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

<b>let </b><font color="Maroon" title="c1">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font> is <a href="altcat_1.html#V2" title="ALTCAT_1:attr.2">transitive</a> &amp;  <a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font> is <a href="altcat_1.html#V11" title="ALTCAT_1:attr.11">associative</a> &amp;  <a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font> is <a href="altcat_1.html#V12" title="ALTCAT_1:attr.12">with_units</a> )</span><br/>

<b>thus </b><a NAME="E1:42"/><i><font color="Green" title="E30">A1</font></i>: 
 <a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font> is <a href="altcat_1.html#V2" title="ALTCAT_1:attr.2">transitive</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font> is <a href="altcat_1.html#V11" title="ALTCAT_1:attr.11">associative</a> &amp;  <a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font> is <a href="altcat_1.html#V12" title="ALTCAT_1:attr.12">with_units</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">o1</font>, <font color="Maroon" title="c3">o2</font>, <font color="Maroon" title="c4">o3</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D4" target="_self" title="ALTCAT_1:def.4">ALTCAT_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c4">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c4">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<b>assume </b><a NAME="E1:42_1"/>
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c4">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c4">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E2:42_1"/><b>then </b>
(  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c4">o3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="altcat_1.html#D16" target="_self" title="ALTCAT_1:def.16">Def16</a></i>;<br/>
<a NAME="E3:42_1"/><b>then </b>
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c4">o3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="altcat_1.html#T5" target="_self" title="ALTCAT_1:th.5">Th5</a></i>;<br/>
<b>hence </b><a NAME="E4:42_1"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c4">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="altcat_1.html#D16" target="_self" title="ALTCAT_1:def.16">Def16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>set </b><font color="Maroon" title="c2">G</font> = the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c3">C</font> = the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span>;<br/>
<b>thus </b><a NAME="E2:42"/>
 <a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font> is <a href="altcat_1.html#V11" title="ALTCAT_1:attr.11">associative</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font> is <a href="altcat_1.html#V12" title="ALTCAT_1:attr.12">with_units</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">j</font>, <font color="Maroon" title="c6">k</font>, <font color="Maroon" title="c7">l</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D7" target="_self" title="ALTCAT_1:def.7">ALTCAT_1:def 7</a>,<a class="ref" href="altcat_1.html#D17" target="_self" title="ALTCAT_1:def.17">ALTCAT_1:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font> &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">l</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b3">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b3">h</font>,<font color="Olive" title="b2">g</font></span>)</span>,<font color="Olive" title="b1">f</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c8">i'</font> = <font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c9">j'</font> = <font color="Maroon" title="c5">j</font>, <font color="Maroon" title="c10">k'</font> = <font color="Maroon" title="c6">k</font>, <font color="Maroon" title="c11">l'</font> = <font color="Maroon" title="c7">l</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> ;<br/>
<b>let </b><font color="Maroon" title="c12">f</font>, <font color="Maroon" title="c13">g</font>, <font color="Maroon" title="c14">h</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> &amp; <font color="Maroon" title="c13">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c14">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">l</font> implies <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c12">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">h</font>,<font color="Maroon" title="c13">g</font></span>)</span>,<font color="Maroon" title="c12">f</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:42_2"/><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>
 <b>and </b><br/><a NAME="E3:42_2"/><i><font color="Green" title="E32">A3</font></i>: 
<font color="Maroon" title="c13">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font>
 <b>and </b><br/><a NAME="E4:42_2"/><i><font color="Green" title="E33">A4</font></i>: 
<font color="Maroon" title="c14">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c12">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">h</font>,<font color="Maroon" title="c13">g</font></span>)</span>,<font color="Maroon" title="c12">f</font></span><br/>

<a NAME="E5:42_2"/><i><font color="Green" title="E34">A5</font></i>: 
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">i'</font>,<font color="Maroon" title="c9">j'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
;<br/>
<a NAME="E6:42_2"/><i><font color="Green" title="E35">A6</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">i'</font>,<font color="Maroon" title="c9">j'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D16" target="_self" title="ALTCAT_1:def.16">Def16</a></i>;<br/>
<a NAME="E7:42_2"/><i><font color="Green" title="E36">A7</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c9">j'</font>,<font color="Maroon" title="c10">k'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D16" target="_self" title="ALTCAT_1:def.16">Def16</a></i>;<br/>
<a NAME="E8:42_2"/><i><font color="Green" title="E37">A8</font></i>: 
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c10">k'</font>,<font color="Maroon" title="c11">l'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
;<br/>
<a NAME="E9:42_2"/><i><font color="Green" title="E38">A9</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c10">k'</font>,<font color="Maroon" title="c11">l'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">l</font>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D16" target="_self" title="ALTCAT_1:def.16">Def16</a></i>;<br/>
<a NAME="E10:42_2"/><i><font color="Green" title="E39">A10</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c9">j'</font>,<font color="Maroon" title="c10">k'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="altcat_1.html#E3:42_2"><i><font color="Green" title="E32">A3</font></i></a></i>;<br/>
<a NAME="E11:42_2"/><b>then </b><i><font color="Green" title="E40">A11</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">i'</font>,<font color="Maroon" title="c10">k'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="altcat_1.html#E1:42"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="altcat_1.html#E2:42_2"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="altcat_1.html#E5:42_2"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="altcat_1.html#D4" target="_self" title="ALTCAT_1:def.4">Def4</a></i>;<br/>
<a NAME="E12:42_2"/><i><font color="Green" title="E41">A12</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c9">j'</font>,<font color="Maroon" title="c11">l'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="altcat_1.html#E1:42"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="altcat_1.html#E4:42_2"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="altcat_1.html#E8:42_2"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="altcat_1.html#E10:42_2"><i><font color="Green" title="E39">A10</font></i></a>, <a class="ref" href="altcat_1.html#D4" target="_self" title="ALTCAT_1:def.4">Def4</a></i>;<br/>
<a NAME="E13:42_2"/><i><font color="Green" title="E42">A13</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">i'</font>,<font color="Maroon" title="c11">l'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="altcat_1.html#E1:42"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="altcat_1.html#E4:42_2"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="altcat_1.html#E8:42_2"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="altcat_1.html#E11:42_2"><i><font color="Green" title="E40">A11</font></i></a>, <a class="ref" href="altcat_1.html#D4" target="_self" title="ALTCAT_1:def.4">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">f'</font> = <font color="Maroon" title="c12">f</font>, <font color="Maroon" title="c16">g'</font> = <font color="Maroon" title="c13">g</font>, <font color="Maroon" title="c17">h'</font> = <font color="Maroon" title="c14">h</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="altcat_1.html#E2:42_2"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="altcat_1.html#E3:42_2"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="altcat_1.html#E4:42_2"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="altcat_1.html#E6:42_2"><i><font color="Green" title="E35">A6</font></i></a>, <a class="txt" href="altcat_1.html#E7:42_2"><i><font color="Green" title="E36">A7</font></i></a>, <a class="txt" href="altcat_1.html#E9:42_2"><i><font color="Green" title="E38">A9</font></i></a>, <a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">f''</font> = <font color="Maroon" title="c12">f</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">i'</font>,<font color="Maroon" title="c9">j'</font> <b>by </b><i><a class="txt" href="altcat_1.html#E2:42_2"><i><font color="Green" title="E31">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">g''</font> = <font color="Maroon" title="c13">g</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">j'</font>,<font color="Maroon" title="c10">k'</font> <b>by </b><i><a class="txt" href="altcat_1.html#E3:42_2"><i><font color="Green" title="E32">A3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">h''</font> = <font color="Maroon" title="c14">h</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c10">k'</font>,<font color="Maroon" title="c11">l'</font> <b>by </b><i><a class="txt" href="altcat_1.html#E4:42_2"><i><font color="Green" title="E33">A4</font></i></a></i>;<br/>
<a NAME="E18:42_2"/><i><font color="Green" title="E43">A14</font></i>: 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">g''</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c18">f''</font>
 
<b>by </b><i><a class="txt" href="altcat_1.html#E2:42_2"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="altcat_1.html#E3:42_2"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="altcat_1.html#D10" target="_self" title="ALTCAT_1:def.10">Def10</a></i>;<br/>
<a NAME="E19:42_2"/><i><font color="Green" title="E44">A15</font></i>: 
<font color="Maroon" title="c19">g''</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c18">f''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g'</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c15">f'</font>
 
<b>by </b><i><a class="txt" href="altcat_1.html#E2:42_2"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="altcat_1.html#E3:42_2"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="altcat_1.html#E11:42_2"><i><font color="Green" title="E40">A11</font></i></a>, <a class="ref" href="altcat_1.html#T18" target="_self" title="ALTCAT_1:th.18">Th18</a></i>;<br/>
<a NAME="E20:42_2"/><i><font color="Green" title="E45">A16</font></i>: 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">h</font>,<font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">h''</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c19">g''</font>
 
<b>by </b><i><a class="txt" href="altcat_1.html#E3:42_2"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="altcat_1.html#E4:42_2"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="altcat_1.html#D10" target="_self" title="ALTCAT_1:def.10">Def10</a></i>;<br/>
<a NAME="E21:42_2"/><i><font color="Green" title="E46">A17</font></i>: 
<font color="Maroon" title="c20">h''</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c19">g''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">h'</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c16">g'</font>
 
<b>by </b><i><a class="txt" href="altcat_1.html#E3:42_2"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="altcat_1.html#E4:42_2"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="altcat_1.html#E12:42_2"><i><font color="Green" title="E41">A12</font></i></a>, <a class="ref" href="altcat_1.html#T18" target="_self" title="ALTCAT_1:th.18">Th18</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c12">f</font></span>)</span> = 
<font color="Maroon" title="c20">h''</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">g''</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c18">f''</font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_1.html#E4:42_2"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="altcat_1.html#E11:42_2"><i><font color="Green" title="E40">A11</font></i></a>, <a class="txt" href="altcat_1.html#E18:42_2"><i><font color="Green" title="E43">A14</font></i></a>, <a class="ref" href="altcat_1.html#D10" target="_self" title="ALTCAT_1:def.10">Def10</a></i>
<br/>.= 
<font color="Maroon" title="c17">h'</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">g'</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c15">f'</font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_1.html#E4:42_2"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="altcat_1.html#E11:42_2"><i><font color="Green" title="E40">A11</font></i></a>, <a class="txt" href="altcat_1.html#E13:42_2"><i><font color="Green" title="E42">A13</font></i></a>, <a class="txt" href="altcat_1.html#E19:42_2"><i><font color="Green" title="E44">A15</font></i></a>, <a class="ref" href="altcat_1.html#T18" target="_self" title="ALTCAT_1:th.18">Th18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">h'</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c16">g'</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c15">f'</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">h''</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c19">g''</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c18">f''</font>
<b>by </b><i><a class="txt" href="altcat_1.html#E2:42_2"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="altcat_1.html#E12:42_2"><i><font color="Green" title="E41">A12</font></i></a>, <a class="txt" href="altcat_1.html#E13:42_2"><i><font color="Green" title="E42">A13</font></i></a>, <a class="txt" href="altcat_1.html#E21:42_2"><i><font color="Green" title="E46">A17</font></i></a>, <a class="ref" href="altcat_1.html#T18" target="_self" title="ALTCAT_1:th.18">Th18</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">h</font>,<font color="Maroon" title="c13">g</font></span>)</span>,<font color="Maroon" title="c12">f</font>
<b>by </b><i><a class="txt" href="altcat_1.html#E2:42_2"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="altcat_1.html#E12:42_2"><i><font color="Green" title="E41">A12</font></i></a>, <a class="txt" href="altcat_1.html#E20:42_2"><i><font color="Green" title="E45">A16</font></i></a>, <a class="ref" href="altcat_1.html#D10" target="_self" title="ALTCAT_1:def.10">Def10</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:42"/>
the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> is <a href="altcat_1.html#V5" title="ALTCAT_1:attr.5">with_left_units</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D18" target="_self" title="ALTCAT_1:def.18">ALTCAT_1:def 18</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> is <a href="altcat_1.html#V4" title="ALTCAT_1:attr.4">with_right_units</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D9" target="_self" title="ALTCAT_1:def.9">ALTCAT_1:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font> &amp; ( for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b3">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">i</font>,<font color="Maroon" title="c4">i</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Olive" title="b2">i</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">e</font>,<font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> ) )</span><br/>

<b>reconsider </b><font color="Maroon" title="c5">i'</font> = <font color="Maroon" title="c4">i</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> ;<br/>
<b>take </b>
 <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b2">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">i</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> ) )</span><br/>

<a NAME="E2:42_3"/><i><font color="Green" title="E31">A18</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">i'</font>,<font color="Maroon" title="c5">i'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D16" target="_self" title="ALTCAT_1:def.16">Def16</a></i>;<br/>
<b>hence </b><a NAME="E3:42_3"/>
 <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="ref" href="altcat_1.html#T2" target="_self" title="ALTCAT_1:th.2">Th2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b2">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">i</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font></span><br/>

<b>let </b><font color="Maroon" title="c6">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c4">i</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font></span><br/><b>let </b><font color="Maroon" title="c7">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c4">i</font> implies <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> )</span><br/>



<b>reconsider </b><font color="Maroon" title="c8">j'</font> = <font color="Maroon" title="c6">j</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> ;<br/>
<b>assume </b><a NAME="E5:42_3"/><i><font color="Green" title="E32">A19</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c4">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font></span><br/>

<a NAME="E6:42_3"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">j'</font>,<font color="Maroon" title="c5">i'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c4">i</font>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D16" target="_self" title="ALTCAT_1:def.16">Def16</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">f'</font> = <font color="Maroon" title="c7">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c4">i</font> <b>by </b><i><a class="txt" href="altcat_1.html#E5:42_3"><i><font color="Green" title="E32">A19</font></i></a>, <a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">f''</font> = <font color="Maroon" title="c7">f</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">j'</font>,<font color="Maroon" title="c5">i'</font> <b>by </b><i><a class="txt" href="altcat_1.html#E5:42_3"><i><font color="Green" title="E32">A19</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">I</font> =  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">i'</font>,<font color="Maroon" title="c5">i'</font> <b>by </b><i><a class="txt" href="altcat_1.html#E2:42_3"><i><font color="Green" title="E31">A18</font></i></a>, <a class="ref" href="altcat_1.html#T2" target="_self" title="ALTCAT_1:th.2">Th2</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c7">f</font> = 
<font color="Maroon" title="c11">I</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c10">f''</font>
<b>by </b><i><a class="txt" href="altcat_1.html#E2:42_3"><i><font color="Green" title="E31">A18</font></i></a>, <a class="txt" href="altcat_1.html#E5:42_3"><i><font color="Green" title="E32">A19</font></i></a>, <a class="ref" href="altcat_1.html#D10" target="_self" title="ALTCAT_1:def.10">Def10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c9">f'</font>
<b>by </b><i><a class="txt" href="altcat_1.html#E2:42_3"><i><font color="Green" title="E31">A18</font></i></a>, <a class="txt" href="altcat_1.html#E5:42_3"><i><font color="Green" title="E32">A19</font></i></a>, <a class="ref" href="altcat_1.html#T18" target="_self" title="ALTCAT_1:th.18">Th18</a></i>
<br/>.= 
<font color="Maroon" title="c7">f</font>
<b>by </b><i><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c4">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D8" target="_self" title="ALTCAT_1:def.8">ALTCAT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font> &amp; ( for <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b3">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Olive" title="b2">j</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font>,<font color="Olive" title="b2">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> ) )</span><br/>

<b>reconsider </b><font color="Maroon" title="c5">i'</font> = <font color="Maroon" title="c4">i</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> ;<br/>
<b>take </b>
 <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font> &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b2">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Olive" title="b1">j</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font>,<font color="Olive" title="b1">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">f</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> ) )</span><br/>

<a NAME="E5:42"/><i><font color="Green" title="E31">A20</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">i'</font>,<font color="Maroon" title="c5">i'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D16" target="_self" title="ALTCAT_1:def.16">Def16</a></i>;<br/>
<b>hence </b><a NAME="E6:42"/>
 <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="ref" href="altcat_1.html#T2" target="_self" title="ALTCAT_1:th.2">Th2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b2">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Olive" title="b1">j</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font>,<font color="Olive" title="b1">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">f</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font></span><br/>

<b>let </b><font color="Maroon" title="c6">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">j</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font></span><br/><b>let </b><font color="Maroon" title="c7">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">j</font> implies <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">f</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> )</span><br/>



<b>reconsider </b><font color="Maroon" title="c8">j'</font> = <font color="Maroon" title="c6">j</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> ;<br/>
<b>assume </b><a NAME="E8:42"/><i><font color="Green" title="E32">A21</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">f</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font></span><br/>

<a NAME="E9:42"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">i'</font>,<font color="Maroon" title="c8">j'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">j</font>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D16" target="_self" title="ALTCAT_1:def.16">Def16</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">f'</font> = <font color="Maroon" title="c7">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">j</font> <b>by </b><i><a class="txt" href="altcat_1.html#E8:42"><i><font color="Green" title="E32">A21</font></i></a>, <a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">f''</font> = <font color="Maroon" title="c7">f</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">i'</font>,<font color="Maroon" title="c8">j'</font> <b>by </b><i><a class="txt" href="altcat_1.html#E8:42"><i><font color="Green" title="E32">A21</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">I</font> =  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">i'</font>,<font color="Maroon" title="c5">i'</font> <b>by </b><i><a class="txt" href="altcat_1.html#E5:42"><i><font color="Green" title="E31">A20</font></i></a>, <a class="ref" href="altcat_1.html#T2" target="_self" title="ALTCAT_1:th.2">Th2</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">f</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span> = 
<font color="Maroon" title="c10">f''</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c11">I</font>
<b>by </b><i><a class="txt" href="altcat_1.html#E5:42"><i><font color="Green" title="E31">A20</font></i></a>, <a class="txt" href="altcat_1.html#E8:42"><i><font color="Green" title="E32">A21</font></i></a>, <a class="ref" href="altcat_1.html#D10" target="_self" title="ALTCAT_1:def.10">Def10</a></i>
<br/>.= 
<font color="Maroon" title="c9">f'</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_1.html#E5:42"><i><font color="Green" title="E31">A20</font></i></a>, <a class="txt" href="altcat_1.html#E8:42"><i><font color="Green" title="E32">A21</font></i></a>, <a class="ref" href="altcat_1.html#T18" target="_self" title="ALTCAT_1:th.18">Th18</a></i>
<br/>.= 
<font color="Maroon" title="c7">f</font>
<b>by </b><i><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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