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<span class="kw">assume </span><a NAME="E1:81_1_1"/><span class="lab"><font color="Green" title="E40">AS</font></span>: 
 <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><a href="zmodul07.html#K1" title="ZMODUL07:func.1">torsion_part</a> <font color="Maroon" title="c1">V</font></span>)</span>) is  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">v</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E3:81_1_1"/><span class="lab"><font color="Green" title="E41">P1</font></span>: 
 not <font color="Maroon" title="c2">v</font> is  <a href="zmodul06.html#V1" title="ZMODUL06:attr.1">torsion</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul06.html#D2" onmouseover="rs('zmodul06/D2')" onmouseout="rh()">ZMODUL06:def 2</a></span>;<br/>
<a NAME="E4:81_1_1"/>
<font color="Maroon" title="c2">v</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="zmodul07.html#K1" title="ZMODUL07:func.1">torsion_part</a> <font color="Maroon" title="c1">V</font></span>)</span> is    <a href="vectsp_4.html#M2" title="VECTSP_4:mode.2">Coset</a> of  <a href="zmodul07.html#K1" title="ZMODUL07:func.1">torsion_part</a> <font color="Maroon" title="c1">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#D6" onmouseover="rs('vectsp_4/D6')" onmouseout="rh()">VECTSP_4:def 6</a></span>;<br/>
<a NAME="E5:81_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">v</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="zmodul07.html#K1" title="ZMODUL07:func.1">torsion_part</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="vectsp10.html#K2" title="VECTSP10:func.2">CosetSet</a> (<font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><a href="zmodul07.html#K1" title="ZMODUL07:func.1">torsion_part</a> <font color="Maroon" title="c1">V</font></span>)</span>)
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">B</font> = <font color="Maroon" title="c2">v</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="zmodul07.html#K1" title="ZMODUL07:func.1">torsion_part</a> <font color="Maroon" title="c1">V</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="vectsp10.html#K2" title="VECTSP10:func.2">CosetSet</a> (<font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><a href="zmodul07.html#K1" title="ZMODUL07:func.1">torsion_part</a> <font color="Maroon" title="c1">V</font></span>)</span>) ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">u</font> = <font color="Maroon" title="c3">B</font> as   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c1">V</font>,<span class="p2">(<span class="default"><a href="zmodul07.html#K1" title="ZMODUL07:func.1">torsion_part</a> <font color="Maroon" title="c1">V</font></span>)</span>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#D6" onmouseover="rs('vectsp10/D6')" onmouseout="rh()">VECTSP10:def 6</a></span>;<br/>
<font color="Maroon" title="c4">u</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c1">V</font>,<span class="p2">(<span class="default"><a href="zmodul07.html#K1" title="ZMODUL07:func.1">torsion_part</a> <font color="Maroon" title="c1">V</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1"><span class="lab"><font color="Green" title="E40">AS</font></span></a></span>
<br/>.= 
 <a href="vectsp10.html#K4" title="VECTSP10:func.4">zeroCoset</a> (<font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><a href="zmodul07.html#K1" title="ZMODUL07:func.1">torsion_part</a> <font color="Maroon" title="c1">V</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#D6" onmouseover="rs('vectsp10/D6')" onmouseout="rh()">VECTSP10:def 6</a></span>
<br/>.= 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="zmodul07.html#K1" title="ZMODUL07:func.1">torsion_part</a> <font color="Maroon" title="c1">V</font></span>)</span>

;<br/>
<a NAME="E9:81_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="zmodul07.html#K1" title="ZMODUL07:func.1">torsion_part</a> <font color="Maroon" title="c1">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul01.html#T63" onmouseover="rs('zmodul01/T63')" onmouseout="rh()">ZMODUL01:63</a></span>;<br/>
<a NAME="E10:81_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">v</font> is  <a href="zmodul06.html#V1" title="ZMODUL06:attr.1">torsion</a>  </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul07.html#D1" target="_self" onmouseover="rs('zmodul07/D1')" onmouseout="rh()">defTorsionPart</a></span>;<br/>
<a NAME="E11:81_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">u</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is  <a href="zmodul06.html#V1" title="ZMODUL06:attr.1">torsion</a>  )
 
;<br/>
<span class="kw">hence </span><a NAME="E12:81_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:81_1_1"><span class="lab"><font color="Green" title="E41">P1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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