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<div class="add">

<b>let </b><font color="Maroon" title="c2">D</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="altcat_2.html#M1" title="ALTCAT_2:mode.1">SubCatStr</a> of <font color="Maroon" title="c1">C</font>;<br/>

<b>assume </b><a NAME="E1:71_1_1"/>
<font color="Maroon" title="c2">D</font> is <a href="altcat_1.html#V2" title="ALTCAT_1:attr.2">transitive</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c3">D</font> = <font color="Maroon" title="c2">D</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="altcat_1.html#V2" title="ALTCAT_1:attr.2">transitive</a>  <a href="altcat_2.html#M1" title="ALTCAT_2:mode.1">SubCatStr</a> of <font color="Maroon" title="c1">C</font> ;<br/>
<a NAME="E3:71_1_1"/>
<font color="Maroon" title="c3">D</font> is <a href="altcat_1.html#V11" title="ALTCAT_1:attr.11">associative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">o1</font>, <font color="Maroon" title="c5">o2</font>, <font color="Maroon" title="c6">o3</font>, <font color="Maroon" title="c7">o4</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">D</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_2.html#D8" target="_self" title="ALTCAT_2:def.8">ALTCAT_2:def 8</a><br/>

<a NAME="E1:71_1_1_1"/><i><font color="Green" title="E34">A1</font></i>: 
( <font color="Maroon" title="c4">o1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">D</font> &amp; <font color="Maroon" title="c5">o2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">D</font> &amp; <font color="Maroon" title="c6">o3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">D</font> &amp; <font color="Maroon" title="c7">o4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">D</font> )
 
;<br/>
<a NAME="E2:71_1_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="altcat_2.html#D11" target="_self" title="ALTCAT_2:def.11">Def11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">p1</font> = <font color="Maroon" title="c4">o1</font>, <font color="Maroon" title="c9">p2</font> = <font color="Maroon" title="c5">o2</font>, <font color="Maroon" title="c10">p3</font> = <font color="Maroon" title="c6">o3</font>, <font color="Maroon" title="c11">p4</font> = <font color="Maroon" title="c7">o4</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="txt" href="altcat_2.html#E1:71_1_1_1"><i><font color="Green" title="E34">A1</font></i></a></i>;<br/>
<b>let </b><font color="Maroon" title="c12">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">o1</font>,<font color="Maroon" title="c5">o2</font>;<br/><b>let </b><font color="Maroon" title="c13">g</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">o2</font>,<font color="Maroon" title="c6">o3</font>;<br/><b>let </b><font color="Maroon" title="c14">h</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c6">o3</font>,<font color="Maroon" title="c7">o4</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E4:71_1_1_1"/><i><font color="Green" title="E35">A2</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o1</font>,<font color="Maroon" title="c5">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</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> 
 <b>and </b><br/><a NAME="E5:71_1_1_1"/><i><font color="Green" title="E36">A3</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">o2</font>,<font color="Maroon" title="c6">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</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> 
 <b>and </b><br/><a NAME="E6:71_1_1_1"/><i><font color="Green" title="E37">A4</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c6">o3</font>,<font color="Maroon" title="c7">o4</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</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="E7:71_1_1_1"/><i><font color="Green" title="E38">A5</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o1</font>,<font color="Maroon" title="c5">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
<b>by </b><i><a class="ref" href="altcat_2.html#T32" target="_self" title="ALTCAT_2:th.32">Th32</a></i>;<br/>
<a NAME="E8:71_1_1_1"/><b>then </b><i><font color="Green" title="E39">A6</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</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> 
 
<b>by </b><i><a class="txt" href="altcat_2.html#E4:71_1_1_1"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E9:71_1_1_1"/><i><font color="Green" title="E40">A7</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">o2</font>,<font color="Maroon" title="c6">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c10">p3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
<b>by </b><i><a class="ref" href="altcat_2.html#T32" target="_self" title="ALTCAT_2:th.32">Th32</a></i>;<br/>
<a NAME="E10:71_1_1_1"/><b>then </b><i><font color="Green" title="E41">A8</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c10">p3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</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> 
 
<b>by </b><i><a class="txt" href="altcat_2.html#E5:71_1_1_1"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E11:71_1_1_1"/><i><font color="Green" title="E42">A9</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c6">o3</font>,<font color="Maroon" title="c7">o4</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c10">p3</font>,<font color="Maroon" title="c11">p4</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
<b>by </b><i><a class="ref" href="altcat_2.html#T32" target="_self" title="ALTCAT_2:th.32">Th32</a></i>;<br/>
<a NAME="E12:71_1_1_1"/><b>then </b><i><font color="Green" title="E43">A10</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c10">p3</font>,<font color="Maroon" title="c11">p4</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</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> 
 
<b>by </b><i><a class="txt" href="altcat_2.html#E6:71_1_1_1"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E13:71_1_1_1"/><i><font color="Green" title="E44">A11</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o1</font>,<font color="Maroon" title="c6">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</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> 
 
<b>by </b><i><a class="txt" href="altcat_2.html#E4:71_1_1_1"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="altcat_2.html#E5:71_1_1_1"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="altcat_1.html#D4" title="ALTCAT_1:def.4">ALTCAT_1:def 4</a></i>;<br/>
<a NAME="E14:71_1_1_1"/><i><font color="Green" title="E45">A12</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">o2</font>,<font color="Maroon" title="c7">o4</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</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> 
 
<b>by </b><i><a class="txt" href="altcat_2.html#E5:71_1_1_1"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="altcat_2.html#E6:71_1_1_1"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="altcat_1.html#D4" title="ALTCAT_1:def.4">ALTCAT_1:def 4</a></i>;<br/>
<a NAME="E15:71_1_1_1"/>
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o1</font>,<font color="Maroon" title="c5">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E4:71_1_1_1"><i><font color="Green" title="E35">A2</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">ff</font> = <font color="Maroon" title="c12">f</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font> <b>by </b><i><a class="txt" href="altcat_2.html#E7:71_1_1_1"><i><font color="Green" title="E38">A5</font></i></a></i>;<br/>
<a NAME="E17:71_1_1_1"/>
<font color="Maroon" title="c13">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">o2</font>,<font color="Maroon" title="c6">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E5:71_1_1_1"><i><font color="Green" title="E36">A3</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">gg</font> = <font color="Maroon" title="c13">g</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c10">p3</font> <b>by </b><i><a class="txt" href="altcat_2.html#E9:71_1_1_1"><i><font color="Green" title="E40">A7</font></i></a></i>;<br/>
<a NAME="E19:71_1_1_1"/>
<font color="Maroon" title="c14">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c6">o3</font>,<font color="Maroon" title="c7">o4</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E6:71_1_1_1"><i><font color="Green" title="E37">A4</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c17">hh</font> = <font color="Maroon" title="c14">h</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c10">p3</font>,<font color="Maroon" title="c11">p4</font> <b>by </b><i><a class="txt" href="altcat_2.html#E11:71_1_1_1"><i><font color="Green" title="E42">A9</font></i></a></i>;<br/>
<a NAME="E21:71_1_1_1"/><i><font color="Green" title="E46">A13</font></i>: 
<font color="Maroon" title="c13">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">gg</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c15">ff</font>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E4:71_1_1_1"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="altcat_2.html#E5:71_1_1_1"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="altcat_2.html#T33" target="_self" title="ALTCAT_2:th.33">Th33</a></i>;<br/>
<a NAME="E22:71_1_1_1"/>
<font color="Maroon" title="c14">h</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">hh</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c16">gg</font>
 
<b>by </b><i><a class="txt" href="altcat_2.html#E5:71_1_1_1"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="altcat_2.html#E6:71_1_1_1"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="altcat_2.html#T33" target="_self" title="ALTCAT_2:th.33">Th33</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c14">h</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c12">f</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">hh</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c16">gg</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c15">ff</font>
<b>by </b><i><a class="txt" href="altcat_2.html#E4:71_1_1_1"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="altcat_2.html#E14:71_1_1_1"><i><font color="Green" title="E45">A12</font></i></a>, <a class="ref" href="altcat_2.html#T33" target="_self" title="ALTCAT_2:th.33">Th33</a></i>
<br/>.= 
<font color="Maroon" title="c17">hh</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">gg</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c15">ff</font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_2.html#E8:71_1_1_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="altcat_2.html#E10:71_1_1_1"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="altcat_2.html#E12:71_1_1_1"><i><font color="Green" title="E43">A10</font></i></a>, <a class="ref" href="altcat_2.html#D8" target="_self" title="ALTCAT_2:def.8">Def8</a></i>
<br/>.= 
<font color="Maroon" title="c14">h</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c12">f</font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_2.html#E6:71_1_1_1"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="altcat_2.html#E13:71_1_1_1"><i><font color="Green" title="E44">A11</font></i></a>, <a class="txt" href="altcat_2.html#E21:71_1_1_1"><i><font color="Green" title="E46">A13</font></i></a>, <a class="ref" href="altcat_2.html#T33" target="_self" title="ALTCAT_2:th.33">Th33</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:71_1_1"/>
<font color="Maroon" title="c2">D</font> is <a href="altcat_1.html#V11" title="ALTCAT_1:attr.11">associative</a>
 ;<br/>


</div>
