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<div class="add">

<b>let </b><font color="Maroon" title="c1">R</font> be   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>;<br/>

<b>assume </b><a NAME="E1:64_1_1"/><i><font color="Green" title="E56">A1</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>

<b>thus </b><a NAME="E2:64_1_1"/>
<font color="Maroon" title="c1">R</font> is <a href="rewrite1.html#V2" title="REWRITE1:attr.2">weakly-normalizing</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D14" target="_self" title="REWRITE1:def.14">REWRITE1:def 14</a><br/>

<b>thus </b><a NAME="E1:64_1_1_1"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> implies <font color="Maroon" title="c2">x</font> <a href="rewrite1.html#R9" title="REWRITE1:pred.9">has_a_normal_form_wrt</a> <font color="Maroon" title="c1">R</font> )
 <b>by </b><i><a class="txt" href="rewrite1.html#E1:64_1_1"><i><font color="Green" title="E56">A1</font></i></a>, <a class="ref" href="relat_1.html#T63" title="RELAT_1:th.63">RELAT_1:63</a></i>;<br/>


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

<b>thus </b><a NAME="E3:64_1_1"/>
<font color="Maroon" title="c1">R</font> is <a href="rewrite1.html#V3" title="REWRITE1:attr.3">strongly-normalizing</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">f</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rewrite1.html#D15" target="_self" title="REWRITE1:def.15">REWRITE1:def 15</a><br/>

<b>take </b>
0
;<br/>

<b>thus </b><a NAME="E1:64_1_1_2"/>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="rewrite1.html#E1:64_1_1"><i><font color="Green" title="E56">A1</font></i></a></i>;<br/>


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


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