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<b>consider </b><font color="Maroon" title="c1">A</font> being   <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">AbGroup</a>;<br/>
<b>take </b>
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> is <a href="prvect_1.html#V1" title="PRVECT_1:attr.1">AbGroup-yielding</a> )</span><br/>

<b>thus </b><a NAME="E2:49_1_1"/>
not <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> is <a href="prvect_1.html#V1" title="PRVECT_1:attr.1">AbGroup-yielding</a></span><br/>

<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="prvect_1.html#D11" target="_self" title="PRVECT_1:def.11">PRVECT_1:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> implies <font color="Maroon" title="c2">x</font> is   <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">AbGroup</a> )</span><br/>

<b>assume </b><a NAME="E3:49_1_1"/><i><font color="Green" title="E32">A1</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Maroon" title="c2">x</font> is  not   <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">AbGroup</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:49_1_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">A</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>;<br/>
<b>hence </b><a NAME="E5:49_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="prvect_1.html#E3:49_1_1"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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