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

<span class="kw">set </span><font color="Maroon" title="c4">a</font> =  the   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>;<br/>
<a NAME="E1:16_1_1"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E2:16_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">O</font> =  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span> as   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>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">O</font> = <font color="Maroon" title="c5">O</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">D</font> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">f</font> =  the   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>;<br/>
<a NAME="E5:16_1_1"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E6:16_1_1"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c2">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E7:16_1_1"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c2">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E8:16_1_1"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c2">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> $1 &amp;  ex <font color="Olive" title="b1">E</font> being    <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of <font color="Maroon" title="c2">D</font> st <br/>(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">O</font> &amp;  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Olive" title="b1">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 ) );<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">MM</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:16_1_1"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">MM</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></span>
<span class="kw">set </span><font color="Maroon" title="c9">HH</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b1">a0</font>,<font color="Olive" title="b2">b</font>)</span>)</span> where <font color="Olive" title="b1">a0</font>, <font color="Olive" title="b2">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">D</font> : ( <font color="Olive" title="b1">a0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O</font> ) </span> } </span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">M0</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b1">a0</font>,<font color="Olive" title="b2">b</font>)</span>)</span> where <font color="Olive" title="b1">a0</font>, <font color="Olive" title="b2">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">D</font> : ( <font color="Olive" title="b1">a0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O</font> ) </span> } </span>  as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">D</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_2.html#T19" title="CAT_2:th.19">CAT_2:19</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">D0</font> =  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c2">D</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">M0</font>, <font color="Maroon" title="c12">C0</font> =  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c2">D</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">M0</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c10">M0</font>,<font color="Maroon" title="c6">O</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_2.html#T20" title="CAT_2:th.20">CAT_2:20</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">CC</font> =  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c10">M0</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c10">M0</font>,<font color="Maroon" title="c10">M0</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<font color="Maroon" title="c10">M0</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_2.html#T20" title="CAT_2:th.20">CAT_2:20</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">I0</font> =  the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c2">D</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">O</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c6">O</font>,<font color="Maroon" title="c10">M0</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_2.html#T20" title="CAT_2:th.20">CAT_2:20</a></span>;<br/>
<a NAME="E15:16_1_1"/>
 <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c6">O</font>,<font color="Maroon" title="c10">M0</font>,<font color="Maroon" title="c11">D0</font>,<font color="Maroon" title="c12">C0</font>,<font color="Maroon" title="c13">CC</font>,<font color="Maroon" title="c14">I0</font> #) <a href="cat_2.html#R1" title="CAT_2:pred.1">is_full_subcategory_of</a> <font color="Maroon" title="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_2.html#T21" title="CAT_2:th.21">CAT_2:21</a></span>;<br/>
<a NAME="E16:16_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A7</font></span>: 
 <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c6">O</font>,<font color="Maroon" title="c10">M0</font>,<font color="Maroon" title="c11">D0</font>,<font color="Maroon" title="c12">C0</font>,<font color="Maroon" title="c13">CC</font>,<font color="Maroon" title="c14">I0</font> #) is    <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of <font color="Maroon" title="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_2.html#D6" title="CAT_2:def.6">CAT_2:def 6</a></span>;<br/>
<a NAME="E17:16_1_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">M0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> or <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M0</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_1_1_1"/>
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M0</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:16_1_1_1"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F</font>
 <span class="kw">and </span><br/><a NAME="E4:16_1_1_1"/><span class="lab"><font color="Green" title="E17">A9</font></span>: 
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">x</font> = <font color="Maroon" title="c16">x</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a></span>;<br/>
<a NAME="E6:16_1_1_1"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
<span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E7:16_1_1_1"/><span class="lab"><font color="Green" title="E19">A11</font></span>: 
<span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E8:16_1_1_1"/><span class="lab"><font color="Green" title="E20">A12</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_1_1_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="ref" href="cat_1.html#T69" title="CAT_1:th.69">CAT_1:69</a></span>;<br/>
<a NAME="E9:16_1_1_1"/><span class="lab"><font color="Green" title="E21">A13</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_1_1_1"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="ref" href="cat_1.html#T69" title="CAT_1:th.69">CAT_1:69</a></span>;<br/>
<a NAME="E10:16_1_1_1"/><span class="lab"><font color="Green" title="E22">A14</font></span>: 
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x</font></span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_1_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a></span>;<br/>
<a NAME="E11:16_1_1_1"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">x</font></span>)</span></span>)</span>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b1">a0</font>,<font color="Olive" title="b2">b</font>)</span>)</span> where <font color="Olive" title="b1">a0</font>, <font color="Olive" title="b2">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">D</font> : ( <font color="Olive" title="b1">a0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16_1_1_1"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E9:16_1_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:16_1_1_1"/>
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:16_1_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" 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>
<a NAME="E18:16_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A15</font></span>: 
<font color="Maroon" title="c10">M0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">MM</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:16_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E16:16_1_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">M</font> =  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c8">MM</font>;<br/>
<a NAME="E19:16_1_1"/><span class="lab"><font color="Green" title="E17">A16</font></span>: 
 for <font color="Olive" title="b1">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">MM</font> holds <br/> <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:16_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a></span>;<br/>
<a NAME="E20:16_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A17</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c8">MM</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:16_1_1"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="ref" href="setfam_1.html#T5" title="SETFAM_1:th.5">SETFAM_1:5</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">MM</font> implies <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">X</font> )</span><br/><span class="kw">assume </span><a NAME="E1:16_1_1_2"/>
<font color="Maroon" title="c16">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">MM</font>
 ; <a class="txt" 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="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">X</font></span><br/><a NAME="E2:16_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A18</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:16_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a></span>;<br/><a NAME="E3:16_1_1_2"/>
<font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_1_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:16_1_1_2"/>
<font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1_2"><span class="lab"><font color="Green" title="E19">A18</font></span></a></span>; <a class="txt" 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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">M</font> =  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c8">MM</font> as   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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:16_1_1"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></span>;<br/>
<a NAME="E23:16_1_1"/>
<font color="Maroon" title="c16">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">M</font> or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">D</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_1_1_3"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">D</font></span><br/>

<a NAME="E2:16_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:16_1_1"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:16_1_1_3"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">D</font>
 ; <a class="txt" 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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">M</font> = <font color="Maroon" title="c16">M</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">D</font> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">DOM</font> =  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c2">D</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">M</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c19">COD</font> =  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c2">D</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">M</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">COMP</font> =  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c17">M</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">ID</font> =  the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c2">D</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">O</font>;<br/>
<a NAME="E25:16_1_1"/><span class="lab"><font color="Green" title="E19">A19</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c2">D</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_1_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/>
<a NAME="E26:16_1_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c2">D</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">M</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">O</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c2">D</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">M</font></span>)</span> or <font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_1_1_4"/>
<font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c2">D</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">M</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:16_1_1_4"/><span class="lab"><font color="Green" title="E20">A20</font></span>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">M</font>
 <span class="kw">and </span><br/><a NAME="E4:16_1_1_4"/><span class="lab"><font color="Green" title="E21">A21</font></span>: 
<font color="Maroon" title="c22">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c2">D</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:16_1_1"><span class="lab"><font color="Green" title="E19">A19</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c24">x</font> = <font color="Maroon" title="c23">x</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">D</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1_4"><span class="lab"><font color="Green" title="E20">A20</font></span></a></span>;<br/>
<a NAME="E6:16_1_1_4"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:16_1_1"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="txt" href="#E3:16_1_1_4"><span class="lab"><font color="Green" title="E20">A20</font></span></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:16_1_1_4"/><span class="lab"><font color="Green" title="E22">A22</font></span>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">X</font>
 <span class="kw">and </span><br/><a NAME="E9:16_1_1_4"/><span class="lab"><font color="Green" title="E23">A23</font></span>: 
<font color="Maroon" title="c25">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b1">a0</font>,<font color="Olive" title="b2">b</font>)</span>)</span> where <font color="Olive" title="b1">a0</font>, <font color="Olive" title="b2">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">D</font> : ( <font color="Olive" title="b1">a0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c26">a</font>, <font color="Maroon" title="c27">b</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">D</font><span class="kw"> such that </span><br/><a NAME="E11:16_1_1_4"/><span class="lab"><font color="Green" title="E24">A24</font></span>: 
<font color="Maroon" title="c25">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c26">a</font>,<font color="Maroon" title="c27">b</font>)
 <span class="kw">and </span><br/><a NAME="E12:16_1_1_4"/><span class="lab"><font color="Green" title="E25">A25</font></span>: 
<font color="Maroon" title="c26">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O</font>
 <span class="kw">and </span><br/><a NAME="E13:16_1_1_4"/>
<font color="Maroon" title="c27">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:16_1_1_4"><span class="lab"><font color="Green" title="E23">A23</font></span></a></span>;<br/>
<a NAME="E14:16_1_1_4"/>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16_1_1_4"><span class="lab"><font color="Green" title="E22">A22</font></span></a>, <a class="txt" href="#E11:16_1_1_4"><span class="lab"><font color="Green" title="E24">A24</font></span></a>, <a class="ref" href="cat_1.html#T1" title="CAT_1:th.1">CAT_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:16_1_1_4"/>
<font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:16_1_1"><span class="lab"><font color="Green" title="E19">A19</font></span></a>, <a class="txt" href="#E3:16_1_1_4"><span class="lab"><font color="Green" title="E20">A20</font></span></a>, <a class="txt" href="#E4:16_1_1_4"><span class="lab"><font color="Green" title="E21">A21</font></span></a>, <a class="txt" href="#E12:16_1_1_4"><span class="lab"><font color="Green" title="E25">A25</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>; <a class="txt" 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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">DOM</font> =  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c2">D</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">M</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c17">M</font>,<font color="Maroon" title="c6">O</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:16_1_1"><span class="lab"><font color="Green" title="E19">A19</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T4" title="RELSET_1:th.4">RELSET_1:4</a></span>;<br/>
<a NAME="E28:16_1_1"/><span class="lab"><font color="Green" title="E20">A26</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c2">D</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/>
<a NAME="E29:16_1_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c2">D</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">M</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">O</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c2">D</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">M</font></span>)</span> or <font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_1_1_5"/>
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c2">D</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">M</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:16_1_1_5"/><span class="lab"><font color="Green" title="E21">A27</font></span>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">M</font>
 <span class="kw">and </span><br/><a NAME="E4:16_1_1_5"/><span class="lab"><font color="Green" title="E22">A28</font></span>: 
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c2">D</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16_1_1"><span class="lab"><font color="Green" title="E20">A26</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c25">x</font> = <font color="Maroon" title="c24">x</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">D</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1_5"><span class="lab"><font color="Green" title="E21">A27</font></span></a></span>;<br/>
<a NAME="E6:16_1_1_5"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">M0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:16_1_1"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="txt" href="#E3:16_1_1_5"><span class="lab"><font color="Green" title="E21">A27</font></span></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:16_1_1_5"/><span class="lab"><font color="Green" title="E23">A29</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">X</font>
 <span class="kw">and </span><br/><a NAME="E9:16_1_1_5"/><span class="lab"><font color="Green" title="E24">A30</font></span>: 
<font color="Maroon" title="c26">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b1">a0</font>,<font color="Olive" title="b2">b</font>)</span>)</span> where <font color="Olive" title="b1">a0</font>, <font color="Olive" title="b2">b</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">D</font> : ( <font color="Olive" title="b1">a0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c27">a</font>, <font color="Maroon" title="c28">b</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">D</font><span class="kw"> such that </span><br/><a NAME="E11:16_1_1_5"/><span class="lab"><font color="Green" title="E25">A31</font></span>: 
<font color="Maroon" title="c26">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c27">a</font>,<font color="Maroon" title="c28">b</font>)
 <span class="kw">and </span><br/><a NAME="E12:16_1_1_5"/>
<font color="Maroon" title="c27">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O</font>
 <span class="kw">and </span><br/><a NAME="E13:16_1_1_5"/><span class="lab"><font color="Green" title="E26">A32</font></span>: 
<font color="Maroon" title="c28">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:16_1_1_5"><span class="lab"><font color="Green" title="E24">A30</font></span></a></span>;<br/>
<a NAME="E14:16_1_1_5"/>
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16_1_1_5"><span class="lab"><font color="Green" title="E23">A29</font></span></a>, <a class="txt" href="#E11:16_1_1_5"><span class="lab"><font color="Green" title="E25">A31</font></span></a>, <a class="ref" href="cat_1.html#T1" title="CAT_1:th.1">CAT_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:16_1_1_5"/>
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16_1_1"><span class="lab"><font color="Green" title="E20">A26</font></span></a>, <a class="txt" href="#E3:16_1_1_5"><span class="lab"><font color="Green" title="E21">A27</font></span></a>, <a class="txt" href="#E4:16_1_1_5"><span class="lab"><font color="Green" title="E22">A28</font></span></a>, <a class="txt" href="#E13:16_1_1_5"><span class="lab"><font color="Green" title="E26">A32</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>; <a class="txt" 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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c23">COD</font> =  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c2">D</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">M</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c17">M</font>,<font color="Maroon" title="c6">O</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16_1_1"><span class="lab"><font color="Green" title="E20">A26</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T4" title="RELSET_1:th.4">RELSET_1:4</a></span>;<br/>
<a NAME="E31:16_1_1"/><span class="lab"><font color="Green" title="E21">A33</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c17">M</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c17">M</font>,<font color="Maroon" title="c17">M</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T58" title="RELAT_1:th.58">RELAT_1:58</a></span>;<br/>
<a NAME="E32:16_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c17">M</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c24">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c17">M</font></span>)</span> or <font color="Maroon" title="c24">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">M</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_1_1_6"/>
<font color="Maroon" title="c24">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c17">M</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">M</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:16_1_1_6"/><span class="lab"><font color="Green" title="E22">A34</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c17">M</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:16_1_1_6"/><span class="lab"><font color="Green" title="E23">A35</font></span>: 
<font color="Maroon" title="c24">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c17">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c26">x1</font> = <font color="Maroon" title="c25">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> , <font color="Maroon" title="c27">x2</font> = <font color="Maroon" title="c25">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c17">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:16_1_1"><span class="lab"><font color="Green" title="E21">A33</font></span></a>, <a class="txt" href="#E3:16_1_1_6"><span class="lab"><font color="Green" title="E22">A34</font></span></a>, <a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></span>;<br/>
<a NAME="E6:16_1_1_6"/><span class="lab"><font color="Green" title="E24">A36</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c26">x1</font>,<font color="Maroon" title="c27">x2</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:16_1_1"><span class="lab"><font color="Green" title="E21">A33</font></span></a>, <a class="txt" href="#E3:16_1_1_6"><span class="lab"><font color="Green" title="E22">A34</font></span></a>, <a class="ref" href="mcart_1.html#T21" title="MCART_1:th.21">MCART_1:21</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_6_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c28">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c28">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">MM</font> implies <font color="Maroon" title="c24">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">X</font> )</span><br/><span class="kw">assume </span><a NAME="E1:16_1_1_6_1"/><span class="lab"><font color="Green" title="E25">A37</font></span>: 
<font color="Maroon" title="c28">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">MM</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">X</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c29">E</font> being    <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of <font color="Maroon" title="c2">D</font><span class="kw"> such that </span><br/><a NAME="E3:16_1_1_6_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c29">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">O</font>
 <span class="kw">and </span><br/><a NAME="E4:16_1_1_6_1"/><span class="lab"><font color="Green" title="E26">A38</font></span>: 
 the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c29">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:16_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c30">y1</font> = <font color="Maroon" title="c26">x1</font>, <font color="Maroon" title="c31">y2</font> = <font color="Maroon" title="c27">x2</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c29">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1_6_1"><span class="lab"><font color="Green" title="E25">A37</font></span></a>, <a class="txt" href="#E4:16_1_1_6_1"><span class="lab"><font color="Green" title="E26">A38</font></span></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></span>;<br/><a NAME="E6:16_1_1_6_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c17">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c2">D</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c17">M</font>,<font color="Maroon" title="c17">M</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>;<br/><a NAME="E7:16_1_1_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c2">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1_6"><span class="lab"><font color="Green" title="E22">A34</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E8:16_1_1_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A39</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c26">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c27">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_1_1_6"><span class="lab"><font color="Green" title="E24">A36</font></span></a>, <a class="ref" href="cat_1.html#T15" title="CAT_1:th.15">CAT_1:15</a></span>;<br/><a NAME="E9:16_1_1_6_1"/><span class="lab"><font color="Green" title="E28">A40</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c30">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c26">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_2.html#T9" title="CAT_2:th.9">CAT_2:9</a></span>;<br/><a NAME="E10:16_1_1_6_1"/>
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c31">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c27">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_2.html#T9" title="CAT_2:th.9">CAT_2:9</a></span>;<br/><a NAME="E11:16_1_1_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A41</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c29">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_1_1_6"><span class="lab"><font color="Green" title="E24">A36</font></span></a>, <a class="txt" href="#E8:16_1_1_6_1"><span class="lab"><font color="Green" title="E27">A39</font></span></a>, <a class="txt" href="#E9:16_1_1_6_1"><span class="lab"><font color="Green" title="E28">A40</font></span></a>, <a class="ref" href="cat_1.html#T15" title="CAT_1:th.15">CAT_1:15</a></span>;<br/><a NAME="E12:16_1_1_6_1"/>
 the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c29">E</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c2">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_2.html#D4" title="CAT_2:def.4">CAT_2:def 4</a></span>;<br/><span class="kw">then </span> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c29">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font> = 
 the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:16_1_1_6_1"><span class="lab"><font color="Green" title="E29">A41</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>
<br/>.= 
<font color="Maroon" title="c24">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1_6"><span class="lab"><font color="Green" title="E22">A34</font></span></a>, <a class="txt" href="#E4:16_1_1_6"><span class="lab"><font color="Green" title="E23">A35</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
;<br/><a NAME="E14:16_1_1_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c24">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c29">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:16_1_1_6_1"><span class="lab"><font color="Green" title="E29">A41</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E15:16_1_1_6_1"/>
<font color="Maroon" title="c24">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_1_1_6_1"><span class="lab"><font color="Green" title="E26">A38</font></span></a></span>; <a class="txt" 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>
<span class="kw">hence </span><a NAME="E8:16_1_1_6"/>
<font color="Maroon" title="c24">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:16_1_1"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></span>; <a class="txt" 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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">COMP</font> =  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c17">M</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c17">M</font>,<font color="Maroon" title="c17">M</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<font color="Maroon" title="c17">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:16_1_1"><span class="lab"><font color="Green" title="E21">A33</font></span></a>, <a class="ref" href="relset_1.html#T4" title="RELSET_1:th.4">RELSET_1:4</a></span>;<br/>
<a NAME="E34:16_1_1"/><span class="lab"><font color="Green" title="E22">A42</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c2">D</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">O</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/>
<a NAME="E35:16_1_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c2">D</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">O</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c25">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c2">D</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">O</font></span>)</span> or <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">M</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_1_1_7"/>
<font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c2">D</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">O</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">M</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:16_1_1_7"/><span class="lab"><font color="Green" title="E23">A43</font></span>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O</font>
 <span class="kw">and </span><br/><a NAME="E4:16_1_1_7"/><span class="lab"><font color="Green" title="E24">A44</font></span>: 
<font color="Maroon" title="c25">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c2">D</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">O</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:16_1_1"><span class="lab"><font color="Green" title="E22">A42</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c27">x</font> = <font color="Maroon" title="c26">x</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">D</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1_7"><span class="lab"><font color="Green" title="E23">A43</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c28">x9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:16_1_1_7"/><span class="lab"><font color="Green" title="E25">A45</font></span>: 
<font color="Maroon" title="c28">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:16_1_1_7"/><span class="lab"><font color="Green" title="E26">A46</font></span>: 
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1_7"><span class="lab"><font color="Green" title="E23">A43</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c29">x9</font> = <font color="Maroon" title="c28">x9</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_1_1_7"><span class="lab"><font color="Green" title="E25">A45</font></span></a></span>;<br/>
<font color="Maroon" title="c25">y</font> = 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c27">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:16_1_1"><span class="lab"><font color="Green" title="E22">A42</font></span></a>, <a class="txt" href="#E3:16_1_1_7"><span class="lab"><font color="Green" title="E23">A43</font></span></a>, <a class="txt" href="#E4:16_1_1_7"><span class="lab"><font color="Green" title="E24">A44</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
<br/>.= 
<font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c29">x9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16_1_1_7"><span class="lab"><font color="Green" title="E26">A46</font></span></a>, <a class="ref" href="cat_1.html#T68" title="CAT_1:th.68">CAT_1:68</a></span>
;<br/>
<a NAME="E11:16_1_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_1_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:16_1_1_7"/>
<font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:16_1_1"><span class="lab"><font color="Green" title="E18">A17</font></span></a></span>; <a class="txt" 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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c25">ID</font> =  the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c2">D</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">O</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c6">O</font>,<font color="Maroon" title="c17">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:16_1_1"><span class="lab"><font color="Green" title="E22">A42</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T4" title="RELSET_1:th.4">RELSET_1:4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c26">T</font> =  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c6">O</font>,<font color="Maroon" title="c17">M</font>,<font color="Maroon" title="c22">DOM</font>,<font color="Maroon" title="c23">COD</font>,<font color="Maroon" title="c24">COMP</font>,<font color="Maroon" title="c25">ID</font> #) as   <a href="cat_1.html#V1" title="CAT_1:attr.1">strict</a>   <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of <font color="Maroon" title="c2">D</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#T9" title="NATTRA_1:th.9">NATTRA_1:9</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c26">T</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c26">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c26">T</font> &amp; (  for <font color="Olive" title="b1">E</font> being    <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of <font color="Maroon" title="c2">D</font>  st  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Olive" title="b1">E</font> holds <br/><font color="Maroon" title="c26">T</font> is    <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of <font color="Olive" title="b1">E</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E38:16_1_1"/>
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c26">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c26">T</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:16_1_1"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="txt" href="#E19:16_1_1"><span class="lab"><font color="Green" title="E17">A16</font></span></a>, <a class="ref" href="setfam_1.html#T5" title="SETFAM_1:th.5">SETFAM_1:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">E</font> being    <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of <font color="Maroon" title="c2">D</font>  st  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Olive" title="b1">E</font> holds <br/><font color="Maroon" title="c26">T</font> is    <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of <font color="Olive" title="b1">E</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c27">E</font> be    <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of <font color="Maroon" title="c2">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c27">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c27">E</font> implies <font color="Maroon" title="c26">T</font> is    <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of <font color="Maroon" title="c27">E</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E39:16_1_1"/><span class="lab"><font color="Green" title="E23">A47</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c27">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E40:16_1_1"/><span class="lab"><font color="Green" title="E24">A48</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c27">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">T</font> is    <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of <font color="Maroon" title="c27">E</font></span><br/>

<span class="kw">thus </span><a NAME="E41:16_1_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c26">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c27">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:16_1_1"><span class="lab"><font color="Green" title="E23">A47</font></span></a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cat_2.html#D4" title="CAT_2:def.4">CAT_2:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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 <font color="Maroon" title="c26">T</font><br/>  for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></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 <font color="Maroon" title="c27">E</font>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>3</sub></font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>4</sub></font> or  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>) ) ) &amp;  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c26">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c27">E</font> &amp; (  for <font color="Olive">b<sub>1</sub></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 <font color="Maroon" title="c26">T</font><br/>  for <font color="Olive">b<sub>2</sub></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 <font color="Maroon" title="c27">E</font>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> or  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Olive">b<sub>2</sub></font> ) ) )</span><br/>

<a NAME="E42:16_1_1"/>
 the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c27">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_2.html#T7" title="CAT_2:th.7">CAT_2:7</a></span>;<br/>
<a NAME="E43:16_1_1"/><span class="kw">then </span>
 the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c27">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">MM</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:16_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E39:16_1_1"><span class="lab"><font color="Green" title="E23">A47</font></span></a>, <a class="txt" href="#E40:16_1_1"><span class="lab"><font color="Green" title="E24">A48</font></span></a></span>;<br/>
<a NAME="E44:16_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A49</font></span>: 
<font color="Maroon" title="c17">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c27">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#T3" title="SETFAM_1:th.3">SETFAM_1:3</a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c26">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c27">E</font> &amp; (  for <font color="Olive">b<sub>1</sub></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 <font color="Maroon" title="c26">T</font><br/>  for <font color="Olive">b<sub>2</sub></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 <font color="Maroon" title="c27">E</font>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> or  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Olive">b<sub>2</sub></font> ) ) )</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c28">a</font>, <font color="Maroon" title="c29">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c26">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a9</font>, <font color="Olive" title="b2">b9</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c27">E</font>  st <font color="Maroon" title="c28">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a9</font> &amp; <font color="Maroon" title="c29">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b9</font> holds <br/> <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c28">a</font>,<font color="Maroon" title="c29">b</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b1">a9</font>,<font color="Olive" title="b2">b9</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c30">a9</font>, <font color="Maroon" title="c31">b9</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c27">E</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c28">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">a9</font> &amp; <font color="Maroon" title="c29">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">b9</font> implies  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c28">a</font>,<font color="Maroon" title="c29">b</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c30">a9</font>,<font color="Maroon" title="c31">b9</font>) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:16_1_1_8"/><span class="lab"><font color="Green" title="E26">A50</font></span>: 
<font color="Maroon" title="c28">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">a9</font>
 <span class="kw">and </span><br/><a NAME="E2:16_1_1_8"/><span class="lab"><font color="Green" title="E27">A51</font></span>: 
<font color="Maroon" title="c29">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">b9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c28">a</font>,<font color="Maroon" title="c29">b</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c30">a9</font>,<font color="Maroon" title="c31">b9</font>)</span><br/><span class="kw">thus </span><a NAME="E3:16_1_1_8"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c28">a</font>,<font color="Maroon" title="c29">b</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c30">a9</font>,<font color="Maroon" title="c31">b9</font>)
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_8_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c32">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c28">a</font>,<font color="Maroon" title="c29">b</font>) or <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c30">a9</font>,<font color="Maroon" title="c31">b9</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_1_1_8_1"/>
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c28">a</font>,<font color="Maroon" title="c29">b</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c30">a9</font>,<font color="Maroon" title="c31">b9</font>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c33">f</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c26">T</font><span class="kw"> such that </span><br/><a NAME="E3:16_1_1_8_1"/><span class="lab"><font color="Green" title="E28">A52</font></span>: 
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">f</font>
 <span class="kw">and </span><br/><a NAME="E4:16_1_1_8_1"/><span class="lab"><font color="Green" title="E29">A53</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c33">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">a</font>
 <span class="kw">and </span><br/><a NAME="E5:16_1_1_8_1"/><span class="lab"><font color="Green" title="E30">A54</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c33">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">b</font>
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c34">g</font> = <font color="Maroon" title="c33">f</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">D</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c35">f</font> = <font color="Maroon" title="c33">f</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c27">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:16_1_1"><span class="lab"><font color="Green" title="E25">A49</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E8:16_1_1_8_1"/><span class="lab"><font color="Green" title="E31">A55</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c34">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_1_1_8_1"><span class="lab"><font color="Green" title="E29">A53</font></span></a>, <a class="ref" href="cat_2.html#T9" title="CAT_2:th.9">CAT_2:9</a></span>;<br/>
<a NAME="E9:16_1_1_8_1"/><span class="lab"><font color="Green" title="E32">A56</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c34">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_1_1_8_1"><span class="lab"><font color="Green" title="E30">A54</font></span></a>, <a class="ref" href="cat_2.html#T9" title="CAT_2:th.9">CAT_2:9</a></span>;<br/>
<a NAME="E10:16_1_1_8_1"/><span class="lab"><font color="Green" title="E33">A57</font></span>: 
<font color="Maroon" title="c30">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c35">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1_8"><span class="lab"><font color="Green" title="E26">A50</font></span></a>, <a class="txt" href="#E8:16_1_1_8_1"><span class="lab"><font color="Green" title="E31">A55</font></span></a>, <a class="ref" href="cat_2.html#T9" title="CAT_2:th.9">CAT_2:9</a></span>;<br/>
<a NAME="E11:16_1_1_8_1"/>
<font color="Maroon" title="c31">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c35">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1_8"><span class="lab"><font color="Green" title="E27">A51</font></span></a>, <a class="txt" href="#E9:16_1_1_8_1"><span class="lab"><font color="Green" title="E32">A56</font></span></a>, <a class="ref" href="cat_2.html#T9" title="CAT_2:th.9">CAT_2:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:16_1_1_8_1"/>
<font color="Maroon" title="c32">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c30">a9</font>,<font color="Maroon" title="c31">b9</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1_8_1"><span class="lab"><font color="Green" title="E28">A52</font></span></a>, <a class="txt" href="#E10:16_1_1_8_1"><span class="lab"><font color="Green" title="E33">A57</font></span></a></span>; <a class="txt" 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>

<a NAME="E46:16_1_1"/><span class="lab"><font color="Green" title="E26">A58</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c26">T</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c27">E</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c28">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c26">T</font> or <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c27">E</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_1_1_9"/><span class="lab"><font color="Green" title="E27">A59</font></span>: 
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c26">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c27">E</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c29">x1</font> = <font color="Maroon" title="c28">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> , <font color="Maroon" title="c30">x2</font> = <font color="Maroon" title="c28">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c17">M</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c31">y1</font> = <font color="Maroon" title="c29">x1</font>, <font color="Maroon" title="c32">y2</font> = <font color="Maroon" title="c30">x2</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c26">T</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c33">z1</font> = <font color="Maroon" title="c29">x1</font>, <font color="Maroon" title="c34">z2</font> = <font color="Maroon" title="c30">x2</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c27">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:16_1_1"><span class="lab"><font color="Green" title="E25">A49</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E5:16_1_1_9"/><span class="lab"><font color="Green" title="E28">A60</font></span>: 
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c29">x1</font>,<font color="Maroon" title="c30">x2</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1_9"><span class="lab"><font color="Green" title="E27">A59</font></span></a>, <a class="ref" href="mcart_1.html#T21" title="MCART_1:th.21">MCART_1:21</a></span>;<br/>
<a NAME="E6:16_1_1_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A61</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c31">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c32">y2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1_9"><span class="lab"><font color="Green" title="E27">A59</font></span></a>, <a class="ref" href="cat_1.html#T15" title="CAT_1:th.15">CAT_1:15</a></span>;<br/>
<a NAME="E7:16_1_1_9"/><span class="lab"><font color="Green" title="E30">A62</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c31">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c29">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_2.html#T9" title="CAT_2:th.9">CAT_2:9</a></span>;<br/>
<a NAME="E8:16_1_1_9"/><span class="lab"><font color="Green" title="E31">A63</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c33">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c29">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_2.html#T9" title="CAT_2:th.9">CAT_2:9</a></span>;<br/>
<a NAME="E9:16_1_1_9"/><span class="lab"><font color="Green" title="E32">A64</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c32">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c30">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_2.html#T9" title="CAT_2:th.9">CAT_2:9</a></span>;<br/>
<a NAME="E10:16_1_1_9"/>
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c34">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c30">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_2.html#T9" title="CAT_2:th.9">CAT_2:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:16_1_1_9"/>
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c27">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_1_1_9"><span class="lab"><font color="Green" title="E28">A60</font></span></a>, <a class="txt" href="#E6:16_1_1_9"><span class="lab"><font color="Green" title="E29">A61</font></span></a>, <a class="txt" href="#E7:16_1_1_9"><span class="lab"><font color="Green" title="E30">A62</font></span></a>, <a class="txt" href="#E8:16_1_1_9"><span class="lab"><font color="Green" title="E31">A63</font></span></a>, <a class="txt" href="#E9:16_1_1_9"><span class="lab"><font color="Green" title="E32">A64</font></span></a>, <a class="ref" href="cat_1.html#T15" title="CAT_1:th.15">CAT_1:15</a></span>; <a class="txt" 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="hs2(this)" href="javascript:()" title="16_1_1_10"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c28">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c26">T</font> implies  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c26">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c27">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:16_1_1_10"/><span class="lab"><font color="Green" title="E27">A65</font></span>: 
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c26">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c26">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c27">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font></span><br/><a NAME="E2:16_1_1_10"/><span class="lab"><font color="Green" title="E28">A66</font></span>: 
 the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c26">T</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c2">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_2.html#D4" title="CAT_2:def.4">CAT_2:def 4</a></span>;<br/><a NAME="E3:16_1_1_10"/><span class="lab"><font color="Green" title="E29">A67</font></span>: 
 the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c27">E</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c2">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_2.html#D4" title="CAT_2:def.4">CAT_2:def 4</a></span>;<br/><span class="kw">thus </span> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c26">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font> = 
 the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1_10"><span class="lab"><font color="Green" title="E27">A65</font></span></a>, <a class="txt" href="#E2:16_1_1_10"><span class="lab"><font color="Green" title="E28">A66</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>
<br/>.= 
 the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c27">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:16_1_1"><span class="lab"><font color="Green" title="E26">A58</font></span></a>, <a class="txt" href="#E1:16_1_1_10"><span class="lab"><font color="Green" title="E27">A65</font></span></a>, <a class="txt" href="#E3:16_1_1_10"><span class="lab"><font color="Green" title="E29">A67</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>
; <a class="txt" 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>
<span class="kw">hence </span><a NAME="E48:16_1_1"/>
 the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c26">T</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c27">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:16_1_1"><span class="lab"><font color="Green" title="E26">A58</font></span></a>, <a class="ref" href="grfunc_1.html#T2" title="GRFUNC_1:th.2">GRFUNC_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c26">T</font><br/>  for <font color="Olive">b<sub>2</sub></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 <font color="Maroon" title="c27">E</font>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> or  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Olive">b<sub>2</sub></font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c28">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c26">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c27">E</font>  holds <br/>(  not <font color="Maroon" title="c28">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> or  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c28">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c29">a9</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c27">E</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c28">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">a9</font> or  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c28">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c29">a9</font> )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c30">b</font> = <font color="Maroon" title="c28">a</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">D</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E50:16_1_1"/><span class="lab"><font color="Green" title="E27">A68</font></span>: 
<font color="Maroon" title="c28">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">a9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c28">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c29">a9</font></span><br/>

<a NAME="E51:16_1_1"/>
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c28">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c30">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_2.html#D4" title="CAT_2:def.4">CAT_2:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E52:16_1_1"/>
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c28">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c29">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:16_1_1"><span class="lab"><font color="Green" title="E27">A68</font></span></a>, <a class="ref" href="cat_2.html#D4" title="CAT_2:def.4">CAT_2:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
