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<b>let </b><font color="Maroon">a</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> 2</span>)</span>;<br/><b>let </b><font color="Maroon">b</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> 2</span>)</span>;<br/><b>let </b><font color="Maroon">c</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> 2</span>)</span>;<br/><b>let </b><font color="Maroon">d</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> 2</span>)</span>;<br/><b>let </b><font color="Maroon">P</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> 2</span>)</span>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:6"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E2:6"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">P</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>and </b><br/><a NAME="E3:6"/><i><font color="Green">E34</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">P</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 ;<br/>

<a NAME="E4:6"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/>
<a NAME="E5:6"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">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> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span>, <font color="Maroon">rb</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:6"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>and </b><br/><a NAME="E8:6"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E9:6"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 <b>and </b><br/><a NAME="E10:6"/><i><font color="Green">E39</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">rb</font>
 <b>and </b><br/><a NAME="E11:6"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">rb</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>and </b><br/><a NAME="E12:6"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">rb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/>
<i><font color="Green">E44</font></i>:  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> = 
 <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> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span>
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
<br/>.= 
<font color="Maroon">P</font>
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>
;<br/>
<b>then consider </b><font color="Maroon">ra</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:6"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">ra</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E16:6"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ra</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<i><font color="Green">E48</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> = 
 <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a> 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<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="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">ra</font> = <font color="Maroon">ra</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<a NAME="E19:6"/><i><font color="Green">E49</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ra</font>
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E20:6"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">ra</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E21:6"/><b>then </b>
<font color="Maroon">ra</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">rb</font>
 
<b>by </b><i>, , , , , , , , , <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/>
<a NAME="E22:6"/><b>then </b>
<font color="Maroon">ra</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">rb</font>
 
<b>by </b><i>, , , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">re</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E24:6"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">ra</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">re</font>
 <b>and </b><br/><a NAME="E25:6"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">re</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">rb</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T7">XREAL_1:7</a></i>;<br/>
<a NAME="E26:6"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">re</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E27:6"/><i><font color="Green">E55</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">re</font> &amp; <font color="Maroon">re</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i>, , , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>set </b><font color="Maroon">e</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">re</font>;<br/>
<a NAME="E28:6"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">re</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E29:6"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">re</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">re</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">e</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:6_3"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 ;<br/><a NAME="E2:6_3"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><a NAME="E3:6_3"/>
( <font color="Maroon">ra</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">rb</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i>, , , , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>hence </b><a NAME="E4:6_3"/>
contradiction
 <b>by </b><i>, , , , , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E32:6"/>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">e</font> )
 ;<br/>

<b>thus </b><a NAME="E33:6"/>
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">a</font>,<font color="Maroon">e</font>,<font color="Maroon">P</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">e</font>,<font color="Maroon">b</font>,<font color="Maroon">P</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font> )
 <b>by </b><i>, , , , , , , , , , , , , , <a class="ref" href="jordan5c.html#T8">JORDAN5C:8</a></i>;<br/>


</div>
