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<b>let </b><font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">d</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>set </b><font color="Maroon">c</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 4;<br/>
<b>assume </b><a NAME="E1:45"/>
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 4</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">d</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:45"/><b>then </b>
2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 4</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> 1
 
;<br/>
<a NAME="E3:45"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 4</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">d</font>
 
;<br/>
<a NAME="E4:45"/><b>then </b><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 4 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0
 
;<br/>
<a NAME="E5:45"/>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">d</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E6:45"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 3 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 3
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E7:45"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 3 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
;<br/>
<b>hence </b><a NAME="E8:45"/>
contradiction
 <b>by </b><i><a class="ref" href="scmpds_9.html#T3" target="_self">Th3</a></i>;<br/>


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