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<b>let </b><font color="Maroon" title="c1">r</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">seq</font> being  <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>  st <font color="Maroon" title="c1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  &amp; <font color="Olive" title="b1">seq</font> is <a href="comseq_1.html#V1" title="COMSEQ_1:attr.1">non-zero</a> holds <br/><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Olive" title="b1">seq</font> is <a href="comseq_1.html#V1" title="COMSEQ_1:attr.1">non-zero</a></span><br/><b>let </b><font color="Maroon" title="c2">seq</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  &amp; <font color="Maroon" title="c2">seq</font> is <a href="comseq_1.html#V1" title="COMSEQ_1:attr.1">non-zero</a> implies <font color="Maroon" title="c1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">seq</font> is <a href="comseq_1.html#V1" title="COMSEQ_1:attr.1">non-zero</a> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:36"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 
 <b>and </b><br/><a NAME="E2:36"/><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon" title="c2">seq</font> is <a href="comseq_1.html#V1" title="COMSEQ_1:attr.1">non-zero</a>
 <b>and </b><br/><a NAME="E3:36"/><i><font color="Green" title="E24">A3</font></i>: 
not <font color="Maroon" title="c1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">seq</font> is <a href="comseq_1.html#V1" title="COMSEQ_1:attr.1">non-zero</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>consider </b><font color="Maroon" title="c3">n1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:36"/><i><font color="Green" title="E25">A4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 
 <b>by </b><i><a class="txt" href="comseq_1.html#E3:36"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="comseq_1.html#T4" target="_self" title="COMSEQ_1:th.4">Th4</a></i>;<br/>
<a NAME="E6:36"/><i><font color="Green" title="E26">A5</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 
 
<b>by </b><i><a class="txt" href="comseq_1.html#E5:36"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="valued_1.html#T6" title="VALUED_1:th.6">VALUED_1:6</a></i>;<br/>
<a NAME="E7:36"/>
<font color="Maroon" title="c2">seq</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">n1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 
 
<b>by </b><i><a class="txt" href="comseq_1.html#E2:36"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="comseq_1.html#T4" target="_self" title="COMSEQ_1:th.4">Th4</a></i>;<br/>
<b>hence </b><a NAME="E8:36"/>
contradiction
 <b>by </b><i><a class="txt" href="comseq_1.html#E1:36"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="comseq_1.html#E6:36"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</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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