<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">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="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>;<br/><b>let </b><font color="Maroon" title="c4">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="c1">E</font>;<br/>







<a NAME="E1:14"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c3">y</font> <a href="zf_lang.html#NR1" title="ZF_LANG:NR.1">is_equality</a> 
 
<b>by </b><i><a class="ref" href="zf_lang.html#T16" title="ZF_LANG:th.16">ZF_LANG:16</a></i>;<br/>
<a NAME="E2:14"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_lang.html#T53" title="ZF_LANG:th.53">ZF_LANG:53</a></i>;<br/>
<a NAME="E3:14"/><b>then </b><i><font color="Green" title="E9">A2</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c3">y</font></span>)</span> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c3">y</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="zf_lang.html#T6" title="ZF_LANG:th.6">ZF_LANG:6</a></i>;<br/>
<a NAME="E4:14"/><i><font color="Green" title="E10">A3</font></i>: 
 <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c3">y</font></span>)</span>,<font color="Maroon" title="c1">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v1</font> where <font color="Olive" title="b1">v1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zf_model.html#K3" title="ZF_MODEL:func.3">VAL</a> <font color="Maroon" title="c1">E</font> : for <font color="Olive" title="b2">f</font> being  <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">E</font>  st <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v1</font> holds <br/><font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> </span>}</span> 
 
<b>by </b><i><a class="txt" href="zf_model.html#E1:14"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="zf_model.html#E7"><i><font color="Green" title="E6">Lm3</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:14"/>
( <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font> implies <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c3">y</font></span>)</span>,<font color="Maroon" title="c1">E</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1"/><i><font color="Green" title="E11">A4</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">v</font> = <font color="Maroon" title="c4">f</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zf_model.html#K3" title="ZF_MODEL:func.3">VAL</a> <font color="Maroon" title="c1">E</font> <b>by </b><i><a class="ref" href="zf_model.html#D2" target="_self" title="ZF_MODEL:def.2">Def2</a></i>;<br/>
<a NAME="E3:14_1"/>
for <font color="Olive" title="b1">f</font> being  <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">E</font>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_model.html#E3:14"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="zf_model.html#E1:14_1"><i><font color="Green" title="E11">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:14_1"/>
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c3">y</font></span>)</span>,<font color="Maroon" title="c1">E</font>
 <b>by </b><i><a class="txt" href="zf_model.html#E4:14"><i><font color="Green" title="E10">A3</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E6:14"/>
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K5" title="ZF_MODEL:func.5">St</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c3">y</font></span>)</span>,<font color="Maroon" title="c1">E</font>
 ;<br/>

<a NAME="E7:14"/><b>then </b>
 ex <font color="Olive" title="b1">v1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zf_model.html#K3" title="ZF_MODEL:func.3">VAL</a> <font color="Maroon" title="c1">E</font> st <br/>( <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v1</font> &amp; ( for <font color="Olive" title="b2">f</font> being  <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">E</font>  st <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v1</font> holds <br/><font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> ) )
 
<b>by </b><i><a class="txt" href="zf_model.html#E4:14"><i><font color="Green" title="E10">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:14"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="txt" href="zf_model.html#E3:14"><i><font color="Green" title="E9">A2</font></i></a></i>;<br/>


</div>
