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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">j</font>,<font color="Olive">i</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K1">IFEQ</a> <font color="Olive">j</font>,<font color="Olive">i</font>,<span class="p1">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Olive">i</font>,<font color="Olive">j</font></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span></span>)</span></span>)</span> );<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">ij</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:9_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">ij</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P</font>
 ;<br/><b>then reconsider </b><font color="Maroon">i2</font> = <font color="Maroon">ij</font> <a href="mcart_1.html#K1">`1</a> , <font color="Maroon">i1</font> = <font color="Maroon">ij</font> <a href="mcart_1.html#K2">`2</a>  as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font> <b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon">i1</font> = <font color="Maroon">i1</font>, <font color="Maroon">i2</font> = <font color="Maroon">i2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font> ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">i2</font>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon">A</font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P</font><b> such that </b><br/><a NAME="E5:9_1_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">ij</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">ij</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P</font> holds <br/><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">ij</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">ij</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S4">PBOOLE:sch 4</a>();<br/></i><b>take </b><font color="Maroon">x</font> = <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ij</font>;<br/><b>take </b><font color="Maroon">i1</font> = <font color="Maroon">i1</font>;<br/><b>take </b><font color="Maroon">i2</font> = <font color="Maroon">i2</font>;<br/><b>thus </b><a NAME="E6:9_1_1_1"/>
( <font color="Maroon">ij</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">i1</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">i2</font>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:9_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">z</font>,<font color="Olive">y</font>]
 
;<br/>
<b>consider </b><font color="Maroon">A</font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P</font><b> such that </b><br/><a NAME="E4:9_1_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">ij</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">ij</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">ij</font>,<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">ij</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>();<br/></i>
<a NAME="E5:9_1_1"/>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> is   <a href="funct_1.html#NM1">Function</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:9_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:9_1_1_2"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">i1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>, <font color="Maroon">i2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font><b> such that </b><br/><a NAME="E4:9_1_1_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">i1</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">i2</font>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1_1_2_1"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> or <font color="Maroon">i1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:9_1_1_2_1_1"/>
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:9_1_1_2_1_1"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> is   <a href="funct_1.html#NM1">Function</a>
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:9_1_1_2_1_2"/>
<font color="Maroon">i1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:9_1_1_2_1_2"/>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> is   <a href="funct_1.html#NM1">Function</a>
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as   <a href="pboole.html#NM2">ManySortedFunction</a> of the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P</font> <b>by </b><i><a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>;<br/><b>let </b><font color="Maroon">j</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>;<br/><b>let </b><font color="Maroon">k</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>;<br/><b>assume </b><a NAME="E1:9_1_1_3"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> )
 ;<br/><b>consider </b><font color="Maroon">kl</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:9_1_1_3"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">kl</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j</font>,<font color="Maroon">i</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><a NAME="E4:9_1_1_3"/>
<font color="Maroon">kl</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><b>then consider </b><font color="Maroon">i1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>, <font color="Maroon">j1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font><b> such that </b><br/><a NAME="E6:9_1_1_3"/><i><font color="Green">E48</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j1</font>,<font color="Maroon">i1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">kl</font> &amp; <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">kl</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">j1</font>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E7:9_1_1_3"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E8:9_1_1_3"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">j</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E9:9_1_1_3"/>
<font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">i</font> is   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">j</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1_1_3_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:9_1_1_3_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:9_1_1_3_1_1_1"/>
<font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">i</font> is   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:9_1_1_3_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:9_1_1_3_1_1_2"/>
<font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">i</font> is   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">i</font> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">j</font></span>)</span> ;<br/><b>consider </b><font color="Maroon">lm</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:9_1_1_3"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">lm</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">k</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><a NAME="E13:9_1_1_3"/>
<font color="Maroon">lm</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><b>then consider </b><font color="Maroon">i2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>, <font color="Maroon">j2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font><b> such that </b><br/><a NAME="E15:9_1_1_3"/><i><font color="Green">E72</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j2</font>,<font color="Maroon">i2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">lm</font> &amp; <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">lm</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">j2</font>,<font color="Maroon">i2</font>,<span class="p1">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E16:9_1_1_3"/>
( <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E17:9_1_1_3"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">k</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">k</font>,<font color="Maroon">j</font>,<span class="p1">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">j</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E18:9_1_1_3"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><a NAME="E19:9_1_1_3"/>
<font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">k</font>,<font color="Maroon">j</font> is   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">k</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_3_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1_1_3_2_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> or <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_2_1_1"><b>suppose </b></a><a NAME="E1:9_1_1_3_2_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:9_1_1_3_2_1_1"/>
<font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">k</font>,<font color="Maroon">j</font> is   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_2_1_2"><b>suppose </b></a><a NAME="E1:9_1_1_3_2_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:9_1_1_3_2_1_2"/>
<font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">k</font>,<font color="Maroon">j</font> is   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">f2</font> = <font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">k</font>,<font color="Maroon">j</font> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">k</font></span>)</span> ;<br/><a NAME="E21:9_1_1_3"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>  st <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">j</font> holds <br/><font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Olive">j</font>,<font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Olive">i</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>;<br/><b>let </b><font color="Maroon">j</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>;<br/>



<b>assume </b><a NAME="E1:9_1_1_3_3"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 ;<br/>

<a NAME="E2:9_1_1_3_3"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#T24">ORDERS_2:24</a></i>;<br/>
<b>consider </b><font color="Maroon">lm</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:9_1_1_3_3"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">lm</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j</font>,<font color="Maroon">i</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>
<a NAME="E5:9_1_1_3_3"/>
<font color="Maroon">lm</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P</font>
 
<b>by </b><i>, , <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon">i2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>, <font color="Maroon">j2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font><b> such that </b><br/><a NAME="E7:9_1_1_3_3"/><i><font color="Green">E79</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j2</font>,<font color="Maroon">i2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">lm</font> &amp; <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">lm</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">j2</font>,<font color="Maroon">i2</font>,<span class="p1">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E8:9_1_1_3_3"/>
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E9:9_1_1_3_3"/>
<font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E22:9_1_1_3"/><i><font color="Green">E80</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>  st <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">j</font> holds <br/><font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Olive">j</font>,<font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Olive">i</font>,<font color="Olive">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_3_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>;<br/><b>let </b><font color="Maroon">j</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>;<br/>



<b>assume </b><a NAME="E1:9_1_1_3_4"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> )
 ;<br/>

<b>consider </b><font color="Maroon">lm</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:9_1_1_3_4"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">lm</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j</font>,<font color="Maroon">i</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>
<a NAME="E4:9_1_1_3_4"/>
<font color="Maroon">lm</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P</font>
 
<b>by </b><i>, , <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon">i2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>, <font color="Maroon">j2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font><b> such that </b><br/><a NAME="E6:9_1_1_3_4"/><i><font color="Green">E83</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j2</font>,<font color="Maroon">i2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">lm</font> &amp; <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">lm</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">j2</font>,<font color="Maroon">i2</font>,<span class="p1">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E7:9_1_1_3_4"/>
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E8:9_1_1_3_4"/><b>then </b>
<font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/>
<b>hence </b><a NAME="E9:9_1_1_3_4"/>
<font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="msualg_3.html#T3">MSUALG_3:3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E23:9_1_1_3"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">k</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">f1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_3_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1_1_3_5_1"/>
( ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> ) or ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> ) or ( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> ) or ( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_5_1_1"><b>suppose </b></a><a NAME="E1:9_1_1_3_5_1_1"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 ;<br/><div class="add"><a NAME="E2:9_1_1_3_5_1_1"/><b>then </b>
<font color="Maroon">f2</font> <a href="pboole.html#R6">=</a>  <a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><b>hence </b><a NAME="E3:9_1_1_3_5_1_1"/>
<font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">k</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">f1</font>
 <b>by </b><i>, <a class="ref" href="msualg_3.html#T3">MSUALG_3:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_5_1_2"><b>suppose </b></a><a NAME="E1:9_1_1_3_5_1_2"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> )
 ;<br/><div class="add"><a NAME="E2:9_1_1_3_5_1_2"/><b>then </b>
<font color="Maroon">f1</font> <a href="pboole.html#R6">=</a>  <a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><b>hence </b><a NAME="E3:9_1_1_3_5_1_2"/>
<font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">k</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">f1</font>
 <b>by </b><i>, <a class="ref" href="msualg_3.html#T3">MSUALG_3:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_5_1_3"><b>suppose </b></a><a NAME="E1:9_1_1_3_5_1_3"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 ;<br/><div class="add"><a NAME="E2:9_1_1_3_5_1_3"/><b>then </b>
<font color="Maroon">f2</font> <a href="pboole.html#R6">=</a>  <a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><b>hence </b><a NAME="E3:9_1_1_3_5_1_3"/>
<font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">k</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">f1</font>
 <b>by </b><i>, <a class="ref" href="msualg_3.html#T4">MSUALG_3:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_5_1_4"><b>suppose </b></a><a NAME="E1:9_1_1_3_5_1_4"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> )
 ;<br/><div class="add"><a NAME="E2:9_1_1_3_5_1_4"/><b>then </b>
( <font color="Maroon">i</font> <a href="orders_2.html#NR4" target="_self">&gt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="orders_2.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font> )
 <b>by </b><i>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/><a NAME="E3:9_1_1_3_5_1_4"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T29">ORDERS_2:29</a></i>;<br/><a NAME="E4:9_1_1_3_5_1_4"/>
<font color="Maroon">f1</font> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><a NAME="E5:9_1_1_3_5_1_4"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">f1</font> <a href="pboole.html#R6">=</a>  <a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="msualg_3.html#T3">MSUALG_3:3</a></i>;<br/><a NAME="E6:9_1_1_3_5_1_4"/>
<font color="Maroon">f2</font> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="cqc_lang.html#D1">CQC_LANG:def 1</a></i>;<br/><a NAME="E7:9_1_1_3_5_1_4"/><b>then </b>
<font color="Maroon">f2</font> <a href="pboole.html#R6">=</a>  <a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="msualg_3.html#T3">MSUALG_3:3</a></i>;<br/><a NAME="E8:9_1_1_3_5_1_4"/><b>then </b>
<font color="Maroon">f2</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">f1</font> <a href="pboole.html#R6">=</a>  <a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i</font>,<font color="Maroon">k</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E9:9_1_1_3_5_1_4"/>
<font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">k</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">f1</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E24:9_1_1_3"/>
<font color="Maroon">f1</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font>,<font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_3_6"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1_1_3_6_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_6_1_1"><b>suppose </b></a><a NAME="E1:9_1_1_3_6_1_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:9_1_1_3_6_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:9_1_1_3_6_1_1"/>
<font color="Maroon">f1</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font>,<font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="msualg_3.html#T9">MSUALG_3:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_3_6_1_2"><b>suppose </b></a><a NAME="E1:9_1_1_3_6_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:9_1_1_3_6_1_2"/><b>then </b>
<font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:9_1_1_3_6_1_2"/>
<font color="Maroon">f1</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font>,<font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E25:9_1_1_3"/>
 ex <font color="Olive">f1</font> being  <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">j</font></span>)</span> ex <font color="Olive">f2</font> being  <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">k</font></span>)</span> st <br/>( <font color="Olive">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">i</font> &amp; <font color="Olive">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">k</font>,<font color="Maroon">j</font> &amp; <font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">k</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">f2</font> <a href="msualg_3.html#K3">**</a> <font color="Olive">f1</font> &amp; <font color="Olive">f1</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font>,<font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">j</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as    <a href="msalimit.html#M2" target="_self">Binding</a> of <font color="Maroon">OAF</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">A</font>
;<br/>

<b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>;<br/><b>let </b><font color="Maroon">j</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>;<br/>



<b>assume </b><a NAME="E9:9_1_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font>
 ;<br/>

<b>consider </b><font color="Maroon">kl</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:9_1_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">kl</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j</font>,<font color="Maroon">i</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>
<a NAME="E12:9_1_1"/>
<font color="Maroon">kl</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P</font>
 
<b>by </b><i>, , <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon">i1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font>, <font color="Maroon">j1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P</font><b> such that </b><br/><a NAME="E14:9_1_1"/><i><font color="Green">E94</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j1</font>,<font color="Maroon">i1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">kl</font> &amp; <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">kl</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">j1</font>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E15:9_1_1"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E16:9_1_1"/>
<font color="Maroon">A</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K1">IFEQ</a> <font color="Maroon">j</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" target="_self">bind</a> <font color="Maroon">B</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p3">(<span class="default"><font color="Maroon">OAF</font> <a href="pralg_2.html#K6">.</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>


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