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<b>let </b><font color="Maroon">S1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V2">transitive</a> <a href="altcat_1.html#V6">strict</a>  <a href="altcat_2.html#M1">SubCatStr</a> of <font color="Maroon">C</font>, <font color="Maroon">S2</font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V2">transitive</a> <a href="altcat_1.html#V6">strict</a>  <a href="altcat_2.html#M1">SubCatStr</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:48_1_2"/><i><font color="Green">E162</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E2:48_1_2"/><i><font color="Green">E163</font></i>: 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">S1</font> <a href="altcat_2.html#R2">cc=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E3:48_1_2"/><i><font color="Green">E164</font></i>: 
for <font color="Olive">o1</font>, <font color="Olive">o2</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font><br/> for <font color="Olive">m</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">o1</font>,<font color="Olive">o2</font> holds <br/> ( <font color="Olive">m</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">S1</font> <a href="binop_1.html#K1">.</a> <font color="Olive">o1</font>,<font color="Olive">o2</font> iff ( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">o1</font>,<font color="Olive">o2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">m</font> is <a href="altcat_3.html#V5">epi</a> ) )
 <b>and </b><br/><a NAME="E4:48_1_2"/><i><font color="Green">E165</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E5:48_1_2"/><i><font color="Green">E166</font></i>: 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">S2</font> <a href="altcat_2.html#R2">cc=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E6:48_1_2"/><i><font color="Green">E167</font></i>: 
for <font color="Olive">o1</font>, <font color="Olive">o2</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font><br/> for <font color="Olive">m</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">o1</font>,<font color="Olive">o2</font> holds <br/> ( <font color="Olive">m</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">S2</font> <a href="binop_1.html#K1">.</a> <font color="Olive">o1</font>,<font color="Olive">o2</font> iff ( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">o1</font>,<font color="Olive">o2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">m</font> is <a href="altcat_3.html#V5">epi</a> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:48_1_2_1"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon">o1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">o2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:48_1_2_1"/><i><font color="Green">E169</font></i>: 
( <font color="Maroon">o1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font> &amp; <font color="Maroon">o2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/><b>reconsider </b><font color="Maroon">o1</font> = <font color="Maroon">o1</font>, <font color="Maroon">o2</font> = <font color="Maroon">o2</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font> <b>by </b><i/>;<br/><b>thus </b><a NAME="E5:48_1_2_1"/>
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:48_1_2_1_1"/>
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:48_1_2_1_1_1"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E2:48_1_2_1_1_1"/>
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="altcat_4.html#T4" target="_self">Th4</a>, , , <a class="ref" href="altcat_2.html#D2">ALTCAT_2:def 2</a></i>;<br/>
<a NAME="E3:48_1_2_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:48_1_2_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">m</font> = <font color="Maroon">n</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o2</font> ;<br/>
<a NAME="E6:48_1_2_1_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">S1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:48_1_2_1_1_1"/><b>then </b>
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">m</font> is <a href="altcat_3.html#V5">epi</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E8:48_1_2_1_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">S2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:48_1_2_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/>


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

<b>let </b><font color="Maroon">n</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:48_1_2_1_1"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E3:48_1_2_1_1"/>
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, , , <a class="ref" href="altcat_2.html#D2">ALTCAT_2:def 2</a></i>;<br/>
<a NAME="E4:48_1_2_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:48_1_2_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">m</font> = <font color="Maroon">n</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o2</font> ;<br/>
<a NAME="E7:48_1_2_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">S2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:48_1_2_1_1"/><b>then </b>
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">m</font> is <a href="altcat_3.html#V5">epi</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E9:48_1_2_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">S1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:48_1_2_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E8:48_1_2"/><b>then </b>
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">S2</font>
 
<b>by </b><i><a class="ref" href="altcat_4.html#T4" target="_self">Th4</a>, , <a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>
<b>hence </b><a NAME="E9:48_1_2"/>
<font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font>
 <b>by </b><i><a class="ref" href="altcat_4.html#T4" target="_self">Th4</a>, , <a class="ref" href="altcat_2.html#T27">ALTCAT_2:27</a></i>;<br/>


</div>
