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<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   non  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>   <a href="zmodul06.html#V3" title="ZMODUL06:attr.3">torsion-free</a>  <a href="zmodul01.html#NM1" title="ZMODUL01:NM.1">Z_Module</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">  not <font color="Maroon" title="c1">V</font> is  <a href="zmodul06.html#V2" title="ZMODUL06:attr.2">torsion</a> </span><br/>

<a NAME="E1:33_1_1"/>
 <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="vectsp_4.html#K1" title="VECTSP_4:func.1">(0).</a> <font color="Maroon" title="c1">V</font>
 
;<br/>
<span class="kw">then </span><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="E3:33_1_1"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
( <font color="Maroon" title="c2">v</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="vectsp_4.html#K2" title="VECTSP_4:func.2">(Omega).</a> <font color="Maroon" title="c1">V</font> &amp; <font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul04.html#T24" onmouseover="rs('zmodul04/T24')" onmouseout="rh()">ZMODUL04:24</a></span>;<br/>
<span class="kw">thus </span><a NAME="E4:33_1_1"/>
 not <font color="Maroon" title="c1">V</font> is  <a href="zmodul06.html#V2" title="ZMODUL06:attr.2">torsion</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_1_1"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="zmodul06.html#D3" target="_self" onmouseover="rs('zmodul06/D3')" onmouseout="rh()">defTorsionFree</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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