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<span class="kw">set </span><font color="Maroon" title="c1">S</font> =  the   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>;<br/>
<span class="kw">take </span>
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"> the   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"> the   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a></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"> the   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> is  <a href="prvect_2.html#V1" title="PRVECT_2:attr.1">RealLinearSpace-yielding</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E1:11_1_1"/>
 not <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"> the   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a></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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"> the   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> is  <a href="prvect_2.html#V1" title="PRVECT_2:attr.1">RealLinearSpace-yielding</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="prvect_2.html#D3" target="_self" title="PRVECT_2:def.3">PRVECT_2:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"> the   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a></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="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:11_1_1"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
<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"> the   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E3:11_1_1"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
<font color="Maroon" title="c2">x</font> is  not   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:11_1_1"/>
<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"> the   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11_1_1"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:11_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:11_1_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" 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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