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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">W</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:44"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> )
 ;<br/>

<b>assume </b><a NAME="E2:44"/>
<font color="Maroon">W</font> is <a href="tbsp_1.html#V6">bounded</a>
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:44"/><i><font color="Green">E56</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> ) )
 <b>by </b><i><a class="ref" href="tbsp_1.html#D9">TBSP_1:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x0</font> = <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K5" target="_self">1.REAL</a> <font color="Maroon">n</font></span>)</span> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y0</font> =  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> ;<br/>
<a NAME="E7:44"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x0</font>,<font color="Maroon">y0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:44"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><a href="jordan2c.html#K5" target="_self">1.REAL</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T45">JGRAPH_1:45</a></i>;<br/>
<a NAME="E9:44"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><a href="jordan2c.html#K5" target="_self">1.REAL</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:44"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><a href="jordan2c.html#K5" target="_self">1.REAL</a> <font color="Maroon">n</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="toprns_1.html#T8">TOPRNS_1:8</a></i>;<br/>
<a NAME="E11:44"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i/>;<br/>
<a NAME="E12:44"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E13:44"/><b>then </b><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E14:44"/>
 <a href="square_1.html#K7">sqrt</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="square_1.html#T94">SQUARE_1:94</a></i>;<br/>
<a NAME="E15:44"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a>, <a class="ref" href="square_1.html#T83">SQUARE_1:83</a></i>;<br/>
<a NAME="E16:44"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E17:44"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K9">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E18:44"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> 0
 
;<br/>
<b>hence </b><a NAME="E19:44"/>
contradiction
 ;<br/>


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