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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>let </b><font color="Maroon" title="c3">a1</font>, <font color="Maroon" title="c4">a2</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>;<br/><b>let </b><font color="Maroon" title="c5">b1</font>, <font color="Maroon" title="c6">b2</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">B</font>;<br/>







<b>assume </b><a NAME="E1:15"/><i><font color="Green" title="E14">A1</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c5">b1</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<a NAME="E2:15"/><b>then </b><i><font color="Green" title="E15">A2</font></i>: 
(  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</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>  &amp;  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c6">b2</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>  )
 
<b>by </b><i><a class="ref" href="isocat_2.html#T13" target="_self" title="ISOCAT_2:th.13">Th13</a></i>;<br/>
<b>let </b><font color="Maroon" title="c7">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">A</font>;<br/><b>let </b><font color="Maroon" title="c8">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">B</font>;<br/>



<b>thus </b><a NAME="E3:15"/>
( <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c5">b1</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span> implies ( <font color="Maroon" title="c7">f</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font> &amp; <font color="Maroon" title="c8">g</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c6">b2</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1"/>
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c5">b1</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>
 ;<br/>

<a NAME="E2:15_1"/><b>then </b>
( <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c5">b1</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span> &amp;  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c5">b1</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c6">b2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="txt" href="isocat_2.html#E1:15"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="cat_1.html#D7" title="CAT_1:def.7">CAT_1:def 7</a>, <a class="ref" href="cat_2.html#T42" title="CAT_2:th.42">CAT_2:42</a></i>;<br/>
<a NAME="E3:15_1"/><b>then </b>
( <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font> &amp; <font color="Maroon" title="c8">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c6">b2</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E4:15_1"/>
( <font color="Maroon" title="c7">f</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font> &amp; <font color="Maroon" title="c8">g</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c6">b2</font> )
 <b>by </b><i><a class="ref" href="cat_1.html#D7" title="CAT_1:def.7">CAT_1:def 7</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:15"/>
( <font color="Maroon" title="c7">f</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font> &amp; <font color="Maroon" title="c8">g</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c6">b2</font> implies <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c5">b1</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span>,<span class="p1"><a href="cat_2.html#K12" title="CAT_2:func.12">[</a><span class="default"><font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c6">b2</font></span><a href="cat_2.html#K12" title="CAT_2:func.12">]</a></span> )
 <b>by </b><i><a class="txt" href="isocat_2.html#E2:15"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="cat_2.html#T43" title="CAT_2:th.43">CAT_2:43</a></i>;<br/>


</div>
