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

<b>set </b><font color="Maroon">T</font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">msf</font> being  <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U1</font>,<font color="Maroon">U1</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">msf</font> &amp; <font color="Olive">msf</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U1</font> );<br/>
<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:12_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pboole.html#K11">MSFuncs</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:12_1_1"/><i><font color="Green">E34</font></i>: 
 <a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pboole.html#K11">MSFuncs</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="autalg_1.html#T22">AUTALG_1:22</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">F</font> =  <a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>;<br/><b>thus </b><a NAME="E1:12_1_1_1"/>
 <a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">F</font>
 ;<br/><b>thus </b><a NAME="E2:12_1_1_1"/>
<font color="Maroon">F</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U1</font>
 <b>by </b><i><a class="ref" href="msualg_3.html#T9">MSUALG_3:9</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, </i>;<br/>
<a NAME="E6:12_1_1"/>
<font color="Maroon">X</font> is    <a href="autalg_1.html#M1">MSFunctionSet</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:12_1_1_2"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="pzfmisc1.html#R1">is_transformable_to</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="autalg_1.html#D6">AUTALG_1:def 6</a><br/>

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

<b>assume </b><a NAME="E2:12_1_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E3:12_1_1_2"/><b>then </b>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pboole.html#K11">MSFuncs</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font></span>)</span> &amp;  ex <font color="Olive">msf</font> being  <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U1</font>,<font color="Maroon">U1</font> st <br/>( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">msf</font> &amp; <font color="Olive">msf</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U1</font> ) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:12_1_1_2"/>
<font color="Maroon">q</font> is   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U1</font>,<font color="Maroon">U1</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as    <a href="autalg_1.html#M1">MSFunctionSet</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> ;<br/>
<b>take </b>
<font color="Maroon">X</font>
;<br/>

<b>thus </b><a NAME="E8:12_1_1"/>
for <font color="Olive">f</font> being   <a href="autalg_1.html#M2">Element</a> of <font color="Maroon">X</font> holds  <font color="Olive">f</font> is   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U1</font>,<font color="Maroon">U1</font>
 ;<br/>

<b>let </b><font color="Maroon">h</font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U1</font>,<font color="Maroon">U1</font>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:12_1_1_3"/>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:12_1_1_3"/><b>then </b>
( <font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pboole.html#K11">MSFuncs</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font></span>)</span> &amp;  ex <font color="Olive">msf</font> being  <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U1</font>,<font color="Maroon">U1</font> st <br/>( <font color="Maroon">h</font> <a href="pboole.html#R6">=</a> <font color="Olive">msf</font> &amp; <font color="Olive">msf</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U1</font> ) )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:12_1_1_3"/>
<font color="Maroon">h</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U1</font>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E10:12_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">h</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U1</font>
 ;<br/>

<a NAME="E11:12_1_1"/>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pboole.html#K11">MSFuncs</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="autalg_1.html#T22">AUTALG_1:22</a></i>;<br/>
<b>hence </b><a NAME="E12:12_1_1"/>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/>


</div>
