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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="finset_1.html#V1">finite</a> <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="finset_1.html#V1">finite</a> <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>2</sub></font>, <a href="numbers.html#K1">REAL</a> ;<br/>





<b>assume </b><a NAME="E1:64"/><i><font color="Green">E43</font></i>: 
(  <a href="rfunct_3.html#K19" target="_self">max+</a> <font color="Maroon">c<sub>3</sub></font>, <a href="rfunct_3.html#K19" target="_self">max+</a> <font color="Maroon">c<sub>4</sub></font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  &amp;  <a href="rfunct_3.html#K20" target="_self">max-</a> <font color="Maroon">c<sub>3</sub></font>, <a href="rfunct_3.html#K20" target="_self">max-</a> <font color="Maroon">c<sub>4</sub></font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  )
 ;<br/>

<a NAME="E2:64"/><i><font color="Green">E44</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a>  &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a>  &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19" target="_self">max+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a>  &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K20" target="_self">max-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a>  &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19" target="_self">max+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a>  &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K20" target="_self">max-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a>  )
 
;<br/>
<a NAME="E3:64"/><i><font color="Green">E45</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19" target="_self">max+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K20" target="_self">max-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19" target="_self">max+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K20" target="_self">max-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D10" target="_self">Def10</a>, <a class="ref" href="rfunct_3.html#D11" target="_self">Def11</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="limfunc1.html#K2">left_closed_halfline</a> 0;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="limfunc1.html#K3">right_closed_halfline</a> 0;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><a href="rfunct_3.html#K19" target="_self">max+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><a href="rfunct_3.html#K20" target="_self">max-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><a href="rfunct_3.html#K19" target="_self">max+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><a href="rfunct_3.html#K20" target="_self">max-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>;<br/>
<i><font color="Green">E46</font></i>: <span class="p1">(<span class="default"><a href="limfunc1.html#K2">left_closed_halfline</a> 0</span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K3">right_closed_halfline</a> 0</span>)</span> = 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,0</span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="limfunc1.html#T19">LIMFUNC1:19</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="rcomp_1.html#T14">RCOMP_1:14</a></i>
;<br/>
<a NAME="E5:64"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K2">left_closed_halfline</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19" target="_self">max+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K3">right_closed_halfline</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K20" target="_self">max-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K2">left_closed_halfline</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19" target="_self">max+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K3">right_closed_halfline</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K20" target="_self">max-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T39" target="_self">Th39</a>, <a class="ref" href="rfunct_3.html#T42" target="_self">Th42</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default"><a href="rfunct_3.html#K19" target="_self">max+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>, <font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><a href="rfunct_3.html#K20" target="_self">max-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>, <font color="Maroon">c<sub>13</sub></font> = <span class="p1">(<span class="default"><a href="rfunct_3.html#K19" target="_self">max+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>, <font color="Maroon">c<sub>14</sub></font> = <span class="p1">(<span class="default"><a href="rfunct_3.html#K20" target="_self">max-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E7:64"/>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="card_2.html#T64">CARD_2:64</a></i>;<br/>
<a NAME="E8:64"/><b>then </b><i><font color="Green">E48</font></i>: 
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> )
 
;<br/>
<i><font color="Green">E49</font></i>: <span class="p1">(<span class="default"><a href="limfunc1.html#K2">left_closed_halfline</a> 0</span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K3">right_closed_halfline</a> 0</span>)</span> = 
<a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,0</span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="ref" href="limfunc1.html#T25">LIMFUNC1:25</a></i>
<br/>.= 
<a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="rcomp_1.html#T12">RCOMP_1:12</a></i>
<br/>.= 
 <a href="numbers.html#K1">REAL</a> 

;<br/>
<a NAME="E10:64"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E5:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="relat_1.html#T175">RELAT_1:175</a></i>;<br/>
<a NAME="E11:64"/>
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T178">RELAT_1:178</a></i>;<br/>
<a NAME="E12:64"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <a href="numbers.html#K1">REAL</a>  <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <a href="numbers.html#K1">REAL</a>  )
 
<b>by </b><i><a class="ref" href="relat_1.html#T167">RELAT_1:167</a>, <a class="ref" href="relat_1.html#T169">RELAT_1:169</a></i>;<br/>
<a NAME="E13:64"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E10:64"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E14:64"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="limfunc1.html#K2">left_closed_halfline</a> 0</span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K3">right_closed_halfline</a> 0</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E5:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="relat_1.html#T175">RELAT_1:175</a></i>;<br/>
<a NAME="E15:64"/>
<font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T178">RELAT_1:178</a></i>;<br/>
<a NAME="E16:64"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <a href="numbers.html#K1">REAL</a>  <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <a href="numbers.html#K1">REAL</a>  )
 
<b>by </b><i><a class="ref" href="relat_1.html#T167">RELAT_1:167</a>, <a class="ref" href="relat_1.html#T169">RELAT_1:169</a></i>;<br/>
<a NAME="E17:64"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E9:64"><i><font color="Green">E49</font></i></a>, <a class="txt" href="rfunct_3.html#E14:64"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><a NAME="E1:64_3"/><i><font color="Green">E54</font></i>: 
(  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>13</sub></font> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="rfunct_3.html#E1:64"><i><font color="Green">E43</font></i></a>, <a class="ref" href="rfinseq.html#T9">RFINSEQ:9</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_3_1_1"/>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>15</sub></font> or 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> or <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_3_1_1_1"><b>case </b></a><a NAME="E1:64_3_1_1_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><div class="add"><a NAME="E2:64_3_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19" target="_self">max+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19" target="_self">max+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="rfunct_3.html#T38" target="_self">Th38</a></i>;<br/><b>hence </b><a NAME="E3:64_3_1_1_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E1:64"><i><font color="Green">E43</font></i></a>, <a class="txt" href="rfunct_3.html#E2:64"><i><font color="Green">E44</font></i></a>, <a class="ref" href="rfinseq.html#T11">RFINSEQ:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_3_1_1_2"><b>case </b></a><a NAME="E1:64_3_1_1_2"/>
0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><div class="add"><a NAME="E2:64_3_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K2">left_closed_halfline</a> 0</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K3">right_closed_halfline</a> 0</span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K2">left_closed_halfline</a> 0</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2">(<span class="default"><a href="limfunc1.html#K3">right_closed_halfline</a> 0</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="rfunct_3.html#E4:64"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_1.html#T137">FUNCT_1:137</a></i>;<br/><b>hence </b><a NAME="E3:64_3_1_1_2"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E1:64"><i><font color="Green">E43</font></i></a>, <a class="txt" href="rfunct_3.html#E3:64"><i><font color="Green">E45</font></i></a>, <a class="txt" href="rfunct_3.html#E5:64"><i><font color="Green">E47</font></i></a>, <a class="txt" href="rfunct_3.html#E8:64"><i><font color="Green">E48</font></i></a>, <a class="txt" href="rfunct_3.html#E13:64"><i><font color="Green">E51</font></i></a>, <a class="txt" href="rfunct_3.html#E17:64"><i><font color="Green">E53</font></i></a>, <a class="txt" href="rfunct_3.html#E1:64_3"><i><font color="Green">E54</font></i></a>, <a class="ref" href="rfinseq.html#T10">RFINSEQ:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_3_1_1_3"><b>case </b></a><a NAME="E1:64_3_1_1_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:64_3_1_1_3"/><b>then </b>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>15</sub></font> &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E3:64_3_1_1_3"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K20" target="_self">max-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K20" target="_self">max-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="rfunct_3.html#T41" target="_self">Th41</a></i>;<br/><b>hence </b><a NAME="E4:64_3_1_1_3"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E1:64"><i><font color="Green">E43</font></i></a>, <a class="txt" href="rfunct_3.html#E2:64"><i><font color="Green">E44</font></i></a>, <a class="ref" href="rfinseq.html#T11">RFINSEQ:11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:64_3"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E19:64"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 <b>by </b><i><a class="txt" href="rfunct_3.html#E2:64"><i><font color="Green">E44</font></i></a>, <a class="ref" href="rfinseq.html#T11">RFINSEQ:11</a></i>;<br/>


</div>
