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

<b>let </b><font color="Maroon">I</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>, <font color="Maroon">B</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/>



<b>assume </b><a NAME="E1:26"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">A</font> <a href="pzfmisc1.html#R1">is_transformable_to</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:26"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/> <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:26_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<a NAME="E2:26_1"/><b>then </b>
( <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, <a class="ref" href="pzfmisc1.html#D3">PZFMISC1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:26_1"/>
 <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:26"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><span class="p1">(<span class="default"><a href="pboole.html#K11">MSFuncs</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:26_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<a NAME="E2:26_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="pboole.html#K11">MSFuncs</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D22">PBOOLE:def 22</a></i>;<br/>
<b>hence </b><a NAME="E3:26_2"/>
<span class="p1">(<span class="default"><a href="pboole.html#K11">MSFuncs</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:26"/><b>then </b>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/> not <span class="p1">(<span class="default"><a href="pboole.html#K11">MSFuncs</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> is <a href="xboole_0.html#V1">empty</a>
 
;<br/>
<b>hence </b><a NAME="E5:26"/>
 <a href="pboole.html#K11">MSFuncs</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> is <a href="relat_1.html#V2">non-empty</a>
 <b>by </b><i><a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>


</div>
