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

<b>set </b><font color="Maroon" title="c2">G</font> =  <a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font>;<br/>
<b>thus </b><a NAME="E1:2_1_1"/>
 <a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font> is <a href="altcat_1.html#V2" title="ALTCAT_1:attr.2">transitive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">o1</font>, <font color="Maroon" title="c4">o2</font>, <font color="Maroon" title="c5">o3</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D4" title="ALTCAT_1:def.4">ALTCAT_1:def 4</a><br/>

<b>reconsider </b><font color="Maroon" title="c6">o1'</font> = <font color="Maroon" title="c3">o1</font>, <font color="Maroon" title="c7">o2'</font> = <font color="Maroon" title="c4">o2</font>, <font color="Maroon" title="c8">o3'</font> = <font color="Maroon" title="c5">o3</font> as    <a href="msalimit.html#M3" title="MSALIMIT:mode.3">Element</a> of  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="msinst_1.html#D1" target="_self" title="MSINST_1:def.1">Def1</a></i>;<br/>
<b>assume </b><a NAME="E2:2_1_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">o1</font>,<font color="Maroon" title="c4">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>  &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">o2</font>,<font color="Maroon" title="c5">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>  )
 ;<br/>

<a NAME="E3:2_1_1_1"/><b>then </b><i><font color="Green" title="E2">A1</font></i>: 
(  <a href="msalimit.html#K6" title="MSALIMIT:func.6">MSS_morph</a> <font color="Maroon" title="c6">o1'</font>,<font color="Maroon" title="c7">o2'</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="msalimit.html#K6" title="MSALIMIT:func.6">MSS_morph</a> <font color="Maroon" title="c7">o2'</font>,<font color="Maroon" title="c8">o3'</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="msinst_1.html#D1" target="_self" title="MSINST_1:def.1">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msalimit.html#K6" title="MSALIMIT:func.6">MSS_morph</a> <font color="Maroon" title="c6">o1'</font>,<font color="Maroon" title="c7">o2'</font>;<br/>
<b>consider </b><font color="Maroon" title="c10">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msalimit.html#K6" title="MSALIMIT:func.6">MSS_morph</a> <font color="Maroon" title="c7">o2'</font>,<font color="Maroon" title="c8">o3'</font>;<br/>
<b>consider </b><font color="Maroon" title="c11">f</font>, <font color="Maroon" title="c12">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E7:2_1_1_1"/><i><font color="Green" title="E3">A2</font></i>: 
( <font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c12">g</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c12">g</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c6">o1'</font>,<font color="Maroon" title="c7">o2'</font> )
 <b>by </b><i><a class="txt" href="msinst_1.html#E3:2_1_1_1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="msalimit.html#D10" title="MSALIMIT:def.10">MSALIMIT:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">u</font>, <font color="Maroon" title="c14">w</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E9:2_1_1_1"/><i><font color="Green" title="E4">A3</font></i>: 
( <font color="Maroon" title="c10">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">u</font>,<font color="Maroon" title="c14">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c13">u</font>,<font color="Maroon" title="c14">w</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c7">o2'</font>,<font color="Maroon" title="c8">o3'</font> )
 <b>by </b><i><a class="txt" href="msinst_1.html#E3:2_1_1_1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="msalimit.html#D10" title="MSALIMIT:def.10">MSALIMIT:def 10</a></i>;<br/>
<a NAME="E10:2_1_1_1"/>
<font color="Maroon" title="c13">u</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c14">w</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">g</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c6">o1'</font>,<font color="Maroon" title="c8">o3'</font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E7:2_1_1_1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="msinst_1.html#E9:2_1_1_1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="pua2mss1.html#T30" title="PUA2MSS1:th.30">PUA2MSS1:30</a></i>;<br/>
<a NAME="E11:2_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">u</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">w</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">g</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K6" title="MSALIMIT:func.6">MSS_morph</a> <font color="Maroon" title="c6">o1'</font>,<font color="Maroon" title="c8">o3'</font>
 
<b>by </b><i><a class="ref" href="msalimit.html#D10" title="MSALIMIT:def.10">MSALIMIT:def 10</a></i>;<br/>
<b>hence </b><a NAME="E12:2_1_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">o1</font>,<font color="Maroon" title="c5">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="ref" href="msinst_1.html#D1" target="_self" title="MSINST_1:def.1">Def1</a></i>;<br/>


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

<b>set </b><font color="Maroon" title="c3">M</font> =  <a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font>;<br/>
<b>set </b><font color="Maroon" title="c4">G</font> = the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c5">C</font> = the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span>;<br/>
<b>thus </b><a NAME="E2:2_1_1"/>
the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> is <a href="altcat_1.html#V3" title="ALTCAT_1:attr.3">associative</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D17" title="ALTCAT_1:def.17">ALTCAT_1:def 17</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c7">j</font>, <font color="Maroon" title="c8">k</font>, <font color="Maroon" title="c9">l</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D7" title="ALTCAT_1:def.7">ALTCAT_1:def 7</a><br/>

<b>reconsider </b><font color="Maroon" title="c10">i'</font> = <font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c11">j'</font> = <font color="Maroon" title="c7">j</font>, <font color="Maroon" title="c12">k'</font> = <font color="Maroon" title="c8">k</font>, <font color="Maroon" title="c13">l'</font> = <font color="Maroon" title="c9">l</font> as    <a href="msalimit.html#M3" title="MSALIMIT:mode.3">Element</a> of  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="msinst_1.html#D1" target="_self" title="MSINST_1:def.1">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">I</font> = <font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c15">J</font> = <font color="Maroon" title="c7">j</font>, <font color="Maroon" title="c16">K</font> = <font color="Maroon" title="c8">k</font>, <font color="Maroon" title="c17">L</font> = <font color="Maroon" title="c9">l</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> ;<br/>
<a NAME="E3:2_1_1_2"/>
( <font color="Maroon" title="c14">I</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 <span class="p1">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> &amp; <font color="Maroon" title="c15">J</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 <span class="p1">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> &amp; <font color="Maroon" title="c16">K</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 <span class="p1">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> &amp; <font color="Maroon" title="c17">L</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 <span class="p1">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> )
 
;<br/>
<a NAME="E4:2_1_1_2"/><b>then </b><i><font color="Green" title="E2">A4</font></i>: 
( <font color="Maroon" title="c14">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c15">J</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c16">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c17">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="ref" href="msinst_1.html#D1" target="_self" title="MSINST_1:def.1">Def1</a></i>;<br/>
<b>let </b><font color="Maroon" title="c18">f</font>, <font color="Maroon" title="c19">g</font>, <font color="Maroon" title="c20">h</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E5:2_1_1_2"/><i><font color="Green" title="E3">A5</font></i>: 
( <font color="Maroon" title="c18">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c19">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c20">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">l</font> )
 ;<br/>

<a NAME="E6:2_1_1_2"/><b>then </b><i><font color="Green" title="E4">A6</font></i>: 
( <font color="Maroon" title="c18">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K6" title="MSALIMIT:func.6">MSS_morph</a> <font color="Maroon" title="c10">i'</font>,<font color="Maroon" title="c11">j'</font> &amp; <font color="Maroon" title="c19">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K6" title="MSALIMIT:func.6">MSS_morph</a> <font color="Maroon" title="c11">j'</font>,<font color="Maroon" title="c12">k'</font> &amp; <font color="Maroon" title="c20">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K6" title="MSALIMIT:func.6">MSS_morph</a> <font color="Maroon" title="c12">k'</font>,<font color="Maroon" title="c13">l'</font> )
 
<b>by </b><i><a class="ref" href="msinst_1.html#D1" target="_self" title="MSINST_1:def.1">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">f1</font>, <font color="Maroon" title="c22">f2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E8:2_1_1_2"/><i><font color="Green" title="E5">A7</font></i>: 
( <font color="Maroon" title="c18">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">f1</font>,<font color="Maroon" title="c22">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c21">f1</font>,<font color="Maroon" title="c22">f2</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c10">i'</font>,<font color="Maroon" title="c11">j'</font> )
 <b>by </b><i><a class="ref" href="msalimit.html#D10" title="MSALIMIT:def.10">MSALIMIT:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c23">g1</font>, <font color="Maroon" title="c24">g2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E10:2_1_1_2"/><i><font color="Green" title="E6">A8</font></i>: 
( <font color="Maroon" title="c19">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">g1</font>,<font color="Maroon" title="c24">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c23">g1</font>,<font color="Maroon" title="c24">g2</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c11">j'</font>,<font color="Maroon" title="c12">k'</font> )
 <b>by </b><i><a class="txt" href="msinst_1.html#E6:2_1_1_2"><i><font color="Green" title="E4">A6</font></i></a>, <a class="ref" href="msalimit.html#D10" title="MSALIMIT:def.10">MSALIMIT:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c25">h1</font>, <font color="Maroon" title="c26">h2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E12:2_1_1_2"/><i><font color="Green" title="E7">A9</font></i>: 
( <font color="Maroon" title="c20">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">h1</font>,<font color="Maroon" title="c26">h2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c25">h1</font>,<font color="Maroon" title="c26">h2</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c12">k'</font>,<font color="Maroon" title="c13">l'</font> )
 <b>by </b><i><a class="txt" href="msinst_1.html#E6:2_1_1_2"><i><font color="Green" title="E4">A6</font></i></a>, <a class="ref" href="msalimit.html#D10" title="MSALIMIT:def.10">MSALIMIT:def 10</a></i>;<br/>
<a NAME="E13:2_1_1_2"/><i><font color="Green" title="E8">A10</font></i>: 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c19">g</font>,<font color="Maroon" title="c18">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">g1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c21">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c22">f2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E5:2_1_1_2"><i><font color="Green" title="E3">A5</font></i></a>, <a class="txt" href="msinst_1.html#E8:2_1_1_2"><i><font color="Green" title="E5">A7</font></i></a>, <a class="txt" href="msinst_1.html#E10:2_1_1_2"><i><font color="Green" title="E6">A8</font></i></a>, <a class="ref" href="msinst_1.html#D1" target="_self" title="MSINST_1:def.1">Def1</a></i>;<br/>
<a NAME="E14:2_1_1_2"/>
<font color="Maroon" title="c23">g1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c21">f1</font>,<font color="Maroon" title="c24">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c22">f2</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c10">i'</font>,<font color="Maroon" title="c12">k'</font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E8:2_1_1_2"><i><font color="Green" title="E5">A7</font></i></a>, <a class="txt" href="msinst_1.html#E10:2_1_1_2"><i><font color="Green" title="E6">A8</font></i></a>, <a class="ref" href="pua2mss1.html#T30" title="PUA2MSS1:th.30">PUA2MSS1:30</a></i>;<br/>
<a NAME="E15:2_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">g1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c21">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c22">f2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K6" title="MSALIMIT:func.6">MSS_morph</a> <font color="Maroon" title="c10">i'</font>,<font color="Maroon" title="c12">k'</font>
 
<b>by </b><i><a class="ref" href="msalimit.html#D10" title="MSALIMIT:def.10">MSALIMIT:def 10</a></i>;<br/>
<a NAME="E16:2_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">g1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c21">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c22">f2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c8">k</font>
 
<b>by </b><i><a class="ref" href="msinst_1.html#D1" target="_self" title="MSINST_1:def.1">Def1</a></i>;<br/>
<a NAME="E17:2_1_1_2"/><b>then </b><i><font color="Green" title="E9">A11</font></i>: 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c20">h</font>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">g1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c21">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c22">f2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">h1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">g1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c21">f1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">h2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c22">f2</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E4:2_1_1_2"><i><font color="Green" title="E2">A4</font></i></a>, <a class="txt" href="msinst_1.html#E5:2_1_1_2"><i><font color="Green" title="E3">A5</font></i></a>, <a class="txt" href="msinst_1.html#E12:2_1_1_2"><i><font color="Green" title="E7">A9</font></i></a>, <a class="ref" href="msinst_1.html#D1" target="_self" title="MSINST_1:def.1">Def1</a></i>;<br/>
<a NAME="E18:2_1_1_2"/><i><font color="Green" title="E10">A12</font></i>: 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c20">h</font>,<font color="Maroon" title="c19">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">h1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c23">g1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">h2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c24">g2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E5:2_1_1_2"><i><font color="Green" title="E3">A5</font></i></a>, <a class="txt" href="msinst_1.html#E10:2_1_1_2"><i><font color="Green" title="E6">A8</font></i></a>, <a class="txt" href="msinst_1.html#E12:2_1_1_2"><i><font color="Green" title="E7">A9</font></i></a>, <a class="ref" href="msinst_1.html#D1" target="_self" title="MSINST_1:def.1">Def1</a></i>;<br/>
<a NAME="E19:2_1_1_2"/>
<font color="Maroon" title="c25">h1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c23">g1</font>,<font color="Maroon" title="c26">h2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c24">g2</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c11">j'</font>,<font color="Maroon" title="c13">l'</font>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E10:2_1_1_2"><i><font color="Green" title="E6">A8</font></i></a>, <a class="txt" href="msinst_1.html#E12:2_1_1_2"><i><font color="Green" title="E7">A9</font></i></a>, <a class="ref" href="pua2mss1.html#T30" title="PUA2MSS1:th.30">PUA2MSS1:30</a></i>;<br/>
<a NAME="E20:2_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">h1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c23">g1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">h2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c24">g2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K6" title="MSALIMIT:func.6">MSS_morph</a> <font color="Maroon" title="c11">j'</font>,<font color="Maroon" title="c13">l'</font>
 
<b>by </b><i><a class="ref" href="msalimit.html#D10" title="MSALIMIT:def.10">MSALIMIT:def 10</a></i>;<br/>
<a NAME="E21:2_1_1_2"/><b>then </b><i><font color="Green" title="E11">A13</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">h1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c23">g1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">h2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c24">g2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c9">l</font>
 
<b>by </b><i><a class="ref" href="msinst_1.html#D1" target="_self" title="MSINST_1:def.1">Def1</a></i>;<br/>
<a NAME="E22:2_1_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c25">h1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c23">g1</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c21">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">h1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">g1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c21">f1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c26">h2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c24">g2</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c22">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">h2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">g2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c22">f2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>;<br/>
<b>hence </b><a NAME="E23:2_1_1_2"/>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c20">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c19">g</font>,<font color="Maroon" title="c18">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c9">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c9">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c20">h</font>,<font color="Maroon" title="c19">g</font></span>)</span>,<font color="Maroon" title="c18">f</font>
 <b>by </b><i><a class="txt" href="msinst_1.html#E4:2_1_1_2"><i><font color="Green" title="E2">A4</font></i></a>, <a class="txt" href="msinst_1.html#E5:2_1_1_2"><i><font color="Green" title="E3">A5</font></i></a>, <a class="txt" href="msinst_1.html#E8:2_1_1_2"><i><font color="Green" title="E5">A7</font></i></a>, <a class="txt" href="msinst_1.html#E13:2_1_1_2"><i><font color="Green" title="E8">A10</font></i></a>, <a class="txt" href="msinst_1.html#E17:2_1_1_2"><i><font color="Green" title="E9">A11</font></i></a>, <a class="txt" href="msinst_1.html#E18:2_1_1_2"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="msinst_1.html#E21:2_1_1_2"><i><font color="Green" title="E11">A13</font></i></a>, <a class="ref" href="msinst_1.html#D1" target="_self" title="MSINST_1:def.1">Def1</a></i>;<br/>


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

<b>thus </b><a NAME="E3:2_1_1"/>
the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> is <a href="altcat_1.html#V5" title="ALTCAT_1:attr.5">with_left_units</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D18" title="ALTCAT_1:def.18">ALTCAT_1:def 18</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D9" title="ALTCAT_1:def.9">ALTCAT_1:def 9</a><br/>

<b>reconsider </b><font color="Maroon" title="c7">j'</font> = <font color="Maroon" title="c6">j</font> as    <a href="msalimit.html#M3" title="MSALIMIT:mode.3">Element</a> of  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="msinst_1.html#D1" target="_self" title="MSINST_1:def.1">Def1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">e1</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">j'</font>;<br/>
<b>set </b><font color="Maroon" title="c9">e2</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c7">j'</font>;<br/>
<a NAME="E2:2_1_1_3"/><i><font color="Green" title="E2">A14</font></i>: 
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">j'</font>, <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c7">j'</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c7">j'</font>,<font color="Maroon" title="c7">j'</font>
 
<b>by </b><i><a class="ref" href="pua2mss1.html#T29" title="PUA2MSS1:th.29">PUA2MSS1:29</a></i>;<br/>
<b>take </b><font color="Maroon" title="c10">e</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">j'</font></span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c7">j'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>

<a NAME="E3:2_1_1_3"/>
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msalimit.html#K6" title="MSALIMIT:func.6">MSS_morph</a> <font color="Maroon" title="c7">j'</font>,<font color="Maroon" title="c7">j'</font>
 
<b>by </b><i><a class="ref" href="msinst_1.html#D1" target="_self" title="MSINST_1:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E4:2_1_1_3"/><i><font color="Green" title="E3">A15</font></i>: 
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c6">j</font>
 <b>by </b><i><a class="txt" href="msinst_1.html#E2:2_1_1_3"><i><font color="Green" title="E2">A14</font></i></a>, <a class="ref" href="msalimit.html#D10" title="MSALIMIT:def.10">MSALIMIT:def 10</a></i>;<br/>

<b>let </b><font color="Maroon" title="c11">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c12">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>reconsider </b><font color="Maroon" title="c13">i'</font> = <font color="Maroon" title="c11">i</font> as    <a href="msalimit.html#M3" title="MSALIMIT:mode.3">Element</a> of  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="msinst_1.html#D1" target="_self" title="MSINST_1:def.1">Def1</a></i>;<br/>
<b>assume </b><a NAME="E6:2_1_1_3"/><i><font color="Green" title="E4">A16</font></i>: 
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c6">j</font>
 ;<br/>

<a NAME="E7:2_1_1_3"/><b>then </b>
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K6" title="MSALIMIT:func.6">MSS_morph</a> <font color="Maroon" title="c13">i'</font>,<font color="Maroon" title="c7">j'</font>
 
<b>by </b><i><a class="ref" href="msinst_1.html#D1" target="_self" title="MSINST_1:def.1">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">f1</font>, <font color="Maroon" title="c15">f2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E9:2_1_1_3"/><i><font color="Green" title="E5">A17</font></i>: 
( <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">f1</font>,<font color="Maroon" title="c15">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c14">f1</font>,<font color="Maroon" title="c15">f2</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c13">i'</font>,<font color="Maroon" title="c7">j'</font> )
 <b>by </b><i><a class="ref" href="msalimit.html#D10" title="MSALIMIT:def.10">MSALIMIT:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">I</font> = <font color="Maroon" title="c11">i</font>, <font color="Maroon" title="c17">J</font> = <font color="Maroon" title="c6">j</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> ;<br/>
<a NAME="E11:2_1_1_3"/><i><font color="Green" title="E6">A18</font></i>: 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font>,<font color="Maroon" title="c17">J</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">j'</font></span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c7">j'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">f1</font>,<font color="Maroon" title="c15">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">j'</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c14">f1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c7">j'</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c15">f2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E4:2_1_1_3"><i><font color="Green" title="E3">A15</font></i></a>, <a class="txt" href="msinst_1.html#E6:2_1_1_3"><i><font color="Green" title="E4">A16</font></i></a>, <a class="txt" href="msinst_1.html#E9:2_1_1_3"><i><font color="Green" title="E5">A17</font></i></a>, <a class="ref" href="msinst_1.html#D1" target="_self" title="MSINST_1:def.1">Def1</a></i>;<br/>
<a NAME="E12:2_1_1_3"/><i><font color="Green" title="E7">A19</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">f1</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="c7">j'</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">f2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c7">j'</font> )
 
<b>by </b><i><a class="txt" href="msinst_1.html#E9:2_1_1_3"><i><font color="Green" title="E5">A17</font></i></a>, <a class="ref" href="pua2mss1.html#D13" title="PUA2MSS1:def.13">PUA2MSS1:def 13</a></i>;<br/>
<a NAME="E13:2_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">j'</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c14">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f1</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T79" title="RELAT_1:th.79">RELAT_1:79</a></i>;<br/>
<b>hence </b><a NAME="E14:2_1_1_3"/>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">e</font>,<font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font>
 <b>by </b><i><a class="txt" href="msinst_1.html#E9:2_1_1_3"><i><font color="Green" title="E5">A17</font></i></a>, <a class="txt" href="msinst_1.html#E11:2_1_1_3"><i><font color="Green" title="E6">A18</font></i></a>, <a class="txt" href="msinst_1.html#E12:2_1_1_3"><i><font color="Green" title="E7">A19</font></i></a>, <a class="ref" href="relat_1.html#T79" title="RELAT_1:th.79">RELAT_1:79</a></i>;<br/>


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

<b>thus </b><a NAME="E4:2_1_1"/>
the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> is <a href="altcat_1.html#V4" title="ALTCAT_1:attr.4">with_right_units</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D8" title="ALTCAT_1:def.8">ALTCAT_1:def 8</a><br/>

<b>reconsider </b><font color="Maroon" title="c7">i'</font> = <font color="Maroon" title="c6">i</font> as    <a href="msalimit.html#M3" title="MSALIMIT:mode.3">Element</a> of  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="msinst_1.html#D1" target="_self" title="MSINST_1:def.1">Def1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">e1</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">i'</font>;<br/>
<b>set </b><font color="Maroon" title="c9">e2</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c7">i'</font>;<br/>
<a NAME="E2:2_1_1_4"/><i><font color="Green" title="E2">A20</font></i>: 
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">i'</font>, <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c7">i'</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c7">i'</font>,<font color="Maroon" title="c7">i'</font>
 
<b>by </b><i><a class="ref" href="pua2mss1.html#T29" title="PUA2MSS1:th.29">PUA2MSS1:29</a></i>;<br/>
<b>take </b><font color="Maroon" title="c10">e</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">i'</font></span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c7">i'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>

<a NAME="E3:2_1_1_4"/>
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msalimit.html#K6" title="MSALIMIT:func.6">MSS_morph</a> <font color="Maroon" title="c7">i'</font>,<font color="Maroon" title="c7">i'</font>
 
<b>by </b><i><a class="ref" href="msinst_1.html#D1" target="_self" title="MSINST_1:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E4:2_1_1_4"/><i><font color="Green" title="E3">A21</font></i>: 
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c6">i</font>
 <b>by </b><i><a class="txt" href="msinst_1.html#E2:2_1_1_4"><i><font color="Green" title="E2">A20</font></i></a>, <a class="ref" href="msalimit.html#D10" title="MSALIMIT:def.10">MSALIMIT:def 10</a></i>;<br/>

<b>let </b><font color="Maroon" title="c11">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c12">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>reconsider </b><font color="Maroon" title="c13">j'</font> = <font color="Maroon" title="c11">j</font> as    <a href="msalimit.html#M3" title="MSALIMIT:mode.3">Element</a> of  <a href="msalimit.html#K5" title="MSALIMIT:func.5">MSS_set</a> <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="msinst_1.html#D1" target="_self" title="MSINST_1:def.1">Def1</a></i>;<br/>
<b>assume </b><a NAME="E6:2_1_1_4"/><i><font color="Green" title="E4">A22</font></i>: 
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c11">j</font>
 ;<br/>

<a NAME="E7:2_1_1_4"/><b>then </b>
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msalimit.html#K6" title="MSALIMIT:func.6">MSS_morph</a> <font color="Maroon" title="c7">i'</font>,<font color="Maroon" title="c13">j'</font>
 
<b>by </b><i><a class="ref" href="msinst_1.html#D1" target="_self" title="MSINST_1:def.1">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">f1</font>, <font color="Maroon" title="c15">f2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E9:2_1_1_4"/><i><font color="Green" title="E5">A23</font></i>: 
( <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">f1</font>,<font color="Maroon" title="c15">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c14">f1</font>,<font color="Maroon" title="c15">f2</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c7">i'</font>,<font color="Maroon" title="c13">j'</font> )
 <b>by </b><i><a class="ref" href="msalimit.html#D10" title="MSALIMIT:def.10">MSALIMIT:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">I</font> = <font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c17">J</font> = <font color="Maroon" title="c11">j</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> ;<br/>
<a NAME="E11:2_1_1_4"/><i><font color="Green" title="E6">A24</font></i>: 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c16">I</font>,<font color="Maroon" title="c17">J</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">f1</font>,<font color="Maroon" title="c15">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">i'</font></span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c7">i'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">f1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">i'</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">f2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c7">i'</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="msinst_1.html#E4:2_1_1_4"><i><font color="Green" title="E3">A21</font></i></a>, <a class="txt" href="msinst_1.html#E6:2_1_1_4"><i><font color="Green" title="E4">A22</font></i></a>, <a class="txt" href="msinst_1.html#E9:2_1_1_4"><i><font color="Green" title="E5">A23</font></i></a>, <a class="ref" href="msinst_1.html#D1" target="_self" title="MSINST_1:def.1">Def1</a></i>;<br/>
<a NAME="E12:2_1_1_4"/><i><font color="Green" title="E7">A25</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">i'</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c7">i'</font> )
 
<b>by </b><i><a class="txt" href="msinst_1.html#E9:2_1_1_4"><i><font color="Green" title="E5">A23</font></i></a>, <a class="ref" href="pua2mss1.html#D13" title="PUA2MSS1:def.13">PUA2MSS1:def 13</a></i>;<br/>
<a NAME="E13:2_1_1_4"/><b>then </b>
<font color="Maroon" title="c14">f1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">i'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f1</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T78" title="RELAT_1:th.78">RELAT_1:78</a></i>;<br/>
<b>hence </b><a NAME="E14:2_1_1_4"/>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="msinst_1.html#K1" title="MSINST_1:func.1">MSSCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c11">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font>
 <b>by </b><i><a class="txt" href="msinst_1.html#E9:2_1_1_4"><i><font color="Green" title="E5">A23</font></i></a>, <a class="txt" href="msinst_1.html#E11:2_1_1_4"><i><font color="Green" title="E6">A24</font></i></a>, <a class="txt" href="msinst_1.html#E12:2_1_1_4"><i><font color="Green" title="E7">A25</font></i></a>, <a class="ref" href="relat_1.html#T78" title="RELAT_1:th.78">RELAT_1:78</a></i>;<br/>


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


</div>
