<?xml version="1.0"?>
<div class="add">

<span class="kw">set </span><font color="Maroon" title="c1">V</font> =  <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a> ;<br/>
<a NAME="E1:32_1_1"/>
 for <font color="Olive" title="b1">v</font> being   <a href="zmodul01.html#NM2" title="ZMODUL01:NM.2">VECTOR</a> of <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a>  st <font color="Olive" title="b1">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> <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a> holds <br/> not <font color="Olive" title="b1">v</font> is  <a href="zmodul06.html#V1" title="ZMODUL06:attr.1">torsion</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">v</font> be   <a href="zmodul01.html#NM2" title="ZMODUL01:NM.2">VECTOR</a> of <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-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"> ( <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> <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a> implies  not <font color="Maroon" title="c2">v</font> is  <a href="zmodul06.html#V1" title="ZMODUL06:attr.1">torsion</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1_1_1"/><span class="lab"><font color="Green" title="E17">AS</font></span>: 
<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> <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-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="c2">v</font> is  <a href="zmodul06.html#V1" title="ZMODUL06:attr.1">torsion</a> </span><br/>

<span class="kw">assume </span><a NAME="E2:32_1_1_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"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a><span class="kw"> such that </span><br/><a NAME="E4:32_1_1_1"/><span class="lab"><font color="Green" title="E18">P1</font></span>: 
( <font color="Maroon" title="c3">i</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> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a> &amp; <font color="Maroon" title="c3">i</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a> )
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">v1</font> = <font color="Maroon" title="c2">v</font> as   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a> ;<br/>
<a NAME="E6:32_1_1_1"/>
<font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul07.html#T15" target="_self" onmouseover="rs('zmodul07/T15')" onmouseout="rh()">LMTFRat2</a>, <a class="txt" href="#E4:32_1_1_1"><span class="lab"><font color="Green" title="E18">P1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:32_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_1"><span class="lab"><font color="Green" title="E17">AS</font></span></a>, <a class="txt" href="#E4:32_1_1_1"><span class="lab"><font color="Green" title="E18">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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:32_1_1"/>
 <a href="zmodul07.html#K3" title="ZMODUL07:func.3">Rat-Module</a>  is  <a href="zmodul06.html#V3" title="ZMODUL06:attr.3">torsion-free</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"> verum</span><br/>


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