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<b>let </b><font color="Maroon" title="c1">G</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="lang1.html#L1" title="LANG1:struct.1">DTConstrStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">n</font>, <font color="Maroon" title="c3">n1</font>, <font color="Maroon" title="c4">n2</font> be   <a href="lang1.html#NM2" title="LANG1:NM.2">String</a> of <font color="Maroon" title="c1">G</font>;<br/>







<b>given </b><font color="Maroon" title="c5">m1</font>, <font color="Maroon" title="c6">m2</font>, <font color="Maroon" title="c7">m3</font> being   <a href="lang1.html#NM2" title="LANG1:NM.2">String</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c8">s</font> being   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font><b> such that </b><a NAME="E2:17"/><i><font color="Green" title="E5">A1</font></i>: 
( <font color="Maroon" title="c3">n1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m1</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <span class="p2"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c8">s</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span></span>)</span> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c6">m2</font> &amp; <font color="Maroon" title="c4">n2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m1</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c7">m3</font></span>)</span> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c6">m2</font> &amp; <font color="Maroon" title="c8">s</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Maroon" title="c7">m3</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lang1.html#D4" target="_self" title="LANG1:def.4">LANG1:def 4</a><br/>

<b>thus </b><a NAME="E3:17"/>
<font color="Maroon" title="c2">n</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c3">n1</font> <a href="lang1.html#R2" title="LANG1:pred.2">==&gt;</a> <font color="Maroon" title="c2">n</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c4">n2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c2">n</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c5">m1</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lang1.html#D4" target="_self" title="LANG1:def.4">LANG1:def 4</a><br/>

<b>take </b>
<font color="Maroon" title="c6">m2</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c7">m3</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c8">s</font>
;<br/>

<b>thus </b><font color="Maroon" title="c2">n</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c3">n1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m1</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <span class="p3"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c8">s</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span></span>)</span></span>)</span> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c6">m2</font>
<b>by </b><i><a class="txt" href="lang1.html#E2:17"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c5">m1</font></span>)</span> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <span class="p2"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c8">s</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span></span>)</span> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c6">m2</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></i>
;<br/>

<b>thus </b><font color="Maroon" title="c2">n</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c4">n2</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m1</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c7">m3</font></span>)</span></span>)</span> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c6">m2</font>
<b>by </b><i><a class="txt" href="lang1.html#E2:17"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c5">m1</font></span>)</span> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c7">m3</font></span>)</span> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c6">m2</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></i>
;<br/>

<b>thus </b><a NAME="E3:17_1"/>
<font color="Maroon" title="c8">s</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Maroon" title="c7">m3</font>
 <b>by </b><i><a class="txt" href="lang1.html#E2:17"><i><font color="Green" title="E5">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>take </b>
<font color="Maroon" title="c5">m1</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lang1.html#D4" target="_self" title="LANG1:def.4">LANG1:def 4</a><br/>

<b>take </b>
<font color="Maroon" title="c6">m2</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c2">n</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c7">m3</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c8">s</font>
;<br/>

<b>thus </b><a NAME="E4:17"/>
( <font color="Maroon" title="c3">n1</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c2">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m1</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <span class="p2"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c8">s</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span></span>)</span> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m2</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c4">n2</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c2">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m1</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c7">m3</font></span>)</span> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m2</font> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c8">s</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Maroon" title="c7">m3</font> )
 <b>by </b><i><a class="txt" href="lang1.html#E2:17"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></i>;<br/>


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