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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:43"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>2</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="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <a href="borsuk_1.html#K20">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:43"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> is <a href="tops_2.html#V3">being_homeomorphism</a> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="jordan6.html#E1:43"><i><font color="Green">E31</font></i></a>, <a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E6:43"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="tops_2.html#V3">being_homeomorphism</a>
 
<b>by </b><i><a class="txt" href="jordan6.html#E4:43"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tops_2.html#T70">TOPS_2:70</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<a href="borsuk_1.html#K20">I[01]</a>  ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="borsuk_1.html#K20">I[01]</a>  ;<br/>
<a NAME="E10:43"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>11</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="jordan6.html#T41" target="_self">Th41</a></i>;<br/>
<a NAME="E11:43"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="jordan6.html#E4:43"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E12:43"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="jordan6.html#E6:43"><i><font color="Green">E33</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E13:43"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="jordan6.html#E4:43"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E14:43"/><i><font color="Green">E38</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a>  <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E4:43"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E15:43"/>
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E16:43"/><b>then </b>
0 <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/>
<a NAME="E17:43"/><b>then </b><i><font color="Green">E39</font></i>: 
0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E4:43"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E18:43"/>
1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E19:43"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a> 
 
;<br/>
<a NAME="E20:43"/><b>then </b><i><font color="Green">E40</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E4:43"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E21:43"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</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="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan6.html#E4:43"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E22:43"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="tops_2.html#NK3" target="_self">"</a>  <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E11:43"><i><font color="Green">E35</font></i></a>, <a class="ref" href="tops_2.html#T64">TOPS_2:64</a></i>;<br/>
<a NAME="E23:43"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a> 
 
<b>by </b><i><a class="txt" href="jordan6.html#E6:43"><i><font color="Green">E33</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E24:43"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K2">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E12:43"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan6.html#E22:43"><i><font color="Green">E41</font></i></a>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/>
<font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="relat_1.html#K10">"</a> the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a> </span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T138">FUNCT_1:138</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_1.html#K9">.:</a> the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a> </span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="jordan6.html#E12:43"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#T155">FUNCT_1:155</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="relat_1.html#K9">.:</a> the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a> </span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="jordan6.html#E12:43"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan6.html#E24:43"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#T155">FUNCT_1:155</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a> </span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="relat_1.html#K9">.:</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="jordan6.html#E24:43"><i><font color="Green">E42</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="jordan6.html#E14:43"><i><font color="Green">E38</font></i></a>, <a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="jordan6.html#E4:43"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="jordan6.html#E1:43"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan6.html#E4:43"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan6.html#E17:43"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan6.html#E20:43"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_1.html#T118">FUNCT_1:118</a></i>
;<br/>
<b>hence </b><a NAME="E26:43"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="jordan6.html#E10:43"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan6.html#E13:43"><i><font color="Green">E37</font></i></a>, <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>


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