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<a NAME="E1:27_1_1"/>
<font color="Maroon" title="c5">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="cat_7.html#D3" target="_self" onmouseover="rs('cat_7/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">f3</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C</font><span class="kw"> such that </span><br/><a NAME="E3:27_1_1"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f3</font> &amp;  ex <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 <br/>( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f1</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f2</font> &amp; <font color="Maroon" title="c7">f3</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">f1</font> &amp; <font color="Olive" title="b2">f2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c7">f3</font> ) )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">f1</font>, <font color="Maroon" title="c9">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C</font><span class="kw"> such that </span><br/><a NAME="E5:27_1_1"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f1</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f2</font> &amp; <font color="Maroon" title="c7">f3</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c8">f1</font> &amp; <font color="Maroon" title="c9">f2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c7">f3</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27_1_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a></span>;<br/>
<a NAME="E6:27_1_1"/>
<font color="Maroon" title="c6">g</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="cat_7.html#D3" target="_self" onmouseover="rs('cat_7/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">g3</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C</font><span class="kw"> such that </span><br/><a NAME="E8:27_1_1"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
( <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g3</font> &amp;  ex <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="c1">C</font> st <br/>( <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g2</font> &amp; <font color="Maroon" title="c10">g3</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Olive" title="b2">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c10">g3</font> ) )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">g1</font>, <font color="Maroon" title="c12">g2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c1">C</font><span class="kw"> such that </span><br/><a NAME="E10:27_1_1"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
( <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g1</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g2</font> &amp; <font color="Maroon" title="c10">g3</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g1</font> &amp; <font color="Maroon" title="c12">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c10">g3</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:27_1_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a></span>;<br/>
<a NAME="E11:27_1_1"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
( <font color="Maroon" title="c1">C</font> is  <a href="cat_6.html#V6" title="CAT_6:attr.6">left_composable</a>  &amp; <font color="Maroon" title="c1">C</font> is  <a href="cat_6.html#V7" title="CAT_6:attr.7">right_composable</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D11" onmouseover="rs('cat_6/D11')" onmouseout="rh()">CAT_6:def 11</a></span>;<br/>
<a NAME="E12:27_1_1"/>
 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="#E3:27_1_1"><span class="lab"><font color="Green" title="E19">A2</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="E13:27_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A7</font></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="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27_1_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E8:27_1_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="cat_7.html#T3" target="_self" onmouseover="rs('cat_7/T3')" onmouseout="rh()">Th3</a>, <a class="ref" href="cat_6.html#T22" onmouseover="rs('cat_6/T22')" onmouseout="rh()">CAT_6:22</a></span>;<br/>
<a NAME="E14:27_1_1"/>
( <font color="Maroon" title="c12">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c6">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c6">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c5">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c8">f1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27_1_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E5:27_1_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E8:27_1_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E10:27_1_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E13:27_1_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="txt" href="#E11:27_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="cat_6.html#D8" onmouseover="rs('cat_6/D8')" onmouseout="rh()">CAT_6:def 8</a>, <a class="ref" href="cat_6.html#D9" onmouseover="rs('cat_6/D9')" onmouseout="rh()">CAT_6:def 9</a></span>;<br/>
<a NAME="E15:27_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c5">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:27_1_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E10:27_1_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:27_1_1"/>
<font color="Maroon" title="c6">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c5">f</font> is    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_7.html#D3" target="_self" onmouseover="rs('cat_7/D3')" onmouseout="rh()">Def3</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/>


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