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<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   <a href="rewrite1.html#V2" title="REWRITE1:attr.2">weakly-normalizing</a>   <a href="rewrite1.html#V4" title="REWRITE1:attr.4">with_UN_property</a>  <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</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">a</font>, <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font> holds <br/> <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Olive" title="b2">b</font>,<font color="Maroon" title="c1">R</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">a</font>,<font color="Maroon" title="c3">b</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font> implies  <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">R</font> )</span><br/>



<a NAME="E1:79"/><span class="lab"><font color="Green" title="E66">A1</font></span>: 
 <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R4" title="REWRITE1:pred.4">is_a_normal_form_of</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#T55" target="_self" title="REWRITE1:th.55">Th55</a></span>;<br/>
<a NAME="E2:79"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A2</font></span>: 
 <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R3" title="REWRITE1:pred.3">is_a_normal_form_wrt</a> <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#D6" target="_self" title="REWRITE1:def.6">Def6</a></span>;<br/>
<a NAME="E3:79"/>
<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">b</font>, <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="ref" href="rewrite1.html#D6" target="_self" title="REWRITE1:def.6">Def6</a></span>;<br/>
<a NAME="E4:79"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A3</font></span>: 
<font color="Maroon" title="c3">b</font>, <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#T26" target="_self" title="REWRITE1:th.26">Th26</a></span>;<br/>
<a NAME="E5:79"/><span class="lab"><font color="Green" title="E69">A4</font></span>: 
 <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R4" title="REWRITE1:pred.4">is_a_normal_form_of</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#T55" target="_self" title="REWRITE1:th.55">Th55</a></span>;<br/>
<a NAME="E6:79"/><span class="kw">then </span>
<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c2">a</font>, <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#D6" target="_self" title="REWRITE1:def.6">Def6</a></span>;<br/>
<a NAME="E7:79"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A5</font></span>: 
 <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">a</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#T26" target="_self" title="REWRITE1:th.26">Th26</a></span>;<br/>
<span class="kw">assume </span><a NAME="E8:79"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">R</font></span><br/>

<a NAME="E9:79"/><span class="kw">then </span>
 <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c3">b</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:79"><span class="lab"><font color="Green" title="E70">A5</font></span></a>, <a class="ref" href="rewrite1.html#T31" target="_self" title="REWRITE1:th.31">Th31</a></span>;<br/>
<a NAME="E10:79"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A6</font></span>: 
 <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">R</font>, <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a> <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:79"><span class="lab"><font color="Green" title="E68">A3</font></span></a>, <a class="ref" href="rewrite1.html#T31" target="_self" title="REWRITE1:th.31">Th31</a></span>;<br/>
<a NAME="E11:79"/>
 <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R3" title="REWRITE1:pred.3">is_a_normal_form_wrt</a> <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:79"><span class="lab"><font color="Green" title="E69">A4</font></span></a>, <a class="ref" href="rewrite1.html#D6" target="_self" title="REWRITE1:def.6">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:79"/>
 <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:79"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="txt" href="#E10:79"><span class="lab"><font color="Green" title="E71">A6</font></span></a>, <a class="ref" href="rewrite1.html#D19" target="_self" title="REWRITE1:def.19">Def19</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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