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<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   <a href="cat_5.html#V3" title="CAT_5:attr.3">Categorial</a>  <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">D</font> being  <a href="cat_5.html#V3" title="CAT_5:attr.3">Categorial</a>   <a href="cat_5.html#V4" title="CAT_5:attr.4">full</a>  <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>  st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">D</font> holds <br/><font color="Maroon" title="c1">C</font> is    <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of <font color="Olive" title="b1">D</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">D</font> be   <a href="cat_5.html#V3" title="CAT_5:attr.3">Categorial</a>   <a href="cat_5.html#V4" title="CAT_5:attr.4">full</a>  <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"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">D</font> implies <font color="Maroon" title="c1">C</font> is    <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of <font color="Maroon" title="c2">D</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:44"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">D</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="c1">C</font> is    <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of <font color="Maroon" title="c2">D</font></span><br/>

<a NAME="E2:44"/>
the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">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="c3">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">C</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">D</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:44_1"/>
<font color="Maroon" title="c3">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">C</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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">D</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c3">x</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">y1</font> =  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c6">y2</font> =  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c4">x</font> as   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_5.html#T12" target="_self" title="CAT_5:th.12">Th12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">F</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">y2</font><span class="kw"> such that </span><br/><a NAME="E5:44_1"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c4">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="c5">y1</font>,<font color="Maroon" title="c6">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c7">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_5.html#D6" target="_self" title="CAT_5:def.6">Def6</a></span>;<br/>
<a NAME="E6:44_1"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
<font color="Maroon" title="c5">y1</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E7:44_1"/>
<font color="Maroon" title="c6">y2</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E8:44_1"/><span class="kw">then </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="c5">y1</font>,<font color="Maroon" title="c6">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c7">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:44_1"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="cat_5.html#D8" target="_self" title="CAT_5:def.8">Def8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:44_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:44_1"><span class="lab"><font color="Green" title="E29">A2</font></span></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="E3:44"/>
<font color="Maroon" title="c1">C</font> is    <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of <font color="Maroon" title="c2">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_5.html#T15" target="_self" title="CAT_5:th.15">Th15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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