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<b>let </b><font color="Maroon">p</font> be  <a href="yellow_1.html#V1">RelStr-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of 1;<br/><b>let </b><font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of 1;<br/>



<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = 0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E2:58"/><i><font color="Green">E26</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">z</font></span>)</span>
 <b>and </b><br/><a NAME="E3:58"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">z</font></span>)</span> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<a NAME="E4:58"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:58"/><i><font color="Green">E31</font></i>: 
0 <a href="hidden.html#R2">in</a> 1
 
<b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:58_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">x</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">p</font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">z</font></span>)</span>
 ;<br/><a NAME="E2:58_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">g</font> = 0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">x</font>;<br/><i><font color="Green">E34</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">x</font></span>)</span> = 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</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 href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="pboole.html#D3">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">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:58_1_2"/>
<font color="Maroon">y</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">p</font></span>)</span>
 ;<br/><a NAME="E2:58_1_2"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> 1
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:58_1_2"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>consider </b><font color="Maroon">R</font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E5:58_1_2"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> )
 <b>by </b><i><a class="ref" href="dickson.html#D2" target="_self">Def2</a>, <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/><b>thus </b><a NAME="E6:58_1_2"/>
<span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</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">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="dickson.html#T4" target="_self">Th4</a>, , <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/></div><b>end;</b></div><a NAME="E5:58_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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">p</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/><b>hence </b><a NAME="E6:58_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">z</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">p</font></span>)</span> <b>by </b><i>, <a class="ref" href="funct_2.html#T5">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">p</font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">z</font> is <a href="struct_0.html#V3">empty</a> or  not <font color="Maroon">p</font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">z</font> is <a href="struct_0.html#V3">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">E70</font></i>: 
<font color="Maroon">p</font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">z</font> is <a href="struct_0.html#V3">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">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">p</font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><a NAME="E2:58_2_1_1_1"/><b>then </b><i><font color="Green">E71</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> <font color="Maroon">p</font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 <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">x</font> being    <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> <font color="Maroon">p</font></span>)</span>;<br/><b>consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:58_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E6:58_2_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</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">p</font></span>)</span>
 <b>and </b><br/><a NAME="E7:58_2_1_1_1"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</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">p</font></span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</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">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font>
 <b>by </b><i>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/><a NAME="E8:58_2_1_1_1"/><i><font color="Green">E73</font></i>: 
0 <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">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="dickson.html#T3" target="_self">Th3</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>consider </b><font color="Maroon">R</font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E10:58_2_1_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 0
 <b>and </b><br/><a NAME="E11:58_2_1_1_1"/><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="dickson.html#T3" target="_self">Th3</a>, <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/><a NAME="E12:58_2_1_1_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="dickson.html#D2" target="_self">Def2</a>, </i>;<br/><a NAME="E13:58_2_1_1_1"/>
<font color="Maroon">R</font> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i>, <a class="ref" href="dickson.html#T4" target="_self">Th4</a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E14:58_2_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="struct_0.html#D1">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">product</a> <font color="Maroon">p</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/><b>hence </b><a NAME="E4:58_2_1_1"/>
<font color="Maroon">f</font> is <a href="waybel_0.html#V23">isomorphic</a>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D38">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">E90</font></i>: 
not <font color="Maroon">p</font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">z</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">pz</font> = <font color="Maroon">p</font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">z</font> as  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">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">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:58_2_1_2_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 ;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:58_2_1_2_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E4:58_2_1_2_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:58_2_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> 1
 <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E6:58_2_1_2_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 0
 <b>by </b><i><a class="ref" href="dickson.html#D2" target="_self">Def2</a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:58_2_1_2_1"/>
<font color="Maroon">i</font> is  non <a href="struct_0.html#V3">empty</a>  <a href="struct_0.html#L1">1-sorted</a> 
 <b>by </b><i>, <a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">np</font> = <font color="Maroon">p</font> as  <a href="yellow_1.html#V1">RelStr-yielding</a> <a href="waybel_3.html#V4">yielding_non-empty_carriers</a>  <a href="pboole.html#M1">ManySortedSet</a> of 1 <b>by </b><i><a class="ref" href="yellow_6.html#D4">YELLOW_6:def 4</a></i>;<br/><a NAME="E5:58_2_1_2"/>
not  <a href="yellow_1.html#K5">product</a> <font color="Maroon">np</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><b>then reconsider </b><font color="Maroon">pp</font> =  <a href="yellow_1.html#K5">product</a> <font color="Maroon">p</font> as  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">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">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:58_2_1_2_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:58_2_1_2_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E3:58_2_1_2_2"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 ;<br/><i><font color="Green">E97</font></i>: <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> = 
0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">x1</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="funcop_1.html#D2">FUNCOP_1:def 2</a></i>
;<br/><i><font color="Green">E98</font></i>: <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> = 
0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">x2</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x2</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="funcop_1.html#D2">FUNCOP_1:def 2</a></i>
;<br/><a NAME="E6:58_2_1_2_2"/><i><font color="Green">E99</font></i>: 
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E7:58_2_1_2_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">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">[</a><span class="default">0,<font color="Maroon">x1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="ref" href="dickson.html#D2" target="_self">Def2</a>, <a class="ref" href="dickson.html#T4" target="_self">Th4</a>, , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">xa</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">ya</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:58_2_1_2_2"/>
<font color="Maroon">xa</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E11:58_2_1_2_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">ya</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x2</font></span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E12:58_2_1_2_2"/><i><font color="Green">E123</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">x1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">xa</font>,<font color="Maroon">ya</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><a NAME="E13:58_2_1_2_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x2</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="dickson.html#T5" target="_self">Th5</a>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E14:58_2_1_2_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">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">E124</font></i>: 
<font color="Maroon">f</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="58_2_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:58_2_1_2_3"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> implies <font color="Maroon">y</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">p</font></span>)</span> )
 ;<br/><b>assume </b><a NAME="E2:58_2_1_2_3"/>
<font color="Maroon">y</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">p</font></span>)</span>
 ;<br/><a NAME="E3:58_2_1_2_3"/><b>then </b>
<font color="Maroon">y</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">p</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">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:58_2_1_2_3"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E6:58_2_1_2_3"/><i><font color="Green">E126</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</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">p</font></span>)</span>
 <b>and </b><br/><a NAME="E7:58_2_1_2_3"/><i><font color="Green">E127</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <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">p</font></span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</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">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/><a NAME="E8:58_2_1_2_3"/><i><font color="Green">E128</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="dickson.html#D2" target="_self">Def2</a>, <a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E9:58_2_1_2_3"/><i><font color="Green">E129</font></i>: 
0 <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">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="dickson.html#T3" target="_self">Th3</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>consider </b><font color="Maroon">R</font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E11:58_2_1_2_3"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 0
 <b>and </b><br/><a NAME="E12:58_2_1_2_3"/><i><font color="Green">E131</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="dickson.html#T3" target="_self">Th3</a>, <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/><a NAME="E13:58_2_1_2_3"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="dickson.html#T4" target="_self">Th4</a>, , </i>;<br/><a NAME="E14:58_2_1_2_3"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> 0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E15:58_2_1_2_3"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>
 <b>by </b><i>, , <a class="ref" href="dickson.html#T5" target="_self">Th5</a>, </i>;<br/><a NAME="E16:58_2_1_2_3"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="wellfnd1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="dickson.html#T4" target="_self">Th4</a>, , <a class="ref" href="dickson.html#T5" target="_self">Th5</a>, </i>;<br/><b>hence </b><a NAME="E17:58_2_1_2_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D5">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">E135</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</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">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">pz</font>,<font color="Maroon">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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">pz</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">pz</font>;<br/><a NAME="E1:58_2_1_2_4"/>
not  <a href="yellow_1.html#K5">product</a> <font color="Maroon">np</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><a NAME="E2:58_2_1_2_4"/><b>then </b>
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">p</font></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/><a NAME="E3:58_2_1_2_4"/><b>then </b><i><font color="Green">E136</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> <font color="Maroon">p</font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/><a NAME="E4:58_2_1_2_4"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</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> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_1.html#D4">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">E138</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:58_2_1_2_4_1"/>
 ex <font color="Olive">f1</font>, <font color="Olive">g1</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> &amp; ( for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> 1 holds <br/> ex <font color="Olive">R</font> being   <a href="orders_2.html#L1">RelStr</a>  ex <font color="Olive">xi</font>, <font color="Olive">yi</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">R</font> st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">xi</font> <a href="hidden.html#R1">=</a> <font color="Olive">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">yi</font> <a href="hidden.html#R1">=</a> <font color="Olive">g1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">xi</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">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">f1</font> = 0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">x</font>;<br/>
<b>set </b><font color="Maroon">g1</font> = 0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">y</font>;<br/>
<b>reconsider </b><font color="Maroon">f1</font> = 0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">x</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<b>reconsider </b><font color="Maroon">g1</font> = 0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">y</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<b>take </b>
<font color="Maroon">f1</font>
;<br/>

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

<b>thus </b><a NAME="E3:58_2_1_2_4_1_1"/>
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E4:58_2_1_2_4_1_1"/>
<font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/>

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

<b>assume </b><a NAME="E5:58_2_1_2_4_1_1"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> 1
 ;<br/>

<a NAME="E6:58_2_1_2_4_1_1"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:58_2_1_2_4_1_1"/>
0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="dickson.html#T3" target="_self">Th3</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E8:58_2_1_2_4_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</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">p0</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 0 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>set </b><font color="Maroon">R</font> = <font color="Maroon">p0</font>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">p0</font> <b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">p0</font> <b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">p0</font>
;<br/>

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

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

<b>thus </b><a NAME="E12:58_2_1_2_4_1_1"/>
<font color="Maroon">p0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>

<b>thus </b><a NAME="E13:58_2_1_2_4_1_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="dickson.html#T5" target="_self">Th5</a>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/>

<b>thus </b><a NAME="E14:58_2_1_2_4_1_1"/>
<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="dickson.html#T5" target="_self">Th5</a>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/>

<b>thus </b><a NAME="E15:58_2_1_2_4_1_1"/>
<font color="Maroon">x'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y'</font>
 <b>by </b><i>, <a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">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">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="dickson.html#D2" target="_self">Def2</a>, <a class="ref" href="dickson.html#T4" target="_self">Th4</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="E6:58_2_1_2_4"/>
<font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 ;<br/><b>then consider </b><font color="Maroon">f1</font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:58_2_1_2_4"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E9:58_2_1_2_4"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E10:58_2_1_2_4"/><i><font color="Green">E143</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> 1 holds <br/> ex <font color="Olive">R</font> being   <a href="orders_2.html#L1">RelStr</a>  ex <font color="Olive">xi</font>, <font color="Olive">yi</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">R</font> st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">xi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">yi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">xi</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">yi</font> )
 <b>by </b><i><a class="ref" href="dickson.html#D2" target="_self">Def2</a>, <a class="ref" href="dickson.html#T4" target="_self">Th4</a>, <a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/><b>consider </b><font color="Maroon">R</font> being    <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">xi</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">yi</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E12:58_2_1_2_4"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 0
 <b>and </b><br/><a NAME="E13:58_2_1_2_4"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">xi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> 0
 <b>and </b><br/><a NAME="E14:58_2_1_2_4"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">yi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> 0
 <b>and </b><br/><a NAME="E15:58_2_1_2_4"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">xi</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">yi</font>
 <b>by </b><i><a class="ref" href="dickson.html#T3" target="_self">Th3</a>, <a class="ref" href="dickson.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E16:58_2_1_2_4"/>
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> 0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/><a NAME="E17:58_2_1_2_4"/><b>then </b><i><font color="Green">E172</font></i>: 
<font color="Maroon">xi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/><a NAME="E18:58_2_1_2_4"/><i><font color="Green">E173</font></i>: 
<font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> 0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/><a NAME="E19:58_2_1_2_4"/>
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pz</font>
 <b>by </b><i>, <a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E20:58_2_1_2_4"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , , <a class="ref" href="dickson.html#T7" target="_self">Th7</a>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:58_2_1_2"/>
<font color="Maroon">f</font> is <a href="waybel_0.html#V23">isomorphic</a>
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#T66">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">p</font> <a href="yellow_1.html#K4">.</a> <font color="Maroon">z</font>, <a href="yellow_1.html#K5">product</a> <font color="Maroon">p</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>
