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

<b>let </b><font color="Maroon">C1</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> , <font color="Maroon">C2</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">E162</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a>  <a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E2:27_1_2"/><i><font color="Green">E163</font></i>: 
for <font color="Olive">F</font>, <font color="Olive">G</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">A</font>,<font color="Maroon">B</font><br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C1</font> <a href="binop_1.html#K1">.</a> <font color="Olive">F</font>,<font color="Olive">G</font> iff ( <font color="Olive">F</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Olive">G</font> &amp; <font color="Olive">x</font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Olive">F</font>,<font color="Olive">G</font> ) )
 <b>and </b><br/><a NAME="E3:27_1_2"/><i><font color="Green">E164</font></i>: 
for <font color="Olive">F</font>, <font color="Olive">G</font>, <font color="Olive">H</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">A</font>,<font color="Maroon">B</font>  st <font color="Olive">F</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Olive">G</font> &amp; <font color="Olive">G</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Olive">H</font> holds <br/>for <font color="Olive">t1</font> being   <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Olive">F</font>,<font color="Olive">G</font><br/> for <font color="Olive">t2</font> being   <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Olive">G</font>,<font color="Olive">H</font>  ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C1</font> <a href="multop_1.html#K1">.</a> <font color="Olive">F</font>,<font color="Olive">G</font>,<font color="Olive">H</font> &amp; <font color="Olive">f</font> <a href="binop_1.html#K1">.</a> <font color="Olive">t2</font>,<font color="Olive">t1</font> <a href="hidden.html#R1">=</a> <font color="Olive">t2</font> <a href="functor2.html#K5" target="_self">`*`</a> <font color="Olive">t1</font> )
 <b>and </b><br/><a NAME="E4:27_1_2"/><i><font color="Green">E165</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C2</font> <a href="hidden.html#R1">=</a>  <a href="functor2.html#K7" target="_self">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E5:27_1_2"/><i><font color="Green">E166</font></i>: 
for <font color="Olive">F</font>, <font color="Olive">G</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">A</font>,<font color="Maroon">B</font><br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Olive">F</font>,<font color="Olive">G</font> iff ( <font color="Olive">F</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Olive">G</font> &amp; <font color="Olive">x</font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Olive">F</font>,<font color="Olive">G</font> ) )
 <b>and </b><br/><a NAME="E6:27_1_2"/><i><font color="Green">E167</font></i>: 
for <font color="Olive">F</font>, <font color="Olive">G</font>, <font color="Olive">H</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">A</font>,<font color="Maroon">B</font>  st <font color="Olive">F</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Olive">G</font> &amp; <font color="Olive">G</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Olive">H</font> holds <br/>for <font color="Olive">t1</font> being   <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Olive">F</font>,<font color="Olive">G</font><br/> for <font color="Olive">t2</font> being   <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Olive">G</font>,<font color="Olive">H</font>  ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C2</font> <a href="multop_1.html#K1">.</a> <font color="Olive">F</font>,<font color="Olive">G</font>,<font color="Olive">H</font> &amp; <font color="Olive">f</font> <a href="binop_1.html#K1">.</a> <font color="Olive">t2</font>,<font color="Olive">t1</font> <a href="hidden.html#R1">=</a> <font color="Olive">t2</font> <a href="functor2.html#K5" target="_self">`*`</a> <font color="Olive">t1</font> )
 ;<br/>

<b>set </b><font color="Maroon">R</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font>;<br/>
<b>set </b><font color="Maroon">T</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C2</font>;<br/>
<b>set </b><font color="Maroon">N</font> = the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C1</font>;<br/>
<b>set </b><font color="Maroon">M</font> = the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font>;<br/>
<b>set </b><font color="Maroon">O</font> = the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C1</font>;<br/>
<b>set </b><font color="Maroon">P</font> = the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C2</font>;<br/>
<a NAME="E7:27_1_2"/><i><font color="Green">E170</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C2</font> holds <br/>the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C1</font> <a href="binop_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Olive">j</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">i</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">j</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">E171</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font>
 <b>and </b><br/><a NAME="E2:27_1_2_1"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">i</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">A</font>,<font color="Maroon">B</font> <b>by </b><i>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">j</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">A</font>,<font color="Maroon">B</font> <b>by </b><i><a class="ref" href="functor2.html#T4" target="_self">Th4</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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:27_1_2_1_1"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> implies ( <font color="Maroon">f</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">g</font> &amp; <font color="Maroon">x</font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">f</font>,<font color="Maroon">g</font> ) ) &amp; ( <font color="Maroon">f</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">g</font> &amp; <font color="Maroon">x</font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">f</font>,<font color="Maroon">g</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> ) &amp; ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> implies ( <font color="Maroon">f</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">g</font> &amp; <font color="Maroon">x</font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">f</font>,<font color="Maroon">g</font> ) ) &amp; ( <font color="Maroon">f</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">g</font> &amp; <font color="Maroon">x</font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">f</font>,<font color="Maroon">g</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> ) )
 <b>by </b><i>, <a class="ref" href="functor2.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E2:27_1_2_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> iff <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</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">C1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</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">E173</font></i>: 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font>
 
<b>by </b><i>, <a class="ref" href="functor2.html#T4" target="_self">Th4</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">i</font>, <font color="Olive">j</font>, <font color="Olive">k</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font> holds <br/>the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C1</font> <a href="multop_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Olive">j</font>,<font color="Olive">k</font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C2</font> <a href="multop_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Olive">j</font>,<font color="Olive">k</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">i</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">j</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">k</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:27_1_2_2"/><i><font color="Green">E174</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">i</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">A</font>,<font color="Maroon">B</font> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">j</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">A</font>,<font color="Maroon">B</font> <b>by </b><i>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">k</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">A</font>,<font color="Maroon">B</font> <b>by </b><i>, , </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">C1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</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">C1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</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">C1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</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">C1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</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">E175</font></i>: 
( the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</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">C1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</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">C1</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C2</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</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">C1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</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">C1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span><b>by </b><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">C1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</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">E176</font></i>: 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="functor2.html#T4" target="_self">Th4</a>, , <a class="ref" href="altcat_1.html#T8">ALTCAT_1:8</a></i>;<br/><b>reconsider </b><font color="Maroon">T</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C2</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">i'</font> = <font color="Maroon">i</font>, <font color="Maroon">j'</font> = <font color="Maroon">j</font>, <font color="Maroon">k'</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">T</font> <b>by </b><i>, <a class="ref" href="functor2.html#T4" target="_self">Th4</a>, </i>;<br/><a NAME="E5:27_1_2_2_1_1_1_1_1"/><i><font color="Green">E177</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j'</font>,<font color="Maroon">k'</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</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="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">E178</font></i>: 
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C2</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j'</font>,<font color="Maroon">k'</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">k'</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">E179</font></i>: 
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</font>
 
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>;<br/>
<i><font color="Green">E180</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">C2</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</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">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j'</font>,<font color="Maroon">k'</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</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">C2</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">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">k'</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">C2</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j'</font>,<font color="Maroon">k'</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">k'</font>
 <b>by </b><i>, , <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">C1</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j'</font>,<font color="Maroon">k'</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">k'</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">E181</font></i>: 
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C1</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</font>
 
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>;<br/>
<i><font color="Green">E182</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">C2</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</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">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j'</font>,<font color="Maroon">k'</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</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">C2</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">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">k'</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">C1</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j'</font>,<font color="Maroon">k'</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">k'</font>
 <b>by </b><i>, <a class="ref" href="functor2.html#T4" target="_self">Th4</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">C1</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C2</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="functor2.html#D3" target="_self">Def3</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">C1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</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">E183</font></i>: 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="functor2.html#T4" target="_self">Th4</a>, , <a class="ref" href="altcat_1.html#T8">ALTCAT_1:8</a></i>;<br/><b>reconsider </b><font color="Maroon">T</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C2</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">i'</font> = <font color="Maroon">i</font>, <font color="Maroon">j'</font> = <font color="Maroon">j</font>, <font color="Maroon">k'</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">T</font> <b>by </b><i>, <a class="ref" href="functor2.html#T4" target="_self">Th4</a>, </i>;<br/><a NAME="E5:27_1_2_2_1_1_1_1_2"/><i><font color="Green">E184</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j'</font>,<font color="Maroon">k'</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</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="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">E185</font></i>: 
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C2</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j'</font>,<font color="Maroon">k'</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">k'</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">E186</font></i>: 
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</font>
 
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>;<br/>
<i><font color="Green">E187</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">C2</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</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">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j'</font>,<font color="Maroon">k'</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</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">C2</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">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">k'</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">C2</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j'</font>,<font color="Maroon">k'</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">k'</font>
 <b>by </b><i>, , <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">C1</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j'</font>,<font color="Maroon">k'</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">k'</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">E188</font></i>: 
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C1</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</font>
 
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>;<br/>
<i><font color="Green">E189</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">C2</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</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">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j'</font>,<font color="Maroon">k'</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</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">C2</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">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">k'</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">C1</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j'</font>,<font color="Maroon">k'</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">k'</font>
 <b>by </b><i>, <a class="ref" href="functor2.html#T4" target="_self">Th4</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">C1</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C2</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="functor2.html#D3" target="_self">Def3</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">E190</font></i>: 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</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">E191</font></i>: 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</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">E192</font></i>: 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="functor2.html#T4" target="_self">Th4</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">E193</font></i>: 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="functor2.html#T4" target="_self">Th4</a>, , , <a class="ref" href="altcat_1.html#T8">ALTCAT_1:8</a></i>;<br/><b>reconsider </b><font color="Maroon">T</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C2</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">i'</font> = <font color="Maroon">i</font>, <font color="Maroon">j'</font> = <font color="Maroon">j</font>, <font color="Maroon">k'</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">T</font> <b>by </b><i>, <a class="ref" href="functor2.html#T4" target="_self">Th4</a>, </i>;<br/><a NAME="E7:27_1_2_2_1_2"/><i><font color="Green">E194</font></i>: 
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C2</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">k</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">E195</font></i>: 
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</font>
 
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>;<br/>
<i><font color="Green">E196</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">C2</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</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">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j'</font>,<font color="Maroon">k'</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</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">C2</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">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">k'</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">C2</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">k</font>
 <b>by </b><i>, , <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">C1</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j'</font>,<font color="Maroon">k'</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">k'</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">E197</font></i>: 
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C1</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</font>
 
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>;<br/>
<i><font color="Green">E198</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">C2</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</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">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j'</font>,<font color="Maroon">k'</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</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">C2</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">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">k'</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">C1</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font>,<font color="Maroon">k'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j'</font>,<font color="Maroon">k'</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i'</font>,<font color="Maroon">k'</font>
 <b>by </b><i>, <a class="ref" href="functor2.html#T4" target="_self">Th4</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">t1</font> = the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C1</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">k</font> ;<br/><b>reconsider </b><font color="Maroon">t2</font> = the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C2</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</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">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">k</font> <b>by </b><i/>;<br/><a NAME="E11:27_1_2_2_1_2"/>
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font><br/> for <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> holds  <font color="Maroon">t1</font> <a href="binop_1.html#K1">.</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> <a href="binop_1.html#K1">.</a> <font color="Olive">a</font>,<font color="Olive">b</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">a</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font>;<br/><b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>;<br/>



<a NAME="E1:27_1_2_2_1_2_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="functor2.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E2:27_1_2_2_1_2_3"/><b>then </b><i><font color="Green">E199</font></i>: 
( <font color="Maroon">g</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">h</font> &amp; <font color="Maroon">a</font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">g</font>,<font color="Maroon">h</font> )
 
<b>by </b><i><a class="ref" href="functor2.html#D2" target="_self">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">g</font>,<font color="Maroon">h</font> <b>by </b><i><a class="ref" href="functor2.html#D2" target="_self">Def2</a>, <a class="ref" href="functor2.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E4:27_1_2_2_1_2_3"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:27_1_2_2_1_2_3"/><b>then </b><i><font color="Green">E200</font></i>: 
( <font color="Maroon">f</font> <a href="functor2.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">g</font> &amp; <font color="Maroon">b</font> is    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">f</font>,<font color="Maroon">g</font> )
 
<b>by </b><i><a class="ref" href="functor2.html#D2" target="_self">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="functor2.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">f</font>,<font color="Maroon">g</font> <b>by </b><i><a class="ref" href="functor2.html#D2" target="_self">Def2</a>, </i>;<br/>
<b>consider </b><font color="Maroon">t1'</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">E202</font></i>: 
<font color="Maroon">t1'</font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C1</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">h</font>
 <b>and </b><br/><a NAME="E9:27_1_2_2_1_2_3"/><i><font color="Green">E203</font></i>: 
<font color="Maroon">t1'</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="functor2.html#K5" target="_self">`*`</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="functor2.html#D1" target="_self">Def1</a>, , </i>;<br/>
<b>consider </b><font color="Maroon">t2'</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">E205</font></i>: 
<font color="Maroon">t2'</font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C2</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">h</font>
 <b>and </b><br/><a NAME="E12:27_1_2_2_1_2_3"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">t2'</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="functor2.html#K5" target="_self">`*`</a> <font color="Maroon">b</font>
 <b>by </b><i>, , </i>;<br/>
<b>thus </b><a NAME="E13:27_1_2_2_1_2_3"/>
<font color="Maroon">t1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="functor2.html#D4" target="_self">Def4</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">C1</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C2</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</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">C1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C2</font>
 <b>by </b><i>, <a class="ref" href="functor2.html#T4" target="_self">Th4</a>, , <a class="ref" href="altcat_1.html#T10">ALTCAT_1:10</a></i>;<br/>


</div>
