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

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V2">transitive</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="E1:27_1_2"/><i><font color="Green">E21</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:27_1_2"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="functor0.html#V9">strict</a> <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</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> iff ( <font color="Olive">b<sub>1</sub></font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> ) )
 <b>and </b><br/><a NAME="E3:27_1_2"/><i><font color="Green">E23</font></i>: 
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="functor0.html#V9">strict</a> <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Olive">b<sub>3</sub></font> holds <br/>for <font color="Olive">b<sub>4</sub></font> being   <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font><br/>for <font color="Olive">b<sub>5</sub></font> being   <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>ex <font color="Olive">b<sub>6</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>3</sub></font> <a href="multop_1.html#K1">.</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> &amp; <font color="Olive">b<sub>6</sub></font> <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="functor2.html#K5" target="_self">`*`</a> <font color="Olive">b<sub>4</sub></font> )
 <b>and </b><br/><a NAME="E4:27_1_2"/><i><font color="Green">E24</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="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E5:27_1_2"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="functor0.html#V9">strict</a> <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>3</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> iff ( <font color="Olive">b<sub>1</sub></font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> ) )
 <b>and </b><br/><a NAME="E6:27_1_2"/><i><font color="Green">E26</font></i>: 
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="functor0.html#V9">strict</a> <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Olive">b<sub>3</sub></font> holds <br/>for <font color="Olive">b<sub>4</sub></font> being   <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font><br/>for <font color="Olive">b<sub>5</sub></font> being   <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>ex <font color="Olive">b<sub>6</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="multop_1.html#K1">.</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> &amp; <font color="Olive">b<sub>6</sub></font> <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="functor2.html#K5" target="_self">`*`</a> <font color="Olive">b<sub>4</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E7:27_1_2"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> holds <br/>the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</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> 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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><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><b>that </b><br/><a NAME="E1:27_1_2_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E2:27_1_2_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as  <a href="functor0.html#V9">strict</a> <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="functor2.html#E1:27_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="functor2.html#E1:27_1_2_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="functor2.html#D10" target="_self">Def10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font> as  <a href="functor0.html#V9">strict</a> <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="functor2.html#E4:27_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="functor2.html#E2:27_1_2_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="functor2.html#D10" target="_self">Def10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:27_1_2_1_1"/>
( ( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</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> implies ( <font color="Maroon">c<sub>13</sub></font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> ) ) &amp; ( <font color="Maroon">c<sub>13</sub></font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> implies <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</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>15</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="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> implies ( <font color="Maroon">c<sub>13</sub></font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> ) ) &amp; ( <font color="Maroon">c<sub>13</sub></font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> implies <font color="Maroon">c<sub>15</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="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> ) )
 <b>by </b><i><a class="txt" href="functor2.html#E2:27_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="functor2.html#E5:27_1_2"><i><font color="Green">E25</font></i></a></i>;<br/><b>hence </b><a NAME="E2:27_1_2_1_1"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</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> iff <font color="Maroon">c<sub>15</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="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:27_1_2_1"/>
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</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> <a href="hidden.html#R1">=</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="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:27_1_2"/><b>then </b><i><font color="Green">E28</font></i>: 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="functor2.html#E1:27_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="functor2.html#E4:27_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="altcat_1.html#T8">ALTCAT_1:8</a></i>;<br/>
<a NAME="E9:27_1_2"/>
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="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> holds <br/>the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>3</sub></font> <a href="multop_1.html#K1">.</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> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="multop_1.html#K1">.</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:27_1_2_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>11</sub></font> as  <a href="functor0.html#V9">strict</a> <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="functor2.html#E1:27_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="functor2.html#D10" target="_self">Def10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>12</sub></font> as  <a href="functor0.html#V9">strict</a> <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="functor2.html#E1:27_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="functor2.html#E1:27_1_2_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="functor2.html#D10" target="_self">Def10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>13</sub></font> as  <a href="functor0.html#V9">strict</a> <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="functor2.html#E1:27_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="functor2.html#E1:27_1_2_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="functor2.html#D10" target="_self">Def10</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_2_2_1"/>
( the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</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> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:27_1_2_2_1_1"/><i><font color="Green">E30</font></i>: 
( the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</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> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:27_1_2_2_1_1"/>
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>3</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2_2_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_2_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_2_2_1_1_1_1"/>
( the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</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> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span><b>by </b><i><a class="txt" href="functor2.html#E1:27_1_2_2_1_1"><i><font color="Green">E30</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_2_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:27_1_2_2_1_1_1_1_1"/>
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</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> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:27_1_2_2_1_1_1_1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
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="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="functor2.html#E1:27_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="functor2.html#E4:27_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="functor2.html#E7:27_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="altcat_1.html#T8">ALTCAT_1:8</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></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">c<sub>18</sub></font> = <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>17</sub></font> <b>by </b><i><a class="txt" href="functor2.html#E1:27_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="functor2.html#E4:27_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="functor2.html#E1:27_1_2_2"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E5:27_1_2_2_1_1_1_1_1"/><i><font color="Green">E32</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">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="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p2">(<span class="default">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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="functor2.html#E2:27_1_2_2_1_1_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>;<br/><a NAME="E6:27_1_2_2_1_1_1_1_1"/><i><font color="Green">E33</font></i>: 
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">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="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p2">(<span class="default">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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2_2_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:27_1_2_2_1_1_1_1_1_1"/><i><font color="Green">E34</font></i>: 
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K4">]</a></span> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>;<br/>
<i><font color="Green">E35</font></i>: <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K4">]</a></span> = 
<span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">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="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p2">(<span class="default">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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="altcat_1.html#D6">ALTCAT_1:def 6</a></i>
;<br/>
<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K4">]</a></span> = 
<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K2">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
<br/>.= 
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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="ref" href="altcat_1.html#D5">ALTCAT_1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E4:27_1_2_2_1_1_1_1_1_1"/>
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">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="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p2">(<span class="default">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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="functor2.html#E1:27_1_2_2_1_1_1_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="functor2.html#E2:27_1_2_2_1_1_1_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:27_1_2_2_1_1_1_1_1"/>
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>3</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">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="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p2">(<span class="default">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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2_2_1_1_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:27_1_2_2_1_1_1_1_1_2"/><i><font color="Green">E34</font></i>: 
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K4">]</a></span> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>3</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>;<br/>
<i><font color="Green">E35</font></i>: <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K4">]</a></span> = 
<span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">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="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p2">(<span class="default">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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="altcat_1.html#D6">ALTCAT_1:def 6</a></i>
;<br/>
<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K4">]</a></span> = 
<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K2">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
<br/>.= 
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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="ref" href="altcat_1.html#D5">ALTCAT_1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E4:27_1_2_2_1_1_1_1_1_2"/>
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>3</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">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="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p2">(<span class="default">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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="functor2.html#E1:27_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="functor2.html#E4:27_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="functor2.html#E8:27_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="functor2.html#E1:27_1_2_2_1_1_1_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="functor2.html#E2:27_1_2_2_1_1_1_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E8:27_1_2_2_1_1_1_1_1"/>
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>3</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="functor2.html#E5:27_1_2_2_1_1_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="functor2.html#E6:27_1_2_2_1_1_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="partfun1.html#T58">PARTFUN1:58</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_2_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:27_1_2_2_1_1_1_1_2"/>
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:27_1_2_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E31</font></i>: 
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="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="functor2.html#E1:27_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="functor2.html#E4:27_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="functor2.html#E7:27_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="altcat_1.html#T8">ALTCAT_1:8</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></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">c<sub>18</sub></font> = <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>17</sub></font> <b>by </b><i><a class="txt" href="functor2.html#E1:27_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="functor2.html#E4:27_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="functor2.html#E1:27_1_2_2"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E5:27_1_2_2_1_1_1_1_2"/><i><font color="Green">E32</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">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="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p2">(<span class="default">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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="functor2.html#E2:27_1_2_2_1_1_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>;<br/><a NAME="E6:27_1_2_2_1_1_1_1_2"/><i><font color="Green">E33</font></i>: 
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">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="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p2">(<span class="default">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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2_2_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:27_1_2_2_1_1_1_1_2_1"/><i><font color="Green">E34</font></i>: 
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K4">]</a></span> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>;<br/>
<i><font color="Green">E35</font></i>: <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K4">]</a></span> = 
<span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">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="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p2">(<span class="default">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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="altcat_1.html#D6">ALTCAT_1:def 6</a></i>
;<br/>
<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K4">]</a></span> = 
<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K2">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
<br/>.= 
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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="ref" href="altcat_1.html#D5">ALTCAT_1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E4:27_1_2_2_1_1_1_1_2_1"/>
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">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="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p2">(<span class="default">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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="functor2.html#E1:27_1_2_2_1_1_1_1_2_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="functor2.html#E2:27_1_2_2_1_1_1_1_2_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:27_1_2_2_1_1_1_1_2"/>
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>3</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">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="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p2">(<span class="default">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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2_2_1_1_1_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:27_1_2_2_1_1_1_1_2_2"/><i><font color="Green">E34</font></i>: 
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K4">]</a></span> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>3</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>;<br/>
<i><font color="Green">E35</font></i>: <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K4">]</a></span> = 
<span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">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="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p2">(<span class="default">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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="altcat_1.html#D6">ALTCAT_1:def 6</a></i>
;<br/>
<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K4">]</a></span> = 
<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K2">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
<br/>.= 
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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="ref" href="altcat_1.html#D5">ALTCAT_1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E4:27_1_2_2_1_1_1_1_2_2"/>
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>3</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">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="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p2">(<span class="default">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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="functor2.html#E1:27_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="functor2.html#E4:27_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="functor2.html#E8:27_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="functor2.html#E1:27_1_2_2_1_1_1_1_2_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="functor2.html#E2:27_1_2_2_1_1_1_1_2_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E8:27_1_2_2_1_1_1_1_2"/>
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>3</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="functor2.html#E5:27_1_2_2_1_1_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="functor2.html#E6:27_1_2_2_1_1_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="partfun1.html#T58">PARTFUN1:58</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_2_2_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:27_1_2_2_1_2"/><i><font color="Green">E30</font></i>: 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:27_1_2_2_1_2"/><i><font color="Green">E31</font></i>: 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E3:27_1_2_2_1_2"/><i><font color="Green">E32</font></i>: 
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="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <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="functor2.html#E1:27_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="functor2.html#E4:27_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="functor2.html#E7:27_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="functor2.html#E1:27_1_2_2_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="altcat_1.html#T8">ALTCAT_1:8</a></i>;<br/><a NAME="E4:27_1_2_2_1_2"/><i><font color="Green">E33</font></i>: 
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="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <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="functor2.html#E1:27_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="functor2.html#E4:27_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="functor2.html#E7:27_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="functor2.html#E2:27_1_2_2_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="altcat_1.html#T8">ALTCAT_1:8</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></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">c<sub>18</sub></font> = <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>17</sub></font> <b>by </b><i><a class="txt" href="functor2.html#E1:27_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="functor2.html#E4:27_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="functor2.html#E1:27_1_2_2"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E7:27_1_2_2_1_2"/><i><font color="Green">E34</font></i>: 
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">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="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span>,<span class="p2">(<span class="default">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="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,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="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2_2_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:27_1_2_2_1_2_1"/><i><font color="Green">E35</font></i>: 
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K4">]</a></span> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>;<br/>
<i><font color="Green">E36</font></i>: <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K4">]</a></span> = 
<span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">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="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p2">(<span class="default">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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="altcat_1.html#D6">ALTCAT_1:def 6</a></i>
;<br/>
<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K4">]</a></span> = 
<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K2">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
<br/>.= 
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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="ref" href="altcat_1.html#D5">ALTCAT_1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E4:27_1_2_2_1_2_1"/>
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">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="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span>,<span class="p2">(<span class="default">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="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,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="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="functor2.html#E1:27_1_2_2_1_2_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="functor2.html#E2:27_1_2_2_1_2_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:27_1_2_2_1_2"/>
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>3</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">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="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p2">(<span class="default">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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2_2_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:27_1_2_2_1_2_2"/><i><font color="Green">E35</font></i>: 
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K4">]</a></span> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>3</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>;<br/>
<i><font color="Green">E36</font></i>: <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K4">]</a></span> = 
<span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">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="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p2">(<span class="default">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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="altcat_1.html#D6">ALTCAT_1:def 6</a></i>
;<br/>
<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K4">]</a></span> = 
<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K2">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
<br/>.= 
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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="ref" href="altcat_1.html#D5">ALTCAT_1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E4:27_1_2_2_1_2_2"/>
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>3</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">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="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p2">(<span class="default">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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,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="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="functor2.html#E1:27_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="functor2.html#E4:27_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="functor2.html#E8:27_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="functor2.html#E1:27_1_2_2_1_2_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="functor2.html#E2:27_1_2_2_1_2_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>3</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">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="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span>,<span class="p2">(<span class="default">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="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,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="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">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="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span>,<span class="p2">(<span class="default">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="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,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="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font> <b>by </b><i><a class="txt" href="functor2.html#E7:27_1_2_2_1_2"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E11:27_1_2_2_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of 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="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of 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="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> holds <font color="Maroon">c<sub>21</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> <font color="Maroon">c<sub>22</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2_2_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</sub></font> be    <a href="subset_1.html#M1">Element</a> of 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="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>24</sub></font> be    <a href="subset_1.html#M1">Element</a> of 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="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>;<br/>



<a NAME="E1:27_1_2_2_1_2_3"/>
<font color="Maroon">c<sub>23</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="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="functor2.html#E4:27_1_2_2_1_2"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E2:27_1_2_2_1_2_3"/><b>then </b><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> )
 
<b>by </b><i><a class="txt" href="functor2.html#E5:27_1_2"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>23</sub></font> as    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> <b>by </b><i><a class="txt" href="functor2.html#E5:27_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="functor2.html#E4:27_1_2_2_1_2"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E4:27_1_2_2_1_2_3"/>
<font color="Maroon">c<sub>24</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="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="functor2.html#E3:27_1_2_2_1_2"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E5:27_1_2_2_1_2_3"/><b>then </b><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> )
 
<b>by </b><i><a class="txt" href="functor2.html#E5:27_1_2"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>24</sub></font> as    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> <b>by </b><i><a class="txt" href="functor2.html#E5:27_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="functor2.html#E3:27_1_2_2_1_2"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>27</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:27_1_2_2_1_2_3"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>3</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E9:27_1_2_2_1_2_3"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> <a href="functor2.html#K5" target="_self">`*`</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="functor2.html#E3:27_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="functor2.html#E2:27_1_2_2_1_2_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="functor2.html#E5:27_1_2_2_1_2_3"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>28</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E11:27_1_2_2_1_2_3"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E12:27_1_2_2_1_2_3"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> <a href="functor2.html#K5" target="_self">`*`</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="functor2.html#E6:27_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="functor2.html#E2:27_1_2_2_1_2_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="functor2.html#E5:27_1_2_2_1_2_3"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>thus </b><a NAME="E13:27_1_2_2_1_2_3"/>
<font color="Maroon">c<sub>21</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="functor2.html#E8:27_1_2_2_1_2_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="functor2.html#E9:27_1_2_2_1_2_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="functor2.html#E11:27_1_2_2_1_2_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="functor2.html#E12:27_1_2_2_1_2_3"><i><font color="Green">E40</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E12:27_1_2_2_1_2"/>
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>3</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E10:27_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="functor2.html#E1:27_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="functor2.html#E4:27_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="functor2.html#E8:27_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="altcat_1.html#T10">ALTCAT_1:10</a></i>;<br/>


</div>
