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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span>;<br/>
<a NAME="E1:55"/><i><font color="Green">E33</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:55_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:55_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E4:55_1_1"/><i><font color="Green">E35</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E5:55_1_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:55"><i><font color="Green">E33</font></i></a>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/><a NAME="E6:55_1_1"/><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E4:55_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1"><b>now </b></a><div class="add"><a NAME="E1:55_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:55_1_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:55_1_1_1_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E1:55_1_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:55_1_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E4:55_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="bagorder.html#E5:55_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="bagorder.html#E3:55_1_1_1_1"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E6:55_1_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E7:55_1_1_1_1"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E6:55_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="bagorder.html#E3:55_1_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/>
<b>thus </b><a NAME="E8:55_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="bagorder.html#E6:55_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="bagorder.html#E3:55_1_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="bagorder.html#E4:55_1_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="bagorder.html#E6:55_1_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="bagorder.html#E7:55_1_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="dickson.html#D15">DICKSON:def 15</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E2:55_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="seqm_3.html#V6">natural-yielding</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#D8">SEQM_3:def 8</a></i>;<br/><a NAME="E3:55_1_1_1"/>
 <a href="polynom1.html#K11">support</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E6:55_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="polynom1.html#T41">POLYNOM1:41</a></i>;<br/><a NAME="E4:55_1_1_1"/><b>then </b>
 <a href="polynom1.html#K11">support</a> <font color="Maroon">c<sub>4</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><b>hence </b><a NAME="E5:55_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="polynom1.html#V1">finite-support</a>
 <b>by </b><i><a class="ref" href="polynom1.html#D8">POLYNOM1:def 8</a></i>;<br/><a NAME="E6:55_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E4:55_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>hence </b><a NAME="E7:55_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:55_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E3:55_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:55_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  <a href="seqm_3.html#V6">natural-yielding</a> <a href="polynom1.html#V1">finite-support</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><a NAME="E4:55_1"/><i><font color="Green">E34</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_2"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/><b>thus </b><a NAME="E1:55_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>thus </b><a NAME="E2:55_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="bagorder.html#E4:55_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E3:55_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E4:55_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E3:55_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E6:55_1_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E7:55_1_2"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/><a NAME="E8:55_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="dickson.html#K11">OrderedNAT</a> 
 <b>by </b><i><a class="txt" href="bagorder.html#E4:55_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="bagorder.html#E6:55_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E9:55_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E7:55_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="dickson.html#D15">DICKSON:def 15</a></i>;<br/></div><b>end;</b></div><a NAME="E6:55_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></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="E7:55_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></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>hence </b><a NAME="E3:55"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="dickson.html#K11">OrderedNAT</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
