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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:56"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="dickson.html#V4" target="_self">Dickson</a>
 <b>and </b><br/><a NAME="E2:56"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="dickson.html#V4" target="_self">Dickson</a>
 <b>and </b><br/><a NAME="E3:56"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a>
 <b>and </b><br/><a NAME="E4:56"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V3">transitive</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i><a class="txt" href="dickson.html#E3:56"><i><font color="Green">E42</font></i></a>, <a class="ref" href="dickson.html#D3" target="_self">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V3">transitive</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i><a class="txt" href="dickson.html#E4:56"><i><font color="Green">E43</font></i></a>, <a class="ref" href="dickson.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E7:56"/>
( <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="yellow_3.html#K3">:]</a></span> is <a href="orders_2.html#V2">reflexive</a> &amp; <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="yellow_3.html#K3">:]</a></span> is <a href="orders_2.html#V3">transitive</a> )
 
;<br/>
<b>hence </b><a NAME="E8:56"/><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3">:]</a></span> is <a href="dickson.html#V3" target="_self">quasi_ordered</a>
 <b>by </b><i><a class="ref" href="dickson.html#D3" target="_self">Def3</a></i>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1"/>
( ( not <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a> &amp; not <font color="Maroon">c<sub>2</sub></font> is <a href="struct_0.html#V3">empty</a> ) or <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a> or <font color="Maroon">c<sub>2</sub></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="56_1_1"><b>suppose </b></a><a NAME="E1:56_1_1"/>
( not <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a> &amp; not <font color="Maroon">c<sub>2</sub></font> is <a href="struct_0.html#V3">empty</a> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a>  ;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="yellow_3.html#K3">:]</a></span>;<br/><a NAME="E3:56_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="normsp_1.html#NM2">sequence</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="yellow_3.html#K3">:]</a></span>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="normsp_1.html#NM2">sequence</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="yellow_3.html#K3">:]</a></span>;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="yellow_3.html#K8">`1</a> ;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E2:56_1_1_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>5</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E5:56_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>11</sub></font> is    <a href="bhsp_3.html#M1">subsequence</a> of <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E6:56_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>11</sub></font> is <a href="dickson.html#V2" target="_self">weakly-ascending</a>
 <b>by </b><i><a class="txt" href="dickson.html#E1:56"><i><font color="Green">E40</font></i></a>, <a class="ref" href="dickson.html#T36" target="_self">Th36</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being  <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a><b> such that </b><br/><a NAME="E8:56_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E5:56_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="bhsp_3.html#D4">BHSP_3:def 4</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="yellow_3.html#K9">`2</a> ;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E10:56_1_1_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <font color="Maroon">c<sub>13</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>6</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:56_1_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="dickson.html#E2:56"><i><font color="Green">E41</font></i></a>, <a class="txt" href="dickson.html#E4:56"><i><font color="Green">E43</font></i></a>, <a class="ref" href="dickson.html#T30" target="_self">Th30</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font>
;<br/>

<b>thus </b><a NAME="E14:56_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E13:56_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="seqm_3.html#T7">SEQM_3:7</a></i>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>, <font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="yellow_3.html#K3">:]</a></span> ;<br/>
<a NAME="E16:56_1_1_1"/><i><font color="Green">E51</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green">E52</font></i>: <font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font> = 
<font color="Maroon">c<sub>10</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="dickson.html#E8:56_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="yellow_3.html#K8">`1</a> 
<b>by </b><i><a class="txt" href="dickson.html#E2:56_1_1_1"><i><font color="Green">E45</font></i></a></i>
;<br/>
<i><font color="Green">E53</font></i>: <font color="Maroon">c<sub>11</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font> = 
<font color="Maroon">c<sub>10</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="dickson.html#E8:56_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="dickson.html#E16:56_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> <a href="yellow_3.html#K8">`1</a> 
<b>by </b><i><a class="txt" href="dickson.html#E2:56_1_1_1"><i><font color="Green">E45</font></i></a></i>
;<br/>
<a NAME="E19:56_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="orders_2.html#V3">transitive</a>
 
<b>by </b><i><a class="txt" href="dickson.html#E3:56"><i><font color="Green">E42</font></i></a>, <a class="ref" href="dickson.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E20:56_1_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="yellow_3.html#K8">`1</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> <a href="yellow_3.html#K8">`1</a> 
 
<b>by </b><i><a class="txt" href="dickson.html#E6:56_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="dickson.html#E13:56_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="dickson.html#E17:56_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="dickson.html#E18:56_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="dickson.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E21:56_1_1_1"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="yellow_3.html#K9">`2</a>  &amp; <font color="Maroon">c<sub>14</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="yellow_3.html#K9">`2</a>  )
 
<b>by </b><i><a class="txt" href="dickson.html#E10:56_1_1_1"><i><font color="Green">E49</font></i></a></i>;<br/>
<b>hence </b><a NAME="E22:56_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E13:56_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="dickson.html#E20:56_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="yellow_3.html#T12">YELLOW_3:12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:56_1_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="yellow_3.html#K3">:]</a></span> holds <br/>(  <a href="dickson.html#K6" target="_self">min-classes</a> <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; not  <a href="dickson.html#K6" target="_self">min-classes</a> <font color="Olive">b<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 <b>by </b><i><a class="txt" href="dickson.html#E8:56"><i><font color="Green">E44</font></i></a>, <a class="ref" href="dickson.html#T32" target="_self">Th32</a></i>;<br/><b>hence </b><a NAME="E5:56_1_1"/>
<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3">:]</a></span> is <a href="dickson.html#V4" target="_self">Dickson</a>
 <b>by </b><i><a class="txt" href="dickson.html#E8:56"><i><font color="Green">E44</font></i></a>, <a class="ref" href="dickson.html#T33" target="_self">Th33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2"><b>suppose </b></a><a NAME="E1:56_1_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a> or <font color="Maroon">c<sub>2</sub></font> is <a href="struct_0.html#V3">empty</a> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_1_2_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a> or <font color="Maroon">c<sub>2</sub></font> is <a href="struct_0.html#V3">empty</a> )
 </span><b>by </b><i><a class="txt" href="dickson.html#E1:56_1_2"><i><font color="Green">E45</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:56_1_2_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> as  <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E3:56_1_2_1_1_1"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><b>hence </b><a NAME="E4:56_1_2_1_1_1"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:56_1_2_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> as  <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E3:56_1_2_1_1_2"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><b>hence </b><a NAME="E4:56_1_2_1_1_2"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E3:56_1_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3">:]</a></span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/><a NAME="E4:56_1_2"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3">:]</a></span> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><b>hence </b><a NAME="E5:56_1_2"/>
<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3">:]</a></span> is <a href="dickson.html#V4" target="_self">Dickson</a>
 <b>by </b><i><a class="ref" href="dickson.html#T37" target="_self">Th37</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
