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

<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">c</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">d</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">O</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">I</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>





<b>assume </b><a NAME="E1:55"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">O</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">O</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">I</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">O</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">I</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; ( for <font color="Olive">r</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds <br/> ( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> ) ) )
 ;<br/>

<i><font color="Green">E35</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> = 
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="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>
<a NAME="E3:55"/><i><font color="Green">E36</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a>  <a href="hidden.html#R1">=</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="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E4:55"/><b>then </b>
0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">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>then reconsider </b><font color="Maroon">P</font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</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> 2</span>)</span> <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> = 
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="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>
<a NAME="E7:55"/><b>then </b>
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="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Q</font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</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> 2</span>)</span> <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E9:55"/><i><font color="Green">E37</font></i>: 
 <a href="topmetr.html#K5">I[01]</a>  is <a href="compts_1.html#V2">compact</a>
 
<b>by </b><i><a class="ref" href="heine.html#T11">HEINE:11</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<a NAME="E10:55"/><i><font color="Green">E38</font></i>: 
 <a href="euclid.html#K15">TOP-REAL</a> 2 <a href="compts_1.html#NR2" target="_self">is_T2</a> 
 
<b>by </b><i><a class="ref" href="sppol_1.html#T26">SPPOL_1:26</a></i>;<br/>
<b>then consider </b><font color="Maroon">f1</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">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><b> such that </b><br/><a NAME="E12:55"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">f1</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  )
 <b>by </b><i>, , <a class="ref" href="jgraph_1.html#D2" target="_self">Def2</a></i>;<br/>
<b>consider </b><font color="Maroon">g1</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">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">Q</font></span>)</span><b> such that </b><br/><a NAME="E14:55"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">g</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">g1</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  )
 <b>by </b><i>, , <a class="ref" href="jgraph_1.html#T8" target="_self">Th8</a>, <a class="ref" href="jgraph_1.html#D2" target="_self">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">O</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>, <a class="txt" href="jgraph_1.html#E21:55"><i><font color="Green">E54</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">p2</font> = <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">I</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>, <a class="txt" href="jgraph_1.html#E21:55"><i><font color="Green">E54</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">q1</font> = <font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">O</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>, <a class="ref" href="jgraph_1.html#T9" target="_self">Th9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q2</font> = <font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">I</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>, <a class="ref" href="jgraph_1.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E19:55"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">P</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 
<b>by </b><i>, <a class="txt" href="jgraph_1.html#E21:55"><i><font color="Green">E54</font></i></a>, <a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/>
<a NAME="E20:55"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">Q</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font>
 
<b>by </b><i>, <a class="ref" href="jgraph_1.html#T9" target="_self">Th9</a>, <a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/>
<a NAME="E21:55"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">p</font> being  <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>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/>( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</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>assume </b><a NAME="E1:55_3"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:55_3"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E21:55"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">x</font> as   <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="jgraph_1.html#E3:55_3"><i><font color="Green">E59</font></i></a></i>;<br/>
<a NAME="E5:55_3"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E21:55"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jgraph_1.html#E3:55_3"><i><font color="Green">E59</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:55_3"/>
( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, <a class="txt" href="jgraph_1.html#E21:55"><i><font color="Green">E54</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:55"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">p</font> being  <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>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> holds <br/>( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</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>assume </b><a NAME="E1:55_4"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:55_4"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="jgraph_1.html#T9" target="_self">Th9</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">x</font> as   <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="jgraph_1.html#E22:55"><i><font color="Green">E60</font></i></a></i>;<br/>
<a NAME="E5:55_4"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T9" target="_self">Th9</a>, <a class="txt" href="jgraph_1.html#E22:55"><i><font color="Green">E60</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:55_4"/>
( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, <a class="txt" href="jgraph_1.html#E21:55"><i><font color="Green">E54</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:55"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">p</font> being  <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>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/>( <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</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>assume </b><a NAME="E1:55_5"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:55_5"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E21:55"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i/>;<br/>
<a NAME="E5:55_5"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E21:55"><i><font color="Green">E54</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E6:55_5"/>
( <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i>, <a class="ref" href="jgraph_1.html#T9" target="_self">Th9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E24:55"/>
for <font color="Olive">p</font> being  <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>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> holds <br/>( <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</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>assume </b><a NAME="E1:55_6"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:55_6"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="jgraph_1.html#T9" target="_self">Th9</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i/>;<br/>
<a NAME="E5:55_6"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T9" target="_self">Th9</a>, </i>;<br/>
<b>hence </b><a NAME="E6:55_6"/>
( <font color="Maroon">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i>, <a class="ref" href="jgraph_1.html#T9" target="_self">Th9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E25:55"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="jgraph_1.html#T10" target="_self">Th10</a>, , , , <a class="txt" href="jgraph_1.html#E22:55"><i><font color="Green">E60</font></i></a>, <a class="ref" href="jgraph_1.html#T5" target="_self">Th5</a></i>;<br/>


</div>
