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<div class="add">

<span class="kw">thus </span><a NAME="E1:35_1_1"/>
 <a href="functor0.html#K10" title="FUNCTOR0:func.10">incl</a> <font color="Maroon" title="c2">C</font> is  <a href="functor0.html#V12" title="FUNCTOR0:attr.12">id-preserving</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="functor0.html#K10" title="FUNCTOR0:func.10">incl</a> <font color="Maroon" title="c2">C</font> is  <a href="functor0.html#V13" title="FUNCTOR0:attr.13">comp-preserving</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">a</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">C</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="functor0.html#D21" title="FUNCTOR0:def.21">FUNCTOR0:def 21</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> <span class="p2">(<span class="default"><a href="functor0.html#K10" title="FUNCTOR0:func.10">incl</a> <font color="Maroon" title="c2">C</font></span>)</span>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K10" title="FUNCTOR0:func.10">incl</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c3">a</font></span>)</span></span><br/>

<a NAME="E1:35_1_1_1"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<span class="p1">(<span class="default"><a href="functor0.html#K10" title="FUNCTOR0:func.10">incl</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#T28" title="FUNCTOR0:th.28">FUNCTOR0:28</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="functor0.html#K5" title="FUNCTOR0:func.5">Morph-Map</a> <span class="p2">(<span class="default"><a href="functor0.html#K10" title="FUNCTOR0:func.10">incl</a> <font color="Maroon" title="c2">C</font></span>)</span>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">a</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="functor0.html#K10" title="FUNCTOR0:func.10">incl</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D16" title="FUNCTOR0:def.16">FUNCTOR0:def 16</a></span>
<br/>.= 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow20.html#T28" target="_self" title="YELLOW20:th.28">Th28</a></span>
<br/>.= 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K10" title="FUNCTOR0:func.10">incl</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c3">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1_1"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="altcat_2.html#T35" title="ALTCAT_2:th.35">ALTCAT_2:35</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c3">o1</font>, <font color="Maroon" title="c4">o2</font>, <font color="Maroon" title="c5">o3</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">C</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="functor0.html#D24" title="FUNCTOR0:def.24">FUNCTOR0:def 24</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c3">o1</font>,<font color="Maroon" title="c4">o2</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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">M2</a>(<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>)<br/> for <font color="Olive">b<sub>2</sub></font> being  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">M2</a>(<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>) holds  <span class="p1">(<span class="default"><a href="functor0.html#K10" title="FUNCTOR0:func.10">incl</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K10" title="FUNCTOR0:func.10">incl</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K10" title="FUNCTOR0:func.10">incl</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:35_1_1"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">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="c4">o2</font>,<font color="Maroon" title="c5">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">M2</a>(<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>)<br/> for <font color="Olive">b<sub>2</sub></font> being  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">M2</a>(<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>) holds  <span class="p1">(<span class="default"><a href="functor0.html#K10" title="FUNCTOR0:func.10">incl</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K10" title="FUNCTOR0:func.10">incl</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K10" title="FUNCTOR0:func.10">incl</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">M2</a>(<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>) holds  <span class="p1">(<span class="default"><a href="functor0.html#K10" title="FUNCTOR0:func.10">incl</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K10" title="FUNCTOR0:func.10">incl</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K10" title="FUNCTOR0:func.10">incl</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c6">f</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">g</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">o2</font>,<font color="Maroon" title="c5">o3</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="functor0.html#K10" title="FUNCTOR0:func.10">incl</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K10" title="FUNCTOR0:func.10">incl</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c7">g</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K10" title="FUNCTOR0:func.10">incl</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c6">f</font></span>)</span></span><br/>



<a NAME="E3:35_1_1"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c5">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="altcat_1.html#D4" title="ALTCAT_1:def.4">ALTCAT_1:def 4</a></span>;<br/>
<a NAME="E4:35_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A3</font></span>: 
( <span class="p1">(<span class="default"><a href="functor0.html#K10" title="FUNCTOR0:func.10">incl</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">o3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">o3</font> &amp; <span class="p1">(<span class="default"><a href="functor0.html#K10" title="FUNCTOR0:func.10">incl</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow20.html#T28" target="_self" title="YELLOW20:th.28">Th28</a>, <a class="ref" href="functor0.html#T28" title="FUNCTOR0:th.28">FUNCTOR0:28</a></span>;<br/>
<a NAME="E5:35_1_1"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
( <span class="p1">(<span class="default"><a href="functor0.html#K10" title="FUNCTOR0:func.10">incl</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c3">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o1</font> &amp; <span class="p1">(<span class="default"><a href="functor0.html#K10" title="FUNCTOR0:func.10">incl</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c4">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#T28" title="FUNCTOR0:th.28">FUNCTOR0:28</a></span>;<br/>
<a NAME="E6:35_1_1"/>
( <span class="p1">(<span class="default"><a href="functor0.html#K10" title="FUNCTOR0:func.10">incl</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> &amp; <span class="p1">(<span class="default"><a href="functor0.html#K10" title="FUNCTOR0:func.10">incl</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="yellow20.html#T28" target="_self" title="YELLOW20:th.28">Th28</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:35_1_1"/>
<span class="p1">(<span class="default"><a href="functor0.html#K10" title="FUNCTOR0:func.10">incl</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K10" title="FUNCTOR0:func.10">incl</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c7">g</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K10" title="FUNCTOR0:func.10">incl</a> <font color="Maroon" title="c2">C</font></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="txt" href="#E5:35_1_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="txt" href="#E4:35_1_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="altcat_2.html#T33" title="ALTCAT_2:th.33">ALTCAT_2:33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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