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<b>let </b><font color="Maroon" title="c2">B</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">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">A</font>;<br/><b>let </b><font color="Maroon" title="c3">b1</font>, <font color="Maroon" title="c4">b2</font>, <font color="Maroon" title="c5">b3</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D14" title="ALTCAT_1:def.14">ALTCAT_1:def 14</a><br/>



<b>assume </b><a NAME="E1:57_1_1"/><i><font color="Green" title="E24">A1</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="c3">b1</font>,<font color="Maroon" title="c4">b2</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>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c5">b3</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>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c5">b3</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/>

<b>let </b><font color="Maroon" title="c6">f1</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c4">b2</font>;<br/><b>let </b><font color="Maroon" title="c7">f2</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c5">b3</font>;<br/><b>let </b><font color="Maroon" title="c8">f'</font>, <font color="Maroon" title="c9">g'</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>





<b>assume </b><a NAME="E2:57_1_1"/><i><font color="Green" title="E25">A2</font></i>: 
( <font color="Maroon" title="c6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f'</font> &amp; <font color="Maroon" title="c7">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g'</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c10">a1</font> = <font color="Maroon" title="c3">b1</font>, <font color="Maroon" title="c11">a2</font> = <font color="Maroon" title="c4">b2</font>, <font color="Maroon" title="c12">a3</font> = <font color="Maroon" title="c5">b3</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="altcat_2.html#T30" title="ALTCAT_2:th.30">ALTCAT_2:30</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">g1</font> = <font color="Maroon" title="c6">f1</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c10">a1</font>,<font color="Maroon" title="c11">a2</font> <b>by </b><i><a class="txt" href="yellow21.html#E1:57_1_1"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="altcat_2.html#T34" title="ALTCAT_2:th.34">ALTCAT_2:34</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">g2</font> = <font color="Maroon" title="c7">f2</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c11">a2</font>,<font color="Maroon" title="c12">a3</font> <b>by </b><i><a class="txt" href="yellow21.html#E1:57_1_1"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="altcat_2.html#T34" title="ALTCAT_2:th.34">ALTCAT_2:34</a></i>;<br/>
<a NAME="E6:57_1_1"/>
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c4">b2</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">a1</font>,<font color="Maroon" title="c11">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">b2</font>,<font color="Maroon" title="c5">b3</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="c11">a2</font>,<font color="Maroon" title="c12">a3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">b1</font>,<font color="Maroon" title="c5">b3</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">a1</font>,<font color="Maroon" title="c12">a3</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" title="ALTCAT_2:th.32">ALTCAT_2:32</a></i>;<br/>
<a NAME="E7:57_1_1"/><b>then </b>
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c10">a1</font>,<font color="Maroon" title="c11">a2</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>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c11">a2</font>,<font color="Maroon" title="c12">a3</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>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c10">a1</font>,<font color="Maroon" title="c12">a3</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="yellow21.html#E1:57_1_1"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E8:57_1_1"/><b>then </b>
<font color="Maroon" title="c14">g2</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c13">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g'</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f'</font>
 
<b>by </b><i><a class="txt" href="yellow21.html#E2:57_1_1"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="altcat_1.html#D14" title="ALTCAT_1:def.14">ALTCAT_1:def 14</a></i>;<br/>
<b>hence </b><a NAME="E9:57_1_1"/>
<font color="Maroon" title="c7">f2</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f'</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <font color="Maroon" title="c9">g'</font>
 <b>by </b><i><a class="txt" href="yellow21.html#E1:57_1_1"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="altcat_2.html#T33" title="ALTCAT_2:th.33">ALTCAT_2:33</a></i>;<br/>


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