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

<span class="kw">reconsider </span><font color="Maroon" title="c1">V</font> =  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">ModuleStr</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a>,<span class="p1">(<span class="default"><a href="zmodul01.html#K2" title="ZMODUL01:func.2">Int-mult-left</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a></span>)</span> #) as   <a href="zmodul01.html#NM1" title="ZMODUL01:NM.1">Z_Module</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul01.html#T164" onmouseover="rs('zmodul01/T164')" onmouseout="rh()">ZMODUL01:164</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c2">p</font> = 2 as   <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">by</span> <span class="lab"><a class="ref" href="int_1.html#D2" onmouseover="rs('int_1/D2')" onmouseout="rh()">INT_1:def 2</a></span>;<br/>
<span class="kw">take </span>
<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">  ex <font color="Olive" title="b1">p</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> st <br/>( <font color="Olive" title="b1">p</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;  not  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c1">V</font></span>)</span>) is  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>  )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c2">p</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">p</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;  not  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c1">V</font></span>)</span>) is  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E3:69"/>
<font color="Maroon" title="c2">p</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>
 ; <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  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c1">V</font></span>)</span>) is  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> </span><br/>

<span class="kw">thus </span><a NAME="E4:69"/>
 not  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="69_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c3">i</font> =  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a> as   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<a NAME="E2:69_1"/>
<font color="Maroon" title="c3">i</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</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 <font color="Maroon" title="c2">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</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="E3:69_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">i</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</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"><font color="Maroon" title="c2">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</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="c4">B</font> = <font color="Maroon" title="c3">i</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</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"><font color="Maroon" title="c2">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c1">V</font></span>)</span>) ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">u</font> = <font color="Maroon" title="c4">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"><font color="Maroon" title="c2">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</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/>
<a NAME="E6:69_1"/>
<font color="Maroon" title="c5">u</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> <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"><font color="Maroon" title="c2">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c1">V</font></span>)</span>)</span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="69_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:69_1_1"/>
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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"><font color="Maroon" title="c2">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c1">V</font></span>)</span>)</span>)</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"> contradiction</span><br/>

<span class="kw">then </span><font color="Maroon" title="c3">i</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c1">V</font></span>)</span> = 
 <a href="vectsp10.html#K4" title="VECTSP10:func.4">zeroCoset</a> (<font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</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"><font color="Maroon" title="c2">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</a> <font color="Maroon" title="c1">V</font></span>)</span>

;<br/>
<a NAME="E3:69_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">i</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</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/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">w</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="E5:69_1_1"/><span class="lab"><font color="Green" title="E39">P1</font></span>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">w</font>
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">w0</font> = <font color="Maroon" title="c6">w</font> as   <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> ;<br/>
<a NAME="E7:69_1_1"/>
<font color="Maroon" title="c2">p</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c6">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">w0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zmodul06.html#T14" onmouseover="rs('zmodul06/T14')" onmouseout="rh()">ZMODUL06:14</a></span>;<br/>
<a NAME="E8:69_1_1"/><span class="kw">then </span>
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">w0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:69_1_1"><span class="lab"><font color="Green" title="E39">P1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:69_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T33" onmouseover="rs('nat_d/T33')" onmouseout="rh()">NAT_D:33</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="E7:69_1"/>
 not  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="zmodul02.html#K5" title="ZMODUL02:func.5">(*)</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"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
