<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">C</font> be    <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">a</font>, <font color="Olive" title="b2">b</font> being   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c1">C</font><br/>  for <font color="Olive" title="b3">f</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>  st  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</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>  holds <br/> ex <font color="Olive" title="b4">f1</font>, <font color="Olive" title="b5">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="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f1</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">f2</font> &amp; <font color="Olive" title="b3">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b4">f1</font> &amp; <font color="Olive" title="b5">f2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b3">f</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</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">  for <font color="Olive" title="b1">f</font> being    <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="c3">b</font>  st  <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>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> ex <font color="Olive" title="b2">f1</font>, <font color="Olive" title="b3">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="b2">f1</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f2</font> &amp; <font color="Olive" title="b1">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">f1</font> &amp; <font color="Olive" title="b3">f2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">f</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be    <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="c3">b</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_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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>  implies  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="c4">f</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="c4">f</font> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:28"/>
 <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>) <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">  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="c4">f</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="c4">f</font> )</span><br/>

<a NAME="E2:28"/><span class="kw">then </span>
<font color="Maroon" title="c4">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="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="c5">f11</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="E4:28"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
( <font color="Maroon" title="c5">f11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</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="c5">f11</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="c5">f11</font> ) )
 ;<br/>
<span class="kw">thus </span><a NAME="E5:28"/>
 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="c4">f</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="c4">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E20">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"> verum</span><br/>


</div>
