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<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> ;<br/><b>let </b><font color="Maroon">r1</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">r2</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="E1:33"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font>
 ;<br/>

<b>let </b><font color="Maroon">x</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="E2:33"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</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">a</font>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">r1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E4:33"/>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">r2</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r2</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">r1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r2</font> is   <a href="real_1.html#NM1">Real</a> )
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a>, <a class="ref" href="euclid.html#T66">EUCLID:66</a></i>;<br/>
<a NAME="E5:33"/><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"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">r2</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r2</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i><a class="ref" href="topreal6.html#T31">TOPREAL6:31</a></i>;<br/>
<a NAME="E6:33"/><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"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">r2</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r1</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> &amp; <font color="Maroon">r2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a>, <a class="ref" href="complex1.html#T146">COMPLEX1:146</a></i>;<br/>
<a NAME="E7:33"/><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"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">r2</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r1</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">r1</font></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">r1</font> )
 
<b>by </b><i>, <a class="ref" href="topreal9.html#T7">TOPREAL9:7</a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E8:33"/><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">x</font> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span></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"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">r2</font></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">r1</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r1</font></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">x</font> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">r2</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span></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"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">r2</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> )
 
<b>by </b><i><a class="ref" href="toprns_1.html#T35">TOPRNS_1:35</a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E9:33"/><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">x</font> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">r2</font></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">r2</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E10:33"/>
<font color="Maroon">x</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">a</font>,<font color="Maroon">r2</font></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">r2</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T7">TOPREAL9:7</a></i>;<br/>


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