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<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:25_1_1_1"/>
<font color="Maroon">I</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">I'</font> = <font color="Maroon">I</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>reconsider </b><font color="Maroon">A'</font> = <font color="Maroon">A</font> as    <a href="pralg_2.html#M2" target="_self">MSAlgebra-Family</a> of <font color="Maroon">I'</font>,<font color="Maroon">S</font> ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I'</font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">A'</font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>;<br/><b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:25_1_1_1"/><i><font color="Green">E29</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I'</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I'</font> holds  <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S4">FUNCT_1:sch 4</a>();<br/></i><b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>take </b><font color="Maroon">f</font> = <font color="Maroon">f</font>;<br/><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E7:25_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><b>then reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I'</font> ;<br/><b>reconsider </b><font color="Maroon">U0</font> = <font color="Maroon">A'</font> <a href="pralg_2.html#K6" target="_self">.</a> <font color="Maroon">i'</font> as    <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font> ;<br/><b>take </b><font color="Maroon">U0</font> = <font color="Maroon">U0</font>;<br/><b>thus </b><a NAME="E10:25_1_1_1"/>
( <font color="Maroon">U0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:25_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>take </b>
 <a href="pboole.html#K1">[0]</a> <font color="Maroon">I</font>
;<br/>

<b>thus </b><a NAME="E3:25_1_1"/>
 <a href="pboole.html#K1">[0]</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="pboole.html#T134">PBOOLE:134</a></i>;<br/>


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