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<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <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">  for <font color="Olive" title="b1">v</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c1">V</font> holds <br/> ( <font color="Olive" title="b1">v</font> is  <a href="zmodul06.html#V1" title="ZMODUL06:attr.1">torsion</a>  iff <font color="Olive" title="b1">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><br/><span class="kw">let </span><font color="Maroon" title="c2">v</font> be   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c1">V</font>; <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"> ( <font color="Maroon" title="c2">v</font> is  <a href="zmodul06.html#V1" title="ZMODUL06:attr.1">torsion</a>  iff <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><br/>



<span class="kw">set </span><font color="Maroon" title="c3">W</font> =  <a href="zmodul07.html#K1" title="ZMODUL07:func.1">torsion_part</a> <font color="Maroon" title="c1">V</font>;<br/>
<div><span class="kw">hereby </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"> ( <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> implies <font color="Maroon" title="c2">v</font> is  <a href="zmodul06.html#V1" title="ZMODUL06:attr.1">torsion</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:18_1"/>
<font color="Maroon" title="c2">v</font> is  <a href="zmodul06.html#V1" title="ZMODUL06:attr.1">torsion</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"> <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><br/><a NAME="E2:18_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> 
 ;<br/><span class="kw">hence </span><a NAME="E3:18_1"/>
<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="zmodul07.html#D1" target="_self" onmouseover="rs('zmodul07/D1')" onmouseout="rh()">defTorsionPart</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/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:18"/>
<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>
 ; <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"> <font color="Maroon" title="c2">v</font> is  <a href="zmodul06.html#V1" title="ZMODUL06:attr.1">torsion</a> </span><br/>

<a NAME="E3:18"/><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/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">vv</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="E5:18"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
( <font color="Maroon" title="c4">vv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c4">vv</font> is  <a href="zmodul06.html#V1" title="ZMODUL06:attr.1">torsion</a>  )
 ;<br/>
<span class="kw">thus </span><a NAME="E6:18"/>
<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="txt" href="#E5:18"><span class="lab"><font color="Green" title="E13">A2</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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