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<span class="kw">let </span><font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c2">D</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"> ( <font color="Maroon" title="c1">C</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c2">D</font> iff  ex <font color="Olive" title="b1">F</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">D</font> st <br/>( <font color="Olive" title="b1">F</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  ) )</span><br/>

<div><span class="kw">hereby </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"> (  ex <font color="Olive" title="b1">F</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">D</font> st <br/>( <font color="Olive" title="b1">F</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  ) implies <font color="Maroon" title="c1">C</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c2">D</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:18_1"/>
<font color="Maroon" title="c1">C</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c2">D</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">  ex <font color="Olive" title="b1">F</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">D</font> st <br/>( <font color="Olive" title="b1">F</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">F</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">D</font>, <font color="Maroon" title="c4">G</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c1">C</font><span class="kw"> such that </span><br/><a NAME="E3:18_1"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
( <font color="Maroon" title="c3">F</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">G</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">G</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c3">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c4">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">D</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D28" onmouseover="rs('cat_6/D28')" onmouseout="rh()">CAT_6:def 28</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c5">F</font> = <font color="Maroon" title="c3">F</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="c5">F</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  )</span><br/><span class="kw">thus </span><a NAME="E4:18_1"/>
<font color="Maroon" title="c5">F</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_1"><span class="lab"><font color="Green" title="E13">A1</font></span></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"> <font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> </span><br/><font color="Maroon" title="c5">F</font> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c4">G</font> = 
 <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">C</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_1"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="cat_6.html#D27" onmouseover="rs('cat_6/D27')" onmouseout="rh()">CAT_6:def 27</a></span>
<br/>.= 
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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="struct_0.html#D4" onmouseover="rs('struct_0/D4')" onmouseout="rh()">STRUCT_0:def 4</a></span>
;<br/><a NAME="E6:18_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<font color="Maroon" title="c5">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_2.html#T23" onmouseover="rs('funct_2/T23')" onmouseout="rh()">FUNCT_2:23</a></span>;<br/><font color="Maroon" title="c4">G</font> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c5">F</font> = 
 <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">D</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_1"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="cat_6.html#D27" onmouseover="rs('cat_6/D27')" onmouseout="rh()">CAT_6:def 27</a></span>
<br/>.= 
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">D</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D4" onmouseover="rs('struct_0/D4')" onmouseout="rh()">STRUCT_0:def 4</a></span>
;<br/><a NAME="E8:18_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T23" onmouseover="rs('funct_2/T23')" onmouseout="rh()">FUNCT_2:23</a></span>;<br/><span class="kw">hence </span><a NAME="E9:18_1"/>
<font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:18_1"><span class="lab"><font color="Green" title="E14">A2</font></span></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>

<span class="kw">given </span><font color="Maroon" title="c3">F</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">D</font><span class="kw"> such that </span><a NAME="E3:18"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
( <font color="Maroon" title="c3">F</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c3">F</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</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"> <font color="Maroon" title="c1">C</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c2">D</font></span><br/>

<a NAME="E4:18"/><span class="lab"><font color="Green" title="E14">A4</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <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="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">G</font> = <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">D</font>, 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="txt" href="#E3:18"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="funct_2.html#T25" onmouseover="rs('funct_2/T25')" onmouseout="rh()">FUNCT_2:25</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">G</font> = <font color="Maroon" title="c4">G</font> as   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c1">C</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2"><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="E1:18_2"/>
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">D</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1"><span class="kw">suppose </span></a><a NAME="E1:18_2_1"/><span class="lab"><font color="Green" title="E15">A5</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">D</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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"> <font color="Maroon" title="c1">C</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c2">D</font></span><br/><div class="add"><a NAME="E2:18_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A6</font></span>: 
( <font color="Maroon" title="c5">G</font> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c3">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c3">F</font> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c5">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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="txt" href="#E3:18"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="funct_2.html#T29" onmouseover="rs('funct_2/T29')" onmouseout="rh()">FUNCT_2:29</a></span>;<br/><a NAME="E3:18_2_1"/><span class="lab"><font color="Green" title="E17">A7</font></span>: 
 not <font color="Maroon" title="c2">D</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_2_1"><span class="lab"><font color="Green" title="E15">A5</font></span></a></span>;<br/><a NAME="E4:18_2_1"/><span class="lab"><font color="Green" title="E18">A8</font></span>: 
 not <font color="Maroon" title="c1">C</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="txt" href="#E1:18_2_1"><span class="lab"><font color="Green" title="E15">A5</font></span></a></span>;<br/><a NAME="E5:18_2_1"/><span class="lab"><font color="Green" title="E19">A9</font></span>: 
( <font color="Maroon" title="c3">F</font> is  <a href="cat_6.html#V13" title="CAT_6:attr.13">identity-preserving</a>  &amp; <font color="Maroon" title="c3">F</font> is  <a href="cat_6.html#V14" title="CAT_6:attr.14">multiplicative</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</a></span>;<br/><a NAME="E6:18_2_1"/><span class="lab"><font color="Green" title="E20">A10</font></span>: 
 for <font color="Olive" title="b1">g</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">D</font> holds  <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">g</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">D</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="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">x1</font> = <font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g</font>, <font color="Maroon" title="c8">x2</font> = <font color="Maroon" title="c6">g</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  ;<br/>
<a NAME="E2:18_2_1_1"/>
<font color="Maroon" title="c6">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_2_1"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="ref" href="subset_1.html#D1" onmouseover="rs('subset_1/D1')" onmouseout="rh()">SUBSET_1:def 1</a></span>;<br/>
<a NAME="E3:18_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A11</font></span>: 
<font color="Maroon" title="c6">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<a NAME="E4:18_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A12</font></span>: 
<font color="Maroon" title="c8">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18_2_1"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g</font></span>)</span> = 
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18_2_1"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_2_1"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_2_1"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="txt" href="#E4:18_2_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c6">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_2_1_1"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="ref" href="funct_1.html#T18" onmouseover="rs('funct_1/T18')" onmouseout="rh()">FUNCT_1:18</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><a NAME="E7:18_2_1"/><span class="lab"><font color="Green" title="E21">A13</font></span>: 
 for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</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="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">f2</font> holds <br/><font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">f1</font>, <font color="Maroon" title="c7">f2</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="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">f2</font> implies <font color="Maroon" title="c6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:18_2_1_2"/><span class="lab"><font color="Green" title="E22">A14</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">f2</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="c6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f2</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">x1</font> = <font color="Maroon" title="c6">f1</font>, <font color="Maroon" title="c9">x2</font> = <font color="Maroon" title="c7">f2</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  ;<br/>
<a NAME="E3:18_2_1_2"/>
( <font color="Maroon" title="c6">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c7">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c1">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18_2_1"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="subset_1.html#D1" onmouseover="rs('subset_1/D1')" onmouseout="rh()">SUBSET_1:def 1</a></span>;<br/>
<a NAME="E4:18_2_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c6">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c7">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<a NAME="E5:18_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A15</font></span>: 
( <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c9">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_2_1"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x1</font> = 
<font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18_2_1"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_2_1_2"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E4:18_2_1"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:18_2_1_2"/>
<font color="Maroon" title="c6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E5:18_2_1_2"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</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><a NAME="E8:18_2_1"/>
 for <font color="Olive" title="b1">g</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">D</font>  st <font color="Olive" title="b1">g</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  holds <br/><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">g</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_2_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">g</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">D</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="c6">g</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  implies <font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:18_2_1_3"/>
<font color="Maroon" title="c6">g</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"> <font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> </span><br/>

<a NAME="E2:18_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A16</font></span>: 
 for <font color="Olive" title="b1">g1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">D</font>  st <font color="Maroon" title="c6">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">g1</font> holds <br/><font color="Maroon" title="c6">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D4" onmouseover="rs('cat_6/D4')" onmouseout="rh()">CAT_6:def 4</a>, <a class="ref" href="cat_6.html#D14" onmouseover="rs('cat_6/D14')" onmouseout="rh()">CAT_6:def 14</a></span>;<br/>
<a NAME="E3:18_2_1_3"/><span class="lab"><font color="Green" title="E23">A17</font></span>: 
 for <font color="Olive" title="b1">f1</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="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">f1</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_2_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">f1</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="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c7">f1</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:18_2_1_3_1"/>
<font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c7">f1</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f1</font></span><br/>

<a NAME="E2:18_2_1_3_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">f1</font> &amp; <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g</font></span>)</span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">f1</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18_2_1"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</a></span>;<br/>
<a NAME="E3:18_2_1_3_1"/><span class="kw">then </span>
( <font color="Maroon" title="c6">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">f1</font> &amp; <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">f1</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:18_2_1"><span class="lab"><font color="Green" title="E20">A10</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:18_2_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_2_1_3"><span class="lab"><font color="Green" title="E22">A16</font></span></a>, <a class="txt" href="#E7:18_2_1"><span class="lab"><font color="Green" title="E21">A13</font></span></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>
<a NAME="E4:18_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g</font> is  <a href="cat_6.html#V2" title="CAT_6:attr.2">left_identity</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D4" onmouseover="rs('cat_6/D4')" onmouseout="rh()">CAT_6:def 4</a></span>;<br/>
<a NAME="E5:18_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g</font> is  <a href="cat_6.html#V3" title="CAT_6:attr.3">right_identity</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#T9" onmouseover="rs('cat_6/T9')" onmouseout="rh()">CAT_6:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:18_2_1_3"/>
<font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_2_1_3"><span class="lab"><font color="Green" title="E23">A17</font></span></a>, <a class="ref" href="cat_6.html#D4" onmouseover="rs('cat_6/D4')" onmouseout="rh()">CAT_6:def 4</a>, <a class="ref" href="cat_6.html#D14" onmouseover="rs('cat_6/D14')" onmouseout="rh()">CAT_6:def 14</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><a NAME="E9:18_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A18</font></span>: 
<font color="Maroon" title="c5">G</font> is  <a href="cat_6.html#V13" title="CAT_6:attr.13">identity-preserving</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D22" onmouseover="rs('cat_6/D22')" onmouseout="rh()">CAT_6:def 22</a></span>;<br/><a NAME="E10:18_2_1"/><span class="lab"><font color="Green" title="E23">A19</font></span>: 
 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> holds  <font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_2_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">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="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">x1</font> = <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font>, <font color="Maroon" title="c8">x2</font> = <font color="Maroon" title="c6">f</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  ;<br/>
<a NAME="E2:18_2_1_4"/>
<font color="Maroon" title="c6">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18_2_1"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="subset_1.html#D1" onmouseover="rs('subset_1/D1')" onmouseout="rh()">SUBSET_1:def 1</a></span>;<br/>
<a NAME="E3:18_2_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A20</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<a NAME="E4:18_2_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A21</font></span>: 
<font color="Maroon" title="c8">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_2_1"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">f</font></span>)</span> = 
<font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_2_1"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18_2_1"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_2_1"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="txt" href="#E4:18_2_1_4"><span class="lab"><font color="Green" title="E25">A21</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c6">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_2_1_4"><span class="lab"><font color="Green" title="E24">A20</font></span></a>, <a class="ref" href="funct_1.html#T18" onmouseover="rs('funct_1/T18')" onmouseout="rh()">FUNCT_1:18</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><a NAME="E11:18_2_1"/>
 for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">D</font>  st <font color="Olive" title="b1">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">g2</font> holds <br/>( <font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">g2</font> &amp; <font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">g1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">g2</font></span>)</span> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_2_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">g1</font>, <font color="Maroon" title="c7">g2</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">D</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="c6">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c7">g2</font> implies ( <font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font> &amp; <font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:18_2_1_5"/><span class="lab"><font color="Green" title="E24">A22</font></span>: 
<font color="Maroon" title="c6">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c7">g2</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="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font> &amp; <font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font></span>)</span> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">f1</font> = <font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font>, <font color="Maroon" title="c9">f2</font> = <font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C</font> ;<br/>
<a NAME="E3:18_2_1_5"/><span class="lab"><font color="Green" title="E25">A23</font></span>: 
( <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font></span>)</span> &amp; <font color="Maroon" title="c7">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:18_2_1"><span class="lab"><font color="Green" title="E20">A10</font></span></a></span>;<br/>
<a NAME="E4:18_2_1_5"/><span class="lab"><font color="Green" title="E26">A24</font></span>: 
 <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c9">f2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_2_1_5"><span class="lab"><font color="Green" title="E24">A22</font></span></a>, <a class="txt" href="#E3:18_2_1_5"><span class="lab"><font color="Green" title="E25">A23</font></span></a>, <a class="txt" href="#E3:18_2_1"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="ref" href="cat_7.html#T5" target="_self" onmouseover="rs('cat_7/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<a NAME="E5:18_2_1_5"/>
 not  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18_2_1"><span class="lab"><font color="Green" title="E18">A8</font></span></a></span>;<br/>
<a NAME="E6:18_2_1_5"/><span class="kw">then </span>
(  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c8">f1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <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="c9">f2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K5" title="CAT_6:func.5">Ob</a> <font color="Maroon" title="c1">C</font> )
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">d1</font> =  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c8">f1</font>, <font color="Maroon" title="c11">c2</font> =  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c9">f2</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C</font> ;<br/>
<font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c10">d1</font> = 
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c8">f1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18_2_1"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
 <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c8">f1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_2_1"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="txt" href="#E4:18_2_1"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="cat_6.html#T32" onmouseover="rs('cat_6/T32')" onmouseout="rh()">CAT_6:32</a></span>
<br/>.= 
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c9">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18_2_1_5"><span class="lab"><font color="Green" title="E26">A24</font></span></a>, <a class="txt" href="#E3:18_2_1"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="txt" href="#E4:18_2_1"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="cat_6.html#T32" onmouseover="rs('cat_6/T32')" onmouseout="rh()">CAT_6:32</a></span>
<br/>.= 
<font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c11">c2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18_2_1"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:18_2_1_5"/><span class="lab"><font color="Green" title="E27">A25</font></span>: 
<font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:18_2_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E4:18_2_1"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="cat_7.html#T5" target="_self" onmouseover="rs('cat_7/T5')" onmouseout="rh()">Th5</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"> <font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font></span>)</span></span><br/>

<a NAME="E10:18_2_1_5"/><span class="lab"><font color="Green" title="E28">A26</font></span>: 
<font color="Maroon" title="c3">F</font> is  <a href="cat_6.html#V14" title="CAT_6:attr.14">multiplicative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</a></span>;<br/>
<a NAME="E11:18_2_1_5"/>
<font color="Maroon" title="c6">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_2_1_5"><span class="lab"><font color="Green" title="E25">A23</font></span></a>, <a class="txt" href="#E10:18_2_1_5"><span class="lab"><font color="Green" title="E28">A26</font></span></a>, <a class="txt" href="#E9:18_2_1_5"><span class="lab"><font color="Green" title="E27">A25</font></span></a>, <a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:18_2_1_5"/>
<font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c7">g2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:18_2_1"><span class="lab"><font color="Green" title="E23">A19</font></span></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><a NAME="E12:18_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A27</font></span>: 
<font color="Maroon" title="c5">G</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:18_2_1"><span class="lab"><font color="Green" title="E22">A18</font></span></a>, <a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</a></span>;<br/><a NAME="E13:18_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c5">G</font> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c3">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c5">G</font> &amp; <font color="Maroon" title="c3">F</font> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c5">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">G</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c3">F</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="cat_6.html#D27" onmouseover="rs('cat_6/D27')" onmouseout="rh()">CAT_6:def 27</a></span>;<br/><a NAME="E14:18_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c5">G</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c3">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c5">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">D</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_2_1"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="ref" href="struct_0.html#D4" onmouseover="rs('struct_0/D4')" onmouseout="rh()">STRUCT_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E15:18_2_1"/>
<font color="Maroon" title="c1">C</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c2">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E12:18_2_1"><span class="lab"><font color="Green" title="E24">A27</font></span></a>, <a class="ref" href="cat_6.html#D28" onmouseover="rs('cat_6/D28')" onmouseout="rh()">CAT_6:def 28</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="18_2_2"><span class="kw">suppose </span></a><a NAME="E1:18_2_2"/><span class="lab"><font color="Green" title="E15">A28</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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"> <font color="Maroon" title="c1">C</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c2">D</font></span><br/><div class="add"><a NAME="E2:18_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A29</font></span>: 
<font color="Maroon" title="c2">D</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ;<br/><a NAME="E3:18_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A30</font></span>: 
<font color="Maroon" title="c1">C</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="cat_6.html#T31" onmouseover="rs('cat_6/T31')" onmouseout="rh()">CAT_6:31</a></span>;<br/><a NAME="E4:18_2_2"/>
 for <font color="Olive" title="b1">g</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">D</font>  st <font color="Olive" title="b1">g</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  holds <br/><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">g</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_2_2"><span class="lab"><font color="Green" title="E17">A30</font></span></a>, <a class="ref" href="cat_6.html#T10" onmouseover="rs('cat_6/T10')" onmouseout="rh()">CAT_6:10</a></span>;<br/><a NAME="E5:18_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A31</font></span>: 
<font color="Maroon" title="c5">G</font> is  <a href="cat_6.html#V13" title="CAT_6:attr.13">identity-preserving</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D22" onmouseover="rs('cat_6/D22')" onmouseout="rh()">CAT_6:def 22</a></span>;<br/><a NAME="E6:18_2_2"/>
 for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c2">D</font>  st <font color="Olive" title="b1">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">g2</font> holds <br/>( <font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">g2</font> &amp; <font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b1">g1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Olive" title="b2">g2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_2_2"><span class="lab"><font color="Green" title="E16">A29</font></span></a>, <a class="ref" href="cat_6.html#T1" onmouseover="rs('cat_6/T1')" onmouseout="rh()">CAT_6:1</a></span>;<br/><a NAME="E7:18_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A32</font></span>: 
<font color="Maroon" title="c5">G</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18_2_2"><span class="lab"><font color="Green" title="E18">A31</font></span></a>, <a class="ref" href="cat_6.html#D23" onmouseover="rs('cat_6/D23')" onmouseout="rh()">CAT_6:def 23</a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</a></span>;<br/><a NAME="E8:18_2_2"/>
( <font color="Maroon" title="c5">G</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c3">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c5">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">D</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_2_2"><span class="lab"><font color="Green" title="E17">A30</font></span></a>, <a class="txt" href="#E1:18_2_2"><span class="lab"><font color="Green" title="E15">A28</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:18_2_2"/>
<font color="Maroon" title="c1">C</font> <a href="cat_6.html#NR4" title="CAT_6:NR.4">~=</a> <font color="Maroon" title="c2">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E7:18_2_2"><span class="lab"><font color="Green" title="E19">A32</font></span></a>, <a class="ref" href="cat_6.html#D28" onmouseover="rs('cat_6/D28')" onmouseout="rh()">CAT_6:def 28</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>

</div>
