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<b>let </b><font color="Maroon">A</font> be   <a href="altcat_1.html#NM4">category</a>, <font color="Maroon">B</font> be   <a href="altcat_1.html#NM4">category</a>;<br/><b>let </b><font color="Maroon">F</font> be  <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>, <font color="Maroon">G</font> be  <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/>



<b>assume </b><b>that </b><br/><a NAME="E1:10"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">a</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font> holds  <font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="functor0.html#K3">.</a> <font color="Olive">a</font>
 <b>and </b><br/><a NAME="E2:10"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>  st <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">f</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">a</font>,<font color="Olive">b</font> holds  <font color="Maroon">F</font> <a href="functor0.html#K6">.</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="functor0.html#K6">.</a> <font color="Olive">f</font>
 ;<br/>

<a NAME="E3:10"/>
the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font> is <a href="functor0.html#V1">Covariant</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D13">FUNCTOR0:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">ff</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E5:10"/><i><font color="Green">E39</font></i>: 
the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font> <a href="funct_2.html#R2">=</a> <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">ff</font>,<font color="Maroon">ff</font></span><a href="funct_3.html#K16">:]</a></span>
 <b>by </b><i><a class="ref" href="functor0.html#D2">FUNCTOR0:def 2</a></i>;<br/>
<a NAME="E6:10"/>
the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font> is <a href="functor0.html#V1">Covariant</a>
 
<b>by </b><i><a class="ref" href="functor0.html#D13">FUNCTOR0:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">gg</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E8:10"/><i><font color="Green">E42</font></i>: 
the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font> <a href="funct_2.html#R2">=</a> <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">gg</font>,<font color="Maroon">gg</font></span><a href="funct_3.html#K16">:]</a></span>
 <b>by </b><i><a class="ref" href="functor0.html#D2">FUNCTOR0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font>, <font color="Maroon">y</font> = <font color="Maroon">b</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font> ;<br/><a NAME="E2:10_1"/><i><font color="Green">E43</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">ff</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">gg</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:10_1"/>
( the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">gg</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">gg</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">gg</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">gg</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, , , <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>;<br/><a NAME="E4:10_1"/><b>then </b>
( <font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">G</font> <a href="functor0.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">gg</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">gg</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">G</font> <a href="functor0.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">gg</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">gg</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a>  )
 ;<br/><a NAME="E5:10_1"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">G</font> <a href="functor0.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">gg</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">G</font> <a href="functor0.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">gg</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><a NAME="E6:10_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="functor0.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="functor0.html#K3">.</a> <font color="Maroon">y</font> )
 <b>by </b><i/>;<br/><b>thus </b>the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">ff</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>
<br/>.= 
the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
<b>by </b><i>, , , , <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E10:10"/><b>then </b><i><font color="Green">E50</font></i>: 
the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font> <a href="funct_2.html#R2">=</a> the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:10_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:10_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font>, <font color="Maroon">y</font> = <font color="Maroon">b</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/><a NAME="E5:10_2"/><i><font color="Green">E54</font></i>: 
( ( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) &amp; ( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="functor0.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="functor0.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 <b>by </b><i><a class="ref" href="functor0.html#D19">FUNCTOR0:def 19</a></i>;<br/><a NAME="E6:10_2"/><b>then </b><i><font color="Green">E55</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="functor0.html#K5">Morph-Map</a> <font color="Maroon">F</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="functor0.html#K5">Morph-Map</a> <font color="Maroon">G</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:10_2_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 ;<br/><b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">z</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">x</font>,<font color="Maroon">y</font> ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="functor0.html#K5">Morph-Map</a> <font color="Maroon">F</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> = 
<font color="Maroon">F</font> <a href="functor0.html#K6">.</a> <font color="Maroon">f</font>
<b>by </b><i>, , <a class="ref" href="functor0.html#D16">FUNCTOR0:def 16</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="functor0.html#K6">.</a> <font color="Maroon">f</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="functor0.html#K5">Morph-Map</a> <font color="Maroon">G</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
<b>by </b><i>, , <a class="ref" href="functor0.html#D16">FUNCTOR0:def 16</a></i>
;<br/></div><b>end;</b></div><b>thus </b><a NAME="E8:10_2"/>
the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:10"/>
 <a href="functor0.html#G2">FunctorStr</a>(# the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">F</font>,the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">F</font> #) <a href="hidden.html#R1">=</a>  <a href="functor0.html#G2">FunctorStr</a>(# the <a href="functor0.html#U1">ObjectMap</a> of <font color="Maroon">G</font>,the <a href="functor0.html#U2">MorphMap</a> of <font color="Maroon">G</font> #)
 <b>by </b><i>, <a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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