:25_1_1_18"/>
<font color="Maroon">c<sub>19</sub></font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>22</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_18_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="domain_1.html#K1">]</a></span>, <font color="Maroon">c<sub>25</sub></font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="domain_1.html#K1">]</a></span>, <font color="Maroon">c<sub>26</sub></font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="domain_1.html#K1">]</a></span> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>27</sub></font> as   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font>, <font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font>, <font color="Maroon">c<sub>31</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font> as   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_18_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_18_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1_18_2_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_18_2_1_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_18_2_1_1_1"/><i><font color="Green">E58</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:25_1_1_18_2_1_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a> 
 <b>by </b><i><a class="txt" href="nagata_2.html#E24:25_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="nagata_1.html#T23">NAGATA_1:23</a></i>;<br/><a NAME="E3:25_1_1_18_2_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">c<sub>15</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  st <font color="Olive">b<sub>1</sub></font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1_1_18_2_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finsop_1.html#D1">FINSOP_1:def 1</a>, <a class="ref" href="setwiseo.html#D2">SETWISEO:def 2</a></i>;<br/><a NAME="E4:25_1_1_18_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>27</sub></font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="binop_1.html#D8">BINOP_1:def 8</a></i>;<br/><a NAME="E5:25_1_1_18_2_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>28</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>28</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>28</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="nagata_2.html#E35:25_1_1"><i><font color="Green">E55</font></i></a></i>;<br/><b>hence </b><a NAME="E6:25_1_1_18_2_1_1_1"/>
<font color="Maroon">c<sub>28</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_18_2_1_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_18_2_1_1_2"/><i><font color="Green">E58</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:25_1_1_18_2_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>27</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>32</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:25_1_1_18_2_1_1_2"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>32</sub></font> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> 1 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>27</sub></font> holds <br/><font color="Maroon">c<sub>32</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) &amp; <font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>32</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="finsop_1.html#D1">FINSOP_1:def 1</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>9</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>27</sub></font> implies for <font color="Olive">b<sub>1</sub></font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>32</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> );<br/><a NAME="E5:25_1_1_18_2_1_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">S<sub>9</sub></font>[0]
 ;<br/><a NAME="E6:25_1_1_18_2_1_1_2"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>9</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>9</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_18_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>33</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:25_1_1_18_2_1_1_2_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">S<sub>9</sub></font>[<font color="Maroon">c<sub>33</sub></font>]
 ;<br/>

<b>assume </b><a NAME="E2:25_1_1_18_2_1_1_2_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>33</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>27</sub></font> )
 ;<br/>

<a NAME="E3:25_1_1_18_2_1_1_2_1"/><b>then </b><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E4:25_1_1_18_2_1_1_2_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>27</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>34</sub></font>, <font color="Maroon">c<sub>35</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:25_1_1_18_2_1_1_2_1"/><i><font color="Green">E66</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>34</sub></font>,<font color="Maroon">c<sub>35</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>34</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>35</sub></font>)) )
 <b>by </b><i><a class="txt" href="nagata_2.html#E20:25_1_1"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E7:25_1_1_18_2_1_1_2_1"/>
<font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>34</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>35</sub></font>))
 
<b>by </b><i><a class="txt" href="nagata_2.html#E4:25_1_1_18_2_1_1_2_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="nagata_2.html#E6:25_1_1_18_2_1_1_2_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>36</sub></font> being   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:25_1_1_18_2_1_1_2_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>34</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>35</sub></font>) )
 ;<br/>
<a NAME="E10:25_1_1_18_2_1_1_2_1"/>
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<a NAME="E11:25_1_1_18_2_1_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">c<sub>32</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">c<sub>32</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E9:25_1_1_18_2_1_1_2_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>37</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font>, <font color="Maroon">c<sub>38</sub></font> = <font color="Maroon">c<sub>32</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <font color="Maroon">c<sub>39</sub></font> = <font color="Maroon">c<sub>32</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>33</sub></font>, <font color="Maroon">c<sub>40</sub></font> = <font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<a NAME="E13:25_1_1_18_2_1_1_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<a NAME="E14:25_1_1_18_2_1_1_2_1"/><b>then </b><i><font color="Green">E68</font></i>: 
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span></span>)</span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>37</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>37</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>37</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E15:25_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="complex1.html#T142">COMPLEX1:142</a>, <a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>41</sub></font> be   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>

<b>assume </b><a NAME="E15:25_1_1_18_2_1_1_2_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>32</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_18_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1_18_2_1_1_2_1_1"/>
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_18_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_18_2_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:25_1_1_18_2_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>41</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="nagata_2.html#E4:25_1_1_18_2_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="nagata_2.html#E9:25_1_1_18_2_1_1_2_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="nagata_2.html#E14:25_1_1_18_2_1_1_2_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="nagata_2.html#E15:25_1_1_18_2_1_1_2_1"><i><font color="Green">E69</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_18_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_18_2_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:25_1_1_18_2_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>39</sub></font>,<font color="Maroon">c<sub>40</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="nagata_2.html#E4:25_1_1_18_2_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="nagata_2.html#E3:25_1_1_18_2_1_1_2_1"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E3:25_1_1_18_2_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>38</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>39</sub></font> <a href="nagata_1.html#K3">+</a> <font color="Maroon">c<sub>40</sub></font> &amp; <font color="Maroon">c<sub>39</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>39</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>39</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>37</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>40</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E24:25_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="nagata_2.html#E1:25_1_1_18_2_1_1_2_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="nagata_2.html#E2:25_1_1_18_2_1_1_2_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="nagata_2.html#E9:25_1_1_18_2_1_1_2_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:25_1_1_18_2_1_1_2_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>39</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>40</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>39</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>39</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>40</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>40</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>38</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>39</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>40</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>38</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>39</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>40</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>38</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>39</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>40</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E14:25_1_1_18_2_1_1_2_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="nagata_1.html#D7">NAGATA_1:def 7</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E5:25_1_1_18_2_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>41</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="nagata_2.html#E15:25_1_1_18_2_1_1_2_1"><i><font color="Green">E69</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E7:25_1_1_18_2_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>9</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="nagata_2.html#E5:25_1_1_18_2_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="nagata_2.html#E6:25_1_1_18_2_1_1_2"><i><font color="Green">E61</font></i></a>);<br/></i><b>hence </b><a NAME="E8:25_1_1_18_2_1_1_2"/>
<font color="Maroon">c<sub>28</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1_1_18_2_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="nagata_2.html#E4:25_1_1_18_2_1_1_2"><i><font color="Green">E59</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E6:25_1_1_18_2"/><b>then </b>
( <font color="Maroon">c<sub>28</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>30</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>31</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E27:25_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="nagata_2.html#E36:25_1_1"><i><font color="Green">E56</font></i></a></i>;<br/>
<a NAME="E7:25_1_1_18_2"/><b>then </b>
( <font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E8:25_1_1_18_2"/><b>then </b>
( <font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>30</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nagata_1.html#D8">NAGATA_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E9:25_1_1_18_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>22</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nagata_1.html#D8">NAGATA_1:def 8</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E44:25_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="nagata_1.html#R1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="nagata_1.html#T28">NAGATA_1:28</a></i>;<br/>
<a NAME="E45:25_1_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st ex <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">b<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>4</sub></font> &amp; not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>4</sub></font> ) holds <br/><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_19"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>21</sub></font>, <font color="Maroon">c<sub>22</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:25_1_1_19"/><i><font color="Green">E60</font></i>: 
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> &amp; not <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="domain_1.html#K1">]</a></span> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>24</sub></font> as   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<b>assume </b><a NAME="E5:25_1_1_19"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="xxreal_0.html#R1">&lt;</a> 1
 ;<br/>

<a NAME="E6:25_1_1_19"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E27:25_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="nagata_1.html#D8">NAGATA_1:def 8</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>26</sub></font>, <font color="Maroon">c<sub>27</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:25_1_1_19"/><i><font color="Green">E63</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>26</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>27</sub></font>)) )
 <b>by </b><i><a class="txt" href="nagata_2.html#E20:25_1_1"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>28</sub></font>, <font color="Maroon">c<sub>29</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:25_1_1_19"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>29</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>28</sub></font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>28</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>29</sub></font> &amp; not <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>29</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1_1_19"><i><font color="Green">E60</font></i></a></i>;<br/>
<a NAME="E11:25_1_1_19"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="txt" href="nagata_2.html#E10:25_1_1_19"><i><font color="Green">E64</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font> as   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<a NAME="E13:25_1_1_19"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>29</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="nagata_2.html#E10:25_1_1_19"><i><font color="Green">E64</font></i></a></i>;<br/>
<a NAME="E14:25_1_1_19"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>29</sub></font>)
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E15:25_1_1_19"/><b>then </b>
( <font color="Maroon">c<sub>28</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>29</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>29</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E4:25_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="nagata_2.html#E10:25_1_1_19"><i><font color="Green">E64</font></i></a>, <a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E16:25_1_1_19"/><b>then </b><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>29</sub></font> <a href="subset_1.html#K3">`</a>  &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>29</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E5:25_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="nagata_2.html#E10:25_1_1_19"><i><font color="Green">E64</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E17:25_1_1_19"/><b>then </b>
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>29</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E11:25_1_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E18:25_1_1_19"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>29</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E19:25_1_1_19"/><b>then </b>
( <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font>) &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E8:25_1_1_19"><i><font color="Green">E63</font></i></a>, <a class="txt" href="nagata_2.html#E10:25_1_1_19"><i><font color="Green">E64</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E20:25_1_1_19"/><b>then </b>
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>26</sub></font>)
 
<b>by </b><i><a class="txt" href="nagata_2.html#E10:25_1_1_19"><i><font color="Green">E64</font></i></a></i>;<br/>
<a NAME="E21:25_1_1_19"/><b>then </b>
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>26</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>27</sub></font>)
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E22:25_1_1_19"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E8:25_1_1_19"><i><font color="Green">E63</font></i></a></i>;<br/>
<a NAME="E23:25_1_1_19"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T27">FINSEQ_1:27</a></i>;<br/>
<a NAME="E24:25_1_1_19"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E25:25_1_1_19"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>31</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E27:25_1_1_19"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>31</sub></font> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> 1 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font> holds <br/><font color="Maroon">c<sub>31</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) &amp; <font color="Maroon">c<sub>15</sub></font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="finsop_1.html#D1">FINSOP_1:def 1</a></i>;<br/>
<a NAME="E28:25_1_1_19"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>32</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:25_1_1_19_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>32</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>33</sub></font> be   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>

<b>assume </b><a NAME="E2:25_1_1_19_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font>
 ;<br/>

<a NAME="E3:25_1_1_19_1"/>
( <font color="Maroon">c<sub>32</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">c<sub>32</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1_1_19_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E4:25_1_1_19_1"/><b>then </b>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:25_1_1_19_1"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>26</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>27</sub></font>))
 
<b>by </b><i><a class="txt" href="nagata_2.html#E8:25_1_1_19"><i><font color="Green">E63</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>34</sub></font> being   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:25_1_1_19_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>26</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>27</sub></font>) )
 ;<br/>
<a NAME="E8:25_1_1_19_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a>  )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E2:25_1_1_19_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="nagata_2.html#E7:25_1_1_19_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<a NAME="E9:25_1_1_19_1"/><b>then </b>
<font color="Maroon">c<sub>33</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>34</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>34</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E15:25_1_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:25_1_1_19_1"/>
<font color="Maroon">c<sub>33</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>9</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font>;<br/>
<a NAME="E29:25_1_1_19"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">S<sub>9</sub></font>[0]
 
;<br/>
<a NAME="E30:25_1_1_19"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>9</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>9</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_19_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>32</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:25_1_1_19_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">S<sub>9</sub></font>[<font color="Maroon">c<sub>32</sub></font>]
 ;<br/>

<b>let </b><font color="Maroon">c<sub>33</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E2:25_1_1_19_2"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_19_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>34</sub></font>, <font color="Maroon">c<sub>35</sub></font> be   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/><b>assume </b><a NAME="E1:25_1_1_19_2_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>36</sub></font> = <font color="Maroon">c<sub>31</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>32</sub></font> as   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/><a NAME="E3:25_1_1_19_2_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E4:25_1_1_19_2_1"/><b>then </b>
<font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="nagata_2.html#E2:25_1_1_19_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:25_1_1_19_2_1"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>37</sub></font> = <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_19_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1_19_2_1_1"/>
( <font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="nagata_2.html#E2:25_1_1_19_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_19_2_1_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_19_2_1_1_1"/>
<font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:25_1_1_19_2_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>32</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="nagata_2.html#E2:25_1_1_19_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:25_1_1_19_2_1_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>34</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>36</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>32</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1_1_19_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="nagata_2.html#E2:25_1_1_19_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="nagata_2.html#E1:25_1_1_19_2_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:25_1_1_19_2_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>31</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>37</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="nagata_2.html#E27:25_1_1_19"><i><font color="Green">E67</font></i></a>, <a class="txt" href="nagata_2.html#E28:25_1_1_19"><i><font color="Green">E68</font></i></a>, <a class="txt" href="nagata_2.html#E2:25_1_1_19_2"><i><font color="Green">E72</font></i></a></i>;<br/><a NAME="E5:25_1_1_19_2_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>35</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>36</sub></font> <a href="nagata_1.html#K3">+</a> <font color="Maroon">c<sub>37</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>37</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E24:25_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="nagata_2.html#E1:25_1_1_19_2_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E6:25_1_1_19_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>35</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>36</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="ref" href="nagata_1.html#D7">NAGATA_1:def 7</a></i>;<br/><b>hence </b><a NAME="E7:25_1_1_19_2_1_1_1"/>
<font color="Maroon">c<sub>34</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="nagata_2.html#E3:25_1_1_19_2_1_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_19_2_1_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_19_2_1_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_19_2_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1_19_2_1_1_2_1"/>
( <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_19_2_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_19_2_1_1_2_1_1"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:25_1_1_19_2_1_1_2_1_1"/>
<font color="Maroon">c<sub>34</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="nagata_2.html#E27:25_1_1_19"><i><font color="Green">E67</font></i></a>, <a class="txt" href="nagata_2.html#E1:25_1_1_19_2_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="nagata_2.html#E1:25_1_1_19_2_1_1_2"><i><font color="Green">E74</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_19_2_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_19_2_1_1_2_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:25_1_1_19_2_1_1_2_1_2"/>
<font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E3:25_1_1_19_2_1_1_2_1_2"/><b>then </b><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c<sub>32</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>32</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E2:25_1_1_19_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="nagata_2.html#E1:25_1_1_19_2_1_1_2_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E4:25_1_1_19_2_1_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:25_1_1_19_2_1_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>38</sub></font> = <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> as   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/><a NAME="E7:25_1_1_19_2_1_1_2_1_2"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>38</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>38</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>36</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E28:25_1_1_19"><i><font color="Green">E68</font></i></a>, <a class="txt" href="nagata_2.html#E1:25_1_1_19_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="nagata_2.html#E3:25_1_1_19_2_1_1_2_1_2"><i><font color="Green">E76</font></i></a></i>;<br/><a NAME="E8:25_1_1_19_2_1_1_2_1_2"/><b>then </b>
( <font color="Maroon">c<sub>31</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>37</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>36</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E27:25_1_1_19"><i><font color="Green">E67</font></i></a>, <a class="txt" href="nagata_2.html#E1:25_1_1_19_2_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="nagata_2.html#E1:25_1_1_19_2_1_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="nagata_2.html#E3:25_1_1_19_2_1_1_2_1_2"><i><font color="Green">E76</font></i></a></i>;<br/><a NAME="E9:25_1_1_19_2_1_1_2_1_2"/><b>then </b>
( <font color="Maroon">c<sub>35</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>36</sub></font> <a href="nagata_1.html#K3">+</a> <font color="Maroon">c<sub>34</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>36</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>34</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>34</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E24:25_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="nagata_2.html#E1:25_1_1_19_2_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E10:25_1_1_19_2_1_1_2_1_2"/>
<font color="Maroon">c<sub>34</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>35</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="ref" href="nagata_1.html#D7">NAGATA_1:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:25_1_1_19_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E31:25_1_1_19"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>9</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="nagata_2.html#E29:25_1_1_19"><i><font color="Green">E69</font></i></a>, <a class="txt" href="nagata_2.html#E30:25_1_1_19"><i><font color="Green">E70</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>32</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> as   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="txt" href="nagata_2.html#E22:25_1_1_19"><i><font color="Green">E66</font></i></a>, <a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<a NAME="E33:25_1_1_19"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font>]
 
<b>by </b><i><a class="txt" href="nagata_2.html#E8:25_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="nagata_2.html#E10:25_1_1_19"><i><font color="Green">E64</font></i></a></i>;<br/>
<a NAME="E34:25_1_1_19"/><b>then </b>
( <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">c<sub>30</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>30</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E10:25_1_1_19"><i><font color="Green">E64</font></i></a>, <a class="txt" href="nagata_2.html#E16:25_1_1_19"><i><font color="Green">E65</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E35:25_1_1_19"/><b>then </b><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c<sub>32</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span> &amp;  <a href="prepower.html#K1">abs</a> 1 <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E15:25_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>33</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E37:25_1_1_19"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E22:25_1_1_19"><i><font color="Green">E66</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E38:25_1_1_19"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E37:25_1_1_19"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>34</sub></font> = <font color="Maroon">c<sub>33</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E40:25_1_1_19"/>
( <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>34</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E27:25_1_1_19"><i><font color="Green">E67</font></i></a>, <a class="txt" href="nagata_2.html#E37:25_1_1_19"><i><font color="Green">E73</font></i></a>, <a class="txt" href="nagata_2.html#E38:25_1_1_19"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E41:25_1_1_19"/>
contradiction
 <b>by </b><i><a class="txt" href="nagata_2.html#E5:25_1_1_19"><i><font color="Green">E61</font></i></a>, <a class="txt" href="nagata_2.html#E6:25_1_1_19"><i><font color="Green">E62</font></i></a>, <a class="txt" href="nagata_2.html#E31:25_1_1_19"><i><font color="Green">E71</font></i></a>, <a class="txt" href="nagata_2.html#E35:25_1_1_19"><i><font color="Green">E72</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>take </b><font color="Maroon">c<sub>21</sub></font> =  <a href="nagata_1.html#K5">min</a> 1,<font color="Maroon">c<sub>20</sub></font>;<br/>

<a NAME="E46:25_1_1"/><i><font color="Green">E60</font></i>: 
(  <a href="nagata_1.html#K5">min</a> 1,<font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#R1">=</a>  <a href="nagata_1.html#K5">min</a> 1,<font color="Maroon">c<sub>19</sub></font> &amp;  <a href="nagata_1.html#K5">min</a> 1,<font color="Maroon">c<sub>20</sub></font> is <a href="pscomp_1.html#V9">continuous</a> &amp;  <a href="nagata_1.html#K5">min</a> 1,<font color="Maroon">c<sub>19</sub></font> <a href="nagata_1.html#R1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E42:25_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="nagata_2.html#E44:25_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a>, <a class="ref" href="nagata_1.html#T27">NAGATA_1:27</a>, <a class="ref" href="nagata_1.html#T30">NAGATA_1:30</a></i>;<br/>
<a NAME="E47:25_1_1"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><a href="nagata_1.html#K5">min</a> 1,<font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_20"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>22</sub></font>, <font color="Maroon">c<sub>23</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:25_1_1_20"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E2:25_1_1_20"/><b>then </b>
<span class="p1">(<span class="default"><a href="nagata_1.html#K5">min</a> 1,<font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K3">min</a> 1,<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="nagata_1.html#D9">NAGATA_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E3:25_1_1_20"/>
<span class="p1">(<span class="default"><a href="nagata_1.html#K5">min</a> 1,<font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_21"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>22</sub></font>, <font color="Maroon">c<sub>23</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:25_1_1_21"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> &amp; not <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> )
 ;<br/><a NAME="E2:25_1_1_21"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="nagata_2.html#E45:25_1_1"><i><font color="Green">E59</font></i></a></i>;<br/><a NAME="E3:25_1_1_21"/><b>then </b>
( 1 <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K3">min</a> 1,<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> )
 <b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a>, <a class="ref" href="xxreal_0.html#D8">XXREAL_0:def 8</a></i>;<br/><b>hence </b><a NAME="E4:25_1_1_21"/>
1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nagata_1.html#K5">min</a> 1,<font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="nagata_1.html#D9">NAGATA_1:def 9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E49:25_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>21</sub></font>]
 <b>by </b><i><a class="txt" href="nagata_2.html#E46:25_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="nagata_2.html#E47:25_1_1"><i><font color="Green">E61</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <span class="p1">(<span class="default"><a href="nagata_2.html#K2" target="_self">PairFunc</a>  <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] ) );<br/>
<a NAME="E8:25_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:25_1_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<a NAME="E2:25_1_2"/><i><font color="Green">E32</font></i>: 
(  <a href="nagata_2.html#K2" target="_self">PairFunc</a>  is <a href="funct_2.html#V2">onto</a> &amp;  <a href="nagata_2.html#K2" target="_self">PairFunc</a>  is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i><a class="ref" href="nagata_2.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E3:25_1_2"/><b>then </b>
 <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <a href="nagata_2.html#K2" target="_self">PairFunc</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:25_1_2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <a href="nagata_2.html#K2" target="_self">PairFunc</a>  &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <a href="nagata_2.html#K2" target="_self">PairFunc</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:25_1_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E5:25_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E9:25_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>]
 <b>by </b><i><a class="txt" href="nagata_2.html#E7:25_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="nagata_2.html#E7:25_1_2"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>11</sub></font>
;<br/>

<b>thus </b><a NAME="E10:25_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>

<b>take </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font>;<br/>

<b>thus </b><a NAME="E11:25_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E12:25_1_2"/>
<span class="p1">(<span class="default"><a href="nagata_2.html#K2" target="_self">PairFunc</a>  <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K1">]</a></span>
 ;<br/>

<a NAME="E13:25_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E2:25_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="nagata_2.html#E5:25_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="nagata_2.html#E7:25_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/>
<a NAME="E14:25_1_2"/><b>then </b>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E15:25_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>12</sub></font>]
 <b>by </b><i><a class="txt" href="nagata_2.html#E9:25_1_2"><i><font color="Green">E35</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E10:25_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="nagata_2.html#E8:25_1"><i><font color="Green">E30</font></i></a>);<br/></i>
<a NAME="E11:25_1"/><i><font color="Green">E32</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:25_1_3"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="numbers.html#K1">REAL</a>  &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> )
 <b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/><b>hence </b><a NAME="E2:25_1_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> is   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="seqfunc.html#M1">Functional_Sequence</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="txt" href="nagata_2.html#E11:25_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="seqfunc.html#T1">SEQFUNC:1</a></i>;<br/>
<a NAME="E14:25_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="nagata_1.html#R1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> holds <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> st <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> is <a href="pscomp_1.html#V9">continuous</a> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:25_1_4"/><i><font color="Green">E34</font></i>: 
(  <a href="nagata_2.html#K2" target="_self">PairFunc</a>  is <a href="funct_2.html#V2">onto</a> &amp;  <a href="nagata_2.html#K2" target="_self">PairFunc</a>  is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i><a class="ref" href="nagata_2.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E2:25_1_4"/><b>then </b>
 <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <a href="nagata_2.html#K2" target="_self">PairFunc</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:25_1_4"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <a href="nagata_2.html#K2" target="_self">PairFunc</a>  &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <a href="nagata_2.html#K2" target="_self">PairFunc</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:25_1_4"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E4:25_1_4"><i><font color="Green">E35</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E7:25_1_4"/><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K2" target="_self">PairFunc</a>  <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1_4"><i><font color="Green">E34</font></i></a>, <a class="txt" href="nagata_2.html#E4:25_1_4"><i><font color="Green">E35</font></i></a>, <a class="txt" href="nagata_2.html#E6:25_1_4"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E9:25_1_4"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <span class="p1">(<span class="default"><a href="nagata_2.html#K2" target="_self">PairFunc</a>  <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>13</sub></font>] ) )
 <b>by </b><i><a class="txt" href="nagata_2.html#E10:25_1"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E11:25_1_4"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> is <a href="pscomp_1.html#V9">continuous</a> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="nagata_1.html#R1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; ( for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st ex <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>5</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>5</sub></font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">b<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>6</sub></font> &amp; not <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>6</sub></font> ) holds <br/><font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> 1 ) ) ) )
 <b>by </b><i><a class="txt" href="nagata_2.html#E6:25_1_4"><i><font color="Green">E36</font></i></a>, <a class="txt" href="nagata_2.html#E7:25_1_4"><i><font color="Green">E37</font></i></a>, <a class="txt" href="nagata_2.html#E9:25_1_4"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>14</sub></font>
;<br/>

<b>thus </b><a NAME="E12:25_1_4"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="nagata_1.html#R1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E9:25_1_4"><i><font color="Green">E38</font></i></a>, <a class="txt" href="nagata_2.html#E11:25_1_4"><i><font color="Green">E39</font></i></a></i>;<br/>

<b>thus </b><a NAME="E13:25_1_4"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> holds <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>

<b>consider </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:25_1_4_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>thus </b><a NAME="E3:25_1_4_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="nagata_2.html#E11:25_1_4"><i><font color="Green">E39</font></i></a>, <a class="txt" href="nagata_2.html#E2:25_1_4_1"><i><font color="Green">E40</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E14:25_1_4"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> st <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="pscomp_1.html#V9">continuous</a>
 <b>by </b><i><a class="txt" href="nagata_2.html#E11:25_1_4"><i><font color="Green">E39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:25_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>4</sub></font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a>  st <font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> holds <br/><span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>10</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:25_1_5"/><i><font color="Green">E34</font></i>: 
( not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K3">`</a> ;<br/>
<a NAME="E2:25_1_5"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1_5"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:25_1_5"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1_5"><i><font color="Green">E34</font></i></a>, <a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:25_1_5"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="nagata_2.html#E6:25_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="nagata_1.html#T19">NAGATA_1:19</a></i>;<br/>
<a NAME="E6:25_1_5"/>
 <a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E6:25_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="cantor_1.html#D2">CANTOR_1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="nagata_2.html#E5:25_1_5"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:25_1_5"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:25_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="nagata_2.html#E6:25_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="nagata_2.html#E5:25_1_5"><i><font color="Green">E35</font></i></a>, <a class="ref" href="nagata_1.html#T19">NAGATA_1:19</a></i>;<br/>
<a NAME="E10:25_1_5"/>
 <a href="nagata_1.html#K2">Union</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E6:25_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="cantor_1.html#D2">CANTOR_1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="nagata_2.html#E9:25_1_5"><i><font color="Green">E36</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:25_1_5"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="nagata_2.html#E5:25_1_5"><i><font color="Green">E35</font></i></a>, <a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E15:25_1_5"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="nagata_2.html#E9:25_1_5"><i><font color="Green">E36</font></i></a>, <a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>18</sub></font> = <a href="nagata_2.html#K2" target="_self">PairFunc</a>  <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<b>take </b>
<a href="nagata_2.html#K2" target="_self">PairFunc</a>  <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="domain_1.html#K1">]</a></span>
;<br/>

<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K2" target="_self">PairFunc</a>  <a href="funct_2.html#K3">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> as   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<a NAME="E17:25_1_5"/>
 <a href="relset_1.html#K4">dom</a> <a href="nagata_2.html#K2" target="_self">PairFunc</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E18:25_1_5"/><b>then </b>
(  <a href="nagata_2.html#K2" target="_self">PairFunc</a>  is <a href="funct_1.html#V2">one-to-one</a> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <a href="nagata_2.html#K2" target="_self">PairFunc</a>  )
 
<b>by </b><i><a class="ref" href="nagata_2.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E19:25_1_5"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K2" target="_self">PairFunc</a>  <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K2" target="_self">PairFunc</a>  <a href="funct_2.html#K3">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="pscomp_1.html#NM1">RealMap</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E21:25_1_5"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K2" target="_self">PairFunc</a>  <a href="funct_2.html#K3">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <span class="p1">(<span class="default"><a href="nagata_2.html#K2" target="_self">PairFunc</a>  <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K2" target="_self">PairFunc</a>  <a href="funct_2.html#K3">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>20</sub></font>] ) )
 <b>by </b><i><a class="txt" href="nagata_2.html#E10:25_1"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E23:25_1_5"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> is <a href="pscomp_1.html#V9">continuous</a> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nagata_1.html#R1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>21</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; ( for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st ex <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>5</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>5</sub></font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">b<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>6</sub></font> &amp; not <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>6</sub></font> ) holds <br/><font color="Maroon">c<sub>21</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> 1 ) ) ) )
 <b>by </b><i><a class="txt" href="nagata_2.html#E19:25_1_5"><i><font color="Green">E39</font></i></a>, <a class="txt" href="nagata_2.html#E21:25_1_5"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>22</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E25:25_1_5"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E26:25_1_5"/>
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E25:25_1_5"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E27:25_1_5"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E28:25_1_5"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</sub></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="E1:25_1_5_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>24</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:25_1_5_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>24</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="nagata_2.html#E3:25_1_5_1"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E5:25_1_5_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>13</sub></font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E5:25_1_5"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E6:25_1_5_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E7:25_1_5_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/>
<a NAME="E8:25_1_5_1"/><b>then </b>
not <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E3:25_1_5_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E9:25_1_5_1"/><b>then </b>
( <font color="Maroon">c<sub>19</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E9:25_1_5"><i><font color="Green">E36</font></i></a>, <a class="txt" href="nagata_2.html#E13:25_1_5"><i><font color="Green">E37</font></i></a>, <a class="txt" href="nagata_2.html#E15:25_1_5"><i><font color="Green">E38</font></i></a>, <a class="txt" href="nagata_2.html#E21:25_1_5"><i><font color="Green">E40</font></i></a>, <a class="txt" href="nagata_2.html#E23:25_1_5"><i><font color="Green">E41</font></i></a>, <a class="ref" href="nagata_2.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E10:25_1_5_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="nagata_2.html#E21:25_1_5"><i><font color="Green">E40</font></i></a>, <a class="txt" href="nagata_2.html#E23:25_1_5"><i><font color="Green">E41</font></i></a>, <a class="txt" href="nagata_2.html#E3:25_1_5_1"><i><font color="Green">E44</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E29:25_1_5"/><b>then </b>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> is <a href="seq_2.html#V2">bounded_below</a> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E23:25_1_5"><i><font color="Green">E41</font></i></a>, <a class="txt" href="nagata_2.html#E27:25_1_5"><i><font color="Green">E43</font></i></a>, <a class="txt" href="nagata_2.html#E8"><i><font color="Green">Lemma8</font></i></a>, <a class="ref" href="seq_4.html#T60">SEQ_4:60</a></i>;<br/>
<b>hence </b><a NAME="E30:25_1_5"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a>  st <font color="Maroon">c<sub>8</sub></font> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K2" target="_self">PairFunc</a>  <a href="funct_2.html#K3">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="nagata_2.html#E21:25_1_5"><i><font color="Green">E40</font></i></a>, <a class="txt" href="nagata_2.html#E23:25_1_5"><i><font color="Green">E41</font></i></a>, <a class="ref" href="nagata_2.html#D3" target="_self">Def3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:25_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="pcomps_1.html#V3">metrizable</a>
 <b>by </b><i><a class="txt" href="nagata_2.html#E2:25_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="nagata_2.html#E14:25_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="nagata_2.html#T17" target="_self">Th17</a></i>;<br/>


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

<b>thus </b><a NAME="E2:25"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="pcomps_1.html#V3">metrizable</a> implies ( <font color="Maroon">c<sub>1</sub></font> <a href="compts_1.html#NR3" target="_self">is_T3</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="urysohn1.html#NR1" target="_self">is_T1</a>  &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="nagata_1.html#NM1">FamilySequence</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="nagata_1.html#V6">Basis_sigma_locally_finite</a> ) )
 <b>by </b><i><a class="ref" href="nagata_1.html#T15">NAGATA_1:15</a>, <a class="ref" href="nagata_1.html#T16">NAGATA_1:16</a></i>;<br/>


</div>
