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

<a NAME="E1:40_1_2"/>
for <font color="Olive" title="b1">PC1</font>, <font color="Olive" title="b2">PC2</font> being   <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">S</font>  st ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/> ex <font color="Olive" title="b4">Ji</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b5">Cs</font> being   <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Olive" title="b4">Ji</font>,<font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b4">Ji</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b5">Cs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b1">PC1</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Olive" title="b5">Cs</font> ) ) &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/> ex <font color="Olive" title="b4">Ji</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b5">Cs</font> being   <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Olive" title="b4">Ji</font>,<font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b4">Ji</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b5">Cs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b2">PC2</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Olive" title="b5">Cs</font> ) ) holds <br/><font color="Olive" title="b1">PC1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">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" title="c5">PC1</font>, <font color="Maroon" title="c6">PC2</font> be    <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">S</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:40_1_2_1"/><i><font color="Green" title="E30">A7</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/> ex <font color="Olive" title="b2">Ji</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b3">Cs</font> being   <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Olive" title="b2">Ji</font>,<font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b2">Ji</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">Cs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c5">PC1</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Olive" title="b3">Cs</font> )
 <b>and </b><br/><a NAME="E2:40_1_2_1"/><i><font color="Green" title="E31">A8</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/> ex <font color="Olive" title="b2">Ji</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b3">Cs</font> being   <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Olive" title="b2">Ji</font>,<font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b2">Ji</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">Cs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c6">PC2</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Olive" title="b3">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" title="c7">i1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:40_1_2_1_1"/>
<font color="Maroon" title="c7">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c8">i'</font> = <font color="Maroon" title="c7">i1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> ;<br/><b>consider </b><font color="Maroon" title="c9">Ji1</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c10">Cs1</font> being    <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c9">Ji1</font>,<font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E4:40_1_2_1_1"/><i><font color="Green" title="E32">A9</font></i>: 
<font color="Maroon" title="c9">Ji1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i'</font>
 <b>and </b><br/><a NAME="E5:40_1_2_1_1"/><i><font color="Green" title="E33">A10</font></i>: 
<font color="Maroon" title="c10">Cs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i'</font>
 <b>and </b><br/><a NAME="E6:40_1_2_1_1"/><i><font color="Green" title="E34">A11</font></i>: 
<font color="Maroon" title="c5">PC1</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c8">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c10">Cs1</font>
 <b>by </b><i><a class="txt" href="pralg_3.html#E1:40_1_2_1"><i><font color="Green" title="E30">A7</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c11">Ji2</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c12">Cs2</font> being    <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c11">Ji2</font>,<font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E8:40_1_2_1_1"/><i><font color="Green" title="E35">A12</font></i>: 
<font color="Maroon" title="c11">Ji2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i'</font>
 <b>and </b><br/><a NAME="E9:40_1_2_1_1"/><i><font color="Green" title="E36">A13</font></i>: 
<font color="Maroon" title="c12">Cs2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i'</font>
 <b>and </b><br/><a NAME="E10:40_1_2_1_1"/><i><font color="Green" title="E37">A14</font></i>: 
<font color="Maroon" title="c6">PC2</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c8">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c12">Cs2</font>
 <b>by </b><i><a class="txt" href="pralg_3.html#E2:40_1_2_1"><i><font color="Green" title="E31">A8</font></i></a></i>;<br/><b>thus </b><a NAME="E11:40_1_2_1_1"/>
<font color="Maroon" title="c5">PC1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">PC2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i1</font>
 <b>by </b><i><a class="txt" href="pralg_3.html#E4:40_1_2_1_1"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="pralg_3.html#E5:40_1_2_1_1"><i><font color="Green" title="E33">A10</font></i></a>, <a class="txt" href="pralg_3.html#E6:40_1_2_1_1"><i><font color="Green" title="E34">A11</font></i></a>, <a class="txt" href="pralg_3.html#E8:40_1_2_1_1"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="pralg_3.html#E9:40_1_2_1_1"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="pralg_3.html#E10:40_1_2_1_1"><i><font color="Green" title="E37">A14</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:40_1_2_1"/>
<font color="Maroon" title="c5">PC1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c6">PC2</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">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" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">S</font>  st ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/> ex <font color="Olive" title="b4">Ji</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b5">Cs</font> being   <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Olive" title="b4">Ji</font>,<font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b4">Ji</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b5">Cs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Olive" title="b5">Cs</font> ) ) &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/> ex <font color="Olive" title="b4">Ji</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b5">Cs</font> being   <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Olive" title="b4">Ji</font>,<font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b4">Ji</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b5">Cs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Olive" title="b5">Cs</font> ) ) holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/>


</div>
