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<span class="kw">let </span><font color="Maroon" title="c1">seq</font>, <font color="Maroon" title="c2">cseq</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="c2">cseq</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">seq</font> implies ( <font color="Maroon" title="c1">seq</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  iff <font color="Maroon" title="c2">cseq</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:8"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<font color="Maroon" title="c2">cseq</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">seq</font>
 ; <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">seq</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  iff <font color="Maroon" title="c2">cseq</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E2:8"/>
( <font color="Maroon" title="c1">seq</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  implies <font color="Maroon" title="c2">cseq</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="comseq_1.html#T41" title="COMSEQ_1:th.41">COMSEQ_1:41</a></span>; <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="c2">cseq</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  implies <font color="Maroon" title="c1">seq</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  )</span><br/>

<a NAME="E3:8"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
( <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">seq</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  implies <font color="Maroon" title="c1">seq</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:8_1"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">seq</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</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">seq</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">n</font> be    <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> ; <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">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</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><br/><a NAME="E1:8_1_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">seq</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</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="#E1:8_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="comseq_1.html#T4" title="COMSEQ_1:th.4">COMSEQ_1:4</a></span>;<br/><a NAME="E2:8_1_1"/><span class="kw">then </span>
<a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <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="ref" href="valued_1.html#T6" title="VALUED_1:th.6">VALUED_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E3:8_1_1"/>
<font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:8_1"/>
<font color="Maroon" title="c1">seq</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="comseq_1.html#T4" title="COMSEQ_1:th.4">COMSEQ_1:4</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/>


</div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E4:8"/>
<font color="Maroon" title="c2">cseq</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</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">seq</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> </span><br/>

<span class="kw">hence </span><a NAME="E5:8"/>
<font color="Maroon" title="c1">seq</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E3:8"><span class="lab"><font color="Green" title="E10">A2</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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