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

<b>set </b><font color="Maroon">c</font> =  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font>;<br/>
<b>let </b><font color="Maroon">A1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V6">strict</a>  <a href="altcat_1.html#L2">AltCatStr</a> , <font color="Maroon">A2</font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V6">strict</a>  <a href="altcat_1.html#L2">AltCatStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:1_1_2"/><i><font color="Green">E66</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a>  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="msalimit.html#M3">Element</a> of  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> holds  the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A1</font> <a href="binop_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a>  <a href="msalimit.html#K6">MSS_morph</a> <font color="Olive">i</font>,<font color="Olive">j</font> ) &amp; ( for <font color="Olive">i</font>, <font color="Olive">j</font>, <font color="Olive">k</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A1</font>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> holds <br/>for <font color="Olive">f1</font>, <font color="Olive">f2</font>, <font color="Olive">g1</font>, <font color="Olive">g2</font> being  <a href="funct_1.html#NM1">Function</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">f1</font>,<font color="Olive">f2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A1</font> <a href="binop_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Olive">j</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g1</font>,<font color="Olive">g2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A1</font> <a href="binop_1.html#K1">.</a> <font color="Olive">j</font>,<font color="Olive">k</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">A1</font> <a href="altcat_1.html#K4">.</a> <font color="Olive">i</font>,<font color="Olive">j</font>,<font color="Olive">k</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g1</font>,<font color="Olive">g2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">f1</font>,<font color="Olive">f2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">g1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">f2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> ) )
 <b>and </b><br/><a NAME="E2:1_1_2"/><i><font color="Green">E67</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a>  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="msalimit.html#M3">Element</a> of  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> holds  the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A2</font> <a href="binop_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a>  <a href="msalimit.html#K6">MSS_morph</a> <font color="Olive">i</font>,<font color="Olive">j</font> ) &amp; ( for <font color="Olive">i</font>, <font color="Olive">j</font>, <font color="Olive">k</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A2</font>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> holds <br/>for <font color="Olive">f1</font>, <font color="Olive">f2</font>, <font color="Olive">g1</font>, <font color="Olive">g2</font> being  <a href="funct_1.html#NM1">Function</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">f1</font>,<font color="Olive">f2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A2</font> <a href="binop_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Olive">j</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g1</font>,<font color="Olive">g2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A2</font> <a href="binop_1.html#K1">.</a> <font color="Olive">j</font>,<font color="Olive">k</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">A2</font> <a href="altcat_1.html#K4">.</a> <font color="Olive">i</font>,<font color="Olive">j</font>,<font color="Olive">k</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g1</font>,<font color="Olive">g2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">f1</font>,<font color="Olive">f2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">g1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">f2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> ) )
 ;<br/>

<b>reconsider </b><font color="Maroon">AA1</font> = the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A1</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">AA2</font> = the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A2</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i/>;<br/>
<div><i><font color="Green">E70</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="msalimit.html#M3">Element</a> of  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font>, <font color="Maroon">j</font> be    <a href="msalimit.html#M3">Element</a> of  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font>;<br/><b>thus </b><font color="Maroon">AA1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> = 
 <a href="msalimit.html#K6">MSS_morph</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">AA2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<a NAME="E6:1_1_2"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">AA1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">AA2</font>
 
<b>by </b><i><a class="ref" href="altcat_1.html#T9">ALTCAT_1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">CC1</font> = the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">A1</font>, <font color="Maroon">CC2</font> = the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">A2</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K3">:]</a></span> <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">j</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">k</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:1_1_2_2"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> )
 ;<br/><b>then reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font>, <font color="Maroon">j'</font> = <font color="Maroon">j</font>, <font color="Maroon">k'</font> = <font color="Maroon">k</font> as    <a href="msalimit.html#M3">Element</a> of  <a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font> ;<br/><b>reconsider </b><font color="Maroon">I</font> = <font color="Maroon">i</font>, <font color="Maroon">J</font> = <font color="Maroon">j</font>, <font color="Maroon">K</font> = <font color="Maroon">k</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A1</font> <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">I'</font> = <font color="Maroon">i</font>, <font color="Maroon">J'</font> = <font color="Maroon">j</font>, <font color="Maroon">K'</font> = <font color="Maroon">k</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A2</font> <b>by </b><i>, </i>;<br/><b>set </b><font color="Maroon">ijk</font> = <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="mcart_1.html#K3">]</a></span>;<br/><a NAME="E5:1_1_2_2"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">CC1</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CC1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="mcart_1.html#K3">]</a></span> &amp; <font color="Maroon">CC2</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CC2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="mcart_1.html#K3">]</a></span> )
 <b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>;<br/><a NAME="E6:1_1_2_2"/><i><font color="Green">E82</font></i>: 
<span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="msalimit.html#K5">MSS_set</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K3">:]</a></span>
 <b>by </b><i>, <a class="ref" href="mcart_1.html#T73">MCART_1:73</a></i>;<br/><b>thus </b><a NAME="E7:1_1_2_2"/>
<font color="Maroon">CC1</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CC2</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">Ci</font> = <font color="Maroon">CC1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="mcart_1.html#K3">]</a></span> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">AA1</font>,<font color="Maroon">AA1</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="mcart_1.html#K3">]</a></span>,<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">AA1</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="mcart_1.html#K3">]</a></span> <b>by </b><i>, , <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Cj</font> = <font color="Maroon">CC2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="mcart_1.html#K3">]</a></span> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">AA2</font>,<font color="Maroon">AA2</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="mcart_1.html#K3">]</a></span>,<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">AA2</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="mcart_1.html#K3">]</a></span> <b>by </b><i>, , <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_2_2_1_1"/>
( <span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">AA1</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">AA1</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_2_2_1_1_1"/><i><font color="Green">E85</font></i>: 
<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">AA1</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:1_1_2_2_1_1_1"/><b>then </b>
<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">AA2</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="altcat_1.html#T9">ALTCAT_1:9</a></i>;<br/><a NAME="E3:1_1_2_2_1_1_1"/><b>then </b><i><font color="Green">E86</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Ci</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">AA1</font>,<font color="Maroon">AA1</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="mcart_1.html#K3">]</a></span> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Cj</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">AA2</font>,<font color="Maroon">AA2</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="mcart_1.html#K3">]</a></span> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E4:1_1_2_2_1_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">AA1</font>,<font color="Maroon">AA1</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="mcart_1.html#K3">]</a></span> holds <br/><font color="Maroon">Ci</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Cj</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:1_1_2_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">AA1</font>,<font color="Maroon">AA1</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="mcart_1.html#K3">]</a></span>
 ;<br/>

<a NAME="E2:1_1_2_2_1_1_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">AA1</font>,<font color="Maroon">AA1</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><font color="Maroon">AA2</font>,<font color="Maroon">AA2</font></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font> )
 
<b>by </b><i>, <a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>;<br/>
<a NAME="E3:1_1_2_2_1_1_1_1"/><b>then </b><i><font color="Green">E87</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">AA1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">AA1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">AA2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">AA2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i>, <a class="ref" href="altcat_1.html#D6">ALTCAT_1:def 6</a></i>;<br/>
<a NAME="E4:1_1_2_2_1_1_1_1"/><b>then </b><i><font color="Green">E88</font></i>: 
( <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">AA1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">AA1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">AA2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">AA2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<a NAME="E5:1_1_2_2_1_1_1_1"/><b>then </b><i><font color="Green">E89</font></i>: 
( <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a>  <a href="msalimit.html#K6">MSS_morph</a> <font color="Maroon">j'</font>,<font color="Maroon">k'</font> &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a>  <a href="msalimit.html#K6">MSS_morph</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font> )
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">g1</font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:1_1_2_2_1_1_1_1"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g1</font>,<font color="Maroon">g2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">g1</font>,<font color="Maroon">g2</font> <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">j'</font>,<font color="Maroon">k'</font> )
 <b>by </b><i><a class="ref" href="msalimit.html#D10">MSALIMIT:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon">f1</font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">f2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:1_1_2_2_1_1_1_1"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">f1</font>,<font color="Maroon">f2</font> <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font> )
 <b>by </b><i>, <a class="ref" href="msalimit.html#D10">MSALIMIT:def 10</a></i>;<br/>
<a NAME="E10:1_1_2_2_1_1_1_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g1</font>,<font color="Maroon">g2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, , , <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<font color="Maroon">Ci</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">A1</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">I</font>,<font color="Maroon">J</font>,<font color="Maroon">K</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g1</font>,<font color="Maroon">g2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a>, </i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">A2</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">I'</font>,<font color="Maroon">J'</font>,<font color="Maroon">K'</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g1</font>,<font color="Maroon">g2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">f1</font>,<font color="Maroon">f2</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , , </i>
<br/>.= 
<font color="Maroon">Cj</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E12:1_1_2_2_1_1_1_1"/>
<font color="Maroon">Ci</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Cj</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:1_1_2_2_1_1_1"/>
<font color="Maroon">CC1</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CC2</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_2_2_1_1_2"/>
<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">AA1</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="mcart_1.html#K3">]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:1_1_2_2_1_1_2"/><b>then </b>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Ci</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Cj</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><a NAME="E3:1_1_2_2_1_1_2"/><b>then </b>
( <font color="Maroon">Ci</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Cj</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/><b>hence </b><a NAME="E4:1_1_2_2_1_1_2"/>
<font color="Maroon">CC1</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CC2</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:1_1_2"/>
<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font>
 <b>by </b><i>, , , <a class="ref" href="altcat_1.html#T10">ALTCAT_1:10</a></i>;<br/>


</div>
