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

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="msinst_1.html#K2" target="_self">MSAlg_set</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V6">strict</a>  <a href="altcat_1.html#L2">AltCatStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:12_1_2"/><i><font color="Green">E8</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="msinst_1.html#K2" target="_self">MSAlg_set</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="msinst_1.html#K2" target="_self">MSAlg_set</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> holds the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="msinst_1.html#K3" target="_self">MSAlg_morph</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>4</sub></font><br/>for <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being <a href="funcop_1.html#V1">Function-yielding</a> <a href="funct_1.html#NM1">Function</a> st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="altcat_1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> <a href="pboole.html#K12">**</a> <font color="Olive">b<sub>4</sub></font> ) )
 <b>and </b><br/><a NAME="E3:12_1_2"/><i><font color="Green">E9</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="msinst_1.html#K2" target="_self">MSAlg_set</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="msinst_1.html#K2" target="_self">MSAlg_set</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> holds the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>5</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="msinst_1.html#K3" target="_self">MSAlg_morph</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>5</sub></font><br/>for <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being <a href="funcop_1.html#V1">Function-yielding</a> <a href="funct_1.html#NM1">Function</a> st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>5</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>5</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>5</sub></font> <a href="altcat_1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> <a href="pboole.html#K12">**</a> <font color="Olive">b<sub>4</sub></font> ) )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>7</sub></font> = the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>5</sub></font> as    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i><a class="txt" href="msinst_1.html#E2:12_1_2"><i><font color="Green">E8</font></i></a>, <a class="txt" href="msinst_1.html#E3:12_1_2"><i><font color="Green">E9</font></i></a></i>;<br/>
<div><i><font color="Green">E10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>thus </b><font color="Maroon">c<sub>6</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> = 
 <a href="msinst_1.html#K3" target="_self">MSAlg_morph</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="msinst_1.html#E2:12_1_2"><i><font color="Green">E8</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="msinst_1.html#E3:12_1_2"><i><font color="Green">E9</font></i></a></i>
;<br/></div><b>end;</b></div>
<a NAME="E6:12_1_2"/><b>then </b><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="altcat_1.html#T9">ALTCAT_1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>9</sub></font> = the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>5</sub></font> as    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K3">:]</a></span> <b>by </b><i><a class="txt" href="msinst_1.html#E2:12_1_2"><i><font color="Green">E8</font></i></a>, <a class="txt" href="msinst_1.html#E3:12_1_2"><i><font color="Green">E9</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:12_1_2_2"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="msinst_1.html#E2:12_1_2"><i><font color="Green">E8</font></i></a>, <a class="txt" href="msinst_1.html#E1:12_1_2_2"><i><font color="Green">E12</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="msinst_1.html#E3:12_1_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msinst_1.html#E1:12_1_2_2"><i><font color="Green">E12</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>22</sub></font> = <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="mcart_1.html#K3">]</a></span>;<br/><a NAME="E5:12_1_2_2"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="mcart_1.html#K3">]</a></span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="mcart_1.html#K3">]</a></span> )
 <b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>;<br/><a NAME="E6:12_1_2_2"/><i><font color="Green">E14</font></i>: 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K3">:]</a></span>
 <b>by </b><i><a class="txt" href="msinst_1.html#E1:12_1_2_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="mcart_1.html#T73">MCART_1:73</a></i>;<br/><b>thus </b><a NAME="E7:12_1_2_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_2_2_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="mcart_1.html#K3">]</a></span> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="mcart_1.html#K3">]</a></span>,<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="mcart_1.html#K3">]</a></span> <b>by </b><i><a class="txt" href="msinst_1.html#E2:12_1_2"><i><font color="Green">E8</font></i></a>, <a class="txt" href="msinst_1.html#E6:12_1_2_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="mcart_1.html#K3">]</a></span> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="mcart_1.html#K3">]</a></span>,<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="mcart_1.html#K3">]</a></span> <b>by </b><i><a class="txt" href="msinst_1.html#E3:12_1_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msinst_1.html#E6:12_1_2_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_1_2_2_1_1"/>
( <span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:12_1_2_2_1_1_1"/><i><font color="Green">E15</font></i>: 
<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:12_1_2_2_1_1_1"/><b>then </b>
<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="msinst_1.html#E5:12_1_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="altcat_1.html#T9">ALTCAT_1:9</a></i>;<br/><a NAME="E3:12_1_2_2_1_1_1"/><b>then </b><i><font color="Green">E16</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="mcart_1.html#K3">]</a></span> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="mcart_1.html#K3">]</a></span> )
 <b>by </b><i><a class="txt" href="msinst_1.html#E1:12_1_2_2_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E4:12_1_2_2_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="mcart_1.html#K3">]</a></span> holds <br/><font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_2_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>25</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:12_1_2_2_1_1_1_1"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="mcart_1.html#K3">]</a></span>
 ;<br/>

<a NAME="E2:12_1_2_2_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="msinst_1.html#E6:12_1_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>;<br/>
<a NAME="E3:12_1_2_2_1_1_1_1"/><b>then </b><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="txt" href="msinst_1.html#E1:12_1_2_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="altcat_1.html#D6">ALTCAT_1:def 6</a></i>;<br/>
<a NAME="E4:12_1_2_2_1_1_1_1"/><b>then </b><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<a NAME="E5:12_1_2_2_1_1_1_1"/><b>then </b><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a>  <a href="msinst_1.html#K3" target="_self">MSAlg_morph</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a>  <a href="msinst_1.html#K3" target="_self">MSAlg_morph</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> )
 
<b>by </b><i><a class="txt" href="msinst_1.html#E2:12_1_2"><i><font color="Green">E8</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>26</sub></font>, <font color="Maroon">c<sub>27</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_6.html#V1">feasible</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>28</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>27</sub></font><b> such that </b><br/><a NAME="E7:12_1_2_2_1_1_1_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> <a href="mcart_1.html#K2">`2</a>  &amp; the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>26</sub></font> <a href="pzfmisc1.html#R1">is_transformable_to</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>27</sub></font> )
 <b>by </b><i><a class="ref" href="msinst_1.html#D3" target="_self">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>29</sub></font>, <font color="Maroon">c<sub>30</sub></font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_6.html#V1">feasible</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>31</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>30</sub></font><b> such that </b><br/><a NAME="E9:12_1_2_2_1_1_1_1"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> <a href="mcart_1.html#K1">`1</a>  &amp; the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>29</sub></font> <a href="pzfmisc1.html#R1">is_transformable_to</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>30</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>30</sub></font> )
 <b>by </b><i><a class="txt" href="msinst_1.html#E5:12_1_2_2_1_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="msinst_1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E10:12_1_2_2_1_1_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>28</sub></font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E3:12_1_2_2_1_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="msinst_1.html#E7:12_1_2_2_1_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="msinst_1.html#E9:12_1_2_2_1_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> = 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="txt" href="msinst_1.html#E10:12_1_2_2_1_1_1_1"><i><font color="Green">E22</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>31</sub></font> <a href="pboole.html#K12">**</a> <font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="txt" href="msinst_1.html#E2:12_1_2"><i><font color="Green">E8</font></i></a>, <a class="txt" href="msinst_1.html#E4:12_1_2_2_1_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="msinst_1.html#E7:12_1_2_2_1_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="msinst_1.html#E9:12_1_2_2_1_1_1_1"><i><font color="Green">E21</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>5</sub></font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="txt" href="msinst_1.html#E3:12_1_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msinst_1.html#E4:12_1_2_2_1_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="msinst_1.html#E7:12_1_2_2_1_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="msinst_1.html#E9:12_1_2_2_1_1_1_1"><i><font color="Green">E21</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>5</sub></font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="msinst_1.html#E10:12_1_2_2_1_1_1_1"><i><font color="Green">E22</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E12:12_1_2_2_1_1_1_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:12_1_2_2_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="msinst_1.html#E6:12_1_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="msinst_1.html#E5:12_1_2_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="msinst_1.html#E3:12_1_2_2_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:12_1_2_2_1_1_2"/>
<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:12_1_2_2_1_1_2"/><b>then </b>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="msinst_1.html#E6:12_1_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><a NAME="E3:12_1_2_2_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/><b>hence </b><a NAME="E4:12_1_2_2_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="msinst_1.html#E5:12_1_2_2"><i><font color="Green">E13</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:12_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="msinst_1.html#E2:12_1_2"><i><font color="Green">E8</font></i></a>, <a class="txt" href="msinst_1.html#E3:12_1_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="msinst_1.html#E6:12_1_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="altcat_1.html#T10">ALTCAT_1:10</a></i>;<br/>


</div>
