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<a NAME="E1:40_1_2"/>
for <font color="Olive">PC1</font>, <font color="Olive">PC2</font> being   <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">I</font>,<font color="Maroon">S</font>  st ( 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">PC1</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> ) ) &amp; ( 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">PC2</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> ) ) holds <br/><font color="Olive">PC1</font> <a href="pboole.html#R6">=</a> <font color="Olive">PC2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">PC1</font> be    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">I</font>,<font color="Maroon">S</font>, <font color="Maroon">PC2</font> be    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">I</font>,<font color="Maroon">S</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:40_1_2_1"/><i><font color="Green">E69</font></i>: 
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">PC1</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>and </b><br/><a NAME="E2:40_1_2_1"/><i><font color="Green">E73</font></i>: 
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">PC2</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> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:40_1_2_1_1"/>
<font color="Maroon">i1</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">i1</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/><b>consider </b><font color="Maroon">Ji1</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">Cs1</font> being    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">Ji1</font>,<font color="Maroon">S</font><b> such that </b><br/><a NAME="E4:40_1_2_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">Ji1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font>
 <b>and </b><br/><a NAME="E5:40_1_2_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">Cs1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font>
 <b>and </b><br/><a NAME="E6:40_1_2_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">PC1</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i'</font> <a href="hidden.html#R1">=</a>  <a href="pralg_2.html#K15">product</a> <font color="Maroon">Cs1</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">Ji2</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">Cs2</font> being    <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">Ji2</font>,<font color="Maroon">S</font><b> such that </b><br/><a NAME="E8:40_1_2_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">Ji2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font>
 <b>and </b><br/><a NAME="E9:40_1_2_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">Cs2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font>
 <b>and </b><br/><a NAME="E10:40_1_2_1_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">PC2</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i'</font> <a href="hidden.html#R1">=</a>  <a href="pralg_2.html#K15">product</a> <font color="Maroon">Cs2</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E11:40_1_2_1_1"/>
<font color="Maroon">PC1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">PC2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font>
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:40_1_2_1"/>
<font color="Maroon">PC1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">PC2</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:40_1_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Maroon">I</font>,<font color="Maroon">S</font>  st ( 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">b<sub>1</sub></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> ) ) &amp; ( 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">b<sub>2</sub></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> ) ) holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/>


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