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

<b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">A</font> be    <a href="pralg_2.html#M2" target="_self">MSAlgebra-Family</a> of <font color="Maroon">I</font>,<font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">o</font> be   <a href="msualg_1.html#NM2">OperSymbol</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>









<b>assume </b><a NAME="E1:35"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:35"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:35"/><i><font color="Green">E35</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K3" target="_self">Frege</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:35"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span> &amp; ( 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>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) )
 <b>by </b><i>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E7:35"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">o</font></span>)</span> <a href="pralg_1.html#K15">..</a> <font color="Maroon">g</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b> <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">o</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_1.html#D17">PRALG_1:def 17</a></i>
<br/>.= 
<font color="Maroon">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/><br/>

<a NAME="E9:35"/><i><font color="Green">E38</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">o</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/>

<a NAME="E10:35"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E11:35"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="pralg_1.html#D17">PRALG_1:def 17</a></i>;<br/>
<a NAME="E12:35"/><i><font color="Green">E50</font></i>: 
 <a href="funct_6.html#K1">SubFuncs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">o</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:35_2"/>
 <a href="funct_6.html#K1">SubFuncs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">o</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_6.html#T27">FUNCT_6:27</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:35_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">o</font></span>)</span>
 ;<br/>

<a NAME="E3:35_2"/><b>then </b>
 ex <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">o</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">o</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:35_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="funct_6.html#K1">SubFuncs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pralg_2.html#D1" target="_self">Def1</a>, <a class="ref" href="funct_6.html#D1">FUNCT_6:def 1</a></i>;<br/>


</div><b>end;</b></div>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">o</font></span>)</span> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1">SubFuncs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_6.html#D2">FUNCT_6:def 2</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">o</font></span>)</span>
<b>by </b><i>, <a class="ref" href="relat_1.html#T169">RELAT_1:169</a></i>
;<br/>
<a NAME="E14:35"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K13" target="_self">?.</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E15:35"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>;<br/>
<a NAME="E16:35"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <font color="Maroon">o</font>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">i</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<b>hence </b><a NAME="E17:35"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="msualg_1.html#K4">Result</a> <font color="Maroon">o</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">i</font></span>)</span>
 ;<br/>


</div>
