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<span class="kw">let </span><font color="Maroon" title="c3">F</font> be   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">F</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a> </span><br/>

<a NAME="E1:12_1_1"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
 for <font color="Olive" title="b1">f</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C1</font>  st <font color="Olive" title="b1">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  holds <br/><font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  implies <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_1_1_1"/>
<font color="Maroon" title="c4">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> </span><br/>

<a NAME="E2:12_1_1_1"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c2">C2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:12_1_1_1_1"/>
( <font color="Maroon" title="c2">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  or  not <font color="Maroon" title="c2">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:12_1_1_1_1_1"/>
<font color="Maroon" title="c2">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:12_1_1_1_1_1"/>
<font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#T10" onmouseover="rs('cat_6/T10')" onmouseout="rh()">CAT_6:10</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:12_1_1_1_1_2"/>
 not <font color="Maroon" title="c2">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:12_1_1_1_1_2"/>
<font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c4">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12_1_1_1"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="cat_6.html#T22" onmouseover="rs('cat_6/T22')" onmouseout="rh()">CAT_6:22</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E2:12_1_1"/>
 for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C1</font>  st <font color="Olive" title="b1">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">f2</font> holds <br/>( <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f2</font> &amp; <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#T1" onmouseover="rs('cat_6/T1')" onmouseout="rh()">CAT_6:1</a></span>;<br/>
<a NAME="E3:12_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">F</font> is  <a href="cat_6.html#V14" title="CAT_6:attr.14">multiplicative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:12_1_1"/>
<font color="Maroon" title="c3">F</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_1_1"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</a>, <a class="ref" href="cat_6.html#D22" onmouseover="rs('cat_6/D22')" onmouseout="rh()">CAT_6:def 22</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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