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<b>let </b><font color="Maroon">x1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">x2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">x3</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">x4</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y3</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y4</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:26"/><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="quaterni.html#K6" target="_self">[*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font>,<font color="Maroon">x4</font></span><a href="quaterni.html#K6" target="_self">*]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="quaterni.html#K6" target="_self">[*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">y3</font>,<font color="Maroon">y4</font></span><a href="quaterni.html#K6" target="_self">*]</a></span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1"/>
( ( <font color="Maroon">x3</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">x4</font> <a href="hidden.html#R1">=</a> 0 ) or <font color="Maroon">x3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">x4</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1"><b>suppose </b></a><a NAME="E1:26_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">x3</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">x4</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:26_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="quaterni.html#K6" target="_self">[*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font>,<font color="Maroon">x4</font></span><a href="quaterni.html#K6" target="_self">*]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i/>;<br/><div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_1_1_1"/>
( <font color="Maroon">y3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">y4</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><a NAME="E2:26_1_1_1"/><b>then </b>
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> 0,1,2,3 <a href="quaterni.html#K3" target="_self">--&gt;</a> <font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">y3</font>,<font color="Maroon">y4</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:26_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E4:26_1_1"/>
<span class="p1"><a href="quaterni.html#K6" target="_self">[*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">y3</font>,<font color="Maroon">y4</font></span><a href="quaterni.html#K6" target="_self">*]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:26_1_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> &amp; <font color="Maroon">x3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y3</font> &amp; <font color="Maroon">x4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y4</font> )
 <b>by </b><i>, , , , <a class="ref" href="arytm_0.html#T12">ARYTM_0:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_2"><b>suppose </b></a><a NAME="E1:26_1_2"/>
( <font color="Maroon">x3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">x4</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:26_1_2"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="quaterni.html#K6" target="_self">[*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">y3</font>,<font color="Maroon">y4</font></span><a href="quaterni.html#K6" target="_self">*]</a></span> <a href="hidden.html#R1">=</a> 0,1,2,3 <a href="quaterni.html#K3" target="_self">--&gt;</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font>,<font color="Maroon">x4</font>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_1_2_1"/>
( <font color="Maroon">y3</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">y4</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><a NAME="E2:26_1_2_1"/><b>then </b>
<span class="p1"><a href="quaterni.html#K6" target="_self">[*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">y3</font>,<font color="Maroon">y4</font></span><a href="quaterni.html#K6" target="_self">*]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:26_1_2_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E4:26_1_2"/><b>then </b>
<span class="p1"><a href="quaterni.html#K6" target="_self">[*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">y3</font>,<font color="Maroon">y4</font></span><a href="quaterni.html#K6" target="_self">*]</a></span> <a href="hidden.html#R1">=</a> 0,1,2,3 <a href="quaterni.html#K3" target="_self">--&gt;</a> <font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">y3</font>,<font color="Maroon">y4</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:26_1_2"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> &amp; <font color="Maroon">x3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y3</font> &amp; <font color="Maroon">x4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y4</font> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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