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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="lattice3.html#V3">complete</a> <a href="orders_2.html#NM1">Poset</a>;<br/>

<b>assume </b><a NAME="E1:98"/><i><font color="Green">E65</font></i>: 
 <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">c<sub>1</sub></font> is <a href="yellow_6.html#V7">(ITERATED_LIMITS)</a>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:98_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K2">the_universe_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="relat_1.html#V2">V3</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>assume </b><a NAME="E2:98_1"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> holds <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>  <a href="yellow_6.html#K2">the_universe_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E3:98_1"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> holds  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="waybel_5.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> is <a href="waybel_0.html#V1">directed</a>
 ;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="yellow_2.html#NK7" target="_self">Inf</a> ;<br/><a NAME="E4:98_1"/><i><font color="Green">E69</font></i>: 
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 <b>by </b><i><a class="ref" href="waybel_5.html#T16">WAYBEL_5:16</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> as   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="relset_1.html#D1">RELSET_1:def 1</a></i>;<br/><a NAME="E6:98_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> =  <a href="waybel_5.html#NK6" target="_self">Sups</a>  as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> #);<br/><a NAME="E8:98_1"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> #) holds <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> #);<br/>

<a NAME="E1:98_1_1"/>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> #)
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E2:98_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:98_1"/><i><font color="Green">E71</font></i>: 
 <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> #) is <a href="orders_2.html#V3">transitive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a><br/>

<b>assume </b><a NAME="E1:98_1_2"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/>

<b>thus </b><a NAME="E2:98_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="waybel11.html#E8:98_1"><i><font color="Green">E70</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:98_1"/>
 <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> #) is <a href="waybel_0.html#V7">directed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D5">YELLOW_6:def 5</a><br/>

<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>

<b>thus </b><a NAME="E1:98_1_3"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="waybel11.html#E8:98_1"><i><font color="Green">E70</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> #) as  <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel11.html#E9:98_1"><i><font color="Green">E71</font></i></a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font>) <b>-&gt; </b> non <a href="struct_0.html#V3">empty</a> <a href="waybel_0.html#V6">strict</a>  <a href="waybel_0.html#L1">NetStr</a> of <font color="Maroon">c<sub>1</sub></font> =  <a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E13:98_1"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> holds <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#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="pboole.html#S5">PBOOLE:sch 5</a>();<br/></i><a NAME="E14:98_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font> holds <br/><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:98_1_4"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> =  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E4:98_1_4"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>14</sub></font> is <a href="waybel_0.html#V1">directed</a>
 
<b>by </b><i><a class="txt" href="waybel11.html#E3:98_1"><i><font color="Green">E68</font></i></a></i>;<br/>
<a NAME="E5:98_1_4"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel11.html#E13:98_1"><i><font color="Green">E72</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:98_1_4"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> is   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel11.html#E4:98_1_4"><i><font color="Green">E73</font></i></a>, <a class="ref" href="waybel11.html#T20" target="_self">Th20</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="yellow_6.html#M3">net_set</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="yellow_6.html#T33">YELLOW_6:33</a></i>;<br/><a NAME="E16:98_1"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <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#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:98_1_5"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/>
<b>take </b><font color="Maroon">c<sub>15</sub></font> =  <a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>;<br/>

<b>thus </b><a NAME="E3:98_1_5"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="waybel11.html#E13:98_1"><i><font color="Green">E72</font></i></a></i>;<br/>

<b>thus </b><a NAME="E4:98_1_5"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="waybel11.html#D10" target="_self">Def10</a></i>;<br/>


</div><b>end;</b></div><i><font color="Green">E74</font></i>:  <a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="yellow_7.html#T45">YELLOW_7:45</a></i>
<br/>.= 
 <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="waybel11.html#E16:98_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>
;<br/><a NAME="E18:98_1"/><i><font color="Green">E75</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E19:98_1"/><b>then </b><i><font color="Green">E76</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E20:98_1"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> holds <font color="Maroon">c<sub>12</sub></font> <a href="yellow_6.html#K8">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/>
<a NAME="E2:98_1_7"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="yellow_6.html#K8">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel11.html#E13:98_1"><i><font color="Green">E72</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="yellow_6.html#K8">.</a> <font color="Maroon">c<sub>13</sub></font> as  <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E4:98_1_7"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K2">the_universe_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E2:98_1"><i><font color="Green">E67</font></i></a></i>;<br/>
<a NAME="E5:98_1_7"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K2">the_universe_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E2:98_1_7"><i><font color="Green">E78</font></i></a>, <a class="ref" href="waybel11.html#D10" target="_self">Def10</a></i>;<br/>
<b>hence </b><a NAME="E6:98_1_7"/>
<font color="Maroon">c<sub>12</sub></font> <a href="yellow_6.html#K8">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D14">YELLOW_6:def 14</a></i>;<br/>


</div><b>end;</b></div><a NAME="E21:98_1"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> holds <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="yellow_6.html#K8">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/>
<a NAME="E2:98_1_8"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="yellow_6.html#K8">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel11.html#E13:98_1"><i><font color="Green">E72</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="yellow_6.html#K8">.</a> <font color="Maroon">c<sub>13</sub></font> as  <a href="orders_2.html#V2">reflexive</a> <a href="waybel_0.html#V8">monotone</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E4:98_1_8"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="yellow_6.html#K8">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E20:98_1"><i><font color="Green">E77</font></i></a></i>;<br/>
<a NAME="E5:98_1_8"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<a NAME="E6:98_1_8"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<font color="Maroon">c<sub>10</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>

<br/>.= 
 <a href="yellow_2.html#NK6" target="_self">Sup</a> 
<b>by </b><i><a class="txt" href="waybel11.html#E6:98_1_8"><i><font color="Green">E81</font></i></a>, <a class="ref" href="waybel_5.html#D1">WAYBEL_5:def 1</a></i>
<br/>.= 
 <a href="yellow_2.html#NK6" target="_self">Sup</a> 
<b>by </b><i><a class="txt" href="waybel11.html#E2:98_1_8"><i><font color="Green">E79</font></i></a>, <a class="ref" href="waybel11.html#D10" target="_self">Def10</a></i>
<br/>.= 
 <a href="waybel_2.html#K1">sup</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="ref" href="waybel_2.html#D1">WAYBEL_2:def 1</a></i>
<br/>.= 
 <a href="waybel11.html#K1" target="_self">lim_inf</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="ref" href="waybel11.html#T22" target="_self">Th22</a></i>
;<br/>
<a NAME="E8:98_1_8"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="waybel11.html#R1" target="_self">is_S-limit_of</a> <font color="Maroon">c<sub>12</sub></font> <a href="yellow_6.html#K8">.</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="waybel11.html#D7" target="_self">Def7</a></i>;<br/>
<b>hence </b><a NAME="E9:98_1_8"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="yellow_6.html#K8">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel11.html#E2:98_1_8"><i><font color="Green">E79</font></i></a>, <a class="txt" href="waybel11.html#E4:98_1_8"><i><font color="Green">E80</font></i></a>, <a class="ref" href="waybel11.html#D8" target="_self">Def8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E22:98_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel11.html#E1:98_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="yellow_6.html#D14">YELLOW_6:def 14</a></i>;<br/><a NAME="E23:98_1"/><b>then </b><i><font color="Green">E80</font></i>: 
 <a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel11.html#E20:98_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="yellow_6.html#T34">YELLOW_6:34</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a<sub>1</sub></font> </span>}</span> ;<br/><a NAME="E24:98_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="yellow_6.html#T35">YELLOW_6:35</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel11.html#E17:98_1"><i><font color="Green">E74</font></i></a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font>,   <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="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a<sub>1</sub></font> </span>}</span> ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means verum;<br/><b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <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="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>) <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>1</sub></font> =  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font>;<br/><b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font>,   <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="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>) <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>1</sub></font> =  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>),<font color="Maroon">c<sub>1</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>),<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> : verum </span>}</span> ;<br/><b>set </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>) where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font>, B 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="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] </span>}</span> ;<br/><b>set </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B 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="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> ;<br/><a NAME="E26:98_1"/><i><font color="Green">E81</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font><br/>for <font color="Olive">b<sub>2</sub></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="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> holds <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>2</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>18</sub></font> be    <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="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>;<br/>



<a NAME="E1:98_1_9"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font>) iff ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:98_1_9_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel11.html#E17:98_1"><i><font color="Green">E74</font></i></a></i>;<br/>
<a NAME="E2:98_1_9_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>12</sub></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="E1:98_1_9_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font>)
 ;<br/><b>then consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font><b> such that </b><br/><a NAME="E3:98_1_9_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>18</sub></font>
 <b>and </b><br/><a NAME="E4:98_1_9_1_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>20</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/><a NAME="E7:98_1_9_1_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E8:98_1_9_1_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>hence </b><a NAME="E9:98_1_9_1_1"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel11.html#E18:98_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="waybel_5.html#T8">WAYBEL_5:8</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel11.html#E18:98_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="waybel11.html#E8:98_1_9_1_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="waybel_5.html#T9">WAYBEL_5:9</a></i>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="waybel11.html#D10" target="_self">Def10</a></i>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="yellow_6.html#K8">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel11.html#E13:98_1"><i><font color="Green">E72</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="waybel11.html#E2:98_1_9_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="waybel11.html#E3:98_1_9_1_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="yellow_6.html#D16">YELLOW_6:def 16</a></i>
;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">c<sub>20</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E4:98_1_9_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E5:98_1_9_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/>

<a NAME="E6:98_1_9_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E18:98_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="waybel11.html#E4:98_1_9_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="waybel_5.html#T8">WAYBEL_5:8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> ;<br/>
<a NAME="E9:98_1_9_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E8:98_1"><i><font color="Green">E70</font></i></a></i>;<br/>
<font color="Maroon">c<sub>19</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel11.html#E18:98_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="waybel11.html#E5:98_1_9_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="waybel11.html#E6:98_1_9_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="waybel_5.html#T9">WAYBEL_5:9</a></i>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="waybel11.html#D10" target="_self">Def10</a></i>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="yellow_6.html#K8">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel11.html#E13:98_1"><i><font color="Green">E72</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="waybel11.html#E2:98_1_9_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="yellow_6.html#D16">YELLOW_6:def 16</a></i>
;<br/>
<b>hence </b><a NAME="E11:98_1_9_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font>)
 <b>by </b><i><a class="txt" href="waybel11.html#E9:98_1_9_1"><i><font color="Green">E86</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:98_1_9"/>
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font>),<font color="Maroon">c<sub>1</sub></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="E27:98_1"/><i><font color="Green">E82</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>),<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> : verum </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>) where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font>, B 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="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_10"><b>proof </b></a><div class="add">

<a NAME="E1:98_1_10"/><i><font color="Green">E83</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T35">YELLOW_6:35</a></i>;<br/>
<a NAME="E2:98_1_10"/><i><font color="Green">E84</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font><br/>for <font color="Olive">b<sub>3</sub></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="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> holds <br/> <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>),<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>),<font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>19</sub></font> be    <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="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:98_1_10_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="domain_1.html#K1">]</a></span>
 ;<br/>

<a NAME="E2:98_1_10_1"/><i><font color="Green">E86</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# 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>10</sub></font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span> #)
 
<b>by </b><i><a class="ref" href="yellow_6.html#D16">YELLOW_6:def 16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel11.html#E17:98_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E4:98_1_10_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 
;<br/>
<a NAME="E5:98_1_10_1"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> ;<br/>
<a NAME="E7:98_1_10_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="orders_2.html#L1">RelStr</a> ex <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">b<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>4</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_10_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>22</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:98_1_10_1_1"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>22</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/>
<a NAME="E4:98_1_10_1_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel11.html#E13:98_1"><i><font color="Green">E72</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>25</sub></font> =  <a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>;<br/>
<a NAME="E5:98_1_10_1_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="waybel_3.html#K4">.</a> <font color="Maroon">c<sub>23</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="waybel11.html#K3" target="_self">Net-Str</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel11.html#E4:98_1_10_1_1"><i><font color="Green">E88</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span> ;<br/>
<b>take </b>
 <a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>26</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>26</sub></font>
;<br/>

<a NAME="E7:98_1_10_1_1"/>
<font color="Maroon">c<sub>26</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>26</sub></font>
 
;<br/>
<b>hence </b><a NAME="E8:98_1_10_1_1"/>
(  <a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>26</sub></font> )
 <b>by </b><i><a class="txt" href="waybel11.html#E13:98_1"><i><font color="Green">E72</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:98_1_10_1"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E5:98_1_10_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E9:98_1_10_1"/>
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>17</sub></font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_10_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>22</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="E1:98_1_10_1_2"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>)
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>23</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font><b> such that </b><br/><a NAME="E3:98_1_10_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E4:98_1_10_1_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="domain_1.html#K1">]</a></span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel11.html#E17:98_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="waybel11.html#E1:98_1_10"><i><font color="Green">E83</font></i></a></i>;<br/>
<a NAME="E7:98_1_10_1_2"/>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="yellow_3.html#K7">]</a></span>
 
<b>by </b><i><a class="txt" href="waybel11.html#E8:98_1_10_1"><i><font color="Green">E88</font></i></a>, <a class="txt" href="waybel11.html#E4:98_1_10_1_2"><i><font color="Green">E90</font></i></a>, <a class="ref" href="yellow_3.html#T11">YELLOW_3:11</a></i>;<br/>
<a NAME="E8:98_1_10_1_2"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E1:98_1_10_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="waybel11.html#E2:98_1_10_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="waybel11.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<font color="Maroon">c<sub>22</sub></font> = 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="waybel11.html#E3:98_1_10_1_2"><i><font color="Green">E89</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>25</sub></font>

;<br/>
<b>hence </b><a NAME="E10:98_1_10_1_2"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>17</sub></font>)
 <b>by </b><i><a class="txt" href="waybel11.html#E8:98_1_10_1_2"><i><font color="Green">E91</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:98_1_10_1"/><b>then </b><i><font color="Green">E89</font></i>: 
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>17</sub></font>),<font color="Maroon">c<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>),<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_7.html#T3">WAYBEL_7:3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font>] means <font color="Maroon">a<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>18</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>6</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font>) <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>1</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">c<sub>19</sub></font>;<br/>
<a NAME="E11:98_1_10_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>6</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S8">DOMAIN_1:sch 8</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>) as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>] means <font color="Maroon">a<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>17</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>7</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>) <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>1</sub></font> = <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E13:98_1_10_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>7</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S8">DOMAIN_1:sch 8</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>17</sub></font>) as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E15:98_1_10_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="yellow_4.html#R4">is_coarser_than</a> <font color="Maroon">c<sub>22</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_10_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>24</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_4.html#D2">YELLOW_4:def 2</a><br/>

<b>assume </b><a NAME="E1:98_1_10_1_3"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>25</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:98_1_10_1_3"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>25</sub></font>
 <b>and </b><br/><a NAME="E4:98_1_10_1_3"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>25</sub></font>
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>26</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>27</sub></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">c<sub>12</sub></font></span>)</span><b> such that </b><br/><a NAME="E6:98_1_10_1_3"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="txt" href="waybel11.html#E1:98_1_10"><i><font color="Green">E83</font></i></a>, <a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>26</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> ;<br/>
<b>set </b><font color="Maroon">c<sub>29</sub></font> = the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>19</sub></font>;<br/>
<a NAME="E8:98_1_10_1_3"/>
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>19</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>30</sub></font> = the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>19</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>30</sub></font>
;<br/>

<b>reconsider </b><font color="Maroon">c<sub>31</sub></font> = <font color="Maroon">c<sub>27</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E11:98_1_10_1_3"/><i><font color="Green">E93</font></i>: 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>31</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="yellow_3.html#K7">]</a></span>
 
<b>by </b><i><a class="txt" href="waybel11.html#E1:98_1_10_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="waybel11.html#E2:98_1_10_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="waybel11.html#E4:98_1_10_1_3"><i><font color="Green">E91</font></i></a>, <a class="txt" href="waybel11.html#E6:98_1_10_1_3"><i><font color="Green">E92</font></i></a>, <a class="txt" href="waybel11.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<a NAME="E12:98_1_10_1_3"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>26</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_3.html#T11">YELLOW_3:11</a></i>;<br/>
<b>hence </b><a NAME="E13:98_1_10_1_3"/>
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>32</sub></font> = <font color="Maroon">c<sub>26</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/>
<b>set </b><font color="Maroon">c<sub>33</sub></font> =  <a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span>;<br/>
<a NAME="E15:98_1_10_1_3"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="yellow_6.html#K8">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel11.html#E13:98_1"><i><font color="Green">E72</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>34</sub></font> = <font color="Maroon">c<sub>20</sub></font> <a href="waybel_3.html#K4">.</a> <font color="Maroon">c<sub>26</sub></font>, <font color="Maroon">c<sub>35</sub></font> = <font color="Maroon">c<sub>31</sub></font> <a href="waybel_3.html#K4">.</a> <font color="Maroon">c<sub>26</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="waybel11.html#E13:98_1"><i><font color="Green">E72</font></i></a></i>;<br/>
<i><font color="Green">E95</font></i>: <font color="Maroon">c<sub>24</sub></font> = 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="yellow_6.html#K8">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="waybel_3.html#K4">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel11.html#E3:98_1_10_1_3"><i><font color="Green">E90</font></i></a>, <a class="txt" href="waybel11.html#E6:98_1_10_1_3"><i><font color="Green">E92</font></i></a>, <a class="ref" href="yellow_6.html#T36">YELLOW_6:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>35</sub></font>
<b>by </b><i><a class="txt" href="waybel11.html#E15:98_1_10_1_3"><i><font color="Green">E94</font></i></a></i>
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>36</sub></font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="domain_1.html#K1">]</a></span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel11.html#E17:98_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="waybel11.html#E1:98_1_10"><i><font color="Green">E83</font></i></a></i>;<br/>
<i><font color="Green">E96</font></i>: <font color="Maroon">c<sub>30</sub></font> = 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>36</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>36</sub></font>

<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="yellow_6.html#K8">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="waybel_3.html#K4">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_6.html#T36">YELLOW_6:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel11.html#K3" target="_self">Net-Str</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>34</sub></font>
<b>by </b><i><a class="txt" href="waybel11.html#E15:98_1_10_1_3"><i><font color="Green">E94</font></i></a></i>
;<br/>
<a NAME="E20:98_1_10_1_3"/>
<font color="Maroon">c<sub>20</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>31</sub></font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E11:98_1_10_1_3"><i><font color="Green">E93</font></i></a>, <a class="ref" href="yellow_3.html#T11">YELLOW_3:11</a></i>;<br/>
<a NAME="E21:98_1_10_1_3"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font> <a href="waybel_3.html#K4">.</a> <font color="Maroon">c<sub>26</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>31</sub></font> <a href="waybel_3.html#K4">.</a> <font color="Maroon">c<sub>26</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T28">WAYBEL_3:28</a></i>;<br/>
<b>hence </b><a NAME="E22:98_1_10_1_3"/>
<font color="Maroon">c<sub>30</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="waybel11.html#E15:98_1_10_1_3"><i><font color="Green">E94</font></i></a>, <a class="txt" href="waybel11.html#E17:98_1_10_1_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="waybel11.html#E19:98_1_10_1_3"><i><font color="Green">E96</font></i></a>, <a class="ref" href="waybel11.html#D10" target="_self">Def10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:98_1_10_1"/><b>then </b>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>),<font color="Maroon">c<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>17</sub></font>),<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel11.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E17:98_1_10_1"/>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>17</sub></font>),<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>),<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel11.html#E10:98_1_10_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:98_1_10"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>),<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> : verum </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>) where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font>, B 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="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_10_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</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="E1:98_1_10_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>),<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:98_1_10_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>18</sub></font>),<font color="Maroon">c<sub>1</sub></font>
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>20</sub></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="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>5</sub></font></span>)</span><b> such that </b><br/><a NAME="E5:98_1_10_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="txt" href="waybel11.html#E17:98_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="waybel11.html#E1:98_1_10"><i><font color="Green">E83</font></i></a>, <a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E6:98_1_10_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>),<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E2:98_1_10"><i><font color="Green">E84</font></i></a>, <a class="txt" href="waybel11.html#E3:98_1_10_2"><i><font color="Green">E85</font></i></a>, <a class="txt" href="waybel11.html#E5:98_1_10_2"><i><font color="Green">E86</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:98_1_10_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>) where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font>, B 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="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] </span>}</span> 
 ;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>17</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="E4:98_1_10"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>) where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font>, B 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="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>19</sub></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="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>5</sub></font></span>)</span><b> such that </b><br/><a NAME="E6:98_1_10"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>),<font color="Maroon">c<sub>1</sub></font>
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="domain_1.html#K1">]</a></span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel11.html#E17:98_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="waybel11.html#E1:98_1_10"><i><font color="Green">E83</font></i></a></i>;<br/>
<a NAME="E8:98_1_10"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>20</sub></font>),<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E2:98_1_10"><i><font color="Green">E84</font></i></a>, <a class="txt" href="waybel11.html#E6:98_1_10"><i><font color="Green">E85</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:98_1_10"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>),<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> : verum </span>}</span> 
 ;<br/>


</div><b>end;</b></div><a NAME="E28:98_1"/><i><font color="Green">E83</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B 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="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>) where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font>, B 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="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] </span>}</span> 
 <b>from </b><i><a class="ref" href="waybel11.html#S1" target="_self">WAYBEL11:sch 1</a>(<a class="txt" href="waybel11.html#E26:98_1"><i><font color="Green">E81</font></i></a>);<br/></i><a NAME="E29:98_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>2</sub></font>) where B 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="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] </span>}</span>  iff ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:98_1_11"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B 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="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_11_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:98_1_11_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B 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="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>18</sub></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="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>5</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:98_1_11_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/>
<b>take </b>
<font color="Maroon">c<sub>18</sub></font>
;<br/>

<b>thus </b><a NAME="E4:98_1_11_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span>
 <b>by </b><i><a class="txt" href="waybel11.html#E19:98_1"><i><font color="Green">E76</font></i></a></i>;<br/>

<a NAME="E5:98_1_11_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>18</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="waybel_5.html#K2">Frege</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
;<br/>
<b>thus </b><font color="Maroon">c<sub>17</sub></font> = 
 <a href="yellow_2.html#K5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="waybel11.html#E3:98_1_11_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="waybel11.html#E5:98_1_11_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="waybel_5.html#D2">WAYBEL_5:def 2</a></i>
;<br/>


</div><b>end;</b></div>

<b>given </b><font color="Maroon">c<sub>18</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:98_1_11"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span>
 <b>and </b><br/><a NAME="E3:98_1_11"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as    <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="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel11.html#E18:98_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="waybel11.html#E2:98_1_11"><i><font color="Green">E84</font></i></a></i>;<br/>
<a NAME="E5:98_1_11"/>
<font color="Maroon">c<sub>19</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="waybel_5.html#K2">Frege</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel11.html#E2:98_1_11"><i><font color="Green">E84</font></i></a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>17</sub></font> = 
 <a href="yellow_2.html#K5">//\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="waybel11.html#E3:98_1_11"><i><font color="Green">E85</font></i></a>, <a class="ref" href="waybel_5.html#D2">WAYBEL_5:def 2</a></i>
<br/>.= 
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_5.html#K2">Frege</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="waybel_5.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>
;<br/>
<b>hence </b><a NAME="E7:98_1_11"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B 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="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 ;<br/>


</div><b>end;</b></div><a NAME="E30:98_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B 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="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then </b><i><font color="Green">E84</font></i>:  <a href="yellow_2.html#NK6" target="_self">Sup</a>  = 
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>),<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> : verum </span>}</span> ,<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="waybel11.html#E27:98_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="waybel11.html#E28:98_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>
<br/>.= 
 <a href="waybel11.html#K1" target="_self">lim_inf</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>

;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> =  <a href="yellow_2.html#NK7" target="_self">Inf</a>  as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>set </b><font color="Maroon">c<sub>18</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_2.html#NK7" target="_self">Inf</a> </span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : verum </span>}</span> ;<br/><b>set </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : verum </span>}</span> ;<br/><a NAME="E33:98_1"/><i><font color="Green">E85</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : verum </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_2.html#NK7" target="_self">Inf</a> </span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : verum </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_13"><b>proof </b></a><div class="add">

<a NAME="E1:98_1_13"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> holds  <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> ,<font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_13_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>20</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font>;<br/>

<b>set </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>20</sub></font> </span>}</span> ;<br/>
<a NAME="E1:98_1_13_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>20</sub></font> </span>}</span>  iff ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_13_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>22</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:98_1_13_1_1_1"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>20</sub></font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>23</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font><b> such that </b><br/><a NAME="E3:98_1_13_1_1_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font>
 <b>and </b><br/><a NAME="E4:98_1_13_1_1_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>24</sub></font>;<br/><a NAME="E6:98_1_13_1_1_1"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>hence </b><a NAME="E7:98_1_13_1_1_1"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>thus </b><a NAME="E8:98_1_13_1_1_1"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="waybel11.html#E3:98_1_13_1_1_1"><i><font color="Green">E87</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">c<sub>23</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:98_1_13_1_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span>
 <b>and </b><br/><a NAME="E3:98_1_13_1_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="txt" href="waybel11.html#E2:98_1_13_1_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:98_1_13_1_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E8:98_1"><i><font color="Green">E70</font></i></a></i>;<br/>
<a NAME="E6:98_1_13_1_1"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E3:98_1_13_1_1"><i><font color="Green">E88</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:98_1_13_1_1"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>20</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel11.html#E5:98_1_13_1_1"><i><font color="Green">E89</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:98_1_13_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>20</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:98_1_13_1"/>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>20</sub></font> </span>}</span> ,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E2:98_1_13"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : verum </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_2.html#NK7" target="_self">Inf</a> </span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : verum </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_13_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>20</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="E1:98_1_13_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font><b> such that </b><br/><a NAME="E3:98_1_13_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>21</sub></font> </span>}</span> ,<font color="Maroon">c<sub>1</sub></font>
 ;<br/>
<a NAME="E4:98_1_13_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a> 
 
<b>by </b><i><a class="txt" href="waybel11.html#E1:98_1_13"><i><font color="Green">E86</font></i></a>, <a class="txt" href="waybel11.html#E3:98_1_13_2"><i><font color="Green">E87</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:98_1_13_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_2.html#NK7" target="_self">Inf</a> </span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : verum </span>}</span> 
 ;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>20</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>consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font>;<br/>
<b>assume </b><a NAME="E4:98_1_13"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_2.html#NK7" target="_self">Inf</a> </span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : verum </span>}</span> 
 ;<br/>

<a NAME="E5:98_1_13"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> st <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a> 
 
;<br/>
<a NAME="E6:98_1_13"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>21</sub></font> </span>}</span> ,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E1:98_1_13"><i><font color="Green">E86</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:98_1_13"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : verum </span>}</span> 
 ;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_14"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>20</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:98_1_14"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_2.html#NK7" target="_self">Inf</a> </span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : verum </span>}</span>  iff ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> st <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a>  )
 ;<br/><a NAME="E2:98_1_14"/><b>then </b>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_2.html#NK7" target="_self">Inf</a> </span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : verum </span>}</span>  iff <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a>  )
 ;<br/><b>hence </b><a NAME="E3:98_1_14"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_2.html#NK7" target="_self">Inf</a> </span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : verum </span>}</span>  iff <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#NK7" target="_self">Inf</a> </span>)</span></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/></div><b>end;</b></div><b>then </b> <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> ,<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>10</sub></font> : verum </span>}</span> ,<font color="Maroon">c<sub>1</sub></font> = 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>17</sub></font></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="txt" href="waybel11.html#E33:98_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>
<br/>.= 
 <a href="yellow_2.html#NK7" target="_self">Inf</a> 
<b>by </b><i><a class="ref" href="yellow_0.html#T39">YELLOW_0:39</a></i>
;<br/><a NAME="E36:98_1"/><b>then </b>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel11.html#K1" target="_self">lim_inf</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><a NAME="E37:98_1"/><b>then </b>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="waybel11.html#R1" target="_self">is_S-limit_of</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="waybel11.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E38:98_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><a href="yellow_2.html#NK7" target="_self">Inf</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel11.html#E22:98_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="waybel11.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E39:98_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_2.html#NK7" target="_self">Inf</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K2" target="_self">Scott-Convergence</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel11.html#E1:98"><i><font color="Green">E65</font></i></a>, <a class="txt" href="waybel11.html#E21:98_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="yellow_6.html#D26">YELLOW_6:def 26</a></i>;<br/><a NAME="E40:98_1"/><b>then </b>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="waybel11.html#R1" target="_self">is_S-limit_of</a>  <a href="yellow_6.html#K9">Iterated</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="waybel11.html#E23:98_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="waybel11.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E41:98_1"/><b>then </b>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 <b>by </b><i><a class="txt" href="waybel11.html#E31:98_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="waybel11.html#D7" target="_self">Def7</a></i>;<br/><b>hence </b><a NAME="E42:98_1"/>
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 <b>by </b><i><a class="txt" href="waybel11.html#E4:98_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:98"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_3.html#V3">continuous</a>
 <b>by </b><i><a class="ref" href="waybel_5.html#T18">WAYBEL_5:18</a></i>;<br/>


</div>
