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<div class="add">

<b>let </b><font color="Maroon">U</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">r</font> be  <a href="xxreal_0.html#V2">positive</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>





<b>assume </b><a NAME="E1:92"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">U</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1">Ball</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">r</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<b>take </b><font color="Maroon">f</font> =  <a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>;<br/>

<b>thus </b><a NAME="E2:92"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,0</span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<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> ;<br/>

<b>thus </b><a NAME="E3:92"/>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> <a href="subset_1.html#K3">`</a>  implies <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:92_1"/><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<a NAME="E2:92_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:92_1"/>
not <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font>
 
<b>by </b><i><a class="ref" href="topgen_5.html#T11" target="_self">Th11</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:92_1"/><b>then </b>
( not <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">r</font> &amp; ( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> or <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) )
 
<b>by </b><i><a class="ref" href="topgen_5.html#T10" target="_self">Th10</a>, <a class="ref" href="setwiseo.html#T6">SETWISEO:6</a></i>;<br/>
<b>hence </b><a NAME="E5:92_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="topgen_5.html#T12" target="_self">Th12</a>, </i>;<br/>


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

<b>assume </b><a NAME="E4:92"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E5:92"/><b>then </b>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> &amp; not <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,0</span><a href="euclid.html#K23">]|</a></span></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E6:92"/><b>then </b>
( <font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">r</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">r</font> )
 
<b>by </b><i><a class="ref" href="topgen_5.html#T10" target="_self">Th10</a>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:92"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,0</span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i/>;<br/>


</div>
