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

<b>set </b><font color="Maroon">T</font> =  <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a> ;<br/>
<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a> , <font color="Maroon">q</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="urysohn1.html#D9">URYSOHN1:def 9</a><br/>

<b>assume </b><a NAME="E1:57"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E2:57"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a>  &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a>  &amp; the <a href="struct_0.html#U1">carrier</a> of <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a>  <a href="hidden.html#R1">=</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  )
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</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> ;<br/>
<a NAME="E4:57"/>
<font color="Maroon">p'</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> 2
 
<b>by </b><i><a class="txt" href="topgen_5.html#E1:57"><i><font color="Green">E42</font></i></a>, <a class="ref" href="euclid.html#T47">EUCLID:47</a></i>;<br/>
<a NAME="E5:57"/><b>then </b>
( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q'</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q'</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i><a class="ref" href="euclid_2.html#T60">EUCLID_2:60</a>, <a class="ref" href="euclid_2.html#T64">EUCLID_2:64</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">r</font> = <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q'</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> as  <a href="xxreal_0.html#V2">positive</a> <a href="real_1.html#NM1">Real</a> ;<br/>
<b>consider </b><font color="Maroon">ap</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>, <font color="Maroon">Up</font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a> <b> such that </b><br/><a NAME="E8:57"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Up</font> &amp; <font color="Maroon">ap</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Up</font> &amp; ( for <font color="Olive">b</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">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Up</font> holds <br/><span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b</font> <a href="euclid.html#K20">-</a> <font color="Maroon">ap</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2 ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">aq</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>, <font color="Maroon">Uq</font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a> <b> such that </b><br/><a NAME="E10:57"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Uq</font> &amp; <font color="Maroon">aq</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Uq</font> &amp; ( for <font color="Olive">b</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">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Uq</font> holds <br/><span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b</font> <a href="euclid.html#K20">-</a> <font color="Maroon">aq</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2 ) )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">Up</font>
;<br/>

<b>take </b>
<font color="Maroon">Uq</font>
;<br/>

<b>thus </b><a NAME="E11:57"/>
( <font color="Maroon">Up</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">Uq</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Up</font> )
 <b>by </b><i><a class="txt" href="topgen_5.html#E8:57"><i><font color="Green">E43</font></i></a></i>;<br/>

<b>thus </b><a NAME="E12:57"/>
not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Up</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Up</font>
 ;<br/>

<a NAME="E2:57_1"/><b>then </b>
( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q'</font> <a href="euclid.html#K20">-</a> <font color="Maroon">ap</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ap</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q'</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q'</font> <a href="euclid.html#K20">-</a> <font color="Maroon">ap</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2 &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="euclid.html#K20">-</a> <font color="Maroon">ap</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2 )
 
<b>by </b><i><a class="txt" href="topgen_5.html#E8:57"><i><font color="Green">E43</font></i></a>, <a class="ref" href="toprns_1.html#T28">TOPRNS_1:28</a></i>;<br/>
<a NAME="E3:57_1"/><b>then </b>
( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="euclid.html#K20">-</a> <font color="Maroon">ap</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ap</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q'</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="euclid.html#K20">-</a> <font color="Maroon">ap</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ap</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q'</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a>, <a class="ref" href="toprns_1.html#T35">TOPRNS_1:35</a></i>;<br/>
<b>hence </b><a NAME="E4:57_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E13:57"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Uq</font>
 <b>by </b><i/>;<br/>

<b>assume </b><a NAME="E14:57"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Uq</font>
 ;<br/>

<a NAME="E15:57"/><b>then </b>
( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q'</font> <a href="euclid.html#K20">-</a> <font color="Maroon">aq</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">aq</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q'</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q'</font> <a href="euclid.html#K20">-</a> <font color="Maroon">aq</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2 &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="euclid.html#K20">-</a> <font color="Maroon">aq</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2 )
 
<b>by </b><i>, <a class="ref" href="toprns_1.html#T28">TOPRNS_1:28</a></i>;<br/>
<a NAME="E16:57"/><b>then </b>
( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="euclid.html#K20">-</a> <font color="Maroon">aq</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">aq</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q'</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="euclid.html#K20">-</a> <font color="Maroon">aq</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">aq</font> <a href="euclid.html#K20">-</a> <font color="Maroon">q'</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a>, <a class="ref" href="toprns_1.html#T35">TOPRNS_1:35</a></i>;<br/>
<b>hence </b><a NAME="E17:57"/>
contradiction
 ;<br/>


</div>
