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<span class="kw">let </span><font color="Maroon" title="c1">x1</font>, <font color="Maroon" title="c2">x2</font>, <font color="Maroon" title="c3">x3</font>, <font color="Maroon" title="c4">x4</font>, <font color="Maroon" title="c5">y1</font>, <font color="Maroon" title="c6">y2</font>, <font color="Maroon" title="c7">y3</font>, <font color="Maroon" title="c8">y4</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font>,<font color="Maroon" title="c4">x4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default"><font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c8">y4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> implies ( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y2</font> &amp; <font color="Maroon" title="c3">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y3</font> &amp; <font color="Maroon" title="c4">x4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y4</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:27"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font>,<font color="Maroon" title="c4">x4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default"><font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c8">y4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y2</font> &amp; <font color="Maroon" title="c3">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y3</font> &amp; <font color="Maroon" title="c4">x4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y4</font> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:27_1"/>
( ( <font color="Maroon" title="c3">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">x4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or <font color="Maroon" title="c3">x3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">x4</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1"><span class="kw">suppose </span></a><a NAME="E1:27_1_1"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
( <font color="Maroon" title="c3">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">x4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y2</font> &amp; <font color="Maroon" title="c3">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y3</font> &amp; <font color="Maroon" title="c4">x4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y4</font> )</span><br/><div class="add"><a NAME="E2:27_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A3</font></span>: 
<span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font>,<font color="Maroon" title="c4">x4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="quaterni.html#E15"><span class="lab"><font color="Green" title="E9">Lm4</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E22">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:27_1_1_1"/>
( <font color="Maroon" title="c7">y3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c8">y4</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:27_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1,2,3) <a href="quaterni.html#K3" title="QUATERNI:func.3">--&gt;</a> (<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c8">y4</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:27_1_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="quaterni.html#D6" target="_self" title="QUATERNI:def.6">Def6</a></span>;<br/><span class="kw">hence </span><a NAME="E3:27_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="quaterni.html#T10" target="_self" title="QUATERNI:th.10">Th10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:27_1_1"/><span class="kw">then </span>
<span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default"><font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c8">y4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">y2</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="quaterni.html#E15"><span class="lab"><font color="Green" title="E9">Lm4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:27_1_1"/>
( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y2</font> &amp; <font color="Maroon" title="c3">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y3</font> &amp; <font color="Maroon" title="c4">x4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y4</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E1:27_1_1"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E2:27_1_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E3:27_1_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="arytm_0.html#T10" title="ARYTM_0:th.10">ARYTM_0:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_2"><span class="kw">suppose </span></a><a NAME="E1:27_1_2"/>
( <font color="Maroon" title="c3">x3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">x4</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y2</font> &amp; <font color="Maroon" title="c3">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y3</font> &amp; <font color="Maroon" title="c4">x4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y4</font> )</span><br/><div class="add"><a NAME="E2:27_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default"><font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c8">y4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1,2,3) <a href="quaterni.html#K3" title="QUATERNI:func.3">--&gt;</a> (<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">x3</font>,<font color="Maroon" title="c4">x4</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="quaterni.html#D6" target="_self" title="QUATERNI:def.6">Def6</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:27_1_2_1"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<font color="Maroon" title="c7">y3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:27_1_2_1"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
<font color="Maroon" title="c8">y4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:27_1_2_1"/>
<span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default"><font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c8">y4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">y2</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_2_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E2:27_1_2_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="quaterni.html#E15"><span class="lab"><font color="Green" title="E9">Lm4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:27_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1_2"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="quaterni.html#T10" target="_self" title="QUATERNI:th.10">Th10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:27_1_2"/><span class="kw">then </span>
<span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default"><font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c8">y4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1,2,3) <a href="quaterni.html#K3" title="QUATERNI:func.3">--&gt;</a> (<font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">y2</font>,<font color="Maroon" title="c7">y3</font>,<font color="Maroon" title="c8">y4</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="quaterni.html#D6" target="_self" title="QUATERNI:def.6">Def6</a></span>;<br/><span class="kw">hence </span><a NAME="E5:27_1_2"/>
( <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y1</font> &amp; <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y2</font> &amp; <font color="Maroon" title="c3">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y3</font> &amp; <font color="Maroon" title="c4">x4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y4</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1_2"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="quaterni.html#E6"><span class="lab"><font color="Green" title="E4">Lm1</font></span></a>, <a class="ref" href="quaterni.html#T11" target="_self" title="QUATERNI:th.11">Th11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
