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<span class="kw">set </span><font color="Maroon" title="c1">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> 4,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  : ( <font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> ;<br/>
<div><span class="lab"><font color="Green" title="E8">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:14_2_1_1"/>
 <a href="quaterni.html#K4" title="QUATERNI:func.4">&lt;j&gt;</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> 4,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  : ( <font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:14_2_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> 4,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>(  <a href="quaterni.html#K4" title="QUATERNI:func.4">&lt;j&gt;</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><span class="kw">hence </span><a NAME="E3:14_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><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#T3" target="_self" title="QUATERNI:th.3">Th3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c2">z</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <font color="Maroon" title="c3">j</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <font color="Maroon" title="c4">w</font> = 1, <font color="Maroon" title="c5">m</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as    <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>  ;<br/>
<a NAME="E3:14_2_1"/>
 <a href="quaterni.html#K4" title="QUATERNI:func.4">&lt;j&gt;</a>  <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="c2">z</font>,<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c5">m</font>
 
;<br/>
<a NAME="E4:14_2_1"/><span class="kw">then </span>
 <a href="quaterni.html#K4" title="QUATERNI:func.4">&lt;j&gt;</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> 4,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T90" title="CARD_1:th.90">CARD_1:90</a>, <a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></span>;<br/>
<a NAME="E5:14_2_1"/><span class="kw">then </span>
 <a href="quaterni.html#K4" title="QUATERNI:func.4">&lt;j&gt;</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> 4,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> 4,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  : ( <font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_2_1"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:14_2_1"/>
 <a href="quaterni.html#K4" title="QUATERNI:func.4">&lt;j&gt;</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quaterni.html#K1" title="QUATERNI:func.1">QUATERNION</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="quaterni.html#D2" target="_self" title="QUATERNI:def.2">QUATERNI:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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