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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">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="Olive" title="b1">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">f2</font> holds <br/>(  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Olive" title="b2">f2</font> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">f2</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> <font color="Olive" title="b1">f1</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">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="c2">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c3">f2</font> implies (  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c3">f2</font> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c3">f2</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> <font color="Maroon" title="c2">f1</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:10"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
<font color="Maroon" title="c2">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c3">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"> (  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c3">f2</font> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c3">f2</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> <font color="Maroon" title="c2">f1</font> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_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="E1:10_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="10_1_1"><span class="kw">suppose </span></a><a NAME="E1:10_1_1"/><span class="lab"><font color="Green" title="E6">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c3">f2</font> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c3">f2</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> <font color="Maroon" title="c2">f1</font> )</span><br/><div class="add"><span class="kw">thus </span> <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c3">f2</font></span>)</span> = 
 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="#E1:10_1_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="cat_6.html#D18" onmouseover="rs('cat_6/D18')" onmouseout="rh()">CAT_6:def 18</a></span>
<br/>.= 
 <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c3">f2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_1_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="cat_6.html#D18" onmouseover="rs('cat_6/D18')" onmouseout="rh()">CAT_6:def 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">  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c3">f2</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> <font color="Maroon" title="c2">f1</font></span><br/><span class="kw">thus </span> <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c3">f2</font></span>)</span> = 
 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="#E1:10_1_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="cat_6.html#D19" onmouseover="rs('cat_6/D19')" onmouseout="rh()">CAT_6:def 19</a></span>
<br/>.= 
 <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c2">f1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_1_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="cat_6.html#D19" onmouseover="rs('cat_6/D19')" onmouseout="rh()">CAT_6:def 19</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="10_1_2"><span class="kw">suppose </span></a><a NAME="E1:10_1_2"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
 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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c3">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c3">f2</font> &amp;  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c3">f2</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> <font color="Maroon" title="c2">f1</font> )</span><br/><div class="add"><a NAME="E2:10_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="cat_6.html#D2" onmouseover="rs('cat_6/D2')" onmouseout="rh()">CAT_6:def 2</a></span>;<br/><a NAME="E3:10_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="cat_6.html#K13" title="CAT_6:func.13">CompMap</a> <font color="Maroon" title="c1">C</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D29" onmouseover="rs('cat_6/D29')" onmouseout="rh()">CAT_6:def 29</a></span>;<br/><a NAME="E4:10_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A4</font></span>: 
<span class="p1">(<span class="default"><a href="cat_6.html#K14" title="CAT_6:func.14">SourceMap</a> <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="c2">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_6.html#K15" title="CAT_6:func.15">TargetMap</a> <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="c3">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_1_2"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="ref" href="cat_6.html#T36" onmouseover="rs('cat_6/T36')" onmouseout="rh()">CAT_6:36</a></span>;<br/><span class="lab"><font color="Green" title="E8">A5</font></span>: <font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c3">f2</font> = 
 the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="cat_6.html#D3" onmouseover="rs('cat_6/D3')" onmouseout="rh()">CAT_6:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_6.html#K13" title="CAT_6:func.13">CompMap</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c3">f2</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D29" onmouseover="rs('cat_6/D29')" onmouseout="rh()">CAT_6:def 29</a></span>
;<br/><span class="kw">thus </span> <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c3">f2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="cat_6.html#K14" title="CAT_6:func.14">SourceMap</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c3">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_1_2"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="ref" href="cat_6.html#D30" onmouseover="rs('cat_6/D30')" onmouseout="rh()">CAT_6:def 30</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_6.html#K14" title="CAT_6:func.14">SourceMap</a> <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="c3">f2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_1_2"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E5:10_1_2"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="ref" href="cat_6.html#T37" onmouseover="rs('cat_6/T37')" onmouseout="rh()">CAT_6:37</a></span>
<br/>.= 
 <a href="cat_6.html#K6" title="CAT_6:func.6">dom</a> <font color="Maroon" title="c3">f2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_1_2"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="ref" href="cat_6.html#D30" onmouseover="rs('cat_6/D30')" onmouseout="rh()">CAT_6:def 30</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">  <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c3">f2</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> <font color="Maroon" title="c2">f1</font></span><br/><span class="kw">thus </span> <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c3">f2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="cat_6.html#K15" title="CAT_6:func.15">TargetMap</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c3">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_1_2"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="ref" href="cat_6.html#D31" onmouseover="rs('cat_6/D31')" onmouseout="rh()">CAT_6:def 31</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_6.html#K15" title="CAT_6:func.15">TargetMap</a> <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="c2">f1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_1_2"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E5:10_1_2"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="ref" href="cat_6.html#T37" onmouseover="rs('cat_6/T37')" onmouseout="rh()">CAT_6:37</a></span>
<br/>.= 
 <a href="cat_6.html#K7" title="CAT_6:func.7">cod</a> <font color="Maroon" title="c2">f1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_1_2"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="ref" href="cat_6.html#D31" onmouseover="rs('cat_6/D31')" onmouseout="rh()">CAT_6:def 31</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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