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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">p1</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">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">p2</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">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">g</font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">p1</font>,<font color="Maroon">p2</font>;<br/>







<b>assume </b><a NAME="E1:7"/><i><font color="Green">E39</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/>

<a NAME="E2:7"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 )
 
<b>by </b><i><a class="ref" href="borsuk_2.html#D4">BORSUK_2:def 4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:7_1"/><i><font color="Green">E41</font></i>: 
not  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 ;<br/><a NAME="E2:7_1"/>
 <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a>  is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="connsp_1.html#T28">CONNSP_1:28</a></i>;<br/><a NAME="E3:7_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a> </span>)</span> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T75">TOPS_2:75</a></i>;<br/><a NAME="E4:7_1"/>
0 <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#T86">BORSUK_1:86</a></i>;<br/><a NAME="E5:7_1"/><b>then </b><i><font color="Green">E46</font></i>: 
0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E6:7_1"/>
the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a>  <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="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/><a NAME="E7:7_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a> </span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E8:7_1"/>
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#T86">BORSUK_1:86</a></i>;<br/><a NAME="E9:7_1"/><b>then </b><i><font color="Green">E48</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E10:7_1"/>
the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a>  <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="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/><a NAME="E11:7_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a> </span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E12:7_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:7_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a> </span>)</span>
 ;<br/>

<a NAME="E2:7_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>thus </b><a NAME="E3:7_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:7_1"/><b>then </b>
<font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E14:7_1"/>
contradiction
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:7"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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