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<b>let </b><font color="Maroon" title="c1">M</font>, <font color="Maroon" title="c2">E</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> ;<br/><b>let </b><font color="Maroon" title="c3">e</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">E</font>;<br/><b>let </b><font color="Maroon" title="c4">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font>;<br/><b>let </b><font color="Maroon" title="c5">v</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c1">M</font>;<br/><b>let </b><font color="Maroon" title="c6">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c2">E</font>;<br/><b>let </b><font color="Maroon" title="c7">x</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>;<br/>













<a NAME="E1:7"/><i><font color="Green" title="E8">A1</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c3">e</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c4">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>assume </b><a NAME="E2:7"/><i><font color="Green" title="E9">A2</font></i>: 
( <font color="Maroon" title="c5">v</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:7_1"/>
<font color="Maroon" title="c8">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> 
 ;<br/><b>then reconsider </b><font color="Maroon" title="c9">u'</font> = <font color="Maroon" title="c8">u</font> as   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a> ;<br/><a NAME="E3:7_1"/>
( <font color="Maroon" title="c9">u'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> or <font color="Maroon" title="c9">u'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">x</font> )
 ;<br/><a NAME="E4:7_1"/><b>then </b>
( ( <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c4">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">u'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c3">e</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">u'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> ) or ( <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c4">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">u'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">u'</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c3">e</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">u'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">u'</font> ) )
 <b>by </b><i><a class="ref" href="zf_lang1.html#D1" title="ZF_LANG1:def.1">ZF_LANG1:def 1</a></i>;<br/><b>hence </b><a NAME="E5:7_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c3">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c4">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">u</font>
 <b>by </b><i><a class="txt" href="zf_fund2.html#E2:7"><i><font color="Green" title="E9">A2</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:7"/>
<font color="Maroon" title="c5">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c4">m</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c3">e</font>
 <b>by </b><i><a class="txt" href="zf_fund2.html#E1:7"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


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