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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">R</font> being  <a href="relset_1.html#NM3" title="RELSET_1:NM.3">Relation</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">R</font> is <a href="rewrite1.html#V4" title="REWRITE1:attr.4">with_UN_property</a> &amp; <font color="Olive" title="b1">R</font> is <a href="rewrite1.html#V2" title="REWRITE1:attr.2">weakly-normalizing</a> holds <br/>for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds   <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/><b>let </b><font color="Maroon" title="c2">R</font> be   <a href="relset_1.html#NM3" title="RELSET_1:NM.3">Relation</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">R</font> is <a href="rewrite1.html#V4" title="REWRITE1:attr.4">with_UN_property</a> &amp; <font color="Maroon" title="c2">R</font> is <a href="rewrite1.html#V2" title="REWRITE1:attr.2">weakly-normalizing</a> implies for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds   <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )</span><br/>



<b>assume </b><a NAME="E1:124"/><i><font color="Green" title="E78">A1</font></i>: 
( <font color="Maroon" title="c2">R</font> is <a href="rewrite1.html#V4" title="REWRITE1:attr.4">with_UN_property</a> &amp; <font color="Maroon" title="c2">R</font> is <a href="rewrite1.html#V2" title="REWRITE1:attr.2">weakly-normalizing</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds   <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/>

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E2:124"/>
 <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">R</font> <a href="rewrite1.html#R4" title="REWRITE1:pred.4">is_a_normal_form_of</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">R</font>
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E1:124"><i><font color="Green" title="E78">A1</font></i></a>, <a class="ref" href="rewrite1.html#T55" title="REWRITE1:th.55">REWRITE1:55</a></i>;<br/>
<a NAME="E3:124"/><b>then </b>
<font color="Maroon" title="c2">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">x</font>, <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">R</font>
 
<b>by </b><i><a class="ref" href="rewrite1.html#D6" title="REWRITE1:def.6">REWRITE1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E4:124"/>
 <a href="rewrite1.html#K2" title="REWRITE1:func.2">nf</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="abcmiz_0.html#T75" target="_self" title="ABCMIZ_0:th.75">Th75</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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