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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   <a href="nattra_1.html#V2" title="NATTRA_1:attr.2">discrete</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">B</font> being   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a><br/>  for <font color="Olive" title="b2">F1</font>, <font color="Olive" title="b3">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Olive" title="b1">B</font>,<font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b2">F1</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Olive" title="b3">F2</font> holds <br/><font color="Olive" title="b2">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b3">F2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">B</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">F1</font>, <font color="Olive" title="b2">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">F1</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Olive" title="b2">F2</font> holds <br/><font color="Olive" title="b1">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">F2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">F1</font>, <font color="Maroon" title="c4">F2</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c1">A</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">F1</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c4">F2</font> implies <font color="Maroon" title="c3">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">F2</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:70"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
<font color="Maroon" title="c3">F1</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c4">F2</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">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">F2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">m</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span><br/><a NAME="E1:70_1"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">m</font></span>)</span></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="#E1:70"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="nattra_1.html#D2" target="_self" title="NATTRA_1:def.2">Def2</a></span>;<br/><a NAME="E2:70_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A2</font></span>: 
<font color="Maroon" title="c3">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</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">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</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">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#T38" target="_self" title="NATTRA_1:th.38">Th45</a></span>;<br/><a NAME="E3:70_1"/><span class="lab"><font color="Green" title="E57">A3</font></span>: 
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">m</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">m</font></span>)</span>)
 ;<br/><a NAME="E4:70_1"/><span class="kw">then </span>
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">m</font></span>)</span></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#T81" title="CAT_1:th.81">CAT_1:81</a></span>;<br/><a NAME="E5:70_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</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">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</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">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#T38" target="_self" title="NATTRA_1:th.38">Th45</a></span>;<br/><a NAME="E6:70_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A4</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="c3">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><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"><font color="Maroon" title="c3">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p4">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#T37" target="_self" title="NATTRA_1:th.37">Th44</a></span>;<br/><a NAME="E7:70_1"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">m</font></span>)</span></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="#E3:70_1"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="ref" href="cat_1.html#T81" title="CAT_1:th.81">CAT_1:81</a></span>;<br/><a NAME="E8:70_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</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">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</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">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#T38" target="_self" title="NATTRA_1:th.38">Th45</a></span>;<br/><a NAME="E9:70_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A5</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="c4">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><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"><font color="Maroon" title="c4">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p4">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nattra_1.html#T37" target="_self" title="NATTRA_1:th.37">Th44</a></span>;<br/><a NAME="E10:70_1"/>
<font color="Maroon" title="c3">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</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="c3">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_1"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="ref" href="cat_1.html#T81" title="CAT_1:th.81">CAT_1:81</a></span>;<br/><a NAME="E11:70_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A6</font></span>: 
<font color="Maroon" title="c3">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font> <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"><font color="Maroon" title="c3">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70_1"><span class="lab"><font color="Green" title="E58">A4</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E12:70_1"/>
<font color="Maroon" title="c4">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</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="c4">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p2">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_1"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="ref" href="cat_1.html#T81" title="CAT_1:th.81">CAT_1:81</a></span>;<br/><span class="kw">hence </span><a NAME="E13:70_1"/>
<font color="Maroon" title="c3">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="txt" href="#E11:70_1"><span class="lab"><font color="Green" title="E60">A6</font></span></a>, <a class="txt" href="#E9:70_1"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:70"/>
<font color="Maroon" title="c3">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" title="FUNCT_2:th.63">FUNCT_2:63</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>
