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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">o</font>, <font color="Olive" title="b2">m</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds   <a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">m</font></span>)</span>,<font color="Maroon" title="c1">A</font> <a href="isocat_1.html#NR2" title="ISOCAT_1:NR.2">~=</a> <font color="Maroon" title="c1">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">o</font>, <font color="Maroon" title="c3">m</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="Red">::  thesis: </font></span></a><span class="hide">  <a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c1">A</font> <a href="isocat_1.html#NR2" title="ISOCAT_1:NR.2">~=</a> <font color="Maroon" title="c1">A</font></span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c4">a</font> = <font color="Maroon" title="c2">o</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c5">F</font> = <font color="Maroon" title="c4">a</font> <a href="isocat_2.html#K2" title="ISOCAT_2:func.2">|-&gt;</a> <font color="Maroon" title="c1">A</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="isocat_1.html#D4" title="ISOCAT_1:def.4">ISOCAT_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">F</font> is  <a href="cat_1.html#V8" title="CAT_1:attr.8">isomorphic</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">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 <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c1">A</font></span>)</span> &amp; <font color="Maroon" title="c7">y</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 <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c1">A</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font> implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> )</span><br/><a NAME="E1:13_1"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></span>;<br/><span class="kw">assume </span><a NAME="E2:13_1"/>
<font color="Maroon" title="c6">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 <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c1">A</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">y</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 <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c1">A</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font> implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">F1</font>, <font color="Maroon" title="c9">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of  <a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c10">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c8">F1</font>,<font color="Maroon" title="c9">F2</font><span class="kw"> such that </span><br/><a NAME="E4:13_1"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F1</font>,<font color="Maroon" title="c9">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c10">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:13_1"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
<font color="Maroon" title="c8">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c9">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="nattra_1.html#D16" title="NATTRA_1:def.16">NATTRA_1:def 16</a></span>;<br/><span class="kw">assume </span><a NAME="E6:13_1"/>
<font color="Maroon" title="c7">y</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 <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c1">A</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font> implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">F19</font>, <font color="Maroon" title="c12">F29</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of  <a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c13">t9</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c11">F19</font>,<font color="Maroon" title="c12">F29</font><span class="kw"> such that </span><br/><a NAME="E8:13_1"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">F19</font>,<font color="Maroon" title="c12">F29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c13">t9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E9:13_1"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
<font color="Maroon" title="c11">F19</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c12">F29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="nattra_1.html#D16" title="NATTRA_1:def.16">NATTRA_1:def 16</a></span>;<br/><span class="kw">assume </span><a NAME="E10:13_1"/>
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font></span><br/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A6</font></span>: <font color="Maroon" title="c10">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">a</font> = 
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:13_1"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E5:13_1"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="isocat_2.html#D1" target="_self" title="ISOCAT_2:def.1">Def1</a></span>
<br/>.= 
<font color="Maroon" title="c13">t9</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:13_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E9:13_1"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="isocat_2.html#D1" target="_self" title="ISOCAT_2:def.1">Def1</a></span>
;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">G1</font> = <font color="Maroon" title="c8">F1</font>, <font color="Maroon" title="c15">G19</font> = <font color="Maroon" title="c11">F19</font>, <font color="Maroon" title="c16">G2</font> = <font color="Maroon" title="c9">F2</font>, <font color="Maroon" title="c17">G29</font> = <font color="Maroon" title="c12">F29</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">A</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">s</font> = <font color="Maroon" title="c10">t</font>, <font color="Maroon" title="c19">s9</font> = <font color="Maroon" title="c13">t9</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E14:13_1"/><span class="lab"><font color="Green" title="E16">A7</font></span>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E15:13_1"/><span class="lab"><font color="Green" title="E17">A8</font></span>: 
<font color="Maroon" title="c8">F1</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c9">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:13_1"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="nattra_1.html#D7" title="NATTRA_1:def.7">NATTRA_1:def 7</a></span>;<br/><a NAME="E16:13_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A9</font></span>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D2" title="NATTRA_1:def.2">NATTRA_1:def 2</a></span>;<br/><a NAME="E17:13_1"/><span class="lab"><font color="Green" title="E19">A10</font></span>: 
<font color="Maroon" title="c11">F19</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c12">F29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:13_1"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="nattra_1.html#D7" title="NATTRA_1:def.7">NATTRA_1:def 7</a></span>;<br/><a NAME="E18:13_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A11</font></span>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F19</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">F29</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D2" title="NATTRA_1:def.2">NATTRA_1:def 2</a></span>;<br/><a NAME="E19:13_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">F19</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:13_1"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="txt" href="#E16:13_1"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="ref" href="cat_1.html#T24" title="CAT_1:th.24">CAT_1:24</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c14">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">a</font></span>)</span> = 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F19</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T108" title="CAT_1:th.108">CAT_1:108</a></span>
<br/>.= 
<font color="Maroon" title="c15">G19</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T108" title="CAT_1:th.108">CAT_1:108</a></span>
;<br/><a NAME="E21:13_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A12</font></span>: 
<font color="Maroon" title="c8">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c11">F19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:13_1"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="ref" href="isocat_2.html#T3" target="_self" title="ISOCAT_2:th.3">Th3</a></span>;<br/><a NAME="E22:13_1"/>
<font color="Maroon" title="c9">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F29</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:13_1"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="txt" href="#E16:13_1"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="txt" href="#E18:13_1"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="ref" href="cat_1.html#T24" title="CAT_1:th.24">CAT_1:24</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c16">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">a</font></span>)</span> = 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F29</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c4">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T108" title="CAT_1:th.108">CAT_1:108</a></span>
<br/>.= 
<font color="Maroon" title="c17">G29</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c4">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T108" title="CAT_1:th.108">CAT_1:108</a></span>
;<br/><a NAME="E24:13_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A13</font></span>: 
<font color="Maroon" title="c9">F2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c12">F29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:13_1"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="ref" href="isocat_2.html#T3" target="_self" title="ISOCAT_2:th.3">Th3</a></span>;<br/><font color="Maroon" title="c18">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> = 
<font color="Maroon" title="c13">t9</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:13_1"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="txt" href="#E11:13_1"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c19">s9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:13_1"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></span>
;<br/><span class="kw">hence </span><a NAME="E26:13_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:13_1"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E8:13_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E21:13_1"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="txt" href="#E24:13_1"><span class="lab"><font color="Green" title="E22">A13</font></span></a>, <a class="ref" href="isocat_2.html#T3" target="_self" title="ISOCAT_2:th.3">Th3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:13"/>
<font color="Maroon" title="c5">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="isocat_1.html#D3" title="ISOCAT_1:def.3">ISOCAT_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">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">A</font></span><br/>

<span class="kw">thus </span><a NAME="E4:13"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">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="c1">A</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">F</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: 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="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">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="c1">A</font> or <font color="Maroon" title="c6">x</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="c5">F</font> )</span><br/>

<span class="kw">assume </span><a NAME="E5:13"/>
<font color="Maroon" title="c6">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="c1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x</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="c5">F</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c6">x</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">A</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">F1</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span>, <font color="Maroon" title="c9">F2</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">A</font> ;<br/>
<div><span class="lab"><font color="Green" title="E10">A14</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">F1</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> <span class="p2">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</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="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c8">F1</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> <span class="p2">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <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="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> )</span><br/><span class="kw">thus </span><font color="Maroon" title="c8">F1</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> <span class="p2">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> = 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c7">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</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"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p3">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></span>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</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="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">F1</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> <span class="p2">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <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="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span></span><br/><span class="kw">thus </span><font color="Maroon" title="c8">F1</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> <span class="p2">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> = 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c7">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <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"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p3">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></span>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <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="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E11">A15</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">h</font>, <font color="Maroon" title="c11">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c11">g</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="c10">h</font> implies <font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c10">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">h</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:13_3"/>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c11">g</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="c10">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c10">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">h</font></span>)</span></span><br/><a NAME="E2:13_3"/><span class="lab"><font color="Green" title="E12">A16</font></span>: 
 <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> <font color="Maroon" title="c7">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T56" title="CAT_1:th.56">CAT_1:56</a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c10">h</font></span>)</span> = 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c7">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T59" title="CAT_1:th.59">CAT_1:59</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13_3"><span class="lab"><font color="Green" title="E12">A16</font></span></a>, <a class="ref" href="cat_1.html#D13" title="CAT_1:def.13">CAT_1:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">h</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">h</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E12">A17</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">h</font>, <font color="Maroon" title="c11">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c11">g</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="c10">h</font> implies <font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c10">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">h</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:13_4"/>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c11">g</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="c10">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c10">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">h</font></span>)</span></span><br/><a NAME="E2:13_4"/><span class="lab"><font color="Green" title="E13">A18</font></span>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c7">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T56" title="CAT_1:th.56">CAT_1:56</a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c10">h</font></span>)</span> = 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c7">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T59" title="CAT_1:th.59">CAT_1:59</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13_4"><span class="lab"><font color="Green" title="E13">A18</font></span></a>, <a class="ref" href="cat_1.html#D13" title="CAT_1:def.13">CAT_1:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">h</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">h</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E13">A19</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">F2</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> <span class="p2">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</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="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c9">F2</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> <span class="p2">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <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="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> )</span><br/><span class="kw">thus </span><font color="Maroon" title="c9">F2</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> <span class="p2">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> = 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c7">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</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"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p3">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></span>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</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="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">F2</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> <span class="p2">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <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="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span></span><br/><span class="kw">thus </span><font color="Maroon" title="c9">F2</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> <span class="p2">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> = 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c7">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <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"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p3">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></span>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <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="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E12:13"/>
( ( for <font color="Olive" title="b1">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>  ex <font color="Olive" title="b2">d</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font> st <font color="Maroon" title="c8">F1</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="Olive" title="b1">c</font></span>)</span> <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" title="b2">d</font> ) &amp; ( for <font color="Olive" title="b1">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>  ex <font color="Olive" title="b2">d</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font> st <font color="Maroon" title="c9">F2</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="Olive" title="b1">c</font></span>)</span> <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" title="b2">d</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">F1</font> = <font color="Maroon" title="c8">F1</font>, <font color="Maroon" title="c11">F2</font> = <font color="Maroon" title="c9">F2</font> as    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of  <a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:13"><span class="lab"><font color="Green" title="E10">A14</font></span></a>, <a class="txt" href="#E9:13"><span class="lab"><font color="Green" title="E11">A15</font></span></a>, <a class="txt" href="#E11:13"><span class="lab"><font color="Green" title="E13">A19</font></span></a>, <a class="txt" href="#E10:13"><span class="lab"><font color="Green" title="E12">A17</font></span></a>, <a class="ref" href="cat_1.html#T96" title="CAT_1:th.96">CAT_1:96</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">t</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c7">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">A</font> ;<br/>
<a NAME="E15:13"/><span class="lab"><font color="Green" title="E14">A20</font></span>: 
for <font color="Olive" title="b1">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span> holds <br/> ( <font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b1">b</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="c7">f</font> &amp; <font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Olive" title="b1">b</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="c7">f</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c13">b</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="c7">f</font> &amp; <font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c13">b</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="c7">f</font> )</span><br/>

<a NAME="E1:13_6"/>
<font color="Maroon" title="c11">F2</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="c13">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c7">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<a NAME="E2:13_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A21</font></span>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c7">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T108" title="CAT_1:th.108">CAT_1:108</a></span>;<br/>
<a NAME="E3:13_6"/>
<font color="Maroon" title="c10">F1</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="c13">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c7">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<a NAME="E4:13_6"/><span class="kw">then </span>
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c7">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T108" title="CAT_1:th.108">CAT_1:108</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:13_6"/>
( <font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c13">b</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="c7">f</font> &amp; <font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c13">b</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="c7">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13_6"><span class="lab"><font color="Green" title="E15">A21</font></span></a>, <a class="ref" href="cat_1.html#T45" title="CAT_1:th.45">CAT_1:45</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E15">A22</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_7"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c13">b</font></span><br/><a NAME="E1:13_7"/><span class="lab"><font color="Green" title="E16">A23</font></span>: 
<font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c13">b</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="c7">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:13"><span class="lab"><font color="Green" title="E14">A20</font></span></a></span>;<br/><a NAME="E2:13_7"/>
( <font color="Maroon" title="c12">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c13">b</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="c7">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:13"><span class="lab"><font color="Green" title="E14">A20</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><a NAME="E3:13_7"/><span class="kw">then </span>
<font color="Maroon" title="c12">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b</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"><font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c13">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c13">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_7"><span class="lab"><font color="Green" title="E16">A23</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:13_7"/>
<font color="Maroon" title="c12">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c13">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D7" title="CAT_1:def.7">CAT_1:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E16">A24</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_8"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c13">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><a NAME="E1:13_8"/>
( <font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c13">b</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="c7">f</font> &amp; <font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c13">b</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="c7">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:13"><span class="lab"><font color="Green" title="E14">A20</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E2:13_8"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c13">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T19" title="CAT_1:th.19">CAT_1:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E18:13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A25</font></span>: 
<font color="Maroon" title="c10">F1</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c11">F2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D2" title="NATTRA_1:def.2">NATTRA_1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">t</font> = <font color="Maroon" title="c12">t</font> as    <a href="nattra_1.html#M1" title="NATTRA_1:mode.1">transformation</a> of <font color="Maroon" title="c10">F1</font>,<font color="Maroon" title="c11">F2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:13"><span class="lab"><font color="Green" title="E15">A22</font></span></a>, <a class="ref" href="nattra_1.html#D3" title="NATTRA_1:def.3">NATTRA_1:def 3</a></span>;<br/>
<a NAME="E20:13"/><span class="lab"><font color="Green" title="E18">A26</font></span>: 
for <font color="Olive" title="b1">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span> holds  <font color="Maroon" title="c13">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c7">f</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c13">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:13"><span class="lab"><font color="Green" title="E17">A25</font></span></a>, <a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E19">A27</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_10"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">b1</font>, <font color="Maroon" title="c15">b2</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c14">b1</font>,<font color="Maroon" title="c15">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies for <font color="Olive" title="b1">g</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c14">b1</font>,<font color="Maroon" title="c15">b2</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c13">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c15">b2</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">b1</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:13_10"/><span class="lab"><font color="Green" title="E20">A28</font></span>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c14">b1</font>,<font color="Maroon" title="c15">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">g</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c14">b1</font>,<font color="Maroon" title="c15">b2</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c13">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c15">b2</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">b1</font></span>)</span></span><br/><a NAME="E2:13_10"/><span class="lab"><font color="Green" title="E21">A29</font></span>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c14">b1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c15">b2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_10"><span class="lab"><font color="Green" title="E20">A28</font></span></a>, <a class="ref" href="cat_1.html#T124" title="CAT_1:th.124">CAT_1:124</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c16">g</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c14">b1</font>,<font color="Maroon" title="c15">b2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c13">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c15">b2</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">b1</font></span>)</span></span><br/><a NAME="E3:13_10"/><span class="lab"><font color="Green" title="E22">A30</font></span>: 
( <font color="Maroon" title="c13">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c14">b1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c14">b1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:13"><span class="lab"><font color="Green" title="E16">A24</font></span></a>, <a class="txt" href="#E20:13"><span class="lab"><font color="Green" title="E18">A26</font></span></a></span>;<br/><a NAME="E4:13_10"/><span class="lab"><font color="Green" title="E23">A31</font></span>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c14">b1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c15">b2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_10"><span class="lab"><font color="Green" title="E20">A28</font></span></a>, <a class="ref" href="cat_1.html#T124" title="CAT_1:th.124">CAT_1:124</a></span>;<br/><a NAME="E5:13_10"/><span class="lab"><font color="Green" title="E24">A32</font></span>: 
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E6:13_10"/><span class="lab"><font color="Green" title="E25">A33</font></span>: 
<font color="Maroon" title="c16">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A34</font></span>: <font color="Maroon" title="c11">F2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c16">g</font> = 
<font color="Maroon" title="c11">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_10"><span class="lab"><font color="Green" title="E20">A28</font></span></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></span>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c7">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:13_10"><span class="lab"><font color="Green" title="E24">A32</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
;<br/><span class="lab"><font color="Green" title="E27">A35</font></span>: <font color="Maroon" title="c10">F1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c16">g</font> = 
<font color="Maroon" title="c10">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_10"><span class="lab"><font color="Green" title="E20">A28</font></span></a>, <a class="txt" href="#E6:13_10"><span class="lab"><font color="Green" title="E25">A33</font></span></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></span>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c7">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:13_10"><span class="lab"><font color="Green" title="E24">A32</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
;<br/><a NAME="E9:13_10"/>
( <font color="Maroon" title="c13">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c15">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c15">b2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c15">b2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:13"><span class="lab"><font color="Green" title="E16">A24</font></span></a>, <a class="txt" href="#E20:13"><span class="lab"><font color="Green" title="E18">A26</font></span></a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c13">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c15">b2</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c16">g</font></span>)</span> = 
<font color="Maroon" title="c7">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c16">g</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:13_10"><span class="lab"><font color="Green" title="E23">A31</font></span></a>, <a class="ref" href="cat_1.html#D13" title="CAT_1:def.13">CAT_1:def 13</a></span>
<br/>.= 
<font color="Maroon" title="c7">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:13_10"><span class="lab"><font color="Green" title="E27">A35</font></span></a>, <a class="ref" href="cat_1.html#T47" title="CAT_1:th.47">CAT_1:47</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c7">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:13_10"><span class="lab"><font color="Green" title="E26">A34</font></span></a>, <a class="ref" href="cat_1.html#T46" title="CAT_1:th.46">CAT_1:46</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">b1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:13_10"><span class="lab"><font color="Green" title="E22">A30</font></span></a>, <a class="txt" href="#E2:13_10"><span class="lab"><font color="Green" title="E21">A29</font></span></a>, <a class="ref" href="cat_1.html#D13" title="CAT_1:def.13">CAT_1:def 13</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E22:13"/>
<font color="Maroon" title="c10">F1</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c11">F2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:13"><span class="lab"><font color="Green" title="E16">A24</font></span></a>, <a class="ref" href="nattra_1.html#D2" title="NATTRA_1:def.2">NATTRA_1:def 2</a></span>;<br/>
<a NAME="E23:13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A36</font></span>: 
<font color="Maroon" title="c10">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c11">F2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:13"><span class="lab"><font color="Green" title="E19">A27</font></span></a>, <a class="ref" href="nattra_1.html#D7" title="NATTRA_1:def.7">NATTRA_1:def 7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">t</font> = <font color="Maroon" title="c13">t</font> as    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c10">F1</font>,<font color="Maroon" title="c11">F2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:13"><span class="lab"><font color="Green" title="E19">A27</font></span></a>, <a class="ref" href="nattra_1.html#D8" title="NATTRA_1:def.8">NATTRA_1:def 8</a></span>;<br/>
<a NAME="E25:13"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F1</font>,<font color="Maroon" title="c11">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c14">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:13"><span class="lab"><font color="Green" title="E20">A36</font></span></a>, <a class="ref" href="nattra_1.html#D16" title="NATTRA_1:def.16">NATTRA_1:def 16</a></span>;<br/>
<a NAME="E26:13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A37</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F1</font>,<font color="Maroon" title="c11">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c14">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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 <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c1">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></span>;<br/>
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F1</font>,<font color="Maroon" title="c11">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c14">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c14">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:13"><span class="lab"><font color="Green" title="E20">A36</font></span></a>, <a class="ref" href="isocat_2.html#D1" target="_self" title="ISOCAT_2:def.1">Def1</a></span>
<br/>.= 
<font color="Maroon" title="c7">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:13"><span class="lab"><font color="Green" title="E18">A26</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E28:13"/>
<font color="Maroon" title="c6">x</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="c5">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:13"><span class="lab"><font color="Green" title="E21">A37</font></span></a>, <a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
