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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="yellow_1.html#V1">RelStr-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:61"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:61"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<a NAME="E3:61"/><i><font color="Green">E46</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="dickson.html#E1:61"><i><font color="Green">E44</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="xboole_0.html#V1">empty</a> or  not the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1"><b>suppose </b></a><a NAME="E1:61_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:61_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E3:61_1_1"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_3.html#V4">non-Empty</a>
 <b>by </b><i><a class="ref" href="dickson.html#T41" target="_self">Th41</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E5:61_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E6:61_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i><a class="ref" href="waybel_3.html#D7">WAYBEL_3:def 7</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:61_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E9:61_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E5:61_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:61_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="dickson.html#E8:61_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E11:61_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="afinsq_1.html#T4">AFINSQ_1:4</a></i>;<br/><a NAME="E12:61_1_1"/><i><font color="Green">E53</font></i>: 
<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span> is <a href="struct_0.html#V3">empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_1_1_1_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 </span><b>by </b><i><a class="txt" href="dickson.html#E11:61_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_1_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:61_1_1_1_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:61"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><a NAME="E3:61_1_1_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E1:61"><i><font color="Green">E44</font></i></a>, <a class="txt" href="dickson.html#E1:61_1_1_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E4:61_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E2:61_1_1_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:61_1_1_1_1_1"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font> is <a href="waybel_3.html#V4">non-Empty</a>
 <b>by </b><i><a class="txt" href="dickson.html#E6:61_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="dickson.html#E9:61_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="waybel_3.html#D7">WAYBEL_3:def 7</a></i>;<br/><a NAME="E6:61_1_1_1_1_1"/><b>then </b>
 <a href="yellow_1.html#K5">product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i><a class="txt" href="dickson.html#E1:61"><i><font color="Green">E44</font></i></a>, <a class="ref" href="dickson.html#T41" target="_self">Th41</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> as  <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E8:61_1_1_1_1_1"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><a NAME="E9:61_1_1_1_1_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/><b>hence </b><a NAME="E10:61_1_1_1_1_1"/>
<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_1_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:61_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i><a class="txt" href="dickson.html#E2:61"><i><font color="Green">E45</font></i></a>, <a class="txt" href="dickson.html#E6:61_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="dickson.html#E9:61_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> as  <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> as    <a href="hidden.html#M1">set</a>  ;<br/><a NAME="E5:61_1_1_1_1_2"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><a NAME="E6:61_1_1_1_1_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/><b>hence </b><a NAME="E7:61_1_1_1_1_2"/>
<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>11</sub></font> being  <a href="xboole_0.html#V1">empty</a> <a href="funct_1.html#NM1">Function</a>;<br/><a NAME="E14:61_1_1"/><i><font color="Green">E54</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span>
 <b>by </b><i><a class="txt" href="dickson.html#E12:61_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E15:61_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="dickson.html#E14:61_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/><a NAME="E18:61_1_1"/>
<font color="Maroon">c<sub>13</sub></font> is <a href="waybel_0.html#V23">isomorphic</a>
 <b>by </b><i><a class="txt" href="dickson.html#E2:61_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="dickson.html#E12:61_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/><b>hence </b><a NAME="E19:61_1_1"/>
<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span>, <a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="waybel_1.html#D8">WAYBEL_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2"><b>suppose </b></a><a NAME="E1:61_1_2"/><i><font color="Green">E47</font></i>: 
not the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font> as  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i><a class="txt" href="dickson.html#E1:61_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E4:61_1_2"/><i><font color="Green">E48</font></i>: 
not  <a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i><a class="txt" href="dickson.html#E1:61_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E5:61_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel_3.html#V4">non-Empty</a>
 <b>by </b><i><a class="ref" href="dickson.html#T41" target="_self">Th41</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  <a href="yellow_1.html#V1">RelStr-yielding</a> <a href="waybel_3.html#V4">non-Empty</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <b>by </b><i><a class="txt" href="dickson.html#E4:61_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="dickson.html#T41" target="_self">Th41</a></i>;<br/><a NAME="E7:61_1_2"/>
not <font color="Maroon">c<sub>11</sub></font> <a href="waybel_3.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="struct_0.html#L1">1-sorted</a> ;<br/><b>assume </b><a NAME="E1:61_1_2_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:61_1_2_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:61_1_2_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:61_1_2_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:61_1_2_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:61_1_2_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E6:61_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E7:61_1_2_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E8:61_1_2_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:61_1_2_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="dickson.html#E4:61_1_2_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><a NAME="E9:61_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E7:61_1_2_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E10:61_1_2_1"/>
not <font color="Maroon">c<sub>13</sub></font> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i><a class="txt" href="dickson.html#E5:61_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="waybel_3.html#D7">WAYBEL_3:def 7</a></i>;<br/></div><b>end;</b></div><a NAME="E10:61_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font> is <a href="waybel_3.html#V4">non-Empty</a>
 <b>by </b><i><a class="ref" href="waybel_3.html#D7">WAYBEL_3:def 7</a></i>;<br/><a NAME="E11:61_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
not  <a href="yellow_1.html#K5">product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i><a class="txt" href="dickson.html#E1:61"><i><font color="Green">E44</font></i></a>, <a class="ref" href="dickson.html#T41" target="_self">Th41</a></i>;<br/><a NAME="E12:61_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
not the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="dickson.html#E11:61_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E14:61_1_2"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E15:61_1_2"/><b>then </b>
not the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span> as  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E17:61_1_2"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><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> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>12</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> &amp; <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">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Olive">b<sub>2</sub></font></span>)</span> );<br/><a NAME="E19:61_1_2"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>15</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>9</sub></font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>15</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/>
<b>set </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="mcart_1.html#K1">`1</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="mcart_1.html#K2">`2</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="mcart_1.html#K1">`1</a>  as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>13</sub></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">c<sub>21</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="mcart_1.html#K2">`2</a>  as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>12</sub></font> <b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<a NAME="E4:61_1_2_2"/><i><font color="Green">E53</font></i>: 
not  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="dickson.html#E12:61_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E5:61_1_2_2"/>
<font color="Maroon">c<sub>20</sub></font> is    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:61_1_2_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E8:61_1_2_2"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> holds <br/><font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E4:61_1_2_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>20</sub></font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="dickson.html#E7:61_1_2_2"><i><font color="Green">E54</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_2_1"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>23</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>23</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> as   <a href="funct_1.html#NM1">Function</a> ;<br/><b>take </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>26</sub></font>;<br/><b>thus </b><a NAME="E2:61_1_2_2_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font>
 ;<br/><a NAME="E3:61_1_2_2_1"/><i><font color="Green">E56</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E7:61_1_2_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E4:61_1_2_2_1"/><i><font color="Green">E57</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><b>thus </b> <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>27</sub></font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="txt" href="dickson.html#E1:61"><i><font color="Green">E44</font></i></a>, <a class="txt" href="dickson.html#E3:61_1_2_2_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="afinsq_1.html#T4">AFINSQ_1:4</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/><b>let </b><font color="Maroon">c<sub>28</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E6:61_1_2_2_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><a NAME="E7:61_1_2_2_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="dickson.html#E6:61_1_2_2_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E8:61_1_2_2_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="afinsq_1.html#T4">AFINSQ_1:4</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_2_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_1_2_2_1_2"/>
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 </span><b>by </b><i><a class="txt" href="dickson.html#E8:61_1_2_2_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_2_1_2_1"><b>suppose </b></a><a NAME="E1:61_1_2_2_1_2_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_2_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E3:61_1_2_2_1_2_1"/><b>then </b>
not <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E4:61_1_2_2_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:61_1_2_2_1_2_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><a NAME="E5:61_1_2_2_1_2_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E1:61"><i><font color="Green">E44</font></i></a>, <a class="txt" href="dickson.html#E1:61_1_2_2_1_2_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>29</sub></font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E7:61_1_2_2_1_2_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>
 <b>and </b><br/><a NAME="E8:61_1_2_2_1_2_1"/><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:61"><i><font color="Green">E44</font></i></a>, <a class="txt" href="dickson.html#E1:61_1_2_2_1_2_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>30</sub></font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E10:61_1_2_2_1_2_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>
 <b>and </b><br/><a NAME="E11:61_1_2_2_1_2_1"/><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>30</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E7:61_1_2_2_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/><a NAME="E12:61_1_2_2_1_2_1"/>
<span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:61"><i><font color="Green">E44</font></i></a>, <a class="txt" href="dickson.html#E1:61_1_2_2_1_2_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="dickson.html#E7:61_1_2_2_1_2_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="dickson.html#E8:61_1_2_2_1_2_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="dickson.html#E10:61_1_2_2_1_2_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="dickson.html#E11:61_1_2_2_1_2_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E13:61_1_2_2_1_2_1"/>
<font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E7:61_1_2_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="dickson.html#E8:61_1_2_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="dickson.html#E4:61_1_2_2_1_2_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="dickson.html#E5:61_1_2_2_1_2_1"><i><font color="Green">E63</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_2_1_2_2"><b>suppose </b></a><a NAME="E1:61_1_2_2_1_2_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:61_1_2_2_1_2_2"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:61_1_2_2_1_2_2"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E1:61_1_2_2_1_2_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><b>then </b><i><font color="Green">E63</font></i>: <font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="dickson.html#E2:61_1_2_2_1_2_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">c<sub>21</sub></font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/><b>consider </b><font color="Maroon">c<sub>29</sub></font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E6:61_1_2_2_1_2_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>29</sub></font> )
 <b>by </b><i><a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/><b>thus </b><a NAME="E7:61_1_2_2_1_2_2"/>
<font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E2:61"><i><font color="Green">E45</font></i></a>, <a class="txt" href="dickson.html#E2:61_1_2_2_1_2_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="dickson.html#E4:61_1_2_2_1_2_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="dickson.html#E6:61_1_2_2_1_2_2"><i><font color="Green">E64</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E11:61_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>23</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>23</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>9</sub></font> <b>by </b><i><a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>25</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>23</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>21</sub></font>
;<br/>

<b>thus </b><a NAME="E13:61_1_2_2"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<b>thus </b><a NAME="E14:61_1_2_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>21</sub></font></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/>

<b>thus </b><a NAME="E15:61_1_2_2"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>9</sub></font><b> such that </b><br/><a NAME="E21:61_1_2"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>15</sub></font> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="dickson.html#E19:61_1_2"><i><font color="Green">E52</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_3"><b>now </b></a><div class="add"><a NAME="E1:61_1_2_3"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/><b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>10</sub></font> ;<br/><b>take </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>20</sub></font>, <font color="Maroon">c<sub>21</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:61_1_2_3_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K1">dom</a> <font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E2:61_1_2_3_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K1">dom</a> <font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E3:61_1_2_3_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/><a NAME="E4:61_1_2_3_1"/>
<font color="Maroon">c<sub>20</sub></font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="dickson.html#E1:61_1_2_3_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">c<sub>23</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>12</sub></font><b> such that </b><br/><a NAME="E6:61_1_2_3_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E7:61_1_2_3_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E8:61_1_2_3_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E21:61_1_2"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E9:61_1_2_3_1"/>
<font color="Maroon">c<sub>21</sub></font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="dickson.html#E2:61_1_2_3_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>24</sub></font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">c<sub>25</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>12</sub></font><b> such that </b><br/><a NAME="E11:61_1_2_3_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E12:61_1_2_3_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E13:61_1_2_3_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E21:61_1_2"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E14:61_1_2_3_1"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E6:61_1_2_3_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>26</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E16:61_1_2_3_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E17:61_1_2_3_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> holds <br/><font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/><a NAME="E18:61_1_2_3_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E11:61_1_2_3_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>27</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E20:61_1_2_3_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E21:61_1_2_3_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> holds <br/><font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/><a NAME="E22:61_1_2_3_1"/><i><font color="Green">E65</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E23:61_1_2_3_1"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E24:61_1_2_3_1"/><i><font color="Green">E67</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:61"><i><font color="Green">E44</font></i></a>, <a class="txt" href="dickson.html#E16:61_1_2_3_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><div><i><font color="Green">E68</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_1_2_3_1_1"/>
not <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R1">misses</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E2:61_1_2_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</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="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>28</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:61_1_2_3_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E5:61_1_2_3_1_1"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="txt" href="dickson.html#E4:61_1_2_3_1_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:61_1_2_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:61_1_2_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E5:61_1_2_3_1_1"><i><font color="Green">E70</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E26:61_1_2_3_1"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>22</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E24:61_1_2_3_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E27:61_1_2_3_1"/><i><font color="Green">E70</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>24</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E16:61_1_2_3_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="dickson.html#E20:61_1_2_3_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="dickson.html#E24:61_1_2_3_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="dickson.html#E25:61_1_2_3_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><div><i><font color="Green">E71</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_3_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>28</sub></font>, <font color="Maroon">c<sub>29</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:61_1_2_3_1_2_1"/><i><font color="Green">E72</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/><a NAME="E2:61_1_2_3_1_2_1"/><b>then </b><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E3:61_1_2_3_1_2_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:61_1_2_3_1_2_1"/><b>then </b>
not <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E26:61_1_2_3_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="dickson.html#E2:61_1_2_3_1_2_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_0.html#T5">XBOOLE_0:5</a></i>;<br/><a NAME="E5:61_1_2_3_1_2_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:61_1_2_3_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="dickson.html#E8:61_1_2_3_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="dickson.html#E13:61_1_2_3_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="dickson.html#E3:61_1_2_3_1_2_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><a NAME="E6:61_1_2_3_1_2_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E16:61_1_2_3_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="dickson.html#E20:61_1_2_3_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="dickson.html#E2:61_1_2_3_1_2_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E7:61_1_2_3_1_2_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E16:61_1_2_3_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="dickson.html#E20:61_1_2_3_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="dickson.html#E1:61_1_2_3_1_2_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E8:61_1_2_3_1_2_1"/><b>then </b>
not <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E27:61_1_2_3_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="dickson.html#E6:61_1_2_3_1_2_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xboole_0.html#T5">XBOOLE_0:5</a></i>;<br/><a NAME="E9:61_1_2_3_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E2:61_1_2_3_1_2_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="dickson.html#E5:61_1_2_3_1_2_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="dickson.html#E6:61_1_2_3_1_2_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><b>hence </b><a NAME="E10:61_1_2_3_1_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E7:61_1_2_3_1_2_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:61_1_2_3_1_2"/><i><font color="Green">E72</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>24</sub></font>
 ;<br/><a NAME="E3:61_1_2_3_1_2"/><b>then </b><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E4:61_1_2_3_1_2"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:61_1_2_3_1_2"/><b>then </b>
not <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E27:61_1_2_3_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="dickson.html#E3:61_1_2_3_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_0.html#T5">XBOOLE_0:5</a></i>;<br/><a NAME="E6:61_1_2_3_1_2"/><b>then </b><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:61_1_2_3_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="dickson.html#E8:61_1_2_3_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="dickson.html#E13:61_1_2_3_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="dickson.html#E4:61_1_2_3_1_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><a NAME="E7:61_1_2_3_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E16:61_1_2_3_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="dickson.html#E20:61_1_2_3_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="dickson.html#E3:61_1_2_3_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E8:61_1_2_3_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E16:61_1_2_3_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="dickson.html#E20:61_1_2_3_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="dickson.html#E2:61_1_2_3_1_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E9:61_1_2_3_1_2"/><b>then </b>
not <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E26:61_1_2_3_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="dickson.html#E7:61_1_2_3_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xboole_0.html#T5">XBOOLE_0:5</a></i>;<br/><a NAME="E10:61_1_2_3_1_2"/><b>then </b>
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:61_1_2_3_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="dickson.html#E6:61_1_2_3_1_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="dickson.html#E7:61_1_2_3_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><b>hence </b><a NAME="E11:61_1_2_3_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E8:61_1_2_3_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E29:61_1_2_3_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E22:61_1_2_3_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E30:61_1_2_3_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E31:61_1_2_3_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E22:61_1_2_3_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="dickson.html#E23:61_1_2_3_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E32:61_1_2_3_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="dickson.html#E3:61_1_2_3_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="dickson.html#E8:61_1_2_3_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="dickson.html#E13:61_1_2_3_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="dickson.html#E31:61_1_2_3_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/><b>then </b><font color="Maroon">c<sub>25</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="dickson.html#E29:61_1_2_3_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="dickson.html#E30:61_1_2_3_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>23</sub></font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/><b>hence </b><a NAME="E35:61_1_2_3_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E7:61_1_2_3_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="dickson.html#E12:61_1_2_3_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="dickson.html#E28:61_1_2_3_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="relat_1.html#D2">RELAT_1:def 2</a></i>;<br/></div><b>end;</b></div><a NAME="E5:61_1_2_3"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>19</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>20</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:61_1_2_3_2"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>19</sub></font> implies <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 ;<br/><b>assume </b><a NAME="E2:61_1_2_3_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><a NAME="E3:61_1_2_3_2"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:61_1_2_3_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>and </b><br/><a NAME="E6:61_1_2_3_2"/><i><font color="Green">E56</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E7:61_1_2_3_2"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/><a NAME="E8:61_1_2_3_2"/><i><font color="Green">E58</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E9:61_1_2_3_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 </span>}</span> 
 <b>by </b><i><a class="ref" href="axioms.html#T30">AXIOMS:30</a></i>;<br/><a NAME="E10:61_1_2_3_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E11:61_1_2_3_2"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="dickson.html#E9:61_1_2_3_2"><i><font color="Green">E59</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>24</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/><a NAME="E12:61_1_2_3_2"/><i><font color="Green">E61</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><i><font color="Green">E62</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="dickson.html#E6:61_1_2_3_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/><a NAME="E14:61_1_2_3_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="dickson.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E15:61_1_2_3_2"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E13:61_1_2_3_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E16:61_1_2_3_2"/><i><font color="Green">E64</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E1:61"><i><font color="Green">E44</font></i></a>, <a class="txt" href="dickson.html#E12:61_1_2_3_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="dickson.html#E13:61_1_2_3_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_3_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>25</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:61_1_2_3_2_2"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:61_1_2_3_2_2"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:61"><i><font color="Green">E44</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:61_1_2_3_2_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="dickson.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E4:61_1_2_3_2_2"/><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E2:61_1_2_3_2_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><a NAME="E5:61_1_2_3_2_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E7:61_1_2_3_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="dickson.html#E8:61_1_2_3_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="dickson.html#E2:61_1_2_3_2_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="dickson.html#E3:61_1_2_3_2_2"><i><font color="Green">E66</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>26</sub></font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E7:61_1_2_3_2_2"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>26</sub></font> )
 <b>by </b><i><a class="txt" href="dickson.html#E2:61_1_2_3_2_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="dickson.html#E3:61_1_2_3_2_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>27</sub></font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E9:61_1_2_3_2_2"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>27</sub></font> )
 <b>by </b><i><a class="txt" href="dickson.html#E1:61"><i><font color="Green">E44</font></i></a>, <a class="txt" href="dickson.html#E2:61_1_2_3_2_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/><b>thus </b><a NAME="E10:61_1_2_3_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:61"><i><font color="Green">E44</font></i></a>, <a class="txt" href="dickson.html#E2:61_1_2_3_2_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="dickson.html#E4:61_1_2_3_2_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="dickson.html#E5:61_1_2_3_2_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="dickson.html#E7:61_1_2_3_2_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="dickson.html#E9:61_1_2_3_2_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div><a NAME="E18:61_1_2_3_2"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E16:61_1_2_3_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E19:61_1_2_3_2"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>25</sub></font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E21:61_1_2_3_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>25</sub></font> )
 <b>by </b><i><a class="txt" href="dickson.html#E11:61_1_2_3_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/><a NAME="E22:61_1_2_3_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E2:61"><i><font color="Green">E45</font></i></a>, <a class="txt" href="dickson.html#E7:61_1_2_3_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="dickson.html#E8:61_1_2_3_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="dickson.html#E21:61_1_2_3_2"><i><font color="Green">E66</font></i></a></i>;<br/><a NAME="E23:61_1_2_3_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="dickson.html#E19:61_1_2_3_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E24:61_1_2_3_2"/><b>then </b><i><font color="Green">E68</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K1">dom</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E25:61_1_2_3_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E19:61_1_2_3_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="dickson.html#E22:61_1_2_3_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>26</sub></font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">c<sub>27</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>12</sub></font><b> such that </b><br/><a NAME="E27:61_1_2_3_2"/>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E28:61_1_2_3_2"/><i><font color="Green">E69</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></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"><font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E29:61_1_2_3_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E21:61_1_2"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E30:61_1_2_3_2"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="dickson.html#E28:61_1_2_3_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_3_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>28</sub></font>, <font color="Maroon">c<sub>29</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:61_1_2_3_2_3_1"/><i><font color="Green">E72</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:61_1_2_3_2_3_1"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E3:61_1_2_3_2_3_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><a NAME="E4:61_1_2_3_2_3_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="dickson.html#E15:61_1_2_3_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E5:61_1_2_3_2_3_1"/><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:61_1_2_3_2_3_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_3_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_1_2_3_2_3_1_1"/>
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> or  not <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_3_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_3_2_3_1_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:61_1_2_3_2_3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E3:61_1_2_3_2_3_1_1_1"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="txt" href="dickson.html#E3:61_1_2_3_2_3_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="dickson.html#E1:61_1_2_3_2_3_1_1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="dickson.html#E3:61_1_2_3_2_3_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/><a NAME="E5:61_1_2_3_2_3_1_1_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:61_1_2_3_2_3_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="dickson.html#E3:61_1_2_3_2_3_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E6:61_1_2_3_2_3_1_1_1"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E6:61_1_2_3_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="dickson.html#E8:61_1_2_3_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="dickson.html#E4:61_1_2_3_2_3_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="afinsq_1.html#T4">AFINSQ_1:4</a></i>;<br/><b>hence </b><a NAME="E7:61_1_2_3_2_3_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E5:61_1_2_3_2_3_1_1_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_3_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_3_2_3_1_1_2"/><i><font color="Green">E76</font></i>: 
not <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:61_1_2_3_2_3_1_1_2"/><b>then </b>
not <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E3:61_1_2_3_2_3_1_1_2"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E4:61_1_2_3_2_3_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:61_1_2_3_2_3_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:61_1_2_3_2_3_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="dickson.html#E1:61_1_2_3_2_3_1_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><a NAME="E5:61_1_2_3_2_3_1_1_2"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E5:61_1_2_3_2_3_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="dickson.html#E3:61_1_2_3_2_3_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><a NAME="E6:61_1_2_3_2_3_1_1_2"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E6:61_1_2_3_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="dickson.html#E8:61_1_2_3_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="dickson.html#E4:61_1_2_3_2_3_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="afinsq_1.html#T4">AFINSQ_1:4</a></i>;<br/><b>hence </b><a NAME="E7:61_1_2_3_2_3_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E5:61_1_2_3_2_3_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div><b>assume </b><a NAME="E2:61_1_2_3_2_3"/><i><font color="Green">E72</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/><a NAME="E3:61_1_2_3_2_3"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="dickson.html#E6:61_1_2_3_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="dickson.html#E8:61_1_2_3_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E4:61_1_2_3_2_3"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="afinsq_1.html#T4">AFINSQ_1:4</a></i>;<br/> <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="txt" href="dickson.html#E15:61_1_2_3_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>
;<br/><a NAME="E6:61_1_2_3_2_3"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E3:61_1_2_3_2_3"><i><font color="Green">E73</font></i></a>, <a class="ref" href="afinsq_1.html#T4">AFINSQ_1:4</a></i>;<br/><a NAME="E7:61_1_2_3_2_3"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_3_2_3_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_1_2_3_2_3_3"/>
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> or  not <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_3_2_3_3_1"><b>suppose </b></a><a NAME="E1:61_1_2_3_2_3_3_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:61_1_2_3_2_3_3_1"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E3:61_1_2_3_2_3_3_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="txt" href="dickson.html#E7:61_1_2_3_2_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="dickson.html#E1:61_1_2_3_2_3_3_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="dickson.html#E3:61_1_2_3_2_3_3_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
<br/>.= 
<font color="Maroon">c<sub>29</sub></font>
<b>by </b><i><a class="txt" href="dickson.html#E2:61_1_2_3_2_3"><i><font color="Green">E72</font></i></a>, <a class="txt" href="dickson.html#E3:61_1_2_3_2_3_3_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>
;<br/><b>hence </b><a NAME="E5:61_1_2_3_2_3_3_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E6:61_1_2_3_2_3"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_3_2_3_3_2"><b>suppose </b></a><a NAME="E1:61_1_2_3_2_3_3_2"/><i><font color="Green">E77</font></i>: 
not <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:61_1_2_3_2_3_3_2"/><b>then </b>
not <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E3:61_1_2_3_2_3_3_2"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E4:61_1_2_3_2_3"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="txt" href="dickson.html#E7:61_1_2_3_2_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="dickson.html#E1:61_1_2_3_2_3_3_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>
<b>by </b><i><a class="txt" href="dickson.html#E3:61_1_2_3_2_3_3_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>29</sub></font>
<b>by </b><i><a class="txt" href="dickson.html#E2:61_1_2_3_2_3"><i><font color="Green">E72</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>
;<br/><b>hence </b><a NAME="E5:61_1_2_3_2_3_3_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E6:61_1_2_3_2_3"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E32:61_1_2_3_2"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E5:61_1_2_3_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="dickson.html#E29:61_1_2_3_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="dickson.html#E30:61_1_2_3_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="relat_1.html#D2">RELAT_1:def 2</a></i>;<br/><b>hence </b><a NAME="E33:61_1_2_3_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E24:61_1_2_3_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><a NAME="E7:61_1_2_3"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_3_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>20</sub></font>, <font color="Maroon">c<sub>21</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>14</sub></font>;<br/><a NAME="E1:61_1_2_3_3"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>20</sub></font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">c<sub>23</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>12</sub></font><b> such that </b><br/><a NAME="E3:61_1_2_3_3"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E4:61_1_2_3_3"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E5:61_1_2_3_3"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E21:61_1_2"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E6:61_1_2_3_3"/>
 <a href="wellfnd1.html#K1">dom</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E7:61_1_2_3_3"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E7:61_1_2_3"><i><font color="Green">E55</font></i></a>, <a class="txt" href="dickson.html#E1:61_1_2_3_3"><i><font color="Green">E56</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:61_1_2_3_3"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/><a NAME="E9:61_1_2_3_3"/>
<font color="Maroon">c<sub>21</sub></font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>24</sub></font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">c<sub>25</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>12</sub></font><b> such that </b><br/><a NAME="E11:61_1_2_3_3"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E12:61_1_2_3_3"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E13:61_1_2_3_3"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E21:61_1_2"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E14:61_1_2_3_3"/><i><font color="Green">E64</font></i>: 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K8">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/><a NAME="E15:61_1_2_3_3"/><i><font color="Green">E65</font></i>: 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K9">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/><a NAME="E16:61_1_2_3_3"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E4:61_1_2_3_3"><i><font color="Green">E58</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/><a NAME="E17:61_1_2_3_3"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E4:61_1_2_3_3"><i><font color="Green">E58</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/><a NAME="E18:61_1_2_3_3"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E3:61_1_2_3_3"><i><font color="Green">E57</font></i></a>, <a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>26</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E20:61_1_2_3_3"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E21:61_1_2_3_3"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> holds <br/><font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/><a NAME="E22:61_1_2_3_3"/><i><font color="Green">E71</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:61"><i><font color="Green">E44</font></i></a>, <a class="txt" href="dickson.html#E20:61_1_2_3_3"><i><font color="Green">E69</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E23:61_1_2_3_3"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E24:61_1_2_3_3"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E11:61_1_2_3_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>27</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E26:61_1_2_3_3"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E27:61_1_2_3_3"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> holds <br/><font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/><a NAME="E28:61_1_2_3_3"/><i><font color="Green">E75</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:61"><i><font color="Green">E44</font></i></a>, <a class="txt" href="dickson.html#E26:61_1_2_3_3"><i><font color="Green">E73</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E29:61_1_2_3_3"/><b>then </b><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>22</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="dickson.html#E3:61_1_2_3_3"><i><font color="Green">E57</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>24</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="dickson.html#E11:61_1_2_3_3"><i><font color="Green">E61</font></i></a></i>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:61_1_2_3_3_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/><a NAME="E2:61_1_2_3_3_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E3:61_1_2_3_3_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow_3.html#K2">["</a><span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="yellow_3.html#K2">"]</a></span>
 <b>by </b><i><a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/><a NAME="E4:61_1_2_3_3_1"/><b>then </b><i><font color="Green">E77</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K8">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K9">`2</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K8">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K9">`2</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K8">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K9">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="yellow_3.html#T10">YELLOW_3:10</a></i>;<br/><a NAME="E5:61_1_2_3_3_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>24</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E12:61_1_2_3_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="dickson.html#E14:61_1_2_3_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="dickson.html#E15:61_1_2_3_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="dickson.html#E16:61_1_2_3_3"><i><font color="Green">E66</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/><a NAME="E6:61_1_2_3_3_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E3:61_1_2_3_3"><i><font color="Green">E57</font></i></a>, <a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/><a NAME="E7:61_1_2_3_3_1"/>
<font color="Maroon">c<sub>28</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E5:61_1_2_3_3_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>30</sub></font>, <font color="Maroon">c<sub>31</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:61_1_2_3_3_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> )
 <b>and </b><br/><a NAME="E10:61_1_2_3_3_1"/><i><font color="Green">E81</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="orders_2.html#L1">RelStr</a> ex <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">b<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="dickson.html#E1:61"><i><font color="Green">E44</font></i></a>, <a class="txt" href="dickson.html#E6:61_1_2_3_3_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/><a NAME="E11:61_1_2_3_3_1"/><i><font color="Green">E82</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E12:61_1_2_3_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="dickson.html#E14:61_1_2_3_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="dickson.html#E15:61_1_2_3_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="dickson.html#E17:61_1_2_3_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="dickson.html#E4:61_1_2_3_3_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>32</sub></font> = <font color="Maroon">c<sub>23</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> ;<br/><b>reconsider </b><font color="Maroon">c<sub>33</sub></font> = <font color="Maroon">c<sub>25</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> ;<br/><a NAME="E14:61_1_2_3_3_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>33</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E11:61_1_2_3_3_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E15:61_1_2_3_3_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 holds <br/>ex <font color="Olive">b<sub>4</sub></font> being   <a href="orders_2.html#L1">RelStr</a> ex <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">b<sub>4</sub></font> st <br/>( <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>6</sub></font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_3_3_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>34</sub></font> = <font color="Maroon">c<sub>22</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>35</sub></font> = <font color="Maroon">c<sub>24</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>;<br/>
<b>take </b>
<font color="Maroon">c<sub>22</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>24</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
;<br/>

<b>thus </b><a NAME="E1:61_1_2_3_3_1_1"/>
<font color="Maroon">c<sub>22</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E5:61_1_2_3_3"><i><font color="Green">E59</font></i></a></i>;<br/>

<b>thus </b><a NAME="E2:61_1_2_3_3_1_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E13:61_1_2_3_3"><i><font color="Green">E63</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>36</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E3:61_1_2_3_3_1_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<a NAME="E4:61_1_2_3_3_1_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="dickson.html#E3:61_1_2_3_3_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="afinsq_1.html#T4">AFINSQ_1:4</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>37</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>38</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>39</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font>;<br/>
<a NAME="E5:61_1_2_3_3_1_1"/>
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="dickson.html#E3:61_1_2_3_3_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E6:61_1_2_3_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>40</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> as    <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i><a class="ref" href="yellow_1.html#D3">YELLOW_1:def 3</a></i>;<br/>
<a NAME="E8:61_1_2_3_3_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="dickson.html#E23:61_1_2_3_3"><i><font color="Green">E72</font></i></a>, <a class="txt" href="dickson.html#E3:61_1_2_3_3_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="afinsq_1.html#T4">AFINSQ_1:4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_3_3_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_3_3_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_1_2_3_3_1_1_1_1"/>
( <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> or  not <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_3_3_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_3_3_1_1_1_1_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:61_1_2_3_3_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E3:61_1_2_3_3_1_1_1_1_1"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:61_1_2_3_3_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E8:61_1_2_3_3_1_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="dickson.html#E1:61_1_2_3_3_1_1_1_1_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><b>hence </b><a NAME="E5:61_1_2_3_3_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>40</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E2:61"><i><font color="Green">E45</font></i></a>, <a class="txt" href="dickson.html#E3:61_1_2_3_3_1_1_1_1_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_3_3_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_3_3_1_1_1_1_2"/><i><font color="Green">E87</font></i>: 
not <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:61_1_2_3_3_1_1_1_1_2"/><b>then </b><i><font color="Green">E88</font></i>: 
not <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E3:61_1_2_3_3_1_1_1_1_2"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E4:61_1_2_3_3_1_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:61_1_2_3_3_1_1_1_1_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E3:61"><i><font color="Green">E46</font></i></a>, <a class="txt" href="dickson.html#E4:61_1_2_3_3_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="dickson.html#E2:61_1_2_3_3_1_1_1_1_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:61_1_2_3_3_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E8:61_1_2_3_3_1_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="dickson.html#E1:61_1_2_3_3_1_1_1_1_2"><i><font color="Green">E87</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><a NAME="E6:61_1_2_3_3_1_1_1_1_2"/><b>then </b><i><font color="Green">E91</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E20:61_1_2_3_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="dickson.html#E21:61_1_2_3_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="dickson.html#E4:61_1_2_3_3_1_1_1_1_2"><i><font color="Green">E90</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>41</sub></font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E8:61_1_2_3_3_1_1_1_1_2"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>41</sub></font> )
 <b>by </b><i><a class="txt" href="dickson.html#E1:61"><i><font color="Green">E44</font></i></a>, <a class="txt" href="dickson.html#E3:61_1_2_3_3_1_1_1_1_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/><a NAME="E9:61_1_2_3_3_1_1_1_1_2"/>
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:61_1_2_3_3_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E10:61_1_2_3_3_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>42</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> as    <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i><a class="ref" href="yellow_1.html#D3">YELLOW_1:def 3</a></i>;<br/><b>thus </b><a NAME="E12:61_1_2_3_3_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>40</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:61"><i><font color="Green">E44</font></i></a>, <a class="txt" href="dickson.html#E3:61_1_2_3_3_1_1_1_1_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="dickson.html#E6:61_1_2_3_3_1_1_1_1_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="dickson.html#E8:61_1_2_3_3_1_1_1_1_2"><i><font color="Green">E92</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</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">c<sub>41</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>40</sub></font> ;<br/>
<a NAME="E11:61_1_2_3_3_1_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="dickson.html#E29:61_1_2_3_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="dickson.html#E3:61_1_2_3_3_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="afinsq_1.html#T4">AFINSQ_1:4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_3_3_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_3_3_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="E1:61_1_2_3_3_1_1_2_1"/>
( <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> or  not <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_3_3_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:61_1_2_3_3_1_1_2_1_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:61_1_2_3_3_1_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E3:61_1_2_3_3_1_1_2_1_1"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:61_1_2_3_3_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E11:61_1_2_3_3_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="dickson.html#E1:61_1_2_3_3_1_1_2_1_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><b>hence </b><a NAME="E5:61_1_2_3_3_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>40</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E2:61"><i><font color="Green">E45</font></i></a>, <a class="txt" href="dickson.html#E3:61_1_2_3_3_1_1_2_1_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_3_3_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:61_1_2_3_3_1_1_2_1_2"/><i><font color="Green">E88</font></i>: 
not <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:61_1_2_3_3_1_1_2_1_2"/><b>then </b><i><font color="Green">E89</font></i>: 
not <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E3:61_1_2_3_3_1_1_2_1_2"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E4:61_1_2_3_3_1_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:61_1_2_3_3_1_1_2_1_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E3:61"><i><font color="Green">E46</font></i></a>, <a class="txt" href="dickson.html#E4:61_1_2_3_3_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="dickson.html#E2:61_1_2_3_3_1_1_2_1_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:61_1_2_3_3_1_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E11:61_1_2_3_3_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="dickson.html#E1:61_1_2_3_3_1_1_2_1_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><a NAME="E6:61_1_2_3_3_1_1_2_1_2"/><b>then </b><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E26:61_1_2_3_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="dickson.html#E27:61_1_2_3_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="dickson.html#E4:61_1_2_3_3_1_1_2_1_2"><i><font color="Green">E91</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>42</sub></font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E8:61_1_2_3_3_1_1_2_1_2"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>42</sub></font> )
 <b>by </b><i><a class="txt" href="dickson.html#E1:61"><i><font color="Green">E44</font></i></a>, <a class="txt" href="dickson.html#E3:61_1_2_3_3_1_1_2_1_2"><i><font color="Green">E90</font></i></a>, <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/><a NAME="E9:61_1_2_3_3_1_1_2_1_2"/>
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:61_1_2_3_3_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E10:61_1_2_3_3_1_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>43</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> as    <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i><a class="ref" href="yellow_1.html#D3">YELLOW_1:def 3</a></i>;<br/><b>thus </b><a NAME="E12:61_1_2_3_3_1_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>40</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:61"><i><font color="Green">E44</font></i></a>, <a class="txt" href="dickson.html#E3:61_1_2_3_3_1_1_2_1_2"><i><font color="Green">E90</font></i></a>, <a class="txt" href="dickson.html#E6:61_1_2_3_3_1_1_2_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="dickson.html#E8:61_1_2_3_3_1_1_2_1_2"><i><font color="Green">E93</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</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">c<sub>42</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>40</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>40</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>41</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>42</sub></font>
;<br/>

<b>thus </b><a NAME="E14:61_1_2_3_3_1_1"/>
<font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font>
 ;<br/>

<b>thus </b><a NAME="E15:61_1_2_3_3_1_1"/>
<font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font>
 ;<br/>

<b>thus </b><a NAME="E16:61_1_2_3_3_1_1"/>
<font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_3_3_1_1_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_1_2_3_3_1_1_3"/>
( <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 </span><b>by </b><i><a class="txt" href="dickson.html#E4:61_1_2_3_3_1_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_3_3_1_1_3_1"><b>suppose </b></a><a NAME="E1:61_1_2_3_3_1_1_3_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:61_1_2_3_3_1_1_3_1"/><i><font color="Green">E89</font></i>: 
not <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_3_3_1_1_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_1_2_3_3_1_1_3_1_1"/>
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E2:61_1_2_3_3_1_1_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="dickson.html#E1:61_1_2_3_3_1_1_3_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:61_1_2_3_3_1_1_3_1_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div><a NAME="E3:61_1_2_3_3_1_1_3_1"/><b>then </b><i><font color="Green">E90</font></i>: 
not <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E4:61_1_2_3_3_1_1_3_1"/>
not <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E2:61_1_2_3_3_1_1_3_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E5:61_1_2_3_3_1_1_3_1"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>42</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E11:61_1_2_3_3_1_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>43</sub></font> being    <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">c<sub>44</sub></font>, <font color="Maroon">c<sub>45</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>43</sub></font><b> such that </b><br/><a NAME="E7:61_1_2_3_3_1_1_3_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font>
 <b>and </b><br/><a NAME="E8:61_1_2_3_3_1_1_3_1"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> &amp; <font color="Maroon">c<sub>45</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> )
 <b>and </b><br/><a NAME="E9:61_1_2_3_3_1_1_3_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>44</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>45</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E10:61_1_2_3_3_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="dickson.html#E1:61_1_2_3_3_1_1_3_1"><i><font color="Green">E88</font></i></a></i>;<br/><a NAME="E10:61_1_2_3_3_1_1_3_1"/>
<font color="Maroon">c<sub>43</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>40</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:61"><i><font color="Green">E44</font></i></a>, <a class="txt" href="dickson.html#E1:61_1_2_3_3_1_1_3_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="dickson.html#E7:61_1_2_3_3_1_1_3_1"><i><font color="Green">E92</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E11:61_1_2_3_3_1_1_3_1"/>
<font color="Maroon">c<sub>41</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>42</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E9:61_1_2_3_3_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="dickson.html#E8:61_1_2_3_3_1_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="dickson.html#E3:61_1_2_3_3_1_1_3_1"><i><font color="Green">E90</font></i></a>, <a class="txt" href="dickson.html#E5:61_1_2_3_3_1_1_3_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="dickson.html#E8:61_1_2_3_3_1_1_3_1"><i><font color="Green">E93</font></i></a>, <a class="txt" href="dickson.html#E9:61_1_2_3_3_1_1_3_1"><i><font color="Green">E94</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_3_3_1_1_3_2"><b>suppose </b></a><a NAME="E1:61_1_2_3_3_1_1_3_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:61_1_2_3_3_1_1_3_2"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:61_1_2_3_3_1_1_3_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E1:61_1_2_3_3_1_1_3_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E4:61_1_2_3_3_1_1_3_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E1:61_1_2_3_3_1_1_3_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><i><font color="Green">E92</font></i>: <font color="Maroon">c<sub>41</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font>
<b>by </b><i><a class="txt" href="dickson.html#E8:61_1_2_3_3_1_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="dickson.html#E3:61_1_2_3_3_1_1_3_2"><i><font color="Green">E90</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>23</sub></font>
<b>by </b><i><a class="txt" href="dickson.html#E2:61_1_2_3_3_1_1_3_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/><font color="Maroon">c<sub>42</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font>
<b>by </b><i><a class="txt" href="dickson.html#E11:61_1_2_3_3_1_1"><i><font color="Green">E87</font></i></a>, <a class="txt" href="dickson.html#E4:61_1_2_3_3_1_1_3_2"><i><font color="Green">E91</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="dickson.html#E2:61_1_2_3_3_1_1_3_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/><b>hence </b><a NAME="E7:61_1_2_3_3_1_1_3_2"/>
<font color="Maroon">c<sub>41</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>42</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E2:61"><i><font color="Green">E45</font></i></a>, <a class="txt" href="dickson.html#E14:61_1_2_3_3_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="dickson.html#E1:61_1_2_3_3_1_1_3_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="dickson.html#E5:61_1_2_3_3_1_1_3_2"><i><font color="Green">E92</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E16:61_1_2_3_3_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E8:61_1_2_3_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E33:61_1_2_3_3"/>
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>30</sub></font>, <font color="Maroon">c<sub>31</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E35:61_1_2_3_3"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font>
 <b>and </b><br/><a NAME="E36:61_1_2_3_3"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>and </b><br/><a NAME="E37:61_1_2_3_3"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="orders_2.html#L1">RelStr</a> ex <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">b<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="dickson.html#E8:61_1_2_3_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_3_3_2"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">c<sub>32</sub></font> = <font color="Maroon">c<sub>28</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>33</sub></font> = <font color="Maroon">c<sub>29</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>34</sub></font> = <font color="Maroon">c<sub>28</sub></font>, <font color="Maroon">c<sub>35</sub></font> = <font color="Maroon">c<sub>29</sub></font> as   <a href="funct_1.html#NM1">Function</a> ;<br/><b>take </b><font color="Maroon">c<sub>36</sub></font> = <font color="Maroon">c<sub>34</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>37</sub></font> = <font color="Maroon">c<sub>35</sub></font>;<br/><b>thus </b><a NAME="E2:61_1_2_3_3_2"/>
( <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>37</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font> )
 ;<br/><b>let </b><font color="Maroon">c<sub>38</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E3:61_1_2_3_3_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E4:61_1_2_3_3_2"/><i><font color="Green">E81</font></i>: 
not <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_3_3_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_1_2_3_3_2_1"/>
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E2:61_1_2_3_3_2_1"/><b>then </b>
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:61_1_2_3_3_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E1:61"><i><font color="Green">E44</font></i></a>, <a class="txt" href="dickson.html#E3:61_1_2_3_3_2"><i><font color="Green">E80</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:61_1_2_3_3_2"/><b>then </b><i><font color="Green">E82</font></i>: 
not <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E6:61_1_2_3_3_2"/><i><font color="Green">E83</font></i>: 
not <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E4:61_1_2_3_3_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><b>set </b><font color="Maroon">c<sub>39</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>38</sub></font>;<br/><a NAME="E7:61_1_2_3_3_2"/>
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E3:61_1_2_3_3_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E8:61_1_2_3_3_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>40</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>38</sub></font> as    <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i><a class="ref" href="yellow_1.html#D3">YELLOW_1:def 3</a></i>;<br/><b>take </b><font color="Maroon">c<sub>41</sub></font> = <font color="Maroon">c<sub>40</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>42</sub></font> = <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>38</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>43</sub></font> = <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>38</sub></font>;<br/><a NAME="E10:61_1_2_3_3_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E1:61"><i><font color="Green">E44</font></i></a>, <a class="txt" href="dickson.html#E22:61_1_2_3_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="dickson.html#E3:61_1_2_3_3_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E11:61_1_2_3_3_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E3:61_1_2_3_3_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>44</sub></font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E13:61_1_2_3_3_2"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>44</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>38</sub></font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>44</sub></font> )
 <b>by </b><i><a class="txt" href="dickson.html#E3:61_1_2_3_3_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>45</sub></font> = <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>38</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>41</sub></font> <b>by </b><i><a class="txt" href="dickson.html#E20:61_1_2_3_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="dickson.html#E21:61_1_2_3_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="dickson.html#E11:61_1_2_3_3_2"><i><font color="Green">E85</font></i></a>, <a class="txt" href="dickson.html#E13:61_1_2_3_3_2"><i><font color="Green">E86</font></i></a></i>;<br/><a NAME="E15:61_1_2_3_3_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E1:61"><i><font color="Green">E44</font></i></a>, <a class="txt" href="dickson.html#E28:61_1_2_3_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="dickson.html#E3:61_1_2_3_3_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>46</sub></font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E17:61_1_2_3_3_2"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">c<sub>46</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>38</sub></font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>46</sub></font> )
 <b>by </b><i><a class="txt" href="dickson.html#E3:61_1_2_3_3_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>47</sub></font> = <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>38</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>41</sub></font> <b>by </b><i><a class="txt" href="dickson.html#E26:61_1_2_3_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="dickson.html#E27:61_1_2_3_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="dickson.html#E11:61_1_2_3_3_2"><i><font color="Green">E85</font></i></a>, <a class="txt" href="dickson.html#E17:61_1_2_3_3_2"><i><font color="Green">E88</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>48</sub></font> = <font color="Maroon">c<sub>45</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>49</sub></font> = <font color="Maroon">c<sub>47</sub></font>;<br/><b>thus </b><a NAME="E19:61_1_2_3_3_2"/>
( <font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>38</sub></font> &amp; <font color="Maroon">c<sub>48</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>36</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>38</sub></font> &amp; <font color="Maroon">c<sub>49</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>37</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>38</sub></font> )
 ;<br/><b>consider </b><font color="Maroon">c<sub>50</sub></font> being    <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">c<sub>51</sub></font>, <font color="Maroon">c<sub>52</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>50</sub></font><b> such that </b><br/><a NAME="E21:61_1_2_3_3_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>38</sub></font>
 <b>and </b><br/><a NAME="E22:61_1_2_3_3_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>51</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>38</sub></font>
 <b>and </b><br/><a NAME="E23:61_1_2_3_3_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>52</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>38</sub></font>
 <b>and </b><br/><a NAME="E24:61_1_2_3_3_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>51</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>52</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E37:61_1_2_3_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="dickson.html#E3:61_1_2_3_3_2"><i><font color="Green">E80</font></i></a></i>;<br/><a NAME="E25:61_1_2_3_3_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>50</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>41</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:61_1_2_3_3_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="dickson.html#E21:61_1_2_3_3_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><a NAME="E26:61_1_2_3_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>48</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E6:61_1_2_3_3_2"><i><font color="Green">E83</font></i></a>, <a class="txt" href="dickson.html#E10:61_1_2_3_3_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/><b>hence </b><a NAME="E27:61_1_2_3_3_2"/>
<font color="Maroon">c<sub>48</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>49</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E5:61_1_2_3_3"><i><font color="Green">E59</font></i></a>, <a class="txt" href="dickson.html#E13:61_1_2_3_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="dickson.html#E35:61_1_2_3_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="dickson.html#E36:61_1_2_3_3"><i><font color="Green">E78</font></i></a>, <a class="txt" href="dickson.html#E5:61_1_2_3_3_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="dickson.html#E15:61_1_2_3_3_2"><i><font color="Green">E87</font></i></a>, <a class="txt" href="dickson.html#E22:61_1_2_3_3_2"><i><font color="Green">E90</font></i></a>, <a class="txt" href="dickson.html#E23:61_1_2_3_3_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="dickson.html#E24:61_1_2_3_3_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="dickson.html#E25:61_1_2_3_3_2"><i><font color="Green">E93</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E39:61_1_2_3_3"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E18:61_1_2_3_3"><i><font color="Green">E68</font></i></a>, <a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/><a NAME="E40:61_1_2_3_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>24</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E41:61_1_2_3_3"/><b>then </b><i><font color="Green">E80</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K8">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K9">`2</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E12:61_1_2_3_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="dickson.html#E14:61_1_2_3_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="dickson.html#E15:61_1_2_3_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="dickson.html#E16:61_1_2_3_3"><i><font color="Green">E66</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>32</sub></font> being    <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">c<sub>33</sub></font>, <font color="Maroon">c<sub>34</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>32</sub></font><b> such that </b><br/><a NAME="E43:61_1_2_3_3"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E44:61_1_2_3_3"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E45:61_1_2_3_3"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E46:61_1_2_3_3"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>34</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E2:61"><i><font color="Green">E45</font></i></a>, <a class="txt" href="dickson.html#E37:61_1_2_3_3"><i><font color="Green">E79</font></i></a></i>;<br/><a NAME="E47:61_1_2_3_3"/><i><font color="Green">E85</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>33</sub></font>,<font color="Maroon">c<sub>34</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E43:61_1_2_3_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="dickson.html#E46:61_1_2_3_3"><i><font color="Green">E84</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E48:61_1_2_3_3"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E49:61_1_2_3_3"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E50:61_1_2_3_3"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then </b><i><font color="Green">E87</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="dickson.html#E49:61_1_2_3_3"><i><font color="Green">E86</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>23</sub></font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/><a NAME="E52:61_1_2_3_3"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E53:61_1_2_3_3"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E54:61_1_2_3_3"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">c<sub>25</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="dickson.html#E53:61_1_2_3_3"><i><font color="Green">E88</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/><a NAME="E56:61_1_2_3_3"/><b>then </b><i><font color="Green">E89</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K8">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K9">`2</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E5:61_1_2_3_3"><i><font color="Green">E59</font></i></a>, <a class="txt" href="dickson.html#E12:61_1_2_3_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="dickson.html#E13:61_1_2_3_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="dickson.html#E14:61_1_2_3_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="dickson.html#E15:61_1_2_3_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="dickson.html#E17:61_1_2_3_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="dickson.html#E35:61_1_2_3_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="dickson.html#E36:61_1_2_3_3"><i><font color="Green">E78</font></i></a>, <a class="txt" href="dickson.html#E44:61_1_2_3_3"><i><font color="Green">E82</font></i></a>, <a class="txt" href="dickson.html#E45:61_1_2_3_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="dickson.html#E47:61_1_2_3_3"><i><font color="Green">E85</font></i></a>, <a class="txt" href="dickson.html#E51:61_1_2_3_3"><i><font color="Green">E87</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/><a NAME="E57:61_1_2_3_3"/>
( <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K8">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="yellow_3.html#K9">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="dickson.html#E4:61_1_2_3_3"><i><font color="Green">E58</font></i></a>, <a class="txt" href="dickson.html#E12:61_1_2_3_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/><a NAME="E58:61_1_2_3_3"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow_3.html#K2">["</a><span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="yellow_3.html#K2">"]</a></span>
 <b>by </b><i><a class="txt" href="dickson.html#E41:61_1_2_3_3"><i><font color="Green">E80</font></i></a>, <a class="txt" href="dickson.html#E56:61_1_2_3_3"><i><font color="Green">E89</font></i></a>, <a class="ref" href="yellow_3.html#T10">YELLOW_3:10</a></i>;<br/><a NAME="E59:61_1_2_3_3"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/><b>hence </b><a NAME="E60:61_1_2_3_3"/>
<font color="Maroon">c<sub>20</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:61_1_2_3"/>
<font color="Maroon">c<sub>19</sub></font> is <a href="waybel_0.html#V23">isomorphic</a>
 <b>by </b><i><a class="txt" href="dickson.html#E5:61_1_2_3"><i><font color="Green">E54</font></i></a>, <a class="txt" href="dickson.html#E7:61_1_2_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E23:61_1_2"/>
<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span>, <a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="waybel_1.html#D8">WAYBEL_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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