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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> , <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">c</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>assume </b><a NAME="E1:53"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:53"/><i><font color="Green">E40</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="square_1.html#K3">^2</a>  &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="square_1.html#K3">^2</a>  &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> <a href="square_1.html#K3">^2</a>  &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> <a href="square_1.html#K3">^2</a>  )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<b>assume </b><a NAME="E3:53"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1"><b>suppose </b></a><a NAME="E1:53_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:53_1_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> <a href="square_1.html#K3">^2</a> 
 <b>by </b><i><a class="ref" href="square_1.html#T74">SQUARE_1:74</a></i>;<br/><a NAME="E3:53_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="square_1.html#K3">^2</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T36">XREAL_1:36</a></i>;<br/><a NAME="E4:53_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="square_1.html#K3">^2</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T35">XREAL_1:35</a></i>;<br/><a NAME="E5:53_1_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="quaterni.html#T2" target="_self">Th2</a>, <a class="ref" href="xreal_1.html#T36">XREAL_1:36</a></i>;<br/><b>hence </b><a NAME="E6:53_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="quaterni.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_2"><b>suppose </b></a><a NAME="E1:53_1_2"/>
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:53_1_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> <a href="square_1.html#K3">^2</a> 
 <b>by </b><i><a class="ref" href="square_1.html#T74">SQUARE_1:74</a></i>;<br/><a NAME="E3:53_1_2"/><b>then </b><i><font color="Green">E43</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="square_1.html#K3">^2</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T36">XREAL_1:36</a></i>;<br/><a NAME="E4:53_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="square_1.html#K3">^2</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T35">XREAL_1:35</a></i>;<br/><a NAME="E5:53_1_2"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="quaterni.html#T2" target="_self">Th2</a>, <a class="ref" href="xreal_1.html#T36">XREAL_1:36</a></i>;<br/><b>hence </b><a NAME="E6:53_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="quaterni.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_3"><b>suppose </b></a><a NAME="E1:53_1_3"/>
<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:53_1_3"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c</font> <a href="square_1.html#K3">^2</a> 
 <b>by </b><i><a class="ref" href="square_1.html#T74">SQUARE_1:74</a></i>;<br/><a NAME="E3:53_1_3"/><b>then </b><i><font color="Green">E44</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="square_1.html#K3">^2</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T36">XREAL_1:36</a></i>;<br/><a NAME="E4:53_1_3"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="square_1.html#K3">^2</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T35">XREAL_1:35</a></i>;<br/><a NAME="E5:53_1_3"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="quaterni.html#T2" target="_self">Th2</a>, <a class="ref" href="xreal_1.html#T36">XREAL_1:36</a></i>;<br/><b>hence </b><a NAME="E6:53_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="quaterni.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_4"><b>suppose </b></a><a NAME="E1:53_1_4"/>
<font color="Maroon">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:53_1_4"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">d</font> <a href="square_1.html#K3">^2</a> 
 <b>by </b><i><a class="ref" href="square_1.html#T74">SQUARE_1:74</a></i>;<br/><a NAME="E3:53_1_4"/><b>then </b><i><font color="Green">E68</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="square_1.html#K3">^2</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T36">XREAL_1:36</a></i>;<br/><a NAME="E4:53_1_4"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="square_1.html#K3">^2</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T35">XREAL_1:35</a></i>;<br/><a NAME="E5:53_1_4"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="quaterni.html#T2" target="_self">Th2</a>, <a class="ref" href="xreal_1.html#T36">XREAL_1:36</a></i>;<br/><b>hence </b><a NAME="E6:53_1_4"/>
contradiction
 <b>by </b><i><a class="ref" href="quaterni.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
