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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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font> being    <a href="oppcat_1.html#M1" title="OPPCAT_1:mode.1">Contravariant_Functor</a> of <font color="Maroon" title="c1">A</font>, <a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> (<font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span>)  st  <a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Olive" title="b1">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">F</font> is  <a href="yoneda_1.html#V1" title="YONEDA_1:attr.1">faithful</a>  holds <br/><font color="Olive" title="b1">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font> be    <a href="oppcat_1.html#M1" title="OPPCAT_1:mode.1">Contravariant_Functor</a> of <font color="Maroon" title="c1">A</font>, <a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> (<font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="yoneda_1.html#K1" title="YONEDA_1:func.1">EnsHom</a> <font color="Maroon" title="c1">A</font></span>)</span>); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c2">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c2">F</font> is  <a href="yoneda_1.html#V1" title="YONEDA_1:attr.1">faithful</a>  implies <font color="Maroon" title="c2">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:11"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
 <a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c2">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">F</font> is  <a href="yoneda_1.html#V1" title="YONEDA_1:attr.1">faithful</a>  or <font color="Maroon" title="c2">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:11"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
<font color="Maroon" title="c2">F</font> is  <a href="yoneda_1.html#V1" title="YONEDA_1:attr.1">faithful</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> </span><br/>

<a NAME="E3:11"/>
 for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c4">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c4">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x2</font> implies <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:11_1"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
( <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c4">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">F</font> )
 <span class="kw">and </span><br/><a NAME="E2:11_1"/><span class="lab"><font color="Green" title="E11">A4</font></span>: 
<font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x2</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">m1</font> = <font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c6">m2</font> = <font color="Maroon" title="c4">x2</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11_1"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">o1</font> =  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">m1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">o2</font> =  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">m1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">o3</font> =  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">m2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">o4</font> =  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">m2</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">m29</font> = <font color="Maroon" title="c6">m2</font> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">m2</font>, <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">m2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T4" title="CAT_1:th.4">CAT_1:4</a></span>;<br/>
<span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">m1</font></span>)</span> = 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m29</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11_1"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="ref" href="oppcat_1.html#T32" title="OPPCAT_1:th.32">OPPCAT_1:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">m2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T32" title="OPPCAT_1:th.32">OPPCAT_1:32</a></span>
;<br/>
<a NAME="E6:11_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A5</font></span>: 
( <font color="Maroon" title="c6">m2</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">m2</font>, <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">m2</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">m1</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="c6">m2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="cat_1.html#T4" title="CAT_1:th.4">CAT_1:4</a>, <a class="ref" href="funct_2.html#T19" title="FUNCT_2:th.19">FUNCT_2:19</a></span>;<br/>
<a NAME="E7:11_1"/><span class="lab"><font color="Green" title="E13">A6</font></span>: 
( <font color="Maroon" title="c5">m1</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">m1</font>, <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">m1</font> &amp;  <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="c5">m1</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="c5">m1</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#T2" title="CAT_1:th.2">CAT_1:2</a>, <a class="ref" href="cat_1.html#T4" title="CAT_1:th.4">CAT_1:4</a></span>;<br/>
<span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">m1</font></span>)</span> = 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">m29</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11_1"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="ref" href="oppcat_1.html#T32" title="OPPCAT_1:th.32">OPPCAT_1:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">m2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T32" title="OPPCAT_1:th.32">OPPCAT_1:32</a></span>
;<br/>
<a NAME="E9:11_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">m2</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">m1</font>, <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">m1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:11"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E6:11_1"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="ref" href="funct_2.html#T19" title="FUNCT_2:th.19">FUNCT_2:19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:11_1"/>
<font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:11"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="txt" href="#E2:11_1"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="txt" href="#E7:11_1"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="ref" href="yoneda_1.html#D6" target="_self" title="YONEDA_1:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:11"/>
<font color="Maroon" title="c2">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_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
