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

<b>let </b><font color="Maroon" title="c1">p</font> be  <a href="yellow_1.html#V1" title="YELLOW_1:attr.1">RelStr-yielding</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of 1;<br/><b>let </b><font color="Maroon" title="c2">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 1;<br/>



<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> $1;<br/>
<b>consider </b><font color="Maroon" title="c3">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E2:58"/><i><font color="Green" title="E42">A1</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="waybel_3.html#K3" title="WAYBEL_3:func.3">.</a> <font color="Maroon" title="c2">z</font></span>)</span>
 <b>and </b><br/><a NAME="E3:58"/><i><font color="Green" title="E43">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="waybel_3.html#K3" title="WAYBEL_3:func.3">.</a> <font color="Maroon" title="c2">z</font></span>)</span> holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i>
<a NAME="E4:58"/><i><font color="Green" title="E44">A3</font></i>: 
<font color="Maroon" title="c2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:58"/><i><font color="Green" title="E45">A4</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 
<b>by </b><i><a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:58_1"/><i><font color="Green" title="E46">A5</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="waybel_3.html#K3" title="WAYBEL_3:func.3">.</a> <font color="Maroon" title="c2">z</font></span>)</span>
 ;<br/><a NAME="E2:58_1"/><b>then </b><i><font color="Green" title="E47">A6</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:58"><i><font color="Green" title="E43">A2</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c5">g</font> = <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c4">x</font>;<br/><i><font color="Green" title="E48">A7</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c4">x</font></span>)</span> = 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c1">p</font></span>)</span>
<b>by </b><i><a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:58_1_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c1">p</font></span>)</span>
 ;<br/><a NAME="E2:58_1_2"/><b>then </b><i><font color="Green" title="E49">A8</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:58_1_2"/><b>then </b><i><font color="Green" title="E50">A9</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">R</font> being    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> <b> such that </b><br/><a NAME="E5:58_1_2"/><i><font color="Green" title="E51">A10</font></i>: 
( <font color="Maroon" title="c7">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">R</font> )
 <b>by </b><i><a class="txt" href="dickson.html#E2:58_1_2"><i><font color="Green" title="E49">A8</font></i></a>, <a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/><b>thus </b><a NAME="E6:58_1_2"/>
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="dickson.html#E4:58"><i><font color="Green" title="E44">A3</font></i></a>, <a class="txt" href="dickson.html#E1:58_1"><i><font color="Green" title="E46">A5</font></i></a>, <a class="txt" href="dickson.html#E3:58_1_2"><i><font color="Green" title="E50">A9</font></i></a>, <a class="txt" href="dickson.html#E5:58_1_2"><i><font color="Green" title="E51">A10</font></i></a>, <a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>;<br/></div><b>end;</b></div><a NAME="E5:58_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E2:58_1"><i><font color="Green" title="E47">A6</font></i></a>, <a class="txt" href="dickson.html#E3:58_1"><i><font color="Green" title="E48">A7</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/><b>hence </b><a NAME="E6:58_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c3">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="waybel_3.html#K3" title="WAYBEL_3:func.3">.</a> <font color="Maroon" title="c2">z</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c1">p</font></span>)</span> <b>by </b><i><a class="txt" href="dickson.html#E2:58"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:58_2_1"/>
( <font color="Maroon" title="c1">p</font> <a href="waybel_3.html#K3" title="WAYBEL_3:func.3">.</a> <font color="Maroon" title="c2">z</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> or  not <font color="Maroon" title="c1">p</font> <a href="waybel_3.html#K3" title="WAYBEL_3:func.3">.</a> <font color="Maroon" title="c2">z</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_2_1_1"><b>suppose </b></a><a NAME="E1:58_2_1_1"/><i><font color="Green" title="E46">A11</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="waybel_3.html#K3" title="WAYBEL_3:func.3">.</a> <font color="Maroon" title="c2">z</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_2_1_1_1"/>
not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c1">p</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><a NAME="E2:58_2_1_1_1"/><b>then </b><i><font color="Green" title="E47">A12</font></i>: 
not  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c1">p</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c5">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c1">p</font></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c6">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:58_2_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font>
 <b>and </b><br/><a NAME="E6:58_2_1_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>and </b><br/><a NAME="E7:58_2_1_1_1"/><i><font color="Green" title="E48">A13</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c1">p</font></span>)</span> holds <br/><font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font>
 <b>by </b><i><a class="txt" href="dickson.html#E2:58_2_1_1_1"><i><font color="Green" title="E47">A12</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/><a NAME="E8:58_2_1_1_1"/><i><font color="Green" title="E49">A14</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E5:58"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">R</font> being    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> <b> such that </b><br/><a NAME="E10:58_2_1_1_1"/><i><font color="Green" title="E50">A15</font></i>: 
<font color="Maroon" title="c7">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E11:58_2_1_1_1"/><i><font color="Green" title="E51">A16</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">R</font>
 <b>by </b><i><a class="txt" href="dickson.html#E5:58"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/><a NAME="E12:58_2_1_1_1"/><i><font color="Green" title="E52">A17</font></i>: 
<font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">R</font>
 <b>by </b><i><a class="txt" href="dickson.html#E7:58_2_1_1_1"><i><font color="Green" title="E48">A13</font></i></a>, <a class="txt" href="dickson.html#E8:58_2_1_1_1"><i><font color="Green" title="E49">A14</font></i></a>, <a class="txt" href="dickson.html#E11:58_2_1_1_1"><i><font color="Green" title="E51">A16</font></i></a></i>;<br/><a NAME="E13:58_2_1_1_1"/>
<font color="Maroon" title="c7">R</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 <b>by </b><i><a class="txt" href="dickson.html#E1:58_2_1_1"><i><font color="Green" title="E46">A11</font></i></a>, <a class="txt" href="dickson.html#E10:58_2_1_1_1"><i><font color="Green" title="E50">A15</font></i></a>, <a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E14:58_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E12:58_2_1_1_1"><i><font color="Green" title="E52">A17</font></i></a>, <a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E3:58_2_1_1"/><b>then </b>
 <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c1">p</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 <b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/><b>hence </b><a NAME="E4:58_2_1_1"/>
<font color="Maroon" title="c4">f</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 <b>by </b><i><a class="txt" href="dickson.html#E1:58_2_1_1"><i><font color="Green" title="E46">A11</font></i></a>, <a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_2_1_2"><b>suppose </b></a><a NAME="E1:58_2_1_2"/><i><font color="Green" title="E46">A18</font></i>: 
not <font color="Maroon" title="c1">p</font> <a href="waybel_3.html#K3" title="WAYBEL_3:func.3">.</a> <font color="Maroon" title="c2">z</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">pz</font> = <font color="Maroon" title="c1">p</font> <a href="waybel_3.html#K3" title="WAYBEL_3:func.3">.</a> <font color="Maroon" title="c2">z</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:58_2_1_2_1"/><i><font color="Green" title="E47">A19</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">p</font>
 ;<br/><b>consider </b><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:58_2_1_2_1"/><i><font color="Green" title="E48">A20</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">p</font>
 <b>and </b><br/><a NAME="E4:58_2_1_2_1"/><i><font color="Green" title="E49">A21</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:58_2_1_2_1"><i><font color="Green" title="E47">A19</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:58_2_1_2_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 <b>by </b><i><a class="txt" href="dickson.html#E3:58_2_1_2_1"><i><font color="Green" title="E48">A20</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E6:58_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="dickson.html#E4:58_2_1_2_1"><i><font color="Green" title="E49">A21</font></i></a>, <a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:58_2_1_2_1"/>
<font color="Maroon" title="c6">i</font> is  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> 
 <b>by </b><i><a class="txt" href="dickson.html#E1:58_2_1_2"><i><font color="Green" title="E46">A18</font></i></a>, <a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c6">np</font> = <font color="Maroon" title="c1">p</font> as  <a href="yellow_1.html#V1" title="YELLOW_1:attr.1">RelStr-yielding</a> <a href="waybel_3.html#V4" title="WAYBEL_3:attr.4">yielding_non-empty_carriers</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of 1 <b>by </b><i><a class="ref" href="yellow_6.html#D4" title="YELLOW_6:def.4">YELLOW_6:def 4</a></i>;<br/><a NAME="E5:58_2_1_2"/>
not  <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c6">np</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c7">pp</font> =  <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c1">p</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">x1</font>, <font color="Maroon" title="c9">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:58_2_1_2_2"/><i><font color="Green" title="E47">A22</font></i>: 
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellfnd1.html#K1" title="WELLFND1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 <b>and </b><br/><a NAME="E2:58_2_1_2_2"/><i><font color="Green" title="E48">A23</font></i>: 
<font color="Maroon" title="c9">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellfnd1.html#K1" title="WELLFND1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 <b>and </b><br/><a NAME="E3:58_2_1_2_2"/><i><font color="Green" title="E49">A24</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x2</font>
 ;<br/><i><font color="Green" title="E50">A25</font></i>: <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x1</font> = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">x1</font>
<b>by </b><i><a class="txt" href="dickson.html#E3:58"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="dickson.html#E1:58_2_1_2_2"><i><font color="Green" title="E47">A22</font></i></a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">x1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>

;<br/><i><font color="Green" title="E51">A26</font></i>: <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x2</font> = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">x2</font>
<b>by </b><i><a class="txt" href="dickson.html#E3:58"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="dickson.html#E2:58_2_1_2_2"><i><font color="Green" title="E48">A23</font></i></a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>

;<br/><a NAME="E6:58_2_1_2_2"/><i><font color="Green" title="E52">A27</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E7:58_2_1_2_2"/>
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">x1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E8:58_2_1_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c8">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x2</font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:58_2_1_2_2"><i><font color="Green" title="E49">A24</font></i></a>, <a class="txt" href="dickson.html#E4:58_2_1_2_2"><i><font color="Green" title="E50">A25</font></i></a>, <a class="txt" href="dickson.html#E6:58_2_1_2_2"><i><font color="Green" title="E52">A27</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">xa</font>, <font color="Maroon" title="c11">ya</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:58_2_1_2_2"/>
<font color="Maroon" title="c10">xa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E11:58_2_1_2_2"/><i><font color="Green" title="E53">A28</font></i>: 
<font color="Maroon" title="c11">ya</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>and </b><br/><a NAME="E12:58_2_1_2_2"/><i><font color="Green" title="E54">A29</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c8">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">xa</font>,<font color="Maroon" title="c11">ya</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="dickson.html#E5:58_2_1_2_2"><i><font color="Green" title="E51">A26</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><a NAME="E13:58_2_1_2_2"/>
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="dickson.html#E11:58_2_1_2_2"><i><font color="Green" title="E53">A28</font></i></a>, <a class="txt" href="dickson.html#E12:58_2_1_2_2"><i><font color="Green" title="E54">A29</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E14:58_2_1_2_2"/>
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x2</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E8:58_2_1_2"/><b>then </b><i><font color="Green" title="E47">A30</font></i>: 
<font color="Maroon" title="c4">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>thus </b><a NAME="E1:58_2_1_2_3"/>
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> implies <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c1">p</font></span>)</span> )
 ;<br/><b>assume </b><a NAME="E2:58_2_1_2_3"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c1">p</font></span>)</span>
 ;<br/><a NAME="E3:58_2_1_2_3"/><b>then </b>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:58_2_1_2_3"/><i><font color="Green" title="E48">A31</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font>
 <b>and </b><br/><a NAME="E6:58_2_1_2_3"/><i><font color="Green" title="E49">A32</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>and </b><br/><a NAME="E7:58_2_1_2_3"/><i><font color="Green" title="E50">A33</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c1">p</font></span>)</span> holds <br/><font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/><a NAME="E8:58_2_1_2_3"/><i><font color="Green" title="E51">A34</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="dickson.html#E6:58_2_1_2_3"><i><font color="Green" title="E49">A32</font></i></a>, <a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E9:58_2_1_2_3"/><i><font color="Green" title="E52">A35</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E5:58"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">R</font> being    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> <b> such that </b><br/><a NAME="E11:58_2_1_2_3"/><i><font color="Green" title="E53">A36</font></i>: 
<font color="Maroon" title="c10">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E12:58_2_1_2_3"/><i><font color="Green" title="E54">A37</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">R</font>
 <b>by </b><i><a class="txt" href="dickson.html#E5:58"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/><a NAME="E13:58_2_1_2_3"/><i><font color="Green" title="E55">A38</font></i>: 
<font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">R</font>
 <b>by </b><i><a class="txt" href="dickson.html#E7:58_2_1_2_3"><i><font color="Green" title="E50">A33</font></i></a>, <a class="txt" href="dickson.html#E9:58_2_1_2_3"><i><font color="Green" title="E52">A35</font></i></a>, <a class="txt" href="dickson.html#E12:58_2_1_2_3"><i><font color="Green" title="E54">A37</font></i></a></i>;<br/><a NAME="E14:58_2_1_2_3"/><i><font color="Green" title="E56">A39</font></i>: 
<font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E8:58_2_1_2_3"><i><font color="Green" title="E51">A34</font></i></a>, <a class="ref" href="dickson.html#T1" target="_self" title="DICKSON:th.1">Th1</a></i>;<br/><a NAME="E15:58_2_1_2_3"/><i><font color="Green" title="E57">A40</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E3:58"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="dickson.html#E4:58"><i><font color="Green" title="E44">A3</font></i></a>, <a class="txt" href="dickson.html#E11:58_2_1_2_3"><i><font color="Green" title="E53">A36</font></i></a>, <a class="txt" href="dickson.html#E13:58_2_1_2_3"><i><font color="Green" title="E55">A38</font></i></a></i>;<br/><a NAME="E16:58_2_1_2_3"/>
<font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="wellfnd1.html#K1" title="WELLFND1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="dickson.html#E2:58"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="dickson.html#E4:58"><i><font color="Green" title="E44">A3</font></i></a>, <a class="txt" href="dickson.html#E7:58_2_1_2_3"><i><font color="Green" title="E50">A33</font></i></a>, <a class="txt" href="dickson.html#E9:58_2_1_2_3"><i><font color="Green" title="E52">A35</font></i></a>, <a class="txt" href="dickson.html#E11:58_2_1_2_3"><i><font color="Green" title="E53">A36</font></i></a>, <a class="txt" href="dickson.html#E12:58_2_1_2_3"><i><font color="Green" title="E54">A37</font></i></a></i>;<br/><b>hence </b><a NAME="E17:58_2_1_2_3"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="dickson.html#E5:58_2_1_2_3"><i><font color="Green" title="E48">A31</font></i></a>, <a class="txt" href="dickson.html#E14:58_2_1_2_3"><i><font color="Green" title="E56">A39</font></i></a>, <a class="txt" href="dickson.html#E15:58_2_1_2_3"><i><font color="Green" title="E57">A40</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><a NAME="E10:58_2_1_2"/><b>then </b><i><font color="Green" title="E48">A41</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">f'</font> = <font color="Maroon" title="c4">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c5">pz</font>,<font color="Maroon" title="c7">pp</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">pz</font>;<br/><a NAME="E1:58_2_1_2_4"/>
not  <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c6">np</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 ;<br/><a NAME="E2:58_2_1_2_4"/><b>then </b>
not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c1">p</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/><a NAME="E3:58_2_1_2_4"/><b>then </b><i><font color="Green" title="E49">A42</font></i>: 
not  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c1">p</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/><a NAME="E4:58_2_1_2_4"/><i><font color="Green" title="E50">A43</font></i>: 
<font color="Maroon" title="c8">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:58_2_1_2_4_1"/><i><font color="Green" title="E51">A44</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">y</font>
 ;<br/><a NAME="E2:58_2_1_2_4_1"/>
 ex <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">g1</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Olive" title="b2">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1 holds <br/> ex <font color="Olive" title="b4">R</font> being   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ex <font color="Olive" title="b5">xi</font>, <font color="Olive" title="b6">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b4">R</font> st <br/>( <font color="Olive" title="b4">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b5">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b6">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b5">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b6">yi</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2_1_2_4_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c11">f1</font> = <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">x</font>;<br/>
<b>set </b><font color="Maroon" title="c12">g1</font> = <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">y</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">f1</font> = <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">x</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c14">g1</font> = <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">y</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<b>take </b>
<font color="Maroon" title="c13">f1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c14">g1</font>
;<br/>

<b>thus </b><a NAME="E3:58_2_1_2_4_1_1"/>
<font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:58"><i><font color="Green" title="E43">A2</font></i></a></i>;<br/>

<b>thus </b><a NAME="E4:58_2_1_2_4_1_1"/>
<font color="Maroon" title="c14">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:58"><i><font color="Green" title="E43">A2</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c15">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E5:58_2_1_2_4_1_1"/><i><font color="Green" title="E52">A45</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 ;<br/>

<a NAME="E6:58_2_1_2_4_1_1"/><i><font color="Green" title="E53">A46</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="dickson.html#E5:58_2_1_2_4_1_1"><i><font color="Green" title="E52">A45</font></i></a>, <a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:58_2_1_2_4_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="dickson.html#E5:58"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E8:58_2_1_2_4_1_1"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">p0</font> = <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  <b>by </b><i><a class="ref" href="yellow_1.html#D3" title="YELLOW_1:def.3">YELLOW_1:def 3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c17">R</font> = <font color="Maroon" title="c16">p0</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">x'</font> = <font color="Maroon" title="c9">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c16">p0</font> <b>by </b><i><a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">y'</font> = <font color="Maroon" title="c10">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c16">p0</font> <b>by </b><i><a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c16">p0</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c18">x'</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c19">y'</font>
;<br/>

<b>thus </b><a NAME="E12:58_2_1_2_4_1_1"/>
<font color="Maroon" title="c16">p0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
 <b>by </b><i><a class="txt" href="dickson.html#E5:58_2_1_2_4_1_1"><i><font color="Green" title="E52">A45</font></i></a>, <a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>

<b>thus </b><a NAME="E13:58_2_1_2_4_1_1"/>
<font color="Maroon" title="c18">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
 <b>by </b><i><a class="txt" href="dickson.html#E6:58_2_1_2_4_1_1"><i><font color="Green" title="E53">A46</font></i></a>, <a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>;<br/>

<b>thus </b><a NAME="E14:58_2_1_2_4_1_1"/>
<font color="Maroon" title="c19">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
 <b>by </b><i><a class="txt" href="dickson.html#E6:58_2_1_2_4_1_1"><i><font color="Green" title="E53">A46</font></i></a>, <a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>;<br/>

<b>thus </b><a NAME="E15:58_2_1_2_4_1_1"/>
<font color="Maroon" title="c18">x'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c19">y'</font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:58_2_1_2_4_1"><i><font color="Green" title="E51">A44</font></i></a>, <a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:58_2_1_2_4_1"/>
<font color="Maroon" title="c8">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:58_2_1_2_4"><i><font color="Green" title="E49">A42</font></i></a>, <a class="txt" href="dickson.html#E4:58_2_1_2_4"><i><font color="Green" title="E50">A43</font></i></a>, <a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E6:58_2_1_2_4"/>
<font color="Maroon" title="c8">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c11">f1</font>, <font color="Maroon" title="c12">g1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E8:58_2_1_2_4"/><i><font color="Green" title="E51">A47</font></i>: 
<font color="Maroon" title="c11">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font>
 <b>and </b><br/><a NAME="E9:58_2_1_2_4"/><i><font color="Green" title="E52">A48</font></i>: 
<font color="Maroon" title="c12">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font>
 <b>and </b><br/><a NAME="E10:58_2_1_2_4"/><i><font color="Green" title="E53">A49</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1 holds <br/> ex <font color="Olive" title="b2">R</font> being   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ex <font color="Olive" title="b3">xi</font>, <font color="Olive" title="b4">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">R</font> st <br/>( <font color="Olive" title="b2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b4">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b4">yi</font> )
 <b>by </b><i><a class="txt" href="dickson.html#E3:58_2_1_2_4"><i><font color="Green" title="E49">A42</font></i></a>, <a class="txt" href="dickson.html#E4:58_2_1_2_4"><i><font color="Green" title="E50">A43</font></i></a>, <a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">R</font> being    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> , <font color="Maroon" title="c14">xi</font>, <font color="Maroon" title="c15">yi</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">R</font><b> such that </b><br/><a NAME="E12:58_2_1_2_4"/><i><font color="Green" title="E54">A50</font></i>: 
<font color="Maroon" title="c13">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E13:58_2_1_2_4"/><i><font color="Green" title="E55">A51</font></i>: 
<font color="Maroon" title="c14">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E14:58_2_1_2_4"/><i><font color="Green" title="E56">A52</font></i>: 
<font color="Maroon" title="c15">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E15:58_2_1_2_4"/><i><font color="Green" title="E57">A53</font></i>: 
<font color="Maroon" title="c14">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">yi</font>
 <b>by </b><i><a class="txt" href="dickson.html#E5:58"><i><font color="Green" title="E45">A4</font></i></a>, <a class="txt" href="dickson.html#E10:58_2_1_2_4"><i><font color="Green" title="E53">A49</font></i></a></i>;<br/><a NAME="E16:58_2_1_2_4"/>
<font color="Maroon" title="c11">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:58"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="dickson.html#E8:58_2_1_2_4"><i><font color="Green" title="E51">A47</font></i></a></i>;<br/><a NAME="E17:58_2_1_2_4"/><b>then </b><i><font color="Green" title="E58">A54</font></i>: 
<font color="Maroon" title="c14">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="txt" href="dickson.html#E13:58_2_1_2_4"><i><font color="Green" title="E55">A51</font></i></a>, <a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>;<br/><a NAME="E18:58_2_1_2_4"/><i><font color="Green" title="E59">A55</font></i>: 
<font color="Maroon" title="c12">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">y</font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:58"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="dickson.html#E9:58_2_1_2_4"><i><font color="Green" title="E52">A48</font></i></a></i>;<br/><a NAME="E19:58_2_1_2_4"/>
<font color="Maroon" title="c13">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">pz</font>
 <b>by </b><i><a class="txt" href="dickson.html#E12:58_2_1_2_4"><i><font color="Green" title="E54">A50</font></i></a>, <a class="ref" href="card_5.html#T1" title="CARD_5:th.1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E20:58_2_1_2_4"/>
<font color="Maroon" title="c9">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">y</font>
 <b>by </b><i><a class="txt" href="dickson.html#E14:58_2_1_2_4"><i><font color="Green" title="E56">A52</font></i></a>, <a class="txt" href="dickson.html#E15:58_2_1_2_4"><i><font color="Green" title="E57">A53</font></i></a>, <a class="txt" href="dickson.html#E17:58_2_1_2_4"><i><font color="Green" title="E58">A54</font></i></a>, <a class="txt" href="dickson.html#E18:58_2_1_2_4"><i><font color="Green" title="E59">A55</font></i></a>, <a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:58_2_1_2"/>
<font color="Maroon" title="c4">f</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 <b>by </b><i><a class="txt" href="dickson.html#E8:58_2_1_2"><i><font color="Green" title="E47">A30</font></i></a>, <a class="txt" href="dickson.html#E10:58_2_1_2"><i><font color="Green" title="E48">A41</font></i></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:58"/>
<font color="Maroon" title="c1">p</font> <a href="waybel_3.html#K3" title="WAYBEL_3:func.3">.</a> <font color="Maroon" title="c2">z</font>, <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c1">p</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="waybel_1.html#D8" title="WAYBEL_1:def.8">WAYBEL_1:def 8</a></i>;<br/>


</div>
