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<span class="kw">set </span><font color="Maroon" title="c3">S</font> =  the    <a href="index_1.html#M4" title="INDEX_1:mode.4">TargetCat</a> of <font color="Maroon" title="c2">I</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">T</font> =  <a href="cat_5.html#K7" title="CAT_5:func.7">Image</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K9" title="INDEX_1:func.9">-functor</a> (<font color="Maroon" title="c1">C</font>, the    <a href="index_1.html#M4" title="INDEX_1:mode.4">TargetCat</a> of <font color="Maroon" title="c2">I</font>)</span>)</span> as   <a href="cat_1.html#V1" title="CAT_1:attr.1">strict</a>   <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of  the    <a href="index_1.html#M4" title="INDEX_1:mode.4">TargetCat</a> of <font color="Maroon" title="c2">I</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">F</font> = <font color="Maroon" title="c2">I</font> <a href="index_1.html#K9" title="INDEX_1:func.9">-functor</a> (<font color="Maroon" title="c1">C</font>, the    <a href="index_1.html#M4" title="INDEX_1:mode.4">TargetCat</a> of <font color="Maroon" title="c2">I</font>) as    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c4">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_5.html#T8" title="CAT_5:th.8">CAT_5:8</a></span>;<br/>
<a NAME="E3:49_1_2"/>
<font color="Maroon" title="c4">T</font> is    <a href="index_1.html#M4" title="INDEX_1:mode.4">TargetCat</a> of <font color="Maroon" title="c2">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:49_1_2_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K9" title="INDEX_1:func.9">-functor</a> (<font color="Maroon" title="c1">C</font>, the    <a href="index_1.html#M4" title="INDEX_1:mode.4">TargetCat</a> of <font color="Maroon" title="c2">I</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_5.html#D3" title="CAT_5:def.3">CAT_5:def 3</a></span>;<br/>
<a NAME="E2:49_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A3</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="index_1.html#E23"><span class="lab"><font color="Green" title="E17">Lm2</font></span></a></span>;<br/>
<a NAME="E3:49_1_2_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:49_1_2_1"/>
 for <font color="Olive" title="b1">a</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <font color="Olive" title="b1">a</font> is   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c4">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49_1_2_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="index_1.html#D9" target="_self" title="INDEX_1:def.9">INDEX_1:def 9</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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> holds  <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <span class="p4">(<span class="default"> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <span class="p4">(<span class="default"> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">b</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K8" title="INDEX_1:func.8">`2</a></span>)</span> <a href="index_1.html#K7" title="INDEX_1:func.7">.</a> <font color="Olive" title="b1">b</font></span>)</span></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="c4">T</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">b</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <span class="p4">(<span class="default"> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <span class="p4">(<span class="default"> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K8" title="INDEX_1:func.8">`2</a></span>)</span> <a href="index_1.html#K7" title="INDEX_1:func.7">.</a> <font color="Maroon" title="c6">b</font></span>)</span></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="c4">T</font></span><br/>

<a NAME="E5:49_1_2_1"/>
<font color="Maroon" title="c5">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <span class="p4">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <span class="p4">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K8" title="INDEX_1:func.8">`2</a></span>)</span> <a href="index_1.html#K7" title="INDEX_1:func.7">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="index_1.html#D11" target="_self" title="INDEX_1:def.11">Def11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:49_1_2_1"/>
<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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <span class="p4">(<span class="default"> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <span class="p4">(<span class="default"> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K8" title="INDEX_1:func.8">`2</a></span>)</span> <a href="index_1.html#K7" title="INDEX_1:func.7">.</a> <font color="Maroon" title="c6">b</font></span>)</span></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="c4">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">T</font> = <font color="Maroon" title="c4">T</font> as   <a href="cat_1.html#V1" title="CAT_1:attr.1">strict</a>   <a href="index_1.html#M4" title="INDEX_1:mode.4">TargetCat</a> of <font color="Maroon" title="c2">I</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">T</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">T</font> being    <a href="index_1.html#M4" title="INDEX_1:mode.4">TargetCat</a> of <font color="Maroon" title="c2">I</font> holds  <font color="Maroon" title="c6">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_5.html#K7" title="CAT_5:func.7">Image</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K9" title="INDEX_1:func.9">-functor</a> (<font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">T</font>)</span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">T9</font> be    <a href="index_1.html#M4" title="INDEX_1:mode.4">TargetCat</a> of <font color="Maroon" title="c2">I</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="c6">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_5.html#K7" title="CAT_5:func.7">Image</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K9" title="INDEX_1:func.9">-functor</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c7">T9</font>)</span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E5:49_1_2"/>
<font color="Maroon" title="c6">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_5.html#K7" title="CAT_5:func.7">Image</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K9" title="INDEX_1:func.9">-functor</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c7">T9</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="index_1.html#T11" target="_self" title="INDEX_1:th.11">Th11</a>, <a class="ref" href="cat_5.html#T22" title="CAT_5:th.22">CAT_5:22</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>
