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<b>thus </b><a NAME="E1:40_1_1"/>
 ex <font color="Olive">PC</font> being   <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">I</font>,<font color="Maroon">S</font> st <br/>for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/> ex <font color="Olive">Ji</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ex <font color="Olive">Cs</font> being   <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Olive">Ji</font>,<font color="Maroon">S</font> st <br/>( <font color="Olive">Ji</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">Cs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">PC</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="pralg_2.html#K15">product</a> <font color="Olive">Cs</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">Ji</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ex <font color="Olive">Cs</font> being   <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Olive">Ji</font>,<font color="Maroon">S</font> st <br/>( <font color="Olive">Ji</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">Cs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pralg_2.html#K15">product</a> <font color="Olive">Cs</font> );<br/>
<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1_1"><b>now </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="E1:40_1_1_1_1"/><i><font color="Green">E41</font></i>: 
<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">Ji</font> = <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/><b>reconsider </b><font color="Maroon">Cs</font> = <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">Ji</font>,<font color="Maroon">S</font> <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:40_1_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="pralg_2.html#K15">product</a> <font color="Maroon">Cs</font>
 ;<br/><b>consider </b><font color="Maroon">j</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:40_1_1_1_1"/><i><font color="Green">E44</font></i>: 
 ex <font color="Olive">Ji</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ex <font color="Olive">Cs</font> being   <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Olive">Ji</font>,<font color="Maroon">S</font> st <br/>( <font color="Olive">Ji</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">Cs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="pralg_2.html#K15">product</a> <font color="Olive">Cs</font> )
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">j</font> = <font color="Maroon">j</font>;<br/><b>thus </b><a NAME="E8:40_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">j</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">PC</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E3:40_1_1_1"/><i><font color="Green">E45</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/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Maroon">PC</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1_2"><b>now </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="E1:40_1_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><b>then consider </b><font color="Maroon">Ji</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">Cs</font> being    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">Ji</font>,<font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:40_1_1_1_2"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">Ji</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">Cs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">PC</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pralg_2.html#K15">product</a> <font color="Maroon">Cs</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E4:40_1_1_1_2"/>
<font color="Maroon">PC</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">PC</font> = <font color="Maroon">PC</font> as    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">I</font>,<font color="Maroon">S</font> <b>by </b><i><a class="ref" href="pralg_2.html#D12">PRALG_2:def 12</a></i>;<br/>
<b>take </b>
<font color="Maroon">PC</font>
;<br/>

<b>thus </b><a NAME="E6:40_1_1_1"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/> ex <font color="Olive">Ji</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ex <font color="Olive">Cs</font> being   <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Olive">Ji</font>,<font color="Maroon">S</font> st <br/>( <font color="Olive">Ji</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">Cs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">PC</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="pralg_2.html#K15">product</a> <font color="Olive">Cs</font> )
 <b>by </b><i/>;<br/>


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


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