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<b>let </b><font color="Maroon">x</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">a</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> , <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>let </b><font color="Maroon">u</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="E1:82"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</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>,<span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></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/>

<a NAME="E2:82"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">u</font> <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>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> or <font color="Maroon">u</font> <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#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">u</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/>
<a NAME="E4:82"/>
(  <a href="topreal9.html#K1">Ball</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  &amp; <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#R2">in</a>  <a href="topgen_5.html#K2" target="_self">y&gt;=0-plane</a>  )
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topgen_5.html#K3" target="_self">Niemytzki-plane</a>  <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:82"/>
<span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> 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#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E7:82"/><b>then </b><i><font color="Green">E49</font></i>: 
( <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">q</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:82_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 or ( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">u</font> <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>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> ) or <font color="Maroon">u</font> <a href="hidden.html#R1">=</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> )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1"><b>suppose </b></a><a NAME="E1:82_1_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:82_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">q</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:82_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Maroon">a</font></span><a href="rcomp_2.html#K1">.[</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/><b>hence </b><a NAME="E4:82_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Maroon">a</font></span><a href="rcomp_2.html#K1">.[</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_2"><b>suppose </b></a><a NAME="E1:82_1_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">u</font> <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>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> )
 ;<br/><div class="add"><a NAME="E2:82_1_2"/><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> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">x</font>,<span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></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="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T7">TOPREAL9:7</a></i>;<br/><a NAME="E3:82_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:82_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Maroon">a</font></span><a href="rcomp_2.html#K1">.[</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/><b>hence </b><a NAME="E5:82_1_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Maroon">a</font></span><a href="rcomp_2.html#K1">.[</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_3"><b>suppose </b></a><a NAME="E1:82_1_3"/>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</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>
 ;<br/><div class="add"><a NAME="E2:82_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><a NAME="E3:82_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Maroon">a</font></span><a href="rcomp_2.html#K1">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/><b>hence </b><a NAME="E4:82_1_3"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topgen_5.html#K4" target="_self">+</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default">0,<font color="Maroon">a</font></span><a href="rcomp_2.html#K1">.[</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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