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<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   <a href="cat_6.html#V9" title="CAT_6:attr.9">composable</a>   <a href="cat_6.html#V10" title="CAT_6:attr.10">with_identities</a>   <a href="cat_6.html#L1" title="CAT_6:struct.1">CategoryStr</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  holds <br/>(  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  implies (  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:12"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
<font color="Maroon" title="c2">f</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:12_1"/>
( <font color="Maroon" title="c1">C</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  or  not <font color="Maroon" title="c1">C</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1"><span class="kw">suppose </span></a><a NAME="E1:12_1_1"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
<font color="Maroon" title="c1">C</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> )</span><br/><div class="add"><a NAME="E2:12_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A3</font></span>: 
 the   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="subset_1.html#D1" onmouseover="rs('subset_1/D1')" onmouseout="rh()">SUBSET_1:def 1</a></span>;<br/><a NAME="E3:12_1_1"/>
(  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c1">C</font> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_1_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="cat_6.html#D18" onmouseover="rs('cat_6/D18')" onmouseout="rh()">CAT_6:def 18</a>, <a class="ref" href="cat_6.html#D19" onmouseover="rs('cat_6/D19')" onmouseout="rh()">CAT_6:def 19</a></span>;<br/><span class="kw">hence </span><a NAME="E4:12_1_1"/>
(  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12_1_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E1:12_1_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="subset_1.html#D1" onmouseover="rs('subset_1/D1')" onmouseout="rh()">SUBSET_1:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_2"><span class="kw">suppose </span></a><a NAME="E1:12_1_2"/>
 not <font color="Maroon" title="c1">C</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:12_1_2"/>
(  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="cat_6.html#T24" onmouseover="rs('cat_6/T24')" onmouseout="rh()">CAT_6:24</a>, <a class="ref" href="cat_6.html#T26" onmouseover="rs('cat_6/T26')" onmouseout="rh()">CAT_6:26</a>, <a class="ref" href="cat_6.html#T27" onmouseover="rs('cat_6/T27')" onmouseout="rh()">CAT_6:27</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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></div><span class="kw">end;</span></div>

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