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

<b>let </b><font color="Maroon">c<sub>8</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <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>8</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>10</sub></font> be   <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>8</sub></font>,<span class="p2">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:40"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E2:40"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>10</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E3:40"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E4:40"/><i><font color="Green">E41</font></i>: 
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>8</sub></font>,<a href="topmetr.html#K5">I[01]</a> </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>8</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </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="E5:40"/><i><font color="Green">E42</font></i>: 
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>8</sub></font>,<span class="p2">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></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>8</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></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="E6:40"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> <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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E39"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span><b> such that </b><br/><a NAME="E8:40"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="toprealb.html#K8">Tunit_circle</a> 2
 <b>and </b><br/><a NAME="E9:40"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>11</sub></font> is <a href="tops_2.html#V1">open</a>
 <b>and </b><br/><a NAME="E10:40"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being <a href="taxonom2.html#V4">mutually-disjoint</a> <a href="tops_2.html#V1">open</a> <a href="struct_0.html#NM3">Subset-Family</a> of <a href="topalg_2.html#K3">R^1</a>  st <br/>(  <a href="setfam_1.html#K5">union</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K11">"</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> holds <br/>for <font color="Olive">b<sub>4</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Olive">b<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>4</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  ) )
 <b>by </b><i><a class="txt" href="topalg_5.html#E38"><i><font color="Green">Lemma32</font></i></a></i>;<br/>
<i><font color="Green">E47</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</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>8</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<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>8</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </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>defpred </b><font color="Maroon">S<sub>1</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="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>8</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ex <font color="Olive">b<sub>3</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font>ex <font color="Olive">b<sub>4</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Olive">b<sub>3</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="borsuk_1.html#K7">]</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</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; <font color="Olive">b<sub>4</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">b<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="partfun1.html#K2">|</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">b<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> &amp; ( for <font color="Olive">b<sub>5</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Olive">b<sub>3</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  st <font color="Olive">b<sub>5</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">b<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="partfun1.html#K2">|</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">b<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Olive">b<sub>4</sub></font> <a href="funct_2.html#R2">=</a> <font color="Olive">b<sub>5</sub></font> ) );<br/>
<a NAME="E12:40"/><i><font color="Green">E48</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>8</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>ex <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topalg_2.html#K3">R^1</a>  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>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</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>8</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>;<br/>

<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E2:40_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50">BORSUK_1:50</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:40_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> 0
 <b>and </b><br/><a NAME="E5:40_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>and </b><br/><a NAME="E6:40_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>15</sub></font> is <a href="seqm_3.html#V1">increasing</a>
 <b>and </b><br/><a NAME="E7:40_2"/><i><font color="Green">E53</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font> st <br/>( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>3</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:40"><i><font color="Green">E38</font></i></a>, <a class="txt" href="topalg_5.html#E8:40"><i><font color="Green">E44</font></i></a>, <a class="txt" href="topalg_5.html#E9:40"><i><font color="Green">E45</font></i></a>, <a class="ref" href="topalg_5.html#T21" target="_self">Th21</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font><b> such that </b><br/><a NAME="E9:40_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E10:40_2"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="ordinal1.html#V7">natural</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>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font> holds <br/>ex <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 <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="topalg_5.html#E7:40_2"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font> <b>by </b><i><a class="txt" href="topalg_5.html#E9:40_2"><i><font color="Green">E54</font></i></a></i>;<br/>
<a NAME="E12:40_2"/><i><font color="Green">E56</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E13:40_2"/><i><font color="Green">E57</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<div><i><font color="Green">E58</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:40_2_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><a NAME="E2:40_2_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:40_2_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:40_2_1"/><b>then </b>
( 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> or 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:40_2_1"/><i><font color="Green">E60</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E4:40_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="topalg_5.html#E12:40_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="seqm_3.html#D1">SEQM_3:def 1</a></i>;<br/><b>assume </b><a NAME="E5:40_2_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><a NAME="E6:40_2_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>18</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/><b>hence </b><a NAME="E7:40_2_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E6:40_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_2_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="seqm_3.html#D1">SEQM_3:def 1</a></i>;<br/><a NAME="E8:40_2_1"/>
<font color="Maroon">c<sub>18</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>15</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E5:40_2_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:40_2_1"/><b>then </b>
( <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font> or <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E10:40_2_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="topalg_5.html#E5:40_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_2_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="topalg_5.html#E13:40_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="seqm_3.html#D1">SEQM_3:def 1</a></i>;<br/><b>hence </b><a NAME="E11:40_2_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <b>by </b><i><a class="txt" href="topalg_5.html#E7:40_2_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>thus </b><a NAME="E12:40_2_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E4:40_2_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_2_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_2_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="seqm_3.html#D1">SEQM_3:def 1</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:40_2_2"/><i><font color="Green">E60</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><b>thus </b><a NAME="E2:40_2_2"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</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="E1:40_2_2_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E3:40_2_2_1"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="topalg_5.html#E1:40_2_2_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E4:40_2_2_1"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="topalg_5.html#E1:40_2_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:40_2_2_1"/>
<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 <a href="topmetr.html#K5">I[01]</a> 
 <b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font>ex <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 <a href="topmetr.html#K5">I[01]</a> ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="partfun1.html#K2">|</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">b<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> ) );<br/>
<a NAME="E16:40_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="ref" href="finseq_3.html#T26">FINSEQ_3:26</a></i>;<br/>
<a NAME="E17:40_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>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</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="40_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:40_2_3"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>18</sub></font>]
 <b>and </b><br/><a NAME="E2:40_2_3"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_2_3_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font> )
 </span><b>by </b><i><a class="txt" href="topalg_5.html#E2:40_2_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="topreala.html#T23">TOPREALA:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_3_1_1"><b>suppose </b></a><a NAME="E1:40_2_3_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>;<br/><a NAME="E2:40_2_3_1_1"/><i><font color="Green">E65</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E4:40_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_3_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="rcomp_1.html#T14">RCOMP_1:14</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E3"><i><font color="Green">Lemma3</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_3_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="rcomp_1.html#T14">RCOMP_1:14</a></i>;<br/><b>take </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>17</sub></font>;<br/><a NAME="E4:40_2_3_1_1"/><i><font color="Green">E66</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E5:40_2_3_1_1"/><i><font color="Green">E67</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E6:40_2_3_1_1"/><i><font color="Green">E68</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></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 <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></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>set </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/><a NAME="E7:40_2_3_1_1"/><i><font color="Green">E69</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E6:40"><i><font color="Green">E43</font></i></a>, <a class="txt" href="topalg_5.html#E2:40_2_3_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E8:40_2_3_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E2:40_2_3_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E9:40_2_3_1_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topalg_2.html#K3">R^1</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E8:40_2_3_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_2_3_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_2_3_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_2_3_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>20</sub></font>
;<br/><b>take </b>
<font color="Maroon">c<sub>23</sub></font>
;<br/><b>thus </b><a NAME="E11:40_2_3_1_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/><b>thus </b><a NAME="E12:40_2_3_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E9:40_2"><i><font color="Green">E54</font></i></a></i>;<br/><b>thus </b><a NAME="E13:40_2_3_1_1"/>
<font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="txt" href="topalg_5.html#E39"><i><font color="Green">Lemma33</font></i></a>, <a class="txt" href="topalg_5.html#E5:40"><i><font color="Green">E42</font></i></a>, <a class="txt" href="topalg_5.html#E2:40_2_3_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E15:40_2_3_1_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E39"><i><font color="Green">Lemma33</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_3_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="rcomp_1.html#T14">RCOMP_1:14</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>20</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span> ;<br/><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font> = 
 <a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> 
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40_2_3_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="topalg_3.html#T6">TOPALG_3:6</a>, <a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40_2_3_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="topalg_5.html#E40"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="tsep_1.html#T3">TSEP_1:3</a></i>
;<br/><a NAME="E18:40_2_3_1_1"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><a href="tex_2.html#K1">Sspace</a> <a href="borsuk_1.html#K21">0[01]</a> </span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="ref" href="borsuk_3.html#T26">BORSUK_3:26</a></i>;<br/><b>hence </b><a NAME="E19:40_2_3_1_1"/>
<font color="Maroon">c<sub>23</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E2:40"><i><font color="Green">E39</font></i></a>, <a class="txt" href="topalg_5.html#E2:40_2_3_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/><a NAME="E20:40_2_3_1_1"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E11:40"><i><font color="Green">E47</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E21:40_2_3_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E22:40_2_3_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="toprealb.html#K12">CircleMap</a> 
 <b>by </b><i><a class="txt" href="topalg_5.html#E35"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><a NAME="E23:40_2_3_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/><a NAME="E24:40_2_3_1_1"/>
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="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></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"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_3_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>26</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:40_2_3_1_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> = 
<font color="Maroon">c<sub>9</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="topalg_5.html#E1:40_2_3_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topalg_5.html#E21:40_2_3_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40"><i><font color="Green">E40</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_3_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topalg_5.html#E8:40_2_3_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="topalg_5.html#E21:40_2_3_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="topalg_5.html#E39"><i><font color="Green">Lemma33</font></i></a>, <a class="txt" href="topalg_5.html#E5:40"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E1:40_2_3_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topalg_5.html#E21:40_2_3_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="topalg_5.html#E7:40_2_3_1_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E1:40_2_3_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topalg_5.html#E2:40_2_3_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="topalg_5.html#E21:40_2_3_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E8:40_2_3_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="topalg_5.html#E21:40_2_3_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_3_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E25:40_2_3_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E8:40_2_3_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="topalg_5.html#E21:40_2_3_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="topalg_5.html#E23:40_2_3_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><a NAME="E26:40_2_3_1_1"/><i><font color="Green">E73</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><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>8</sub></font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="subset_1.html#K8">/\</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E8:40_2_3_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E27:40_2_3_1_1"/><i><font color="Green">E74</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="subset_1.html#K8">/\</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E2:40_2_3_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="zfmisc_1.html#T124">ZFMISC_1:124</a></i>;<br/><i><font color="Green">E75</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> = 
<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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="txt" href="topalg_5.html#E6:40"><i><font color="Green">E43</font></i></a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40_2_3_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="zfmisc_1.html#T124">ZFMISC_1:124</a></i>
;<br/><a NAME="E29:40_2_3_1_1"/>
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="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><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>8</sub></font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><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>8</sub></font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</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>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_3_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>26</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:40_2_3_1_1_4"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><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>8</sub></font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 ;<br/>

<a NAME="E2:40_2_3_1_1_4"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E1:40_2_3_1_1_4"><i><font color="Green">E76</font></i></a>, <a class="txt" href="topalg_5.html#E26:40_2_3_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topalg_5.html#E2:40_2_3_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:40_2_3_1_1_4"/><i><font color="Green">E78</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3"><i><font color="Green">Lemma3</font></i></a>, <a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E4:40_2_3_1_1_4"/>
<font color="Maroon">c<sub>26</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E1:40_2_3_1_1_4"><i><font color="Green">E76</font></i></a>, <a class="txt" href="topalg_5.html#E26:40_2_3_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><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>8</sub></font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>26</sub></font> = 
<font color="Maroon">c<sub>23</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40_2_3_1_1_4"><i><font color="Green">E77</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40_2_3_1_1_4"><i><font color="Green">E77</font></i></a>, <a class="txt" href="topalg_5.html#E3:40_2_3_1_1_4"><i><font color="Green">E78</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E30:40_2_3_1_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="partfun1.html#K2">|</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E26:40_2_3_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topalg_5.html#E28:40_2_3_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="topalg_5.html#E27:40_2_3_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_3_1_2"><b>suppose </b></a><a NAME="E1:40_2_3_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>19</sub></font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>20</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">c<sub>21</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E3:40_2_3_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>and </b><br/><a NAME="E4:40_2_3_1_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E5:40_2_3_1_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E6:40_2_3_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>21</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E7:40_2_3_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>21</sub></font>
 <b>and </b><br/><a NAME="E8:40_2_3_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K2">|</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:40_2_3"><i><font color="Green">E62</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>19</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font> <b>by </b><i><a class="txt" href="topalg_5.html#E4:40_2_3_1_2"><i><font color="Green">E66</font></i></a></i>;<br/><a NAME="E10:40_2_3_1_2"/><i><font color="Green">E71</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E11:40_2_3_1_2"/><i><font color="Green">E72</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></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 <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></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>set </b><font color="Maroon">c<sub>23</sub></font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>;<br/><b>consider </b><font color="Maroon">c<sub>24</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span><b> such that </b><br/><a NAME="E13:40_2_3_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E14:40_2_3_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E10:40_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="topalg_5.html#E2:40_2_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_3_1_2"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E15:40_2_3_1_2"/><i><font color="Green">E75</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>25</sub></font> being  <a href="taxonom2.html#V4">mutually-disjoint</a> <a href="tops_2.html#V1">open</a> <a href="struct_0.html#NM3">Subset-Family</a> of <a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E17:40_2_3_1_2"/><i><font color="Green">E76</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>24</sub></font>
 <b>and </b><br/><a NAME="E18:40_2_3_1_2"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i><a class="txt" href="topalg_5.html#E13:40_2_3_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topalg_5.html#E10:40"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E19:40_2_3_1_2"/><i><font color="Green">E78</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E20:40_2_3_1_2"/><i><font color="Green">E79</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:40_2_3_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="topalg_5.html#E14:40_2"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E21:40_2_3_1_2"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E3:40_2_3_1_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><a NAME="E22:40_2_3_1_2"/><b>then </b><i><font color="Green">E81</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></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"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E4:40_2_3_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>then </b><i><font color="Green">E82</font></i>: <font color="Maroon">c<sub>9</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>13</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</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>13</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="topalg_5.html#E7:40_2_3_1_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</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>13</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E19:40_2_3_1_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="topalg_5.html#E10:40_2_3_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="topalg_5.html#E22:40_2_3_1_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="topalg_5.html#E11:40_2_3_1_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/><a NAME="E24:40_2_3_1_2"/><i><font color="Green">E83</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E21:40_2_3_1_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="topalg_5.html#E11:40"><i><font color="Green">E47</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E25:40_2_3_1_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E2:40_2_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_3_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="topalg_5.html#E14:40_2"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E26:40_2_3_1_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><a NAME="E27:40_2_3_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>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E9:40_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E28:40_2_3_1_2"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>9</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>13</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E24:40_2_3_1_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/><a NAME="E29:40_2_3_1_2"/>
<font color="Maroon">c<sub>21</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>13</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E28:40_2_3_1_2"><i><font color="Green">E85</font></i></a>, <a class="txt" href="topalg_5.html#E14:40_2_3_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="topalg_5.html#E23:40_2_3_1_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>26</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E31:40_2_3_1_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>21</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>13</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>26</sub></font>
 <b>and </b><br/><a NAME="E32:40_2_3_1_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E17:40_2_3_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>26</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E31:40_2_3_1_2"><i><font color="Green">E86</font></i></a>, <a class="txt" href="topalg_5.html#E32:40_2_3_1_2"><i><font color="Green">E87</font></i></a></i>;<br/><a NAME="E34:40_2_3_1_2"/><i><font color="Green">E88</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font> <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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E35:40_2_3_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E36:40_2_3_1_2"/><b>then </b><i><font color="Green">E90</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E4:40_2_3_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><a NAME="E37:40_2_3_1_2"/><i><font color="Green">E91</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E21:40_2_3_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>28</sub></font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E21:40_2_3_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E39:40_2_3_1_2"/><i><font color="Green">E92</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E40:40_2_3_1_2"/><i><font color="Green">E93</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></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 <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></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>set </b><font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>28</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>;<br/><a NAME="E41:40_2_3_1_2"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E34:40_2_3_1_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="topalg_5.html#E19:40_2_3_1_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="topalg_5.html#E10:40_2_3_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="topalg_5.html#E11:40_2_3_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="topalg_5.html#E37:40_2_3_1_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E42:40_2_3_1_2"/><b>then </b><i><font color="Green">E94</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>28</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E43:40_2_3_1_2"/><i><font color="Green">E95</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>28</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <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>30</sub></font> = <font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>28</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E19:40_2_3_1_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="topalg_5.html#E43:40_2_3_1_2"><i><font color="Green">E95</font></i></a>, <a class="txt" href="topalg_5.html#E39:40_2_3_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="topalg_5.html#E40:40_2_3_1_2"><i><font color="Green">E93</font></i></a>, <a class="txt" href="topalg_5.html#E42:40_2_3_1_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><a NAME="E45:40_2_3_1_2"/>
<font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>31</sub></font> = <font color="Maroon">c<sub>22</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>32</sub></font> = <font color="Maroon">c<sub>28</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <b>by </b><i><a class="txt" href="topalg_5.html#E10:40_2_3_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="topalg_5.html#E21:40_2_3_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E48:40_2_3_1_2"/><i><font color="Green">E96</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>22</sub></font> &amp; <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>28</sub></font> )
 <b>by </b><i><a class="ref" href="goboard9.html#T4">GOBOARD9:4</a></i>;<br/><a NAME="E49:40_2_3_1_2"/>
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>31</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>32</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>32</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_3.html#T26">BORSUK_3:26</a></i>;<br/><a NAME="E50:40_2_3_1_2"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">c<sub>30</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E48:40_2_3_1_2"><i><font color="Green">E96</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_2_3_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/><a NAME="E51:40_2_3_1_2"/>
<font color="Maroon">c<sub>27</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E32:40_2_3_1_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/><a NAME="E52:40_2_3_1_2"/><b>then </b>
<font color="Maroon">c<sub>30</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>27</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E50:40_2_3_1_2"><i><font color="Green">E97</font></i></a>, <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>33</sub></font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>28</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E54:40_2_3_1_2"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>33</sub></font>
 <b>and </b><br/><a NAME="E55:40_2_3_1_2"/><i><font color="Green">E99</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>33</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="borsuk_1.html#T45">BORSUK_1:45</a></i>;<br/><a NAME="E56:40_2_3_1_2"/>
<font color="Maroon">c<sub>30</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>13</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E31:40_2_3_1_2"><i><font color="Green">E86</font></i></a>, <a class="txt" href="topalg_5.html#E36:40_2_3_1_2"><i><font color="Green">E90</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><a NAME="E57:40_2_3_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>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>30</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E36:40_2_3_1_2"><i><font color="Green">E90</font></i></a>, <a class="txt" href="topalg_5.html#E42:40_2_3_1_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>34</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E59:40_2_3_1_2"/><i><font color="Green">E100</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>34</sub></font>
 <b>and </b><br/><a NAME="E60:40_2_3_1_2"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>33</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E54:40_2_3_1_2"><i><font color="Green">E98</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>35</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>, <font color="Maroon">c<sub>36</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>28</sub></font></span>)</span><b> such that </b><br/><a NAME="E62:40_2_3_1_2"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>35</sub></font>,<font color="Maroon">c<sub>36</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>and </b><br/><a NAME="E63:40_2_3_1_2"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">c<sub>35</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E64:40_2_3_1_2"/>
<font color="Maroon">c<sub>36</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E55:40_2_3_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="topalg_5.html#E60:40_2_3_1_2"><i><font color="Green">E101</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>37</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font><b> such that </b><br/><a NAME="E66:40_2_3_1_2"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">c<sub>37</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E67:40_2_3_1_2"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">c<sub>37</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>35</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E63:40_2_3_1_2"><i><font color="Green">E103</font></i></a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>38</sub></font>, <font color="Maroon">c<sub>39</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E69:40_2_3_1_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>35</sub></font>
 <b>and </b><br/><a NAME="E70:40_2_3_1_2"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">c<sub>39</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>36</sub></font>
 <b>and </b><br/><a NAME="E71:40_2_3_1_2"/><i><font color="Green">E108</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></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>38</sub></font>,<font color="Maroon">c<sub>39</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E59:40_2_3_1_2"><i><font color="Green">E100</font></i></a>, <a class="txt" href="topalg_5.html#E62:40_2_3_1_2"><i><font color="Green">E102</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>set </b><font color="Maroon">c<sub>40</sub></font> = <font color="Maroon">c<sub>37</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>22</sub></font>;<br/><a NAME="E72:40_2_3_1_2"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_3_1_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:40_2_3_1_2_2"/>
<font color="Maroon">c<sub>36</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E39:40_2_3_1_2"><i><font color="Green">E92</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">c<sub>41</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="E2:40_2_3_1_2_2"/>
<font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>28</sub></font>
 ;<br/>

<a NAME="E3:40_2_3_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:40_2_3_1_2_2"/>
<font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>36</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E70:40_2_3_1_2"><i><font color="Green">E107</font></i></a>, <a class="txt" href="topalg_5.html#E71:40_2_3_1_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>


</div><b>end;</b></div><a NAME="E73:40_2_3_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>38</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E71:40_2_3_1_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E74:40_2_3_1_2"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>37</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E67:40_2_3_1_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="topalg_5.html#E69:40_2_3_1_2"><i><font color="Green">E106</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>41</sub></font> = <font color="Maroon">c<sub>37</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>22</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font> <b>by </b><i><a class="txt" href="topalg_5.html#E66:40_2_3_1_2"><i><font color="Green">E104</font></i></a>, <a class="ref" href="tops_1.html#T38">TOPS_1:38</a>, <a class="txt" href="topalg_5.html#E4:40_2_3_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E76:40_2_3_1_2"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>41</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E77:40_2_3_1_2"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>27</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_3_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>42</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="E1:40_2_3_1_2_3"/>
<font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>43</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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E3:40_2_3_1_2_3"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E4:40_2_3_1_2_3"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>42</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E5:40_2_3_1_2_3"/>
<font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>34</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E67:40_2_3_1_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a>, <a class="txt" href="topalg_5.html#E3:40_2_3_1_2_3"><i><font color="Green">E113</font></i></a>, <a class="txt" href="topalg_5.html#E62:40_2_3_1_2"><i><font color="Green">E102</font></i></a>, <a class="txt" href="topalg_5.html#E72:40_2_3_1_2"><i><font color="Green">E109</font></i></a></i>;<br/>
<a NAME="E6:40_2_3_1_2_3"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>33</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E60:40_2_3_1_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E7:40_2_3_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>30</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E54:40_2_3_1_2"><i><font color="Green">E98</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E8:40_2_3_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>30</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>43</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E9:40_2_3_1_2_3"/>
<font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E4:40_2_3_1_2_3"><i><font color="Green">E114</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_2_3_1_2_3"><i><font color="Green">E115</font></i></a>, <a class="txt" href="topalg_5.html#E54:40_2_3_1_2"><i><font color="Green">E98</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div><a NAME="E78:40_2_3_1_2"/><i><font color="Green">E113</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>41</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>41</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E79:40_2_3_1_2"/><i><font color="Green">E114</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E5:40_2_3_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E80:40_2_3_1_2"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E76:40_2_3_1_2"><i><font color="Green">E111</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E81:40_2_3_1_2"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E79:40_2_3_1_2"><i><font color="Green">E114</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E82:40_2_3_1_2"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/><b>set </b><font color="Maroon">c<sub>42</sub></font> = <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>27</sub></font>;<br/><a NAME="E83:40_2_3_1_2"/><i><font color="Green">E116</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E84:40_2_3_1_2"/><i><font color="Green">E117</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E35"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E85:40_2_3_1_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>24</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_3_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>43</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="E1:40_2_3_1_2_4"/>
<font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>44</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_2_3_1_2_4"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">c<sub>44</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="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:40_2_3_1_2_4"/><i><font color="Green">E119</font></i>: 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>43</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:40_2_3_1_2_4"/>
<font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40_2_3_1_2_4"><i><font color="Green">E118</font></i></a>, <a class="txt" href="topalg_5.html#E84:40_2_3_1_2"><i><font color="Green">E117</font></i></a>, <a class="txt" href="topalg_5.html#E32:40_2_3_1_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E6:40_2_3_1_2_4"/><b>then </b>
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E17:40_2_3_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<b>hence </b><a NAME="E7:40_2_3_1_2_4"/>
<font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E3:40_2_3_1_2_4"><i><font color="Green">E118</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_2_3_1_2_4"><i><font color="Green">E119</font></i></a>, <a class="txt" href="topalg_5.html#E84:40_2_3_1_2"><i><font color="Green">E117</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>43</sub></font> = <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>27</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <b>by </b><i><a class="txt" href="topalg_5.html#E84:40_2_3_1_2"><i><font color="Green">E117</font></i></a>, <a class="txt" href="topalg_5.html#E83:40_2_3_1_2"><i><font color="Green">E116</font></i></a>, <a class="txt" href="topalg_5.html#E15:40_2_3_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><a NAME="E87:40_2_3_1_2"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">c<sub>43</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i><a class="txt" href="topalg_5.html#E18:40_2_3_1_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="topalg_5.html#E32:40_2_3_1_2"><i><font color="Green">E87</font></i></a></i>;<br/><a NAME="E88:40_2_3_1_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="topalg_5.html#E2:40_2_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_3_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="topalg_5.html#E14:40_2"><i><font color="Green">E58</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>44</sub></font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E25:40_2_3_1_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="topalg_5.html#E20:40_2_3_1_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="topalg_5.html#E15:40_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>41</sub></font>
;<br/><b>take </b><font color="Maroon">c<sub>45</sub></font> = <font color="Maroon">c<sub>20</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>44</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>46</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>43</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">c<sub>47</sub></font> = <font color="Maroon">c<sub>43</sub></font> as   <a href="funct_1.html#NM1">Function</a> ;<br/><a NAME="E91:40_2_3_1_2"/><i><font color="Green">E119</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E11:40"><i><font color="Green">E47</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E92:40_2_3_1_2"/>
<font color="Maroon">c<sub>43</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i><a class="txt" href="topalg_5.html#E87:40_2_3_1_2"><i><font color="Green">E118</font></i></a>, <a class="ref" href="tops_2.html#T70">TOPS_2:70</a></i>;<br/><a NAME="E93:40_2_3_1_2"/><b>then </b><i><font color="Green">E120</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>43</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><a NAME="E94:40_2_3_1_2"/><i><font color="Green">E121</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E87:40_2_3_1_2"><i><font color="Green">E118</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><a NAME="E95:40_2_3_1_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">c<sub>43</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E87:40_2_3_1_2"><i><font color="Green">E118</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><a NAME="E96:40_2_3_1_2"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">c<sub>43</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>47</sub></font> <a href="funct_1.html#K2">"</a> 
 <b>by </b><i><a class="txt" href="topalg_5.html#E94:40_2_3_1_2"><i><font color="Green">E121</font></i></a>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/><a NAME="E97:40_2_3_1_2"/><i><font color="Green">E124</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>44</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>44</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E98:40_2_3_1_2"/><i><font color="Green">E125</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>45</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E99:40_2_3_1_2"/><i><font color="Green">E126</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E91:40_2_3_1_2"><i><font color="Green">E119</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E100:40_2_3_1_2"/><i><font color="Green">E127</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>43</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_3_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>48</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="E1:40_2_3_1_2_5"/>
<font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>49</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_2_3_1_2_5"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">c<sub>49</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:40_2_3_1_2_5"/><i><font color="Green">E129</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>49</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>48</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:40_2_3_1_2_5"/>
<font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>49</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40_2_3_1_2_5"><i><font color="Green">E128</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_2_3_1_2_5"><i><font color="Green">E129</font></i></a>, <a class="txt" href="topalg_5.html#E99:40_2_3_1_2"><i><font color="Green">E126</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E6:40_2_3_1_2_5"/><b>then </b>
<font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40_2_3_1_2_5"><i><font color="Green">E128</font></i></a>, <a class="txt" href="topalg_5.html#E99:40_2_3_1_2"><i><font color="Green">E126</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E7:40_2_3_1_2_5"/><b>then </b>
<font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>44</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E82:40_2_3_1_2"><i><font color="Green">E115</font></i></a></i>;<br/>
<a NAME="E8:40_2_3_1_2_5"/><b>then </b>
<font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E14:40_2_3_1_2"><i><font color="Green">E74</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:40_2_3_1_2_5"/>
<font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>43</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E93:40_2_3_1_2"><i><font color="Green">E120</font></i></a>, <a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>


</div><b>end;</b></div><a NAME="E101:40_2_3_1_2"/><b>then </b><i><font color="Green">E128</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>43</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/><a NAME="E102:40_2_3_1_2"/><i><font color="Green">E129</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</sub></font></span><a href="borsuk_1.html#K6">:]</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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>41</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>44</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E78:40_2_3_1_2"><i><font color="Green">E113</font></i></a>, <a class="txt" href="topalg_5.html#E97:40_2_3_1_2"><i><font color="Green">E124</font></i></a>, <a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/><a NAME="E103:40_2_3_1_2"/><i><font color="Green">E130</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>45</sub></font></span><a href="borsuk_1.html#K6">:]</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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>41</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>45</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E78:40_2_3_1_2"><i><font color="Green">E113</font></i></a>, <a class="txt" href="topalg_5.html#E98:40_2_3_1_2"><i><font color="Green">E125</font></i></a>, <a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/><a NAME="E104:40_2_3_1_2"/><i><font color="Green">E131</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>41</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E78:40_2_3_1_2"><i><font color="Green">E113</font></i></a>, <a class="txt" href="topalg_5.html#E10:40_2_3_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/><a NAME="E105:40_2_3_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> is   <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a> 
 <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/><a NAME="E106:40_2_3_1_2"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>43</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topalg_2.html#K3">R^1</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>48</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>43</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>41</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>44</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E99:40_2_3_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="topalg_5.html#E101:40_2_3_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="topalg_5.html#E102:40_2_3_1_2"><i><font color="Green">E129</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><b>set </b><font color="Maroon">c<sub>49</sub></font> = <font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>;<br/><b>set </b><font color="Maroon">c<sub>50</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>48</sub></font>;<br/><a NAME="E108:40_2_3_1_2"/><i><font color="Green">E132</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</sub></font></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"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>45</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T120">ZFMISC_1:120</a></i>;<br/><a NAME="E109:40_2_3_1_2"/><i><font color="Green">E133</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E76:40_2_3_1_2"><i><font color="Green">E111</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E110:40_2_3_1_2"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E19:40_2_3_1_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="topalg_5.html#E10:40_2_3_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="topalg_5.html#E11:40_2_3_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="topalg_5.html#E34:40_2_3_1_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E111:40_2_3_1_2"/><b>then </b><i><font color="Green">E134</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>48</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E99:40_2_3_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="topalg_5.html#E101:40_2_3_1_2"><i><font color="Green">E128</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><a NAME="E112:40_2_3_1_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>48</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>48</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#T18">FUNCT_4:18</a></i>;<br/><a NAME="E113:40_2_3_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>48</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topalg_2.html#K3">R^1</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>51</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>48</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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>41</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>45</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E111:40_2_3_1_2"><i><font color="Green">E134</font></i></a>, <a class="txt" href="topalg_5.html#E108:40_2_3_1_2"><i><font color="Green">E132</font></i></a>, <a class="txt" href="topalg_5.html#E103:40_2_3_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>51</sub></font>
;<br/><b>thus </b><a NAME="E115:40_2_3_1_2"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E3:40_2_3_1_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="topalg_5.html#E20:40_2_3_1_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="topalg_5.html#E25:40_2_3_1_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="heine.html#T2">HEINE:2</a></i>;<br/><b>thus </b><a NAME="E116:40_2_3_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>41</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E74:40_2_3_1_2"><i><font color="Green">E110</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_2_3_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b><a NAME="E117:40_2_3_1_2"/>
<font color="Maroon">c<sub>41</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E76:40_2_3_1_2"><i><font color="Green">E111</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_2_3_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E118:40_2_3_1_2"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>45</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E119:40_2_3_1_2"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">c<sub>44</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>45</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E120:40_2_3_1_2"/><i><font color="Green">E138</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>41</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E109:40_2_3_1_2"><i><font color="Green">E133</font></i></a>, <a class="txt" href="topalg_5.html#E104:40_2_3_1_2"><i><font color="Green">E131</font></i></a>, <a class="txt" href="topalg_5.html#E19:40_2_3_1_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="topalg_5.html#E10:40_2_3_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="topalg_5.html#E11:40_2_3_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="topalg_5.html#E34:40_2_3_1_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E121:40_2_3_1_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topalg_2.html#K3">R^1</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>52</sub></font> = <font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>41</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E120:40_2_3_1_2"><i><font color="Green">E138</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><a NAME="E123:40_2_3_1_2"/><i><font color="Green">E139</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>45</sub></font></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>53</sub></font> =  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>, <font color="Maroon">c<sub>54</sub></font> =  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>44</sub></font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>45</sub></font></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a>, <a class="txt" href="topalg_5.html#E98:40_2_3_1_2"><i><font color="Green">E125</font></i></a>, <a class="txt" href="topalg_5.html#E118:40_2_3_1_2"><i><font color="Green">E136</font></i></a>, <a class="txt" href="topalg_5.html#E119:40_2_3_1_2"><i><font color="Green">E137</font></i></a></i>;<br/><a NAME="E125:40_2_3_1_2"/><i><font color="Green">E140</font></i>: 
<a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>45</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E10:40_2_3_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="topalg_5.html#E98:40_2_3_1_2"><i><font color="Green">E125</font></i></a>, <a class="txt" href="topalg_5.html#E118:40_2_3_1_2"><i><font color="Green">E136</font></i></a>, <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a></i>;<br/><a NAME="E126:40_2_3_1_2"/><i><font color="Green">E141</font></i>: 
<a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>44</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>45</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E97:40_2_3_1_2"><i><font color="Green">E124</font></i></a>, <a class="txt" href="topalg_5.html#E98:40_2_3_1_2"><i><font color="Green">E125</font></i></a>, <a class="txt" href="topalg_5.html#E119:40_2_3_1_2"><i><font color="Green">E137</font></i></a>, <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>55</sub></font> = <font color="Maroon">c<sub>53</sub></font>, <font color="Maroon">c<sub>56</sub></font> = <font color="Maroon">c<sub>54</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E128:40_2_3_1_2"/>
<font color="Maroon">c<sub>55</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E3:40_2_3_1_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="topalg_5.html#E10:40_2_3_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="borsuk_4.html#T48">BORSUK_4:48</a></i>;<br/><a NAME="E129:40_2_3_1_2"/><b>then </b><i><font color="Green">E142</font></i>: 
<font color="Maroon">c<sub>53</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tsep_1.html#T8">TSEP_1:8</a></i>;<br/><a NAME="E130:40_2_3_1_2"/>
<font color="Maroon">c<sub>56</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E97:40_2_3_1_2"><i><font color="Green">E124</font></i></a>, <a class="ref" href="borsuk_4.html#T48">BORSUK_4:48</a></i>;<br/><a NAME="E131:40_2_3_1_2"/><b>then </b><i><font color="Green">E143</font></i>: 
<font color="Maroon">c<sub>54</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tsep_1.html#T8">TSEP_1:8</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>57</sub></font> = <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="txt" href="topalg_5.html#E10:40_2_3_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="topalg_5.html#E11:40_2_3_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="topalg_5.html#E109:40_2_3_1_2"><i><font color="Green">E133</font></i></a>, <a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>58</sub></font> = <font color="Maroon">c<sub>41</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <b>by </b><i><a class="txt" href="topalg_5.html#E76:40_2_3_1_2"><i><font color="Green">E111</font></i></a>, <a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E134:40_2_3_1_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>59</sub></font> = <font color="Maroon">c<sub>20</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>57</sub></font> = 
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>58</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>59</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
<b>by </b><i><a class="ref" href="borsuk_3.html#T26">BORSUK_3:26</a></i>
<br/>.= 
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>41</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>59</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
<b>by </b><i><a class="ref" href="goboard9.html#T4">GOBOARD9:4</a></i>
<br/>.= 
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>41</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
<b>by </b><i><a class="ref" href="goboard9.html#T4">GOBOARD9:4</a></i>
;<br/><a NAME="E137:40_2_3_1_2"/><b>then </b><i><font color="Green">E144</font></i>: 
<font color="Maroon">c<sub>52</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E6:40_2_3_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>60</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>41</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>44</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <b>by </b><i><a class="txt" href="topalg_5.html#E93:40_2_3_1_2"><i><font color="Green">E120</font></i></a>, <a class="txt" href="topalg_5.html#E99:40_2_3_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="topalg_5.html#E100:40_2_3_1_2"><i><font color="Green">E127</font></i></a>, <a class="txt" href="topalg_5.html#E102:40_2_3_1_2"><i><font color="Green">E129</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>61</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>41</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>44</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <b>by </b><i><a class="txt" href="topalg_5.html#E99:40_2_3_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="topalg_5.html#E100:40_2_3_1_2"><i><font color="Green">E127</font></i></a>, <a class="txt" href="topalg_5.html#E102:40_2_3_1_2"><i><font color="Green">E129</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><a NAME="E140:40_2_3_1_2"/>
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>41</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>44</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_3.html#T26">BORSUK_3:26</a></i>;<br/><a NAME="E141:40_2_3_1_2"/><b>then </b>
<font color="Maroon">c<sub>61</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:40"><i><font color="Green">E38</font></i></a>, <a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/><a NAME="E142:40_2_3_1_2"/><b>then </b><i><font color="Green">E145</font></i>: 
<font color="Maroon">c<sub>60</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T9">TOPMETR:9</a></i>;<br/><a NAME="E143:40_2_3_1_2"/>
<font color="Maroon">c<sub>43</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E87:40_2_3_1_2"><i><font color="Green">E118</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><a NAME="E144:40_2_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>43</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>60</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E142:40_2_3_1_2"><i><font color="Green">E145</font></i></a>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/><a NAME="E145:40_2_3_1_2"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">c<sub>48</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T7">TOPMETR:7</a></i>;<br/><a NAME="E146:40_2_3_1_2"/>
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> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>41</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>41</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>44</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span> holds <br/><font color="Maroon">c<sub>52</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>48</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_3_1_2_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>62</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:40_2_3_1_2_7"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>41</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>41</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>44</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span>
 ;<br/>

<a NAME="E2:40_2_3_1_2_7"/><i><font color="Green">E148</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</sub></font></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"><font color="Maroon">c<sub>41</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>44</sub></font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T122">ZFMISC_1:122</a></i>;<br/>
<a NAME="E3:40_2_3_1_2_7"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40_2_3_1_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="topalg_5.html#E20:40_2_3_1_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="topalg_5.html#E25:40_2_3_1_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="topmetr2.html#T1">TOPMETR2:1</a></i>;<br/>
<a NAME="E4:40_2_3_1_2_7"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>41</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>44</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E1:40_2_3_1_2_7"><i><font color="Green">E147</font></i></a></i>;<br/>
<a NAME="E5:40_2_3_1_2_7"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>41</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>44</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>41</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E1:40_2_3_1_2_7"><i><font color="Green">E147</font></i></a></i>;<br/>
<a NAME="E6:40_2_3_1_2_7"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40_2_3_1_2_7"><i><font color="Green">E148</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_3_1_2_7"><i><font color="Green">E147</font></i></a>, <a class="txt" href="topalg_5.html#E102:40_2_3_1_2"><i><font color="Green">E129</font></i></a>, <a class="txt" href="topalg_5.html#E3:40_2_3_1_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="topalg_5.html#E20:40_2_3_1_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="topalg_5.html#E25:40_2_3_1_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="topmetr2.html#T1">TOPMETR2:1</a>, <a class="txt" href="topalg_5.html#E104:40_2_3_1_2"><i><font color="Green">E131</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>63</sub></font> being    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>41</sub></font>, <font color="Maroon">c<sub>64</sub></font> being    <a href="subset_1.html#M2">Element</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span><b> such that </b><br/><a NAME="E8:40_2_3_1_2_7"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">c<sub>62</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>63</sub></font>,<font color="Maroon">c<sub>64</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E9:40_2_3_1_2_7"/><i><font color="Green">E154</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E10:40_2_3_1_2_7"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>52</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E5:40_2_3_1_2_7"><i><font color="Green">E151</font></i></a>, <a class="txt" href="topalg_5.html#E104:40_2_3_1_2"><i><font color="Green">E131</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E11:40_2_3_1_2_7"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">c<sub>64</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40_2_3_1_2_7"><i><font color="Green">E149</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E12:40_2_3_1_2_7"/>
<font color="Maroon">c<sub>63</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>41</sub></font>
 
;<br/>
<a NAME="E13:40_2_3_1_2_7"/><b>then </b><i><font color="Green">E157</font></i>: 
<font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E8:40_2_3_1_2_7"><i><font color="Green">E153</font></i></a>, <a class="txt" href="topalg_5.html#E11:40_2_3_1_2_7"><i><font color="Green">E156</font></i></a>, <a class="txt" href="topalg_5.html#E76:40_2_3_1_2"><i><font color="Green">E111</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E14:40_2_3_1_2_7"/><b>then </b><i><font color="Green">E158</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E6:40_2_3_1_2_7"><i><font color="Green">E152</font></i></a>, <a class="txt" href="topalg_5.html#E19:40_2_3_1_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="topalg_5.html#E10:40_2_3_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="topalg_5.html#E11:40_2_3_1_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>62</sub></font> = 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>62</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E4:40_2_3_1_2_7"><i><font color="Green">E150</font></i></a>, <a class="txt" href="topalg_5.html#E102:40_2_3_1_2"><i><font color="Green">E129</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>62</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E13:40_2_3_1_2_7"><i><font color="Green">E157</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E7:40_2_3_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="topalg_5.html#E34:40_2_3_1_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="topalg_5.html#E9:40_2_3_1_2_7"><i><font color="Green">E154</font></i></a>, <a class="txt" href="topalg_5.html#E11:40_2_3_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="topalg_5.html#E10:40_2_3_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="topalg_5.html#E13:40_2_3_1_2_7"><i><font color="Green">E157</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E14:40_2_3_1_2_7"><i><font color="Green">E158</font></i></a>, <a class="txt" href="topalg_5.html#E77:40_2_3_1_2"><i><font color="Green">E112</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>47</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>52</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E5:40_2_3_1_2_7"><i><font color="Green">E151</font></i></a>, <a class="txt" href="topalg_5.html#E104:40_2_3_1_2"><i><font color="Green">E131</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>
<b>hence </b><font color="Maroon">c<sub>52</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>62</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E14:40_2_3_1_2_7"><i><font color="Green">E158</font></i></a>, <a class="txt" href="topalg_5.html#E77:40_2_3_1_2"><i><font color="Green">E112</font></i></a>, <a class="txt" href="topalg_5.html#E10:40_2_3_1_2_7"><i><font color="Green">E155</font></i></a>, <a class="txt" href="topalg_5.html#E84:40_2_3_1_2"><i><font color="Green">E117</font></i></a>, <a class="txt" href="topalg_5.html#E95:40_2_3_1_2"><i><font color="Green">E122</font></i></a>, <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>
<br/>.= 
<font color="Maroon">c<sub>48</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E4:40_2_3_1_2_7"><i><font color="Green">E150</font></i></a>, <a class="txt" href="topalg_5.html#E99:40_2_3_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="topalg_5.html#E102:40_2_3_1_2"><i><font color="Green">E129</font></i></a>, <a class="txt" href="topalg_5.html#E96:40_2_3_1_2"><i><font color="Green">E123</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><br/>


</div><b>end;</b></div><a NAME="E147:40_2_3_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>41</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>45</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>52</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>48</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i><a class="txt" href="topalg_5.html#E129:40_2_3_1_2"><i><font color="Green">E142</font></i></a>, <a class="txt" href="topalg_5.html#E131:40_2_3_1_2"><i><font color="Green">E143</font></i></a>, <a class="txt" href="topalg_5.html#E137:40_2_3_1_2"><i><font color="Green">E144</font></i></a>, <a class="txt" href="topalg_5.html#E145:40_2_3_1_2"><i><font color="Green">E146</font></i></a>, <a class="txt" href="topalg_5.html#E125:40_2_3_1_2"><i><font color="Green">E140</font></i></a>, <a class="txt" href="topalg_5.html#E126:40_2_3_1_2"><i><font color="Green">E141</font></i></a>, <a class="txt" href="topalg_5.html#E97:40_2_3_1_2"><i><font color="Green">E124</font></i></a>, <a class="txt" href="topalg_5.html#E10:40_2_3_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="topalg_5.html#E98:40_2_3_1_2"><i><font color="Green">E125</font></i></a>, <a class="txt" href="topalg_5.html#E123:40_2_3_1_2"><i><font color="Green">E139</font></i></a>, <a class="ref" href="topalg_3.html#T20">TOPALG_3:20</a></i>;<br/><b>hence </b><a NAME="E148:40_2_3_1_2"/>
<font color="Maroon">c<sub>51</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/><a NAME="E149:40_2_3_1_2"/><i><font color="Green">E147</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>45</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E11:40"><i><font color="Green">E47</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E150:40_2_3_1_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>51</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="toprealb.html#K12">CircleMap</a> 
 <b>by </b><i><a class="txt" href="topalg_5.html#E35"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><a NAME="E151:40_2_3_1_2"/><b>then </b><i><font color="Green">E148</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>51</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>51</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/><a NAME="E152:40_2_3_1_2"/><i><font color="Green">E149</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>51</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>45</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E108:40_2_3_1_2"><i><font color="Green">E132</font></i></a>, <a class="txt" href="topalg_5.html#E111:40_2_3_1_2"><i><font color="Green">E134</font></i></a>, <a class="txt" href="topalg_5.html#E149:40_2_3_1_2"><i><font color="Green">E147</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E153:40_2_3_1_2"/>
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="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>51</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>51</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> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_3_1_2_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>62</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:40_2_3_1_2_8"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">c<sub>62</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="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>51</sub></font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_3_1_2_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_2_3_1_2_8_1"/>
( <font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>48</sub></font> or  not <font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>48</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_3_1_2_8_1_1"><b>suppose </b></a><a NAME="E1:40_2_3_1_2_8_1_1"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>48</sub></font>
 ;<br/><div class="add"><a NAME="E2:40_2_3_1_2_8_1_1"/><b>then </b><i><font color="Green">E152</font></i>: 
<font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E101:40_2_3_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="topalg_5.html#E91:40_2_3_1_2"><i><font color="Green">E119</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E3:40_2_3_1_2_8_1_1"/><i><font color="Green">E153</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</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>8</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E4:40_2_3_1_2_8_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E11:40"><i><font color="Green">E47</font></i></a>, <a class="txt" href="topalg_5.html#E2:40_2_3_1_2_8_1_1"><i><font color="Green">E152</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E5:40_2_3_1_2_8_1_1"/><b>then </b><i><font color="Green">E154</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>44</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E82:40_2_3_1_2"><i><font color="Green">E115</font></i></a></i>;<br/><a NAME="E6:40_2_3_1_2_8_1_1"/><b>then </b>
<font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E2:40_2_3_1_2_8_1_1"><i><font color="Green">E152</font></i></a>, <a class="txt" href="topalg_5.html#E3:40_2_3_1_2_8_1_1"><i><font color="Green">E153</font></i></a>, <a class="txt" href="topalg_5.html#E11:40"><i><font color="Green">E47</font></i></a>, <a class="txt" href="topalg_5.html#E93:40_2_3_1_2"><i><font color="Green">E120</font></i></a>, <a class="txt" href="topalg_5.html#E14:40_2_3_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="topalg_5.html#E96:40_2_3_1_2"><i><font color="Green">E123</font></i></a>, <a class="txt" href="topalg_5.html#E15:40_2_3_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E7:40_2_3_1_2_8_1_1"/><b>then </b><i><font color="Green">E155</font></i>: 
<font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T182">RELAT_1:182</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>51</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>62</sub></font> = 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>51</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E1:40_2_3_1_2_8"><i><font color="Green">E150</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>48</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E1:40_2_3_1_2_8_1_1"><i><font color="Green">E151</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>43</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p4"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>62</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E101:40_2_3_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_3_1_2_8_1_1"><i><font color="Green">E151</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>43</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>62</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E99:40_2_3_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="topalg_5.html#E101:40_2_3_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_3_1_2_8_1_1"><i><font color="Green">E151</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>62</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E91:40_2_3_1_2"><i><font color="Green">E119</font></i></a>, <a class="txt" href="topalg_5.html#E99:40_2_3_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="topalg_5.html#E101:40_2_3_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_3_1_2_8_1_1"><i><font color="Green">E151</font></i></a>, <a class="txt" href="topalg_5.html#E96:40_2_3_1_2"><i><font color="Green">E123</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>62</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E7:40_2_3_1_2_8_1_1"><i><font color="Green">E155</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>62</sub></font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>62</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E91:40_2_3_1_2"><i><font color="Green">E119</font></i></a>, <a class="txt" href="topalg_5.html#E99:40_2_3_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="topalg_5.html#E101:40_2_3_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_3_1_2_8_1_1"><i><font color="Green">E151</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>62</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E95:40_2_3_1_2"><i><font color="Green">E122</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_2_3_1_2_8_1_1"><i><font color="Green">E154</font></i></a>, <a class="txt" href="topalg_5.html#E14:40_2_3_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="topalg_5.html#E94:40_2_3_1_2"><i><font color="Green">E121</font></i></a>, <a class="txt" href="topalg_5.html#E15:40_2_3_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="topalg_3.html#T2">TOPALG_3:2</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_3_1_2_8_1_2"><b>suppose </b></a><a NAME="E1:40_2_3_1_2_8_1_2"/><i><font color="Green">E151</font></i>: 
not <font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>48</sub></font>
 ;<br/><div class="add"><a NAME="E2:40_2_3_1_2_8_1_2"/><b>then </b><i><font color="Green">E152</font></i>: 
<font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E111:40_2_3_1_2"><i><font color="Green">E134</font></i></a>, <a class="txt" href="topalg_5.html#E101:40_2_3_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="topalg_5.html#E99:40_2_3_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_3_1_2_8"><i><font color="Green">E150</font></i></a>, <a class="txt" href="topalg_5.html#E151:40_2_3_1_2"><i><font color="Green">E148</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>51</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>62</sub></font> = 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>51</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E1:40_2_3_1_2_8"><i><font color="Green">E150</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E1:40_2_3_1_2_8_1_2"><i><font color="Green">E151</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40_2_3_1_2_8_1_2"><i><font color="Green">E152</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>62</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E19:40_2_3_1_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="topalg_5.html#E10:40_2_3_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="topalg_5.html#E11:40_2_3_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="topalg_5.html#E2:40_2_3_1_2_8_1_2"><i><font color="Green">E152</font></i></a>, <a class="txt" href="topalg_5.html#E109:40_2_3_1_2"><i><font color="Green">E133</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>62</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40_2_3_1_2_8_1_2"><i><font color="Green">E152</font></i></a>, <a class="txt" href="topalg_5.html#E7:40_2_3_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="topalg_5.html#E109:40_2_3_1_2"><i><font color="Green">E133</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E154:40_2_3_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>45</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>51</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E151:40_2_3_1_2"><i><font color="Green">E148</font></i></a>, <a class="txt" href="topalg_5.html#E152:40_2_3_1_2"><i><font color="Green">E149</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E155:40_2_3_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E2:40_2_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="topalg_5.html#E14:40_2"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E156:40_2_3_1_2"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>45</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E115:40_2_3_1_2"><i><font color="Green">E135</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E157:40_2_3_1_2"/><b>then </b><i><font color="Green">E150</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>45</sub></font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E158:40_2_3_1_2"/><i><font color="Green">E151</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>51</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><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>8</sub></font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>51</sub></font></span>)</span> <a href="subset_1.html#K8">/\</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><b>then </b><i><font color="Green">E152</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>51</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><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>8</sub></font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> = 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="txt" href="topalg_5.html#E108:40_2_3_1_2"><i><font color="Green">E132</font></i></a>, <a class="txt" href="topalg_5.html#E111:40_2_3_1_2"><i><font color="Green">E134</font></i></a>, <a class="ref" href="zfmisc_1.html#T123">ZFMISC_1:123</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>45</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="txt" href="topalg_5.html#E157:40_2_3_1_2"><i><font color="Green">E150</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><i><font color="Green">E153</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>41</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="txt" href="topalg_5.html#E6:40"><i><font color="Green">E43</font></i></a>, <a class="ref" href="zfmisc_1.html#T123">ZFMISC_1:123</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="txt" href="topalg_5.html#E3"><i><font color="Green">Lemma3</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><a NAME="E161:40_2_3_1_2"/>
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="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>51</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><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>8</sub></font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>51</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><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>8</sub></font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</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>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_3_1_2_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>62</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:40_2_3_1_2_11"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>51</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><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>8</sub></font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 ;<br/>

<a NAME="E2:40_2_3_1_2_11"/><b>then </b><i><font color="Green">E155</font></i>: 
<font color="Maroon">c<sub>62</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E158:40_2_3_1_2"><i><font color="Green">E151</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>63</sub></font>, <font color="Maroon">c<sub>64</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:40_2_3_1_2_11"/>
<font color="Maroon">c<sub>63</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E5:40_2_3_1_2_11"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">c<sub>64</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E6:40_2_3_1_2_11"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">c<sub>62</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>63</sub></font>,<font color="Maroon">c<sub>64</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/>
<a NAME="E7:40_2_3_1_2_11"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">c<sub>64</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="topalg_5.html#E5:40_2_3_1_2_11"><i><font color="Green">E156</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E8:40_2_3_1_2_11"/>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E20:40_2_3_1_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="topalg_5.html#E3:40_2_3_1_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E9:40_2_3_1_2_11"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E10:40_2_3_1_2_11"/><b>then </b><i><font color="Green">E159</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E11:40_2_3_1_2_11"/><b>then </b><i><font color="Green">E160</font></i>: 
<font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E1:40_2_3_1_2_11"><i><font color="Green">E154</font></i></a>, <a class="txt" href="topalg_5.html#E159:40_2_3_1_2"><i><font color="Green">E152</font></i></a></i>;<br/>
<a NAME="E12:40_2_3_1_2_11"/><i><font color="Green">E161</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E13:40_2_3_1_2_11"/><b>then </b><i><font color="Green">E162</font></i>: 
<font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E11:40_2_3_1_2_11"><i><font color="Green">E160</font></i></a></i>;<br/>
<a NAME="E14:40_2_3_1_2_11"/><i><font color="Green">E163</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E76:40_2_3_1_2"><i><font color="Green">E111</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_3_1_2_11_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_2_3_1_2_11_1"/>
(  not <font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>48</sub></font> or <font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>48</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_3_1_2_11_1_1"><b>suppose </b></a><a NAME="E1:40_2_3_1_2_11_1_1"/><i><font color="Green">E164</font></i>: 
not <font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>48</sub></font>
 ;<br/><div class="add"><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>51</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><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>8</sub></font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font> = 
<font color="Maroon">c<sub>51</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40_2_3_1_2_11"><i><font color="Green">E155</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E1:40_2_3_1_2_11_1_1"><i><font color="Green">E164</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">c<sub>21</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E10:40_2_3_1_2_11"><i><font color="Green">E159</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_3_1_2_11"><i><font color="Green">E154</font></i></a>, <a class="txt" href="topalg_5.html#E159:40_2_3_1_2"><i><font color="Green">E152</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E8:40_2_3_1_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="topalg_5.html#E2:40_2_3_1_2_11"><i><font color="Green">E155</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E14:40_2_3_1_2_11"><i><font color="Green">E163</font></i></a>, <a class="txt" href="topalg_5.html#E13:40_2_3_1_2_11"><i><font color="Green">E162</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E12:40_2_3_1_2_11"><i><font color="Green">E161</font></i></a>, <a class="txt" href="topalg_5.html#E11:40_2_3_1_2_11"><i><font color="Green">E160</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_3_1_2_11_1_2"><b>suppose </b></a><a NAME="E1:40_2_3_1_2_11_1_2"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>48</sub></font>
 ;<br/><div class="add"><a NAME="E2:40_2_3_1_2_11_1_2"/><b>then </b>
<font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E101:40_2_3_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="topalg_5.html#E91:40_2_3_1_2"><i><font color="Green">E119</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>65</sub></font>, <font color="Maroon">c<sub>66</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:40_2_3_1_2_11_1_2"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">c<sub>65</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>41</sub></font>
 <b>and </b><br/><a NAME="E5:40_2_3_1_2_11_1_2"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">c<sub>66</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>44</sub></font>
 <b>and </b><br/><a NAME="E6:40_2_3_1_2_11_1_2"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">c<sub>62</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>65</sub></font>,<font color="Maroon">c<sub>66</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>set </b><font color="Maroon">c<sub>67</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font>;<br/><i><font color="Green">E168</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font> = 
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E11:40_2_3_1_2_11"><i><font color="Green">E160</font></i></a>, <a class="txt" href="topalg_5.html#E12:40_2_3_1_2_11"><i><font color="Green">E161</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E14:40_2_3_1_2_11"><i><font color="Green">E163</font></i></a>, <a class="txt" href="topalg_5.html#E13:40_2_3_1_2_11"><i><font color="Green">E162</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>21</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E8:40_2_3_1_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="topalg_5.html#E2:40_2_3_1_2_11"><i><font color="Green">E155</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><a NAME="E8:40_2_3_1_2_11_1_2"/>
<font color="Maroon">c<sub>64</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>66</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E6:40_2_3_1_2_11"><i><font color="Green">E157</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_2_3_1_2_11_1_2"><i><font color="Green">E167</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E9:40_2_3_1_2_11_1_2"/><b>then </b><i><font color="Green">E169</font></i>: 
<font color="Maroon">c<sub>64</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E20:40_2_3_1_2"><i><font color="Green">E79</font></i></a>, <a class="txt" href="topalg_5.html#E7:40_2_3_1_2_11"><i><font color="Green">E158</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_2_3_1_2_11_1_2"><i><font color="Green">E166</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E10:40_2_3_1_2_11_1_2"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">c<sub>63</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>65</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E6:40_2_3_1_2_11"><i><font color="Green">E157</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_2_3_1_2_11_1_2"><i><font color="Green">E167</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E11:40_2_3_1_2_11_1_2"/><b>then </b><i><font color="Green">E171</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>63</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E4:40_2_3_1_2_11_1_2"><i><font color="Green">E165</font></i></a>, <a class="txt" href="topalg_5.html#E21:40_2_3_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E12:40_2_3_1_2_11_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>63</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E4:40_2_3_1_2_11_1_2"><i><font color="Green">E165</font></i></a>, <a class="txt" href="topalg_5.html#E10:40_2_3_1_2_11_1_2"><i><font color="Green">E170</font></i></a>, <a class="txt" href="topalg_5.html#E35:40_2_3_1_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E13:40_2_3_1_2_11_1_2"/><b>then </b><i><font color="Green">E172</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E19:40_2_3_1_2"><i><font color="Green">E78</font></i></a>, <a class="txt" href="topalg_5.html#E7:40_2_3_1_2_11_1_2"><i><font color="Green">E168</font></i></a>, <a class="txt" href="topalg_5.html#E34:40_2_3_1_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="topalg_5.html#E11:40_2_3_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="topalg_5.html#E10:40_2_3_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="topalg_5.html#E9:40_2_3_1_2_11_1_2"><i><font color="Green">E169</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_2_3_1_2_11"><i><font color="Green">E157</font></i></a>, <a class="txt" href="topalg_5.html#E109:40_2_3_1_2"><i><font color="Green">E133</font></i></a>, <a class="txt" href="topalg_5.html#E11:40_2_3_1_2_11_1_2"><i><font color="Green">E171</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><i><font color="Green">E173</font></i>: <font color="Maroon">c<sub>47</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font></span>)</span> = 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E13:40_2_3_1_2_11_1_2"><i><font color="Green">E172</font></i></a>, <a class="txt" href="topalg_5.html#E77:40_2_3_1_2"><i><font color="Green">E112</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E11:40_2_3_1_2_11"><i><font color="Green">E160</font></i></a>, <a class="txt" href="topalg_5.html#E12:40_2_3_1_2_11"><i><font color="Green">E161</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>62</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E6:40"><i><font color="Green">E43</font></i></a>, <a class="txt" href="topalg_5.html#E2:40_2_3_1_2_11"><i><font color="Green">E155</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>62</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40"><i><font color="Green">E40</font></i></a>, <a class="txt" href="topalg_5.html#E2:40_2_3_1_2_11"><i><font color="Green">E155</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>51</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><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>8</sub></font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font> = 
<font color="Maroon">c<sub>51</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40_2_3_1_2_11"><i><font color="Green">E155</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>48</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E1:40_2_3_1_2_11_1_2"><i><font color="Green">E164</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,<font color="Maroon">c<sub>44</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>62</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E101:40_2_3_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_3_1_2_11_1_2"><i><font color="Green">E164</font></i></a>, <a class="txt" href="topalg_5.html#E96:40_2_3_1_2"><i><font color="Green">E123</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>47</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>62</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E99:40_2_3_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="topalg_5.html#E101:40_2_3_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_3_1_2_11_1_2"><i><font color="Green">E164</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>62</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E84:40_2_3_1_2"><i><font color="Green">E117</font></i></a>, <a class="txt" href="topalg_5.html#E13:40_2_3_1_2_11_1_2"><i><font color="Green">E172</font></i></a>, <a class="txt" href="topalg_5.html#E77:40_2_3_1_2"><i><font color="Green">E112</font></i></a>, <a class="txt" href="topalg_5.html#E14:40_2_3_1_2_11_1_2"><i><font color="Green">E173</font></i></a>, <a class="txt" href="topalg_5.html#E95:40_2_3_1_2"><i><font color="Green">E122</font></i></a>, <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E162:40_2_3_1_2"/>
<font color="Maroon">c<sub>51</sub></font> <a href="partfun1.html#K2">|</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>41</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E159:40_2_3_1_2"><i><font color="Green">E152</font></i></a>, <a class="txt" href="topalg_5.html#E160:40_2_3_1_2"><i><font color="Green">E153</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E18:40_2"/><i><font color="Green">E62</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>2</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="topalg_5.html#E16:40_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="topalg_5.html#E17:40_2"><i><font color="Green">E61</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon">c<sub>18</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>19</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">c<sub>20</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E20:40_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>and </b><br/><a NAME="E21:40_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font>
 <b>and </b><br/><a NAME="E22:40_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E23:40_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>20</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E24:40_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>20</sub></font>
 <b>and </b><br/><a NAME="E25:40_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="partfun1.html#K2">|</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E18:40_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="topalg_5.html#E13:40_2"><i><font color="Green">E57</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>21</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>13</sub></font>
;<br/>

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

<b>take </b>
<font color="Maroon">c<sub>18</sub></font>
;<br/>

<a NAME="E27:40_2"/><i><font color="Green">E69</font></i>: 
<a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="txt" href="topalg_5.html#E20:40_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="topalg_5.html#E7"><i><font color="Green">Lemma6</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="tsep_1.html#T3">TSEP_1:3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>20</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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  ;<br/>
<b>take </b>
<font color="Maroon">c<sub>22</sub></font>
;<br/>

<b>thus </b><a NAME="E29:40_2"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="borsuk_1.html#K7">]</a></span> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="txt" href="topalg_5.html#E2:40_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="topalg_5.html#E21:40_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="topalg_5.html#E27:40_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="topalg_5.html#E20:40_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="topalg_5.html#E23:40_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="topalg_5.html#E24:40_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="topalg_5.html#E25:40_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>23</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E30:40_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>23</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E31:40_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>23</sub></font>
 <b>and </b><br/><a NAME="E32:40_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="partfun1.html#K2">|</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<a NAME="E33:40_2"/><i><font color="Green">E73</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>23</sub></font> <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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E34:40_2"/><i><font color="Green">E74</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </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 <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </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="E35:40_2"/><i><font color="Green">E75</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E36:40_2"/><i><font color="Green">E76</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>22</sub></font> <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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> ) );<br/>
<a NAME="E37:40_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<b>by </b><i><a class="ref" href="finseq_3.html#T26">FINSEQ_3:26</a></i>;<br/>
<a NAME="E38:40_2"/><i><font color="Green">E78</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>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>3</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="40_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>24</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:40_2_4"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>24</sub></font>]
 <b>and </b><br/><a NAME="E2:40_2_4"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_2_4_1"/>
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font> )
 </span><b>by </b><i><a class="txt" href="topalg_5.html#E2:40_2_4"><i><font color="Green">E80</font></i></a>, <a class="ref" href="topreala.html#T23">TOPREALA:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_4_1_1"><b>suppose </b></a><a NAME="E1:40_2_4_1_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>25</sub></font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>;<br/><a NAME="E2:40_2_4_1_1"/><i><font color="Green">E82</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E4:40_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_4_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="rcomp_1.html#T14">RCOMP_1:14</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>26</sub></font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E3"><i><font color="Green">Lemma3</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_4_1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="rcomp_1.html#T14">RCOMP_1:14</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>26</sub></font>
;<br/><b>thus </b><a NAME="E4:40_2_4_1_1"/>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/><a NAME="E5:40_2_4_1_1"/><i><font color="Green">E83</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/><a NAME="E6:40_2_4_1_1"/><b>then </b><i><font color="Green">E84</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E36:40_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="topalg_5.html#E34:40_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="topalg_5.html#E35:40_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E7:40_2_4_1_1"/><i><font color="Green">E85</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E33:40_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topalg_5.html#E34:40_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="topalg_5.html#E35:40_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_2_4_1_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E8:40_2_4_1_1"/>
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="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</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>23</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_4_1_1_1"><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> ;<br/>

<b>assume </b><a NAME="E1:40_2_4_1_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 ;<br/>

<a NAME="E2:40_2_4_1_1_1"/><i><font color="Green">E87</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</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>8</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>27</sub></font> = 
<font color="Maroon">c<sub>22</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>27</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E1:40_2_4_1_1_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_2_4_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><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>8</sub></font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>27</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40_2_4_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_4_1_1_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_2_4_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="topalg_5.html#E2:40_2_4_1_1_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>23</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>27</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40_2_4_1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="topalg_5.html#E25:40_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="topalg_5.html#E32:40_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="topalg_5.html#E2:40_2_4_1_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_4_1_1_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_2_4_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>27</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E6:40_2_4_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_4_1_1_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E9:40_2_4_1_1"/>
<font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E6:40_2_4_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="topalg_5.html#E7:40_2_4_1_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_4_1_2"><b>suppose </b></a><a NAME="E1:40_2_4_1_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>25</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E3:40_2_4_1_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>and </b><br/><a NAME="E4:40_2_4_1_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:40_2_4"><i><font color="Green">E79</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>26</sub></font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>;<br/><a NAME="E5:40_2_4_1_2"/><i><font color="Green">E84</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:40_2_4_1_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="topalg_5.html#E14:40_2"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E6:40_2_4_1_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="topalg_5.html#E2:40_2_4"><i><font color="Green">E80</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_4_1_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="topalg_5.html#E14:40_2"><i><font color="Green">E58</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>27</sub></font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E5:40_2_4_1_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="topalg_5.html#E15:40_2"><i><font color="Green">E59</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>25</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>27</sub></font>;<br/><a NAME="E8:40_2_4_1_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E2:40_2_4"><i><font color="Green">E80</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_4_1_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="topalg_5.html#E14:40_2"><i><font color="Green">E58</font></i></a></i>;<br/><b>hence </b><a NAME="E9:40_2_4_1_2"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E3:40_2_4_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_2_4_1_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="heine.html#T2">HEINE:2</a></i>;<br/><a NAME="E10:40_2_4_1_2"/><i><font color="Green">E87</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>28</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/><a NAME="E11:40_2_4_1_2"/><b>then </b><i><font color="Green">E88</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>28</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>28</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E36:40_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="topalg_5.html#E34:40_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="topalg_5.html#E35:40_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E12:40_2_4_1_2"/><i><font color="Green">E89</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>28</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>28</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E33:40_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topalg_5.html#E34:40_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="topalg_5.html#E35:40_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="topalg_5.html#E10:40_2_4_1_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E13:40_2_4_1_2"/>
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="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>28</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>28</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</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>23</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>28</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_4_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>29</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:40_2_4_1_2_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>28</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>30</sub></font>, <font color="Maroon">c<sub>31</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_2_4_1_2_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font>
 <b>and </b><br/><a NAME="E4:40_2_4_1_2_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>28</sub></font>
 <b>and </b><br/><a NAME="E5:40_2_4_1_2_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>29</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>30</sub></font>,<font color="Maroon">c<sub>31</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E11:40_2_4_1_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_4_1_2_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>32</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>30</sub></font></span><a href="tarski.html#K1">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font> <b>by </b><i><a class="txt" href="topalg_5.html#E3:40_2_4_1_2_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E7:40_2_4_1_2_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40_2_4_1_2_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E8:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E95</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>33</sub></font> = <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="txt" href="topalg_5.html#E7:40_2_4_1_2_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="topalg_5.html#E34:40_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="topalg_5.html#E35:40_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>34</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span><b> such that </b><br/><a NAME="E11:40_2_4_1_2_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E12:40_2_4_1_2_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>34</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E10:40_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="topalg_5.html#E2:40_2_4"><i><font color="Green">E80</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_4_1_2"><i><font color="Green">E81</font></i></a></i>;<br/>
<a NAME="E13:40_2_4_1_2_1"/><i><font color="Green">E98</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>34</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>35</sub></font> being  <a href="taxonom2.html#V4">mutually-disjoint</a> <a href="tops_2.html#V1">open</a> <a href="struct_0.html#NM3">Subset-Family</a> of <a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E15:40_2_4_1_2_1"/><i><font color="Green">E99</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>34</sub></font>
 <b>and </b><br/><a NAME="E16:40_2_4_1_2_1"/><i><font color="Green">E100</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>35</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i><a class="txt" href="topalg_5.html#E11:40_2_4_1_2_1"><i><font color="Green">E96</font></i></a>, <a class="txt" href="topalg_5.html#E10:40"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E17:40_2_4_1_2_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40_2_4_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_2_4_1_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<a NAME="E18:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E101</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40_2_4_1_2_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E19:40_2_4_1_2_1"/><i><font color="Green">E102</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<b>then </b><i><font color="Green">E103</font></i>: <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="topalg_5.html#E18:40_2_4_1_2_1"><i><font color="Green">E101</font></i></a>, <a class="txt" href="topalg_5.html#E20:40_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="topalg_5.html#E24:40_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E19:40_2_4_1_2_1"><i><font color="Green">E102</font></i></a>, <a class="txt" href="topalg_5.html#E36:40_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="topalg_5.html#E34:40_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="topalg_5.html#E35:40_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="topalg_5.html#E18:40_2_4_1_2_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<a NAME="E21:40_2_4_1_2_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>27</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E8:40_2_4_1_2"><i><font color="Green">E86</font></i></a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<a NAME="E22:40_2_4_1_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40_2_4_1_2_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="topalg_5.html#E22:40_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E23:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E24:40_2_4_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>34</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E12:40_2_4_1_2_1"><i><font color="Green">E97</font></i></a>, <a class="txt" href="topalg_5.html#E20:40_2_4_1_2_1"><i><font color="Green">E103</font></i></a>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>36</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E26:40_2_4_1_2_1"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>36</sub></font>
 <b>and </b><br/><a NAME="E27:40_2_4_1_2_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>35</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E15:40_2_4_1_2_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>37</sub></font> = <font color="Maroon">c<sub>36</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E26:40_2_4_1_2_1"><i><font color="Green">E106</font></i></a>, <a class="txt" href="topalg_5.html#E27:40_2_4_1_2_1"><i><font color="Green">E107</font></i></a></i>;<br/>
<a NAME="E29:40_2_4_1_2_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40_2_4"><i><font color="Green">E80</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_4_1_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="topalg_5.html#E14:40_2"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E30:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font> is   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<a NAME="E31:40_2_4_1_2_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40_2_4"><i><font color="Green">E80</font></i></a>, <a class="txt" href="topalg_5.html#E14:40_2"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E32:40_2_4_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="txt" href="topalg_5.html#E6:40_2_4_1_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<a NAME="E33:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">c<sub>27</sub></font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E30:40_2_4_1_2_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="topalg_5.html#E8:40_2_4_1_2"><i><font color="Green">E86</font></i></a>, <a class="ref" href="borsuk_4.html#T49">BORSUK_4:49</a></i>;<br/>
<a NAME="E34:40_2_4_1_2_1"/>
<font color="Maroon">c<sub>32</sub></font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40_2_4_1_2_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="connsp_1.html#T29">CONNSP_1:29</a></i>;<br/>
<a NAME="E35:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E110</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E33:40_2_4_1_2_1"><i><font color="Green">E109</font></i></a>, <a class="ref" href="topalg_3.html#T17">TOPALG_3:17</a></i>;<br/>
<a NAME="E36:40_2_4_1_2_1"/>
 <a href="topmetr.html#K5">I[01]</a>  is    <a href="pre_topc.html#M1">SubSpace</a> of  <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="tsep_1.html#T2">TSEP_1:2</a></i>;<br/>
<a NAME="E37:40_2_4_1_2_1"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> is    <a href="pre_topc.html#M1">SubSpace</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_3.html#T25">BORSUK_3:25</a></i>;<br/>
<a NAME="E38:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>33</sub></font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E35:40_2_4_1_2_1"><i><font color="Green">E110</font></i></a>, <a class="ref" href="connsp_1.html#T24">CONNSP_1:24</a></i>;<br/>
<a NAME="E39:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>33</sub></font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E23:40_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="topalg_5.html#E27:40_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="tops_2.html#T75">TOPS_2:75</a></i>;<br/>
<a NAME="E40:40_2_4_1_2_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>32</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E41:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E114</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>33</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E21:40_2_4_1_2_1"><i><font color="Green">E104</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E42:40_2_4_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>33</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E43:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">c<sub>37</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>22</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>33</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E26:40_2_4_1_2_1"><i><font color="Green">E106</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E44:40_2_4_1_2_1"/>
<font color="Maroon">c<sub>35</sub></font> <a href="compts_1.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>22</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>33</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_4_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>38</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D1">COMPTS_1:def 1</a>,<a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:40_2_4_1_2_1_2"/>
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>33</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>39</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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E3:40_2_4_1_2_1_2"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">c<sub>39</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>33</sub></font>
 <b>and </b><br/><a NAME="E4:40_2_4_1_2_1_2"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>39</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>38</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<i><font color="Green">E118</font></i>: <a href="toprealb.html#K12">CircleMap</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>39</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>39</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>39</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E34:40_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="topalg_5.html#E35:40_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="topalg_5.html#E20:40_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="topalg_5.html#E24:40_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>
<a NAME="E6:40_2_4_1_2_1_2"/>
<font color="Maroon">c<sub>32</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E22:40_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="topalg_5.html#E3:40_2_4_1_2_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E7:40_2_4_1_2_1_2"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E8:40_2_4_1_2_1_2"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">c<sub>39</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40_2_4_1_2_1_2"><i><font color="Green">E116</font></i></a></i>;<br/>
<a NAME="E9:40_2_4_1_2_1_2"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</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>8</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E10:40_2_4_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>39</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E11:40"><i><font color="Green">E47</font></i></a>, <a class="txt" href="topalg_5.html#E8:40_2_4_1_2_1_2"><i><font color="Green">E119</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E11:40_2_4_1_2_1_2"/>
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>35</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E12:40_2_4_1_2_1"><i><font color="Green">E97</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_2_4_1_2_1_2"><i><font color="Green">E117</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_2_4_1_2_1_2"><i><font color="Green">E118</font></i></a>, <a class="txt" href="topalg_5.html#E15:40_2_4_1_2_1"><i><font color="Green">E99</font></i></a>, <a class="txt" href="topalg_5.html#E35"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E45:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>33</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>37</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E27:40_2_4_1_2_1"><i><font color="Green">E107</font></i></a>, <a class="txt" href="topalg_5.html#E39:40_2_4_1_2_1"><i><font color="Green">E112</font></i></a>, <a class="txt" href="topalg_5.html#E43:40_2_4_1_2_1"><i><font color="Green">E115</font></i></a>, <a class="ref" href="topalg_3.html#T13">TOPALG_3:13</a></i>;<br/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="topalg_5.html#E31:40_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="topalg_5.html#E19:40_2_4_1_2_1"><i><font color="Green">E102</font></i></a>, <a class="txt" href="topalg_5.html#E18:40_2_4_1_2_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E33:40_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topalg_5.html#E19:40_2_4_1_2_1"><i><font color="Green">E102</font></i></a>, <a class="txt" href="topalg_5.html#E34:40_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="topalg_5.html#E35:40_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="topalg_5.html#E18:40_2_4_1_2_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<a NAME="E47:40_2_4_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>23</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>34</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E23:40_2_4_1_2_1"><i><font color="Green">E105</font></i></a>, <a class="txt" href="topalg_5.html#E12:40_2_4_1_2_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>38</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E49:40_2_4_1_2_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>38</sub></font>
 <b>and </b><br/><a NAME="E50:40_2_4_1_2_1"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>35</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E15:40_2_4_1_2_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>39</sub></font> = <font color="Maroon">c<sub>38</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E49:40_2_4_1_2_1"><i><font color="Green">E117</font></i></a>, <a class="txt" href="topalg_5.html#E50:40_2_4_1_2_1"><i><font color="Green">E118</font></i></a></i>;<br/>
<a NAME="E52:40_2_4_1_2_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>33</sub></font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E38:40_2_4_1_2_1"><i><font color="Green">E111</font></i></a>, <a class="txt" href="topalg_5.html#E30:40_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="tops_2.html#T75">TOPS_2:75</a></i>;<br/>
<a NAME="E53:40_2_4_1_2_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>33</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E41:40_2_4_1_2_1"><i><font color="Green">E114</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E54:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">c<sub>39</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>23</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>33</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E49:40_2_4_1_2_1"><i><font color="Green">E117</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E55:40_2_4_1_2_1"/>
<font color="Maroon">c<sub>35</sub></font> <a href="compts_1.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>23</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>33</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_4_1_2_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>40</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D1">COMPTS_1:def 1</a>,<a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:40_2_4_1_2_1_4"/>
<font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>33</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>41</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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E3:40_2_4_1_2_1_4"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>33</sub></font>
 <b>and </b><br/><a NAME="E4:40_2_4_1_2_1_4"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>40</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<i><font color="Green">E123</font></i>: <a href="toprealb.html#K12">CircleMap</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>41</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>41</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>41</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E31:40_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="topalg_5.html#E34:40_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="topalg_5.html#E35:40_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>
<a NAME="E6:40_2_4_1_2_1_4"/>
<font color="Maroon">c<sub>32</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E22:40_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="topalg_5.html#E3:40_2_4_1_2_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E7:40_2_4_1_2_1_4"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E8:40_2_4_1_2_1_4"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40_2_4_1_2_1_4"><i><font color="Green">E121</font></i></a></i>;<br/>
<a NAME="E9:40_2_4_1_2_1_4"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</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>8</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E10:40_2_4_1_2_1_4"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E11:40"><i><font color="Green">E47</font></i></a>, <a class="txt" href="topalg_5.html#E8:40_2_4_1_2_1_4"><i><font color="Green">E124</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E11:40_2_4_1_2_1_4"/>
<font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>35</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E12:40_2_4_1_2_1"><i><font color="Green">E97</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_2_4_1_2_1_4"><i><font color="Green">E122</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_2_4_1_2_1_4"><i><font color="Green">E123</font></i></a>, <a class="txt" href="topalg_5.html#E15:40_2_4_1_2_1"><i><font color="Green">E99</font></i></a>, <a class="txt" href="topalg_5.html#E35"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E56:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>33</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>39</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E50:40_2_4_1_2_1"><i><font color="Green">E118</font></i></a>, <a class="txt" href="topalg_5.html#E52:40_2_4_1_2_1"><i><font color="Green">E119</font></i></a>, <a class="txt" href="topalg_5.html#E54:40_2_4_1_2_1"><i><font color="Green">E120</font></i></a>, <a class="ref" href="topalg_3.html#T13">TOPALG_3:13</a></i>;<br/>
<a NAME="E57:40_2_4_1_2_1"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>33</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E33:40_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topalg_5.html#E41:40_2_4_1_2_1"><i><font color="Green">E114</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<font color="Maroon">c<sub>22</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="topalg_5.html#E18:40_2_4_1_2_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>23</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="topalg_5.html#E18:40_2_4_1_2_1"><i><font color="Green">E101</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_2_4_1_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>
<a NAME="E59:40_2_4_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>37</sub></font> <a href="subset_1.html#R2">meets</a> <font color="Maroon">c<sub>39</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E26:40_2_4_1_2_1"><i><font color="Green">E106</font></i></a>, <a class="txt" href="topalg_5.html#E56:40_2_4_1_2_1"><i><font color="Green">E121</font></i></a>, <a class="txt" href="topalg_5.html#E57:40_2_4_1_2_1"><i><font color="Green">E122</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E60:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>39</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E27:40_2_4_1_2_1"><i><font color="Green">E107</font></i></a>, <a class="txt" href="topalg_5.html#E50:40_2_4_1_2_1"><i><font color="Green">E118</font></i></a>, <a class="ref" href="taxonom2.html#D5">TAXONOM2:def 5</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>40</sub></font> = <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>37</sub></font>;<br/>
<a NAME="E61:40_2_4_1_2_1"/><i><font color="Green">E124</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>37</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>37</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E62:40_2_4_1_2_1"/><i><font color="Green">E125</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>37</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>37</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E35"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E63:40_2_4_1_2_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>37</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>34</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_4_1_2_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>41</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="E1:40_2_4_1_2_1_6"/>
<font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>37</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>42</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_2_4_1_2_1_6"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">c<sub>42</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="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>37</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:40_2_4_1_2_1_6"/><i><font color="Green">E127</font></i>: 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>37</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>41</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:40_2_4_1_2_1_6"/>
<font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>35</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40_2_4_1_2_1_6"><i><font color="Green">E126</font></i></a>, <a class="txt" href="topalg_5.html#E62:40_2_4_1_2_1"><i><font color="Green">E125</font></i></a>, <a class="txt" href="topalg_5.html#E27:40_2_4_1_2_1"><i><font color="Green">E107</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E6:40_2_4_1_2_1_6"/><b>then </b>
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>34</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E15:40_2_4_1_2_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<b>hence </b><a NAME="E7:40_2_4_1_2_1_6"/>
<font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>34</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E3:40_2_4_1_2_1_6"><i><font color="Green">E126</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_2_4_1_2_1_6"><i><font color="Green">E127</font></i></a>, <a class="txt" href="topalg_5.html#E62:40_2_4_1_2_1"><i><font color="Green">E125</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>41</sub></font> = <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>37</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>37</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <b>by </b><i><a class="txt" href="topalg_5.html#E62:40_2_4_1_2_1"><i><font color="Green">E125</font></i></a>, <a class="txt" href="topalg_5.html#E61:40_2_4_1_2_1"><i><font color="Green">E124</font></i></a>, <a class="txt" href="topalg_5.html#E13:40_2_4_1_2_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E65:40_2_4_1_2_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">c<sub>41</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i><a class="txt" href="topalg_5.html#E16:40_2_4_1_2_1"><i><font color="Green">E100</font></i></a>, <a class="txt" href="topalg_5.html#E27:40_2_4_1_2_1"><i><font color="Green">E107</font></i></a></i>;<br/>
<a NAME="E66:40_2_4_1_2_1"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">c<sub>41</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E65:40_2_4_1_2_1"><i><font color="Green">E126</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E67:40_2_4_1_2_1"/><i><font color="Green">E128</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E8:40_2_4_1_2_1"><i><font color="Green">E95</font></i></a>, <a class="txt" href="topalg_5.html#E36:40_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="topalg_5.html#E34:40_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="topalg_5.html#E35:40_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E68:40_2_4_1_2_1"/><i><font color="Green">E129</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E8:40_2_4_1_2_1"><i><font color="Green">E95</font></i></a>, <a class="txt" href="topalg_5.html#E33:40_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topalg_5.html#E34:40_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="topalg_5.html#E35:40_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E69:40_2_4_1_2_1"/><i><font color="Green">E130</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>41</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_4_1_2_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>42</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="E1:40_2_4_1_2_1_7"/>
<font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>43</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_2_4_1_2_1_7"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:40_2_4_1_2_1_7"/><i><font color="Green">E132</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>42</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:40_2_4_1_2_1_7"/>
<font color="Maroon">c<sub>22</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>42</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E67:40_2_4_1_2_1"><i><font color="Green">E128</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_2_4_1_2_1_7"><i><font color="Green">E132</font></i></a>, <a class="txt" href="topalg_5.html#E3:40_2_4_1_2_1_7"><i><font color="Green">E131</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E6:40_2_4_1_2_1_7"/><b>then </b>
<font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>33</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E36:40_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="topalg_5.html#E67:40_2_4_1_2_1"><i><font color="Green">E128</font></i></a>, <a class="txt" href="topalg_5.html#E3:40_2_4_1_2_1_7"><i><font color="Green">E131</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E7:40_2_4_1_2_1_7"/>
<font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>41</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E62:40_2_4_1_2_1"><i><font color="Green">E125</font></i></a>, <a class="txt" href="topalg_5.html#E45:40_2_4_1_2_1"><i><font color="Green">E116</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E70:40_2_4_1_2_1"/><i><font color="Green">E131</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>41</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_4_1_2_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>42</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="E1:40_2_4_1_2_1_8"/>
<font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>43</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_2_4_1_2_1_8"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:40_2_4_1_2_1_8"/><i><font color="Green">E133</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>42</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:40_2_4_1_2_1_8"/>
<font color="Maroon">c<sub>23</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>42</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E68:40_2_4_1_2_1"><i><font color="Green">E129</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_2_4_1_2_1_8"><i><font color="Green">E133</font></i></a>, <a class="txt" href="topalg_5.html#E3:40_2_4_1_2_1_8"><i><font color="Green">E132</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E6:40_2_4_1_2_1_8"/><b>then </b>
<font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>33</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E33:40_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topalg_5.html#E68:40_2_4_1_2_1"><i><font color="Green">E129</font></i></a>, <a class="txt" href="topalg_5.html#E3:40_2_4_1_2_1_8"><i><font color="Green">E132</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E7:40_2_4_1_2_1_8"/>
<font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>41</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E62:40_2_4_1_2_1"><i><font color="Green">E125</font></i></a>, <a class="txt" href="topalg_5.html#E56:40_2_4_1_2_1"><i><font color="Green">E121</font></i></a>, <a class="txt" href="topalg_5.html#E60:40_2_4_1_2_1"><i><font color="Green">E123</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E71:40_2_4_1_2_1"/><i><font color="Green">E132</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E69:40_2_4_1_2_1"><i><font color="Green">E130</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E72:40_2_4_1_2_1"/><i><font color="Green">E133</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E70:40_2_4_1_2_1"><i><font color="Green">E131</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E73:40_2_4_1_2_1"/>
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="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></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>41</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_4_1_2_1_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>42</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:40_2_4_1_2_1_9"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:40_2_4_1_2_1_9"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E71:40_2_4_1_2_1"><i><font color="Green">E132</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_4_1_2_1_9"><i><font color="Green">E134</font></i></a>, <a class="txt" href="topalg_5.html#E67:40_2_4_1_2_1"><i><font color="Green">E128</font></i></a>, <a class="txt" href="topalg_5.html#E36:40_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E3:40_2_4_1_2_1_9"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E33:40_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topalg_5.html#E71:40_2_4_1_2_1"><i><font color="Green">E132</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_4_1_2_1_9"><i><font color="Green">E134</font></i></a>, <a class="txt" href="topalg_5.html#E67:40_2_4_1_2_1"><i><font color="Green">E128</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>42</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>42</sub></font>
<b>by </b><i><a class="ref" href="relat_1.html#T112">RELAT_1:112</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>42</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E67:40_2_4_1_2_1"><i><font color="Green">E128</font></i></a>, <a class="txt" href="topalg_5.html#E71:40_2_4_1_2_1"><i><font color="Green">E132</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_4_1_2_1_9"><i><font color="Green">E134</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>41</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>42</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E36:40_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_4_1_2_1_9"><i><font color="Green">E134</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>42</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40_2_4_1_2_1_9"><i><font color="Green">E135</font></i></a>, <a class="txt" href="topalg_5.html#E45:40_2_4_1_2_1"><i><font color="Green">E116</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>42</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E1:40_2_4_1_2_1_9"><i><font color="Green">E134</font></i></a>, <a class="txt" href="topalg_5.html#E36:40_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>42</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E31:40_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_4_1_2_1_9"><i><font color="Green">E134</font></i></a>, <a class="txt" href="topalg_5.html#E33:40_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topalg_5.html#E20:40_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="topalg_5.html#E24:40_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>41</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>42</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40_2_4_1_2_1_9"><i><font color="Green">E136</font></i></a>, <a class="txt" href="topalg_5.html#E56:40_2_4_1_2_1"><i><font color="Green">E121</font></i></a>, <a class="txt" href="topalg_5.html#E60:40_2_4_1_2_1"><i><font color="Green">E123</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>42</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E33:40_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_4_1_2_1_9"><i><font color="Green">E134</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>42</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E67:40_2_4_1_2_1"><i><font color="Green">E128</font></i></a>, <a class="txt" href="topalg_5.html#E71:40_2_4_1_2_1"><i><font color="Green">E132</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_4_1_2_1_9"><i><font color="Green">E134</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>42</sub></font>
<b>by </b><i><a class="ref" href="relat_1.html#T112">RELAT_1:112</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E74:40_2_4_1_2_1"/><b>then </b><i><font color="Green">E134</font></i>: 
<font color="Maroon">c<sub>41</sub></font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>41</sub></font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E67:40_2_4_1_2_1"><i><font color="Green">E128</font></i></a>, <a class="txt" href="topalg_5.html#E68:40_2_4_1_2_1"><i><font color="Green">E129</font></i></a>, <a class="txt" href="topalg_5.html#E71:40_2_4_1_2_1"><i><font color="Green">E132</font></i></a>, <a class="txt" href="topalg_5.html#E72:40_2_4_1_2_1"><i><font color="Green">E133</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_4_1_2_1_10"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_2_4_1_2_1_10"/>
( <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>27</sub></font> or  not <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>27</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_4_1_2_1_10_1"><b>suppose </b></a><a NAME="E1:40_2_4_1_2_1_10_1"/>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>27</sub></font>
 ;<br/><div class="add"><a NAME="E2:40_2_4_1_2_1_10_1"/><b>then </b><i><font color="Green">E135</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E5:40_2_4_1_2_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="topalg_5.html#E40:40_2_4_1_2_1"><i><font color="Green">E113</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>28</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>29</sub></font> = 
<font color="Maroon">c<sub>22</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>29</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E11:40_2_4_1_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_4_1_2_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>29</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40_2_4_1_2_1_10_1"><i><font color="Green">E135</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>29</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E74:40_2_4_1_2_1"><i><font color="Green">E134</font></i></a>, <a class="txt" href="topalg_5.html#E66:40_2_4_1_2_1"><i><font color="Green">E127</font></i></a>, <a class="txt" href="topalg_5.html#E69:40_2_4_1_2_1"><i><font color="Green">E130</font></i></a>, <a class="txt" href="topalg_5.html#E70:40_2_4_1_2_1"><i><font color="Green">E131</font></i></a>, <a class="txt" href="topalg_5.html#E67:40_2_4_1_2_1"><i><font color="Green">E128</font></i></a>, <a class="txt" href="topalg_5.html#E68:40_2_4_1_2_1"><i><font color="Green">E129</font></i></a>, <a class="ref" href="funct_1.html#T49">FUNCT_1:49</a></i>
<br/>.= 
<font color="Maroon">c<sub>23</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>29</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40_2_4_1_2_1_10_1"><i><font color="Green">E135</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>28</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>29</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E11:40_2_4_1_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_4_1_2_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_4_1_2_1_10_2"><b>suppose </b></a><a NAME="E1:40_2_4_1_2_1_10_2"/>
not <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>27</sub></font>
 ;<br/><div class="add"><a NAME="E2:40_2_4_1_2_1_10_2"/><b>then </b>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E4:40_2_4_1_2_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:40_2_4_1_2_1_10_2"/><b>then </b><i><font color="Green">E135</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E3:40_2_4_1_2_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_2_4_1_2_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>28</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>29</sub></font> = 
<font color="Maroon">c<sub>22</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>29</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E11:40_2_4_1_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_4_1_2_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>29</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40_2_4_1_2_1_10_2"><i><font color="Green">E135</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>23</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>29</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40_2_4_1_2_1_10_2"><i><font color="Green">E135</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_2_4_1_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>28</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>29</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E11:40_2_4_1_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_2_4_1_2_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E14:40_2_4_1_2"/>
<font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>28</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>28</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E11:40_2_4_1_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="topalg_5.html#E12:40_2_4_1_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E39:40_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>3</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="topalg_5.html#E37:40_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="topalg_5.html#E38:40_2"><i><font color="Green">E78</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon">c<sub>24</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E41:40_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>and </b><br/><a NAME="E42:40_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>24</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>24</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E13:40_2"><i><font color="Green">E57</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>22</sub></font> = 
<font color="Maroon">c<sub>22</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>24</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
<b>by </b><i><a class="txt" href="topalg_5.html#E36:40_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="topalg_5.html#E34:40_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="topalg_5.html#E35:40_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="topalg_5.html#E41:40_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="relat_1.html#T98">RELAT_1:98</a></i>
<br/>.= 
<font color="Maroon">c<sub>23</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E33:40_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topalg_5.html#E34:40_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="topalg_5.html#E35:40_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="topalg_5.html#E42:40_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="topalg_5.html#E41:40_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="relat_1.html#T98">RELAT_1:98</a></i>
;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <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>8</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E14:40"/><i><font color="Green">E49</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>8</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> holds <font color="Maroon">S<sub>1</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="topalg_5.html#E12:40"><i><font color="Green">E48</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/>

<a NAME="E15:40"/><i><font color="Green">E50</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font> <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>8</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E4:40"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E16:40"/><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>8</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> holds <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <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="40_3"><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 <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>;<br/>

<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>15</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">c<sub>16</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>17</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E2:40_3"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E3:40_3"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E4:40_3"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E5:40_3"/>
<font color="Maroon">c<sub>17</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E6:40_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E7:40_3"/>
<font color="Maroon">c<sub>17</sub></font> <a href="partfun1.html#K2">|</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E8:40_3"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="partfun1.html#K2">|</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#R2">=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E14:40"><i><font color="Green">E49</font></i></a></i>;<br/>
<i><font color="Green">E56</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> = 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </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/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>
;<br/>
<a NAME="E10:40_3"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>13</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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_3"><i><font color="Green">E54</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> = 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><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>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="topalg_5.html#E10:40_3"><i><font color="Green">E57</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E6:40_3"><i><font color="Green">E55</font></i></a>, <a class="txt" href="topalg_5.html#E9:40_3"><i><font color="Green">E56</font></i></a>, <a class="txt" href="topalg_5.html#E10:40_3"><i><font color="Green">E57</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> ;<br/><b>thus </b> <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </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/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E18:40"/><i><font color="Green">E53</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>8</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>8</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <br/>for <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font><br/>for <font color="Olive">b<sub>6</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Olive">b<sub>4</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> <br/>for <font color="Olive">b<sub>7</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Olive">b<sub>5</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="borsuk_1.html#K7">]</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>5</sub></font> &amp; <font color="Olive">b<sub>7</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">b<sub>6</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">b<sub>5</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Olive">b<sub>7</sub></font> &amp; <font color="Olive">b<sub>7</sub></font> <a href="partfun1.html#K2">|</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">b<sub>5</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">b<sub>4</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Olive">b<sub>6</sub></font> &amp; <font color="Olive">b<sub>6</sub></font> <a href="partfun1.html#K2">|</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">b<sub>4</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Olive">b<sub>6</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>7</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5"><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 <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>;<br/><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>8</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/><b>let </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> ;<br/><b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> ;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:40_5"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E2:40_5"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E3:40_5"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E4:40_5"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>19</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E5:40_5"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>18</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E6:40_5"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E7:40_5"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="partfun1.html#K2">|</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E8:40_5"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>18</sub></font>
 <b>and </b><br/><a NAME="E9:40_5"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<a NAME="E10:40_5"/><i><font color="Green">E62</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40_5"><i><font color="Green">E54</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E11:40_5"/><i><font color="Green">E63</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E17:40"><i><font color="Green">E52</font></i></a></i>;<br/>
<a NAME="E12:40_5"/><i><font color="Green">E64</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40_5"><i><font color="Green">E55</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E13:40_5"/><i><font color="Green">E65</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E17:40"><i><font color="Green">E52</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E15:40_5"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> 0
 <b>and </b><br/><a NAME="E16:40_5"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>and </b><br/><a NAME="E17:40_5"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>20</sub></font> is <a href="seqm_3.html#V1">increasing</a>
 <b>and </b><br/><a NAME="E18:40_5"/><i><font color="Green">E69</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font> st <br/>( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>3</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:40"><i><font color="Green">E38</font></i></a>, <a class="txt" href="topalg_5.html#E8:40"><i><font color="Green">E44</font></i></a>, <a class="txt" href="topalg_5.html#E9:40"><i><font color="Green">E45</font></i></a>, <a class="ref" href="topalg_5.html#T21" target="_self">Th21</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>21</sub></font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font><b> such that </b><br/><a NAME="E20:40_5"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font>
 <b>and </b><br/><a NAME="E21:40_5"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="ordinal1.html#V7">natural</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>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font> holds <br/>ex <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 <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="topalg_5.html#E18:40_5"><i><font color="Green">E69</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font> <b>by </b><i><a class="txt" href="topalg_5.html#E20:40_5"><i><font color="Green">E70</font></i></a></i>;<br/>
<a NAME="E23:40_5"/><i><font color="Green">E72</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E24:40_5"/><i><font color="Green">E73</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E25:40_5"/><i><font color="Green">E74</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font> <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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E26:40_5"/><i><font color="Green">E75</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </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 <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </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="E27:40_5"/><i><font color="Green">E76</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E28:40_5"/><i><font color="Green">E77</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </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 <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </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="E29:40_5"/><i><font color="Green">E78</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E30:40_5"/><i><font color="Green">E79</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font> <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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Olive">b<sub>1</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Olive">b<sub>1</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> ) );<br/>
<a NAME="E31:40_5"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="ref" href="finseq_3.html#T26">FINSEQ_3:26</a></i>;<br/>
<a NAME="E32:40_5"/><i><font color="Green">E81</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>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</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="40_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:40_5_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>23</sub></font>]
 <b>and </b><br/><a NAME="E2:40_5_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_5_1_1"/>
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font> )
 </span><b>by </b><i><a class="txt" href="topalg_5.html#E2:40_5_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="topreala.html#T23">TOPREALA:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_1_1"><b>suppose </b></a><a NAME="E1:40_5_1_1_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>24</sub></font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>;<br/><a NAME="E2:40_5_1_1_1"/><i><font color="Green">E85</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E15:40_5"><i><font color="Green">E66</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_5_1_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="rcomp_1.html#T14">RCOMP_1:14</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E3"><i><font color="Green">Lemma3</font></i></a>, <a class="txt" href="topalg_5.html#E15:40_5"><i><font color="Green">E66</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_5_1_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="rcomp_1.html#T14">RCOMP_1:14</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>25</sub></font>
;<br/><b>thus </b><a NAME="E4:40_5_1_1_1"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/><a NAME="E5:40_5_1_1_1"/><i><font color="Green">E86</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>25</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E10:40_5"><i><font color="Green">E62</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E6:40_5_1_1_1"/><b>then </b><i><font color="Green">E87</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>25</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>25</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E11:40_5"><i><font color="Green">E63</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E7:40_5_1_1_1"/><i><font color="Green">E88</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>25</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E12:40_5"><i><font color="Green">E64</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E8:40_5_1_1_1"/><b>then </b><i><font color="Green">E89</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>25</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>25</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E13:40_5"><i><font color="Green">E65</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E9:40_5_1_1_1"/>
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="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>25</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>25</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</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>19</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>25</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>26</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:40_5_1_1_1_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>25</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 ;<br/>

<a NAME="E2:40_5_1_1_1_1"/><i><font color="Green">E91</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>25</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</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>8</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>25</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>26</sub></font> = 
<font color="Maroon">c<sub>18</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E1:40_5_1_1_1_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_5_1_1_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><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>8</sub></font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40_5_1_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_5_1_1_1_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_5_1_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="topalg_5.html#E2:40_5_1_1_1_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E9:40_5"><i><font color="Green">E61</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_5_1_1_1_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_5_1_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_5_1_1_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E1:40_5_1_1_1_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_5_1_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="topalg_5.html#E7:40_5_1_1_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40_5_1_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="topalg_5.html#E7:40_5"><i><font color="Green">E59</font></i></a>, <a class="txt" href="topalg_5.html#E2:40_5_1_1_1_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_5_1_1_1_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_5_1_1_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>25</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E6:40_5_1_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_5_1_1_1_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E10:40_5_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>25</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>25</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E6:40_5_1_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="topalg_5.html#E8:40_5_1_1_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_1_2"><b>suppose </b></a><a NAME="E1:40_5_1_1_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>24</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E3:40_5_1_1_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>and </b><br/><a NAME="E4:40_5_1_1_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>24</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>24</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:40_5_1"><i><font color="Green">E82</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>25</sub></font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>;<br/><div><i><font color="Green">E87</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>26</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:40_5_1_1_2_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/><a NAME="E2:40_5_1_1_2_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:40_5_1_1_2_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:40_5_1_1_2_1"/><b>then </b>
( 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> or 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>26</sub></font> )
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:40_5_1_1_2_1"/><i><font color="Green">E89</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E15:40_5"><i><font color="Green">E66</font></i></a>, <a class="txt" href="topalg_5.html#E17:40_5"><i><font color="Green">E68</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_5_1_1_2_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="topalg_5.html#E23:40_5"><i><font color="Green">E72</font></i></a>, <a class="ref" href="seqm_3.html#D1">SEQM_3:def 1</a></i>;<br/><b>assume </b><a NAME="E5:40_5_1_1_2_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/><a NAME="E6:40_5_1_1_2_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>26</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/><b>hence </b><a NAME="E7:40_5_1_1_2_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E17:40_5"><i><font color="Green">E68</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_5_1_1_2_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_5_1_1_2_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="seqm_3.html#D1">SEQM_3:def 1</a></i>;<br/><a NAME="E8:40_5_1_1_2_1"/>
<font color="Maroon">c<sub>26</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>20</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E5:40_5_1_1_2_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:40_5_1_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font> or <font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E10:40_5_1_1_2_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="topalg_5.html#E16:40_5"><i><font color="Green">E67</font></i></a>, <a class="txt" href="topalg_5.html#E17:40_5"><i><font color="Green">E68</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_5_1_1_2_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="topalg_5.html#E24:40_5"><i><font color="Green">E73</font></i></a>, <a class="ref" href="seqm_3.html#D1">SEQM_3:def 1</a></i>;<br/><b>hence </b><a NAME="E11:40_5_1_1_2_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <b>by </b><i><a class="txt" href="topalg_5.html#E7:40_5_1_1_2_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>thus </b><a NAME="E12:40_5_1_1_2_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E6:40_5_1_1_2_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_5_1_1_2_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="topalg_5.html#E17:40_5"><i><font color="Green">E68</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_5_1_1_2_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_5_1_1_2_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="seqm_3.html#D1">SEQM_3:def 1</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E88</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>26</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:40_5_1_1_2_2"/><i><font color="Green">E89</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><b>thus </b><a NAME="E2:40_5_1_1_2_2"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5_1_1_2_2_1"><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="E1:40_5_1_1_2_2_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>27</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E3:40_5_1_1_2_2_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="topalg_5.html#E1:40_5_1_1_2_2_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E4:40_5_1_1_2_2_1"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="topalg_5.html#E1:40_5_1_1_2_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:40_5_1_1_2_2_1"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 <b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><a NAME="E7:40_5_1_1_2"/><i><font color="Green">E89</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:40_5_1_1_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_5_1_1_2"><i><font color="Green">E87</font></i></a></i>;<br/><a NAME="E8:40_5_1_1_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="topalg_5.html#E2:40_5_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_5_1_1_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_5_1_1_2"><i><font color="Green">E87</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>26</sub></font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E7:40_5_1_1_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_5_1_1_2"><i><font color="Green">E88</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>24</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>26</sub></font>;<br/><a NAME="E10:40_5_1_1_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E2:40_5_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_5_1_1_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_5_1_1_2"><i><font color="Green">E87</font></i></a></i>;<br/><b>hence </b><a NAME="E11:40_5_1_1_2"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E3:40_5_1_1_2"><i><font color="Green">E85</font></i></a>, <a class="txt" href="topalg_5.html#E7:40_5_1_1_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="heine.html#T2">HEINE:2</a></i>;<br/><a NAME="E12:40_5_1_1_2"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E10:40_5"><i><font color="Green">E62</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E13:40_5_1_1_2"/><b>then </b><i><font color="Green">E92</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E11:40_5"><i><font color="Green">E63</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E14:40_5_1_1_2"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E12:40_5"><i><font color="Green">E64</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/><a NAME="E15:40_5_1_1_2"/><b>then </b><i><font color="Green">E93</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E25:40_5"><i><font color="Green">E74</font></i></a>, <a class="txt" href="topalg_5.html#E26:40_5"><i><font color="Green">E75</font></i></a>, <a class="txt" href="topalg_5.html#E27:40_5"><i><font color="Green">E76</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E16:40_5_1_1_2"/>
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="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</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>19</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>28</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:40_5_1_1_2_3"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>29</sub></font>, <font color="Maroon">c<sub>30</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_5_1_1_2_3"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E4:40_5_1_1_2_3"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>27</sub></font>
 <b>and </b><br/><a NAME="E5:40_5_1_1_2_3"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">c<sub>28</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>29</sub></font>,<font color="Maroon">c<sub>30</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E13:40_5_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_5_1_1_2_3"><i><font color="Green">E94</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>31</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>29</sub></font></span><a href="tarski.html#K1">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font> <b>by </b><i><a class="txt" href="topalg_5.html#E3:40_5_1_1_2_3"><i><font color="Green">E95</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E7:40_5_1_1_2_3"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E12:40_5"><i><font color="Green">E64</font></i></a>, <a class="txt" href="topalg_5.html#E3:40_5_1_1_2_3"><i><font color="Green">E95</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E8:40_5_1_1_2_3"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E10:40_5"><i><font color="Green">E62</font></i></a>, <a class="txt" href="topalg_5.html#E3:40_5_1_1_2_3"><i><font color="Green">E95</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E9:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E100</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E10:40_5_1_1_2_3"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40_5_1_1_2_3"><i><font color="Green">E95</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E11:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E20:40_5"><i><font color="Green">E70</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>32</sub></font> = <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="txt" href="topalg_5.html#E8:40_5_1_1_2_3"><i><font color="Green">E99</font></i></a>, <a class="txt" href="topalg_5.html#E28:40_5"><i><font color="Green">E77</font></i></a>, <a class="txt" href="topalg_5.html#E29:40_5"><i><font color="Green">E78</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>33</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span><b> such that </b><br/><a NAME="E14:40_5_1_1_2_3"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E15:40_5_1_1_2_3"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>33</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E21:40_5"><i><font color="Green">E71</font></i></a>, <a class="txt" href="topalg_5.html#E2:40_5_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_5_1_1_2"><i><font color="Green">E84</font></i></a></i>;<br/>
<a NAME="E16:40_5_1_1_2_3"/><i><font color="Green">E105</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>33</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>33</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>34</sub></font> being  <a href="taxonom2.html#V4">mutually-disjoint</a> <a href="tops_2.html#V1">open</a> <a href="struct_0.html#NM3">Subset-Family</a> of <a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E18:40_5_1_1_2_3"/><i><font color="Green">E106</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>33</sub></font>
 <b>and </b><br/><a NAME="E19:40_5_1_1_2_3"/><i><font color="Green">E107</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>34</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>33</sub></font></span>)</span> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i><a class="txt" href="topalg_5.html#E14:40_5_1_1_2_3"><i><font color="Green">E103</font></i></a>, <a class="txt" href="topalg_5.html#E10:40"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E20:40_5_1_1_2_3"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40_5_1_1_2"><i><font color="Green">E85</font></i></a>, <a class="txt" href="topalg_5.html#E7:40_5_1_1_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<a NAME="E21:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E109</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span><a href="tarski.html#K4">]</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"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>24</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40_5_1_1_2_3"><i><font color="Green">E95</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E22:40_5_1_1_2_3"/><i><font color="Green">E110</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>24</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E20:40_5_1_1_2_3"><i><font color="Green">E108</font></i></a>, <a class="txt" href="topalg_5.html#E12:40_5"><i><font color="Green">E64</font></i></a>, <a class="txt" href="topalg_5.html#E3:40_5_1_1_2_3"><i><font color="Green">E95</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E23:40_5_1_1_2_3"/><i><font color="Green">E111</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>24</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E24:40_5_1_1_2_3"/><i><font color="Green">E112</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>24</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E10:40_5"><i><font color="Green">E62</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>then </b><i><font color="Green">E113</font></i>: <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="topalg_5.html#E8:40_5"><i><font color="Green">E60</font></i></a>, <a class="txt" href="topalg_5.html#E21:40_5_1_1_2_3"><i><font color="Green">E109</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E11:40_5"><i><font color="Green">E63</font></i></a>, <a class="txt" href="topalg_5.html#E24:40_5_1_1_2_3"><i><font color="Green">E112</font></i></a>, <a class="txt" href="topalg_5.html#E21:40_5_1_1_2_3"><i><font color="Green">E109</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<a NAME="E26:40_5_1_1_2_3"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>26</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E10:40_5_1_1_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<a NAME="E27:40_5_1_1_2_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E10:40_5_1_1_2_3"><i><font color="Green">E101</font></i></a>, <a class="txt" href="topalg_5.html#E20:40_5"><i><font color="Green">E70</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E28:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E29:40_5_1_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>33</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E15:40_5_1_1_2_3"><i><font color="Green">E104</font></i></a>, <a class="txt" href="topalg_5.html#E25:40_5_1_1_2_3"><i><font color="Green">E113</font></i></a>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>35</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E31:40_5_1_1_2_3"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>35</sub></font>
 <b>and </b><br/><a NAME="E32:40_5_1_1_2_3"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>34</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E18:40_5_1_1_2_3"><i><font color="Green">E106</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>36</sub></font> = <font color="Maroon">c<sub>35</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E31:40_5_1_1_2_3"><i><font color="Green">E116</font></i></a>, <a class="txt" href="topalg_5.html#E32:40_5_1_1_2_3"><i><font color="Green">E117</font></i></a></i>;<br/>
<a NAME="E34:40_5_1_1_2_3"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40_5_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_5_1_1_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_5_1_1_2"><i><font color="Green">E87</font></i></a></i>;<br/>
<a NAME="E35:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font> is   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<a NAME="E36:40_5_1_1_2_3"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40_5_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_5_1_1_2"><i><font color="Green">E87</font></i></a></i>;<br/>
<a NAME="E37:40_5_1_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="txt" href="topalg_5.html#E8:40_5_1_1_2"><i><font color="Green">E90</font></i></a>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<a NAME="E38:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">c<sub>26</sub></font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E35:40_5_1_1_2_3"><i><font color="Green">E118</font></i></a>, <a class="txt" href="topalg_5.html#E10:40_5_1_1_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="borsuk_4.html#T49">BORSUK_4:49</a></i>;<br/>
<a NAME="E39:40_5_1_1_2_3"/>
<font color="Maroon">c<sub>31</sub></font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40_5_1_1_2_3"><i><font color="Green">E95</font></i></a>, <a class="ref" href="connsp_1.html#T29">CONNSP_1:29</a></i>;<br/>
<a NAME="E40:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E120</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E38:40_5_1_1_2_3"><i><font color="Green">E119</font></i></a>, <a class="ref" href="topalg_3.html#T17">TOPALG_3:17</a></i>;<br/>
<a NAME="E41:40_5_1_1_2_3"/><i><font color="Green">E121</font></i>: 
 <a href="topmetr.html#K5">I[01]</a>  is    <a href="pre_topc.html#M1">SubSpace</a> of  <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="tsep_1.html#T2">TSEP_1:2</a></i>;<br/>
<a NAME="E42:40_5_1_1_2_3"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> is    <a href="pre_topc.html#M1">SubSpace</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_3.html#T25">BORSUK_3:25</a></i>;<br/>
<a NAME="E43:40_5_1_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>32</sub></font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E40:40_5_1_1_2_3"><i><font color="Green">E120</font></i></a>, <a class="ref" href="connsp_1.html#T24">CONNSP_1:24</a></i>;<br/>
<a NAME="E44:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>32</sub></font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E5:40_5"><i><font color="Green">E57</font></i></a>, <a class="ref" href="tops_2.html#T75">TOPS_2:75</a></i>;<br/>
<a NAME="E45:40_5_1_1_2_3"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>31</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E46:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E124</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>32</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E26:40_5_1_1_2_3"><i><font color="Green">E114</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E47:40_5_1_1_2_3"/>
<font color="Maroon">c<sub>18</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>32</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E46:40_5_1_1_2_3"><i><font color="Green">E124</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E48:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>18</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>32</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E31:40_5_1_1_2_3"><i><font color="Green">E116</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E49:40_5_1_1_2_3"/>
<font color="Maroon">c<sub>34</sub></font> <a href="compts_1.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>18</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>32</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5_1_1_2_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>37</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D1">COMPTS_1:def 1</a>,<a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:40_5_1_1_2_3_2"/>
<font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>32</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>38</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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E3:40_5_1_1_2_3_2"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>32</sub></font>
 <b>and </b><br/><a NAME="E4:40_5_1_1_2_3_2"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>37</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<i><font color="Green">E128</font></i>: <a href="toprealb.html#K12">CircleMap</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>38</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>38</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>38</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E28:40_5"><i><font color="Green">E77</font></i></a>, <a class="txt" href="topalg_5.html#E29:40_5"><i><font color="Green">E78</font></i></a>, <a class="txt" href="topalg_5.html#E8:40_5"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>
<a NAME="E6:40_5_1_1_2_3_2"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E11:40_5_1_1_2_3"><i><font color="Green">E102</font></i></a>, <a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E7:40_5_1_1_2_3_2"/><b>then </b><i><font color="Green">E129</font></i>: 
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40_5_1_1_2_3_2"><i><font color="Green">E126</font></i></a></i>;<br/>
<a NAME="E8:40_5_1_1_2_3_2"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</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>8</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E9:40_5_1_1_2_3_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E11:40"><i><font color="Green">E47</font></i></a>, <a class="txt" href="topalg_5.html#E7:40_5_1_1_2_3_2"><i><font color="Green">E129</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E10:40_5_1_1_2_3_2"/>
<font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>34</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E15:40_5_1_1_2_3"><i><font color="Green">E104</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_5_1_1_2_3_2"><i><font color="Green">E127</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_5_1_1_2_3_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="topalg_5.html#E18:40_5_1_1_2_3"><i><font color="Green">E106</font></i></a>, <a class="txt" href="topalg_5.html#E35"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E50:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>32</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>36</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E32:40_5_1_1_2_3"><i><font color="Green">E117</font></i></a>, <a class="txt" href="topalg_5.html#E44:40_5_1_1_2_3"><i><font color="Green">E122</font></i></a>, <a class="txt" href="topalg_5.html#E48:40_5_1_1_2_3"><i><font color="Green">E125</font></i></a>, <a class="ref" href="topalg_3.html#T13">TOPALG_3:13</a></i>;<br/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="topalg_5.html#E6:40_5"><i><font color="Green">E58</font></i></a>, <a class="txt" href="topalg_5.html#E22:40_5_1_1_2_3"><i><font color="Green">E110</font></i></a>, <a class="txt" href="topalg_5.html#E23:40_5_1_1_2_3"><i><font color="Green">E111</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E25:40_5"><i><font color="Green">E74</font></i></a>, <a class="txt" href="topalg_5.html#E26:40_5"><i><font color="Green">E75</font></i></a>, <a class="txt" href="topalg_5.html#E27:40_5"><i><font color="Green">E76</font></i></a>, <a class="txt" href="topalg_5.html#E23:40_5_1_1_2_3"><i><font color="Green">E111</font></i></a>, <a class="txt" href="topalg_5.html#E22:40_5_1_1_2_3"><i><font color="Green">E110</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<a NAME="E52:40_5_1_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>33</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E28:40_5_1_1_2_3"><i><font color="Green">E115</font></i></a>, <a class="txt" href="topalg_5.html#E15:40_5_1_1_2_3"><i><font color="Green">E104</font></i></a>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>37</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E54:40_5_1_1_2_3"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>37</sub></font>
 <b>and </b><br/><a NAME="E55:40_5_1_1_2_3"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>34</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E18:40_5_1_1_2_3"><i><font color="Green">E106</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>38</sub></font> = <font color="Maroon">c<sub>37</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E54:40_5_1_1_2_3"><i><font color="Green">E127</font></i></a>, <a class="txt" href="topalg_5.html#E55:40_5_1_1_2_3"><i><font color="Green">E128</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>39</sub></font> = <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="txt" href="topalg_5.html#E7:40_5_1_1_2_3"><i><font color="Green">E98</font></i></a>, <a class="txt" href="topalg_5.html#E26:40_5"><i><font color="Green">E75</font></i></a>, <a class="txt" href="topalg_5.html#E27:40_5"><i><font color="Green">E76</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E58:40_5_1_1_2_3"/>
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> is    <a href="pre_topc.html#M1">SubSpace</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E41:40_5_1_1_2_3"><i><font color="Green">E121</font></i></a>, <a class="ref" href="borsuk_3.html#T25">BORSUK_3:25</a></i>;<br/>
<a NAME="E59:40_5_1_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>39</sub></font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E40:40_5_1_1_2_3"><i><font color="Green">E120</font></i></a>, <a class="ref" href="connsp_1.html#T24">CONNSP_1:24</a></i>;<br/>
<a NAME="E60:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E129</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>39</sub></font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E4:40_5"><i><font color="Green">E56</font></i></a>, <a class="ref" href="tops_2.html#T75">TOPS_2:75</a></i>;<br/>
<a NAME="E61:40_5_1_1_2_3"/><i><font color="Green">E130</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E7:40_5_1_1_2_3"><i><font color="Green">E98</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E62:40_5_1_1_2_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40_5_1_1_2_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="topalg_5.html#E10:40_5_1_1_2_3"><i><font color="Green">E101</font></i></a>, <a class="txt" href="topalg_5.html#E26:40_5_1_1_2_3"><i><font color="Green">E114</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E63:40_5_1_1_2_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E13:40_5"><i><font color="Green">E65</font></i></a>, <a class="txt" href="topalg_5.html#E61:40_5_1_1_2_3"><i><font color="Green">E130</font></i></a></i>;<br/>
<a NAME="E64:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E131</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>32</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E46:40_5_1_1_2_3"><i><font color="Green">E124</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E65:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E132</font></i>: 
<font color="Maroon">c<sub>38</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>19</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>32</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E54:40_5_1_1_2_3"><i><font color="Green">E127</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E66:40_5_1_1_2_3"/>
<font color="Maroon">c<sub>34</sub></font> <a href="compts_1.html#R1">is_a_cover_of</a> <font color="Maroon">c<sub>19</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>32</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5_1_1_2_3_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>40</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D1">COMPTS_1:def 1</a>,<a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:40_5_1_1_2_3_4"/>
<font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>32</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>41</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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E3:40_5_1_1_2_3_4"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>32</sub></font>
 <b>and </b><br/><a NAME="E4:40_5_1_1_2_3_4"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>40</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<i><font color="Green">E135</font></i>: <a href="toprealb.html#K12">CircleMap</a>  <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>41</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>41</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>41</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E6:40_5"><i><font color="Green">E58</font></i></a>, <a class="txt" href="topalg_5.html#E26:40_5"><i><font color="Green">E75</font></i></a>, <a class="txt" href="topalg_5.html#E27:40_5"><i><font color="Green">E76</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>
<a NAME="E6:40_5_1_1_2_3_4"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E11:40_5_1_1_2_3"><i><font color="Green">E102</font></i></a>, <a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E7:40_5_1_1_2_3_4"/><b>then </b><i><font color="Green">E136</font></i>: 
<font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40_5_1_1_2_3_4"><i><font color="Green">E133</font></i></a></i>;<br/>
<a NAME="E8:40_5_1_1_2_3_4"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</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>8</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E9:40_5_1_1_2_3_4"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E11:40"><i><font color="Green">E47</font></i></a>, <a class="txt" href="topalg_5.html#E7:40_5_1_1_2_3_4"><i><font color="Green">E136</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E10:40_5_1_1_2_3_4"/>
<font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>34</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E15:40_5_1_1_2_3"><i><font color="Green">E104</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_5_1_1_2_3_4"><i><font color="Green">E134</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_5_1_1_2_3_4"><i><font color="Green">E135</font></i></a>, <a class="txt" href="topalg_5.html#E18:40_5_1_1_2_3"><i><font color="Green">E106</font></i></a>, <a class="txt" href="topalg_5.html#E35"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E67:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>32</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>38</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E55:40_5_1_1_2_3"><i><font color="Green">E128</font></i></a>, <a class="txt" href="topalg_5.html#E60:40_5_1_1_2_3"><i><font color="Green">E129</font></i></a>, <a class="txt" href="topalg_5.html#E65:40_5_1_1_2_3"><i><font color="Green">E132</font></i></a>, <a class="ref" href="topalg_3.html#T13">TOPALG_3:13</a></i>;<br/>
<font color="Maroon">c<sub>18</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>24</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="topalg_5.html#E21:40_5_1_1_2_3"><i><font color="Green">E109</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="topalg_5.html#E21:40_5_1_1_2_3"><i><font color="Green">E109</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_5_1_1_2"><i><font color="Green">E86</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>
<a NAME="E69:40_5_1_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>36</sub></font> <a href="subset_1.html#R2">meets</a> <font color="Maroon">c<sub>38</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E31:40_5_1_1_2_3"><i><font color="Green">E116</font></i></a>, <a class="txt" href="topalg_5.html#E67:40_5_1_1_2_3"><i><font color="Green">E133</font></i></a>, <a class="txt" href="topalg_5.html#E64:40_5_1_1_2_3"><i><font color="Green">E131</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E70:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E134</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>38</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E32:40_5_1_1_2_3"><i><font color="Green">E117</font></i></a>, <a class="txt" href="topalg_5.html#E55:40_5_1_1_2_3"><i><font color="Green">E128</font></i></a>, <a class="ref" href="taxonom2.html#D5">TAXONOM2:def 5</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>40</sub></font> = <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>36</sub></font>;<br/>
<a NAME="E71:40_5_1_1_2_3"/><i><font color="Green">E135</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>36</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>36</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E72:40_5_1_1_2_3"/><i><font color="Green">E136</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>36</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>36</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E35"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E73:40_5_1_1_2_3"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>36</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>33</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5_1_1_2_3_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>41</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="E1:40_5_1_1_2_3_6"/>
<font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>36</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>42</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_5_1_1_2_3_6"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">c<sub>42</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="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>36</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:40_5_1_1_2_3_6"/><i><font color="Green">E138</font></i>: 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>36</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>41</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:40_5_1_1_2_3_6"/>
<font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>34</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40_5_1_1_2_3_6"><i><font color="Green">E137</font></i></a>, <a class="txt" href="topalg_5.html#E72:40_5_1_1_2_3"><i><font color="Green">E136</font></i></a>, <a class="txt" href="topalg_5.html#E32:40_5_1_1_2_3"><i><font color="Green">E117</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E6:40_5_1_1_2_3_6"/><b>then </b>
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>33</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E18:40_5_1_1_2_3"><i><font color="Green">E106</font></i></a>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<b>hence </b><a NAME="E7:40_5_1_1_2_3_6"/>
<font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>33</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E3:40_5_1_1_2_3_6"><i><font color="Green">E137</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_5_1_1_2_3_6"><i><font color="Green">E138</font></i></a>, <a class="txt" href="topalg_5.html#E72:40_5_1_1_2_3"><i><font color="Green">E136</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>41</sub></font> = <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>36</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>36</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>33</sub></font></span>)</span> <b>by </b><i><a class="txt" href="topalg_5.html#E72:40_5_1_1_2_3"><i><font color="Green">E136</font></i></a>, <a class="txt" href="topalg_5.html#E71:40_5_1_1_2_3"><i><font color="Green">E135</font></i></a>, <a class="txt" href="topalg_5.html#E16:40_5_1_1_2_3"><i><font color="Green">E105</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E75:40_5_1_1_2_3"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">c<sub>41</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i><a class="txt" href="topalg_5.html#E19:40_5_1_1_2_3"><i><font color="Green">E107</font></i></a>, <a class="txt" href="topalg_5.html#E32:40_5_1_1_2_3"><i><font color="Green">E117</font></i></a></i>;<br/>
<a NAME="E76:40_5_1_1_2_3"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">c<sub>41</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E75:40_5_1_1_2_3"><i><font color="Green">E137</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E77:40_5_1_1_2_3"/><i><font color="Green">E139</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E9:40_5_1_1_2_3"><i><font color="Green">E100</font></i></a>, <a class="txt" href="topalg_5.html#E30:40_5"><i><font color="Green">E79</font></i></a>, <a class="txt" href="topalg_5.html#E28:40_5"><i><font color="Green">E77</font></i></a>, <a class="txt" href="topalg_5.html#E29:40_5"><i><font color="Green">E78</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E78:40_5_1_1_2_3"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E7:40_5_1_1_2_3"><i><font color="Green">E98</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E79:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E140</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E25:40_5"><i><font color="Green">E74</font></i></a>, <a class="txt" href="topalg_5.html#E26:40_5"><i><font color="Green">E75</font></i></a>, <a class="txt" href="topalg_5.html#E27:40_5"><i><font color="Green">E76</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E80:40_5_1_1_2_3"/><i><font color="Green">E141</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>41</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5_1_1_2_3_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>42</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="E1:40_5_1_1_2_3_7"/>
<font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>43</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_5_1_1_2_3_7"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:40_5_1_1_2_3_7"/><i><font color="Green">E143</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>42</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:40_5_1_1_2_3_7"/>
<font color="Maroon">c<sub>18</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>42</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E77:40_5_1_1_2_3"><i><font color="Green">E139</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_5_1_1_2_3_7"><i><font color="Green">E143</font></i></a>, <a class="txt" href="topalg_5.html#E3:40_5_1_1_2_3_7"><i><font color="Green">E142</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E6:40_5_1_1_2_3_7"/><b>then </b>
<font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>32</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E30:40_5"><i><font color="Green">E79</font></i></a>, <a class="txt" href="topalg_5.html#E77:40_5_1_1_2_3"><i><font color="Green">E139</font></i></a>, <a class="txt" href="topalg_5.html#E3:40_5_1_1_2_3_7"><i><font color="Green">E142</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E7:40_5_1_1_2_3_7"/>
<font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>41</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E72:40_5_1_1_2_3"><i><font color="Green">E136</font></i></a>, <a class="txt" href="topalg_5.html#E50:40_5_1_1_2_3"><i><font color="Green">E126</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E81:40_5_1_1_2_3"/><i><font color="Green">E142</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>41</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5_1_1_2_3_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>42</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="E1:40_5_1_1_2_3_8"/>
<font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>43</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_5_1_1_2_3_8"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:40_5_1_1_2_3_8"/><i><font color="Green">E144</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>42</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:40_5_1_1_2_3_8"/>
<font color="Maroon">c<sub>19</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>42</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E79:40_5_1_1_2_3"><i><font color="Green">E140</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_5_1_1_2_3_8"><i><font color="Green">E144</font></i></a>, <a class="txt" href="topalg_5.html#E3:40_5_1_1_2_3_8"><i><font color="Green">E143</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<a NAME="E6:40_5_1_1_2_3_8"/><b>then </b>
<font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>32</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E25:40_5"><i><font color="Green">E74</font></i></a>, <a class="txt" href="topalg_5.html#E79:40_5_1_1_2_3"><i><font color="Green">E140</font></i></a>, <a class="txt" href="topalg_5.html#E3:40_5_1_1_2_3_8"><i><font color="Green">E143</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E7:40_5_1_1_2_3_8"/>
<font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>41</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E72:40_5_1_1_2_3"><i><font color="Green">E136</font></i></a>, <a class="txt" href="topalg_5.html#E67:40_5_1_1_2_3"><i><font color="Green">E133</font></i></a>, <a class="txt" href="topalg_5.html#E70:40_5_1_1_2_3"><i><font color="Green">E134</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E82:40_5_1_1_2_3"/><i><font color="Green">E143</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E80:40_5_1_1_2_3"><i><font color="Green">E141</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E83:40_5_1_1_2_3"/><i><font color="Green">E144</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E81:40_5_1_1_2_3"><i><font color="Green">E142</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E84:40_5_1_1_2_3"/>
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="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></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>41</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5_1_1_2_3_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>42</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:40_5_1_1_2_3_9"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:40_5_1_1_2_3_9"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E82:40_5_1_1_2_3"><i><font color="Green">E143</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_5_1_1_2_3_9"><i><font color="Green">E145</font></i></a>, <a class="txt" href="topalg_5.html#E77:40_5_1_1_2_3"><i><font color="Green">E139</font></i></a>, <a class="txt" href="topalg_5.html#E30:40_5"><i><font color="Green">E79</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E3:40_5_1_1_2_3_9"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E4:40_5_1_1_2_3_9"/><b>then </b><i><font color="Green">E147</font></i>: 
<font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E1:40_5_1_1_2_3_9"><i><font color="Green">E145</font></i></a>, <a class="txt" href="topalg_5.html#E82:40_5_1_1_2_3"><i><font color="Green">E143</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:40_5_1_1_2_3_9"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E82:40_5_1_1_2_3"><i><font color="Green">E143</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_5_1_1_2_3_9"><i><font color="Green">E145</font></i></a>, <a class="txt" href="topalg_5.html#E77:40_5_1_1_2_3"><i><font color="Green">E139</font></i></a>, <a class="txt" href="topalg_5.html#E13:40_5"><i><font color="Green">E65</font></i></a>, <a class="txt" href="topalg_5.html#E61:40_5_1_1_2_3"><i><font color="Green">E130</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E6:40_5_1_1_2_3_9"/><i><font color="Green">E149</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T89">RELAT_1:89</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>42</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>42</sub></font>
<b>by </b><i><a class="ref" href="relat_1.html#T112">RELAT_1:112</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>42</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E77:40_5_1_1_2_3"><i><font color="Green">E139</font></i></a>, <a class="txt" href="topalg_5.html#E82:40_5_1_1_2_3"><i><font color="Green">E143</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_5_1_1_2_3_9"><i><font color="Green">E145</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>41</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>42</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E30:40_5"><i><font color="Green">E79</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_5_1_1_2_3_9"><i><font color="Green">E145</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>42</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40_5_1_1_2_3_9"><i><font color="Green">E146</font></i></a>, <a class="txt" href="topalg_5.html#E50:40_5_1_1_2_3"><i><font color="Green">E126</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>42</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E1:40_5_1_1_2_3_9"><i><font color="Green">E145</font></i></a>, <a class="txt" href="topalg_5.html#E30:40_5"><i><font color="Green">E79</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>42</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E6:40_5_1_1_2_3_9"><i><font color="Green">E149</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_5_1_1_2_3_9"><i><font color="Green">E145</font></i></a>, <a class="txt" href="topalg_5.html#E82:40_5_1_1_2_3"><i><font color="Green">E143</font></i></a>, <a class="txt" href="topalg_5.html#E11:40_5"><i><font color="Green">E63</font></i></a>, <a class="txt" href="topalg_5.html#E8:40_5"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>42</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E6:40_5"><i><font color="Green">E58</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_5_1_1_2_3_9"><i><font color="Green">E147</font></i></a>, <a class="txt" href="topalg_5.html#E61:40_5_1_1_2_3"><i><font color="Green">E130</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>42</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E4:40_5_1_1_2_3_9"><i><font color="Green">E147</font></i></a>, <a class="txt" href="topalg_5.html#E13:40_5"><i><font color="Green">E65</font></i></a>, <a class="txt" href="topalg_5.html#E61:40_5_1_1_2_3"><i><font color="Green">E130</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>41</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>42</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_5.html#E5:40_5_1_1_2_3_9"><i><font color="Green">E148</font></i></a>, <a class="txt" href="topalg_5.html#E67:40_5_1_1_2_3"><i><font color="Green">E133</font></i></a>, <a class="txt" href="topalg_5.html#E70:40_5_1_1_2_3"><i><font color="Green">E134</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>42</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E4:40_5_1_1_2_3_9"><i><font color="Green">E147</font></i></a>, <a class="txt" href="topalg_5.html#E13:40_5"><i><font color="Green">E65</font></i></a>, <a class="txt" href="topalg_5.html#E61:40_5_1_1_2_3"><i><font color="Green">E130</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>42</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E77:40_5_1_1_2_3"><i><font color="Green">E139</font></i></a>, <a class="txt" href="topalg_5.html#E82:40_5_1_1_2_3"><i><font color="Green">E143</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_5_1_1_2_3_9"><i><font color="Green">E145</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>41</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>42</sub></font>
<b>by </b><i><a class="ref" href="relat_1.html#T112">RELAT_1:112</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E85:40_5_1_1_2_3"/><b>then </b><i><font color="Green">E145</font></i>: 
<font color="Maroon">c<sub>41</sub></font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>41</sub></font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E77:40_5_1_1_2_3"><i><font color="Green">E139</font></i></a>, <a class="txt" href="topalg_5.html#E79:40_5_1_1_2_3"><i><font color="Green">E140</font></i></a>, <a class="txt" href="topalg_5.html#E82:40_5_1_1_2_3"><i><font color="Green">E143</font></i></a>, <a class="txt" href="topalg_5.html#E83:40_5_1_1_2_3"><i><font color="Green">E144</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_1_2_3_10"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_5_1_1_2_3_10"/>
( <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>26</sub></font> or  not <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>26</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_1_2_3_10_1"><b>suppose </b></a><a NAME="E1:40_5_1_1_2_3_10_1"/>
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>26</sub></font>
 ;<br/><div class="add"><a NAME="E2:40_5_1_1_2_3_10_1"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E5:40_5_1_1_2_3"><i><font color="Green">E97</font></i></a>, <a class="txt" href="topalg_5.html#E45:40_5_1_1_2_3"><i><font color="Green">E123</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>28</sub></font> = 
<font color="Maroon">c<sub>18</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E13:40_5_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_5_1_1_2_3"><i><font color="Green">E94</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40_5_1_1_2_3_10_1"><i><font color="Green">E146</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E85:40_5_1_1_2_3"><i><font color="Green">E145</font></i></a>, <a class="txt" href="topalg_5.html#E76:40_5_1_1_2_3"><i><font color="Green">E138</font></i></a>, <a class="txt" href="topalg_5.html#E80:40_5_1_1_2_3"><i><font color="Green">E141</font></i></a>, <a class="txt" href="topalg_5.html#E81:40_5_1_1_2_3"><i><font color="Green">E142</font></i></a>, <a class="txt" href="topalg_5.html#E77:40_5_1_1_2_3"><i><font color="Green">E139</font></i></a>, <a class="txt" href="topalg_5.html#E79:40_5_1_1_2_3"><i><font color="Green">E140</font></i></a>, <a class="ref" href="funct_1.html#T49">FUNCT_1:49</a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40_5_1_1_2_3_10_1"><i><font color="Green">E146</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E13:40_5_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_5_1_1_2_3"><i><font color="Green">E94</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_5_1_1_2_3_10_2"><b>suppose </b></a><a NAME="E1:40_5_1_1_2_3_10_2"/>
not <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>26</sub></font>
 ;<br/><div class="add"><a NAME="E2:40_5_1_1_2_3_10_2"/><b>then </b>
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E4:40_5_1_1_2_3"><i><font color="Green">E96</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:40_5_1_1_2_3_10_2"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>24</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E3:40_5_1_1_2_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_5_1_1_2_3"><i><font color="Green">E97</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>28</sub></font> = 
<font color="Maroon">c<sub>18</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E13:40_5_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_5_1_1_2_3"><i><font color="Green">E94</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>24</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40_5_1_1_2_3_10_2"><i><font color="Green">E146</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E3:40_5_1_1_2_3_10_2"><i><font color="Green">E146</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_5_1_1_2"><i><font color="Green">E86</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E13:40_5_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_5_1_1_2_3"><i><font color="Green">E94</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E17:40_5_1_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>27</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E13:40_5_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="topalg_5.html#E15:40_5_1_1_2"><i><font color="Green">E93</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E33:40_5"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>2</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="topalg_5.html#E31:40_5"><i><font color="Green">E80</font></i></a>, <a class="txt" href="topalg_5.html#E32:40_5"><i><font color="Green">E81</font></i></a>);<br/></i>
<a NAME="E34:40_5"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Olive">b<sub>1</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Olive">b<sub>1</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> )
 
<b>by </b><i><a class="txt" href="topalg_5.html#E24:40_5"><i><font color="Green">E73</font></i></a></i>;<br/>
<b>hence </b><a NAME="E35:40_5"/>
<font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E16:40_5"><i><font color="Green">E67</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:40"/>
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>8</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a>  st <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <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#V3">open</a> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Olive">b<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_6"><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 <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>;<br/><b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topalg_2.html#K3">R^1</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:40_6"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E2:40_6"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>14</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>16</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">c<sub>17</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>18</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E4:40_6"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E5:40_6"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E6:40_6"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E7:40_6"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>18</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E8:40_6"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>18</sub></font>
 <b>and </b><br/><a NAME="E9:40_6"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E10:40_6"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="partfun1.html#K2">|</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#R2">=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E14:40"><i><font color="Green">E49</font></i></a></i>;<br/>
<i><font color="Green">E62</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> = 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </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/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>
;<br/>
<a NAME="E12:40_6"/><b>then </b>
<font color="Maroon">c<sub>13</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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E4:40_6"><i><font color="Green">E56</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_6"><i><font color="Green">E58</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E14:40_6"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E15:40_6"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>19</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E16:40_6"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:40_6"><i><font color="Green">E54</font></i></a>, <a class="txt" href="topalg_5.html#E2:40_6"><i><font color="Green">E55</font></i></a>, <a class="txt" href="topalg_5.html#E7:40_6"><i><font color="Green">E59</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_6"><i><font color="Green">E57</font></i></a>, <a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>
<a NAME="E17:40_6"/><i><font color="Green">E66</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E11:40_6"><i><font color="Green">E62</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E18:40_6"/>
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E8"><i><font color="Green">Lemma7</font></i></a>, <a class="ref" href="borsuk_3.html#T26">BORSUK_3:26</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E20:40_6"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>20</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E21:40_6"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E15:40_6"><i><font color="Green">E64</font></i></a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>;<br/>

<a NAME="E22:40_6"/><i><font color="Green">E69</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E23:40_6"/><b>then </b><i><font color="Green">E70</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_3.html#T1">BORSUK_3:1</a></i>;<br/>
<b>thus </b><a NAME="E24:40_6"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E21:40_6"><i><font color="Green">E68</font></i></a>, <a class="txt" href="topalg_5.html#E14:40_6"><i><font color="Green">E63</font></i></a>, <a class="txt" href="topalg_5.html#E22:40_6"><i><font color="Green">E69</font></i></a>, <a class="ref" href="borsuk_3.html#T1">BORSUK_3:1</a></i>;<br/>

<a NAME="E25:40_6"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K6">:]</a></span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T46">BORSUK_1:46</a></i>;<br/>
<b>hence </b><a NAME="E26:40_6"/>
<font color="Maroon">c<sub>21</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E20:40_6"><i><font color="Green">E67</font></i></a>, <a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>22</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="E27:40_6"/>
<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="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>23</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>8</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E29:40_6"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font>
 <b>and </b><br/><a NAME="E30:40_6"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</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="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>24</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>25</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">c<sub>26</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>27</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E32:40_6"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E33:40_6"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>12</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>27</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font>
 <b>and </b><br/><a NAME="E34:40_6"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>26</sub></font>
 <b>and </b><br/><a NAME="E35:40_6"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>27</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E36:40_6"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>26</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>27</sub></font>
 <b>and </b><br/><a NAME="E37:40_6"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="partfun1.html#K2">|</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>26</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E38:40_6"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>26</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="partfun1.html#K2">|</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>26</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Maroon">c<sub>27</sub></font> <a href="funct_2.html#R2">=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E14:40"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E39:40_6"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E29:40_6"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E40:40_6"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E32:40_6"><i><font color="Green">E73</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E41:40_6"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>24</sub></font></span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E32:40_6"><i><font color="Green">E73</font></i></a>, <a class="ref" href="zfmisc_1.html#T128">ZFMISC_1:128</a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>18</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>24</sub></font></span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>24</sub></font></span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E40:40_6"><i><font color="Green">E79</font></i></a>, <a class="txt" href="topalg_5.html#E32:40_6"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topalg_5.html#E7:40_6"><i><font color="Green">E59</font></i></a>, <a class="txt" href="topalg_5.html#E34:40_6"><i><font color="Green">E75</font></i></a>, <a class="txt" href="topalg_5.html#E35:40_6"><i><font color="Green">E76</font></i></a>, <a class="txt" href="topalg_5.html#E8:40_6"><i><font color="Green">E60</font></i></a>, <a class="txt" href="topalg_5.html#E9:40_6"><i><font color="Green">E61</font></i></a>, <a class="txt" href="topalg_5.html#E36:40_6"><i><font color="Green">E77</font></i></a>, <a class="txt" href="topalg_5.html#E37:40_6"><i><font color="Green">E78</font></i></a>, <a class="txt" href="topalg_5.html#E18:40"><i><font color="Green">E53</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>27</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E41:40_6"><i><font color="Green">E80</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>
<a NAME="E43:40_6"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E30:40_6"><i><font color="Green">E72</font></i></a>, <a class="txt" href="topalg_5.html#E33:40_6"><i><font color="Green">E74</font></i></a>, <a class="txt" href="topalg_5.html#E29:40_6"><i><font color="Green">E71</font></i></a>, <a class="txt" href="topalg_5.html#E21:40_6"><i><font color="Green">E68</font></i></a>, <a class="txt" href="topalg_5.html#E23:40_6"><i><font color="Green">E70</font></i></a>, <a class="txt" href="topalg_5.html#E17:40_6"><i><font color="Green">E66</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E44:40_6"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E16:40_6"><i><font color="Green">E65</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E20:40"/>
<font color="Maroon">c<sub>12</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>

<b>thus </b><a NAME="E21:40"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E16:40"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>

<a NAME="E22:40"/><i><font color="Green">E54</font></i>: 
<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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</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>8</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3"><i><font color="Green">Lemma3</font></i></a>, <a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E23:40"/><i><font color="Green">E55</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="subset_1.html#K8">/\</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E15:40"><i><font color="Green">E50</font></i></a>, <a class="txt" href="topalg_5.html#E22:40"><i><font color="Green">E54</font></i></a>, <a class="txt" href="topalg_5.html#E6:40"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E24:40"/>
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="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> holds <br/><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_7"><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> ;<br/>

<b>assume </b><a NAME="E1:40_7"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E2:40_7"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E15:40"><i><font color="Green">E50</font></i></a>, <a class="txt" href="topalg_5.html#E22:40"><i><font color="Green">E54</font></i></a>, <a class="txt" href="topalg_5.html#E6:40"><i><font color="Green">E43</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>15</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">c<sub>16</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>17</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E4:40_7"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E5:40_7"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E6:40_7"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E7:40_7"/>
( <font color="Maroon">c<sub>17</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>and </b><br/><a NAME="E8:40_7"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="partfun1.html#K2">|</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E9:40_7"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="partfun1.html#K2">|</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#R2">=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E14:40"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E10:40_7"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E4:40_7"><i><font color="Green">E57</font></i></a>, <a class="txt" href="topalg_5.html#E6:40_7"><i><font color="Green">E59</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E39"><i><font color="Green">Lemma33</font></i></a>, <a class="txt" href="topalg_5.html#E5:40"><i><font color="Green">E42</font></i></a>, <a class="txt" href="topalg_5.html#E1:40_7"><i><font color="Green">E56</font></i></a>, <a class="txt" href="topalg_5.html#E8:40_7"><i><font color="Green">E60</font></i></a>, <a class="txt" href="topalg_5.html#E5:40_7"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E25:40"/>
<font color="Maroon">c<sub>12</sub></font> <a href="partfun1.html#K2">|</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E23:40"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>13</sub></font> be   <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>8</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E26:40"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>13</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E27:40"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#R2">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E28:40"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="partfun1.html#K2">|</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E29:40"/>
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>8</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> holds <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</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="40_8"><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 <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>;<br/>

<b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>16</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">c<sub>17</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>18</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a> <b> such that </b><br/><a NAME="E2:40_8"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="borsuk_1.html#K7">]</a></span>
 <b>and </b><br/><a NAME="E3:40_8"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E4:40_8"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E5:40_8"/>
( <font color="Maroon">c<sub>18</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>and </b><br/><a NAME="E6:40_8"/>
<font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K2">|</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E7:40_8"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="partfun1.html#K2">|</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#R2">=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E14:40"><i><font color="Green">E49</font></i></a></i>;<br/>
<i><font color="Green">E63</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> = 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </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/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>
;<br/>
<a NAME="E9:40_8"/><b>then </b><i><font color="Green">E64</font></i>: 
<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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E2:40_8"><i><font color="Green">E59</font></i></a>, <a class="txt" href="topalg_5.html#E4:40_8"><i><font color="Green">E61</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E10:40_8"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font> <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>8</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E11:40_8"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E4:40"><i><font color="Green">E41</font></i></a>, <a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E12:40_8"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E13:40_8"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topalg_2.html#K3">R^1</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<a href="topalg_2.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E12:40_8"><i><font color="Green">E65</font></i></a>, <a class="txt" href="topalg_5.html#E8:40_8"><i><font color="Green">E63</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E15:40_8"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>19</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E26:40"><i><font color="Green">E56</font></i></a>, <a class="ref" href="topalg_3.html#T18">TOPALG_3:18</a></i>;<br/>
<i><font color="Green">E67</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="partfun1.html#K2">|</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>8</sub></font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> = 
<font color="Maroon">c<sub>13</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2"><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>8</sub></font>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T100">RELAT_1:100</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="txt" href="topalg_5.html#E28:40"><i><font color="Green">E58</font></i></a>, <a class="ref" href="relat_1.html#T100">RELAT_1:100</a></i>
;<br/>
<a NAME="E17:40_8"/>
<font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12">CircleMap</a>  <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E27:40"><i><font color="Green">E57</font></i></a>, <a class="ref" href="relat_1.html#T112">RELAT_1:112</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E7:40_8"><i><font color="Green">E62</font></i></a>, <a class="txt" href="topalg_5.html#E3:40_8"><i><font color="Green">E60</font></i></a>, <a class="txt" href="topalg_5.html#E15:40_8"><i><font color="Green">E66</font></i></a>, <a class="txt" href="topalg_5.html#E16:40_8"><i><font color="Green">E67</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E8:40_8"><i><font color="Green">E63</font></i></a>, <a class="txt" href="topalg_5.html#E9:40_8"><i><font color="Green">E64</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E30:40"/>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div>
