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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#NV1" title="COMSEQ_1:NV.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#NV1" title="COMSEQ_1:NV.1">non-zero</a> </span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#NV1" title="COMSEQ_1:NV.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#NV1" title="COMSEQ_1:NV.1">non-zero</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:39"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E2:39"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
<font color="Maroon" title="c2">seq</font> is  <a href="comseq_1.html#NV1" title="COMSEQ_1:NV.1">non-zero</a> 
 <span class="kw">and </span><br/><a NAME="E3:39"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
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#NV1" title="COMSEQ_1:NV.1">non-zero</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E5:39"/><span class="lab"><font color="Green" title="E23">A4</font></span>: 
<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="nat_1.html#K6" title="NAT_1:func.6">.</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="ref" href="comseq_1.html#T4" target="_self" title="COMSEQ_1:th.4">Th4</a></span>;<br/>
<a NAME="E6:39"/><span class="lab"><font color="Green" title="E24">A5</font></span>: 
<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="nat_1.html#K6" title="NAT_1:func.6">.</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="valued_1.html#T6" title="VALUED_1:th.6">VALUED_1:6</a></span>;<br/>
<a NAME="E7:39"/>
<font color="Maroon" title="c2">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="comseq_1.html#T4" target="_self" title="COMSEQ_1:th.4">Th4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:39"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E24">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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