<?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="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:99_1"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="pscomp_1.html#V9" target="_self">continuous</a> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="seq_2.html#V3">bounded</a>
 ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E2:99_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="pscomp_1.html#V9" target="_self">continuous</a>
 ;<br/><b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E3:99_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V3">bounded</a>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:99_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="pscomp_1.html#E2:99_1"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E4:99_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V1">bounded_above</a>
 <b>by </b><i><a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a></i>;<br/><a NAME="E5:99_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V1">bounded_above</a>
 <b>by </b><i><a class="ref" href="pscomp_1.html#D12" target="_self">Def12</a></i>;<br/><b>assume </b><a NAME="E6:99_1"/>
not <font color="Maroon">c<sub>3</sub></font> is <a href="pscomp_1.html#V7" target="_self">with_max</a>
 ;<br/><a NAME="E7:99_1"/><b>then </b><i><font color="Green">E66</font></i>: 
not <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="pscomp_1.html#V1" target="_self">with_max</a>
 <b>by </b><i><a class="ref" href="pscomp_1.html#D14" target="_self">Def14</a></i>;<br/><a NAME="E8:99_1"/><b>then </b><i><font color="Green">E67</font></i>: 
not  <a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</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="pscomp_1.html#E5:99_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="pscomp_1.html#D3" target="_self">Def3</a></i>;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" target="_self">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K7" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="pscomp_1.html#K9" target="_self">Inv</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" target="_self">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K7" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a>  ;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" target="_self">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K7" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a>  ;<br/><div><i><font color="Green">E68</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:99_1_1"/>
0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" target="_self">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K7" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:99_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>10</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="pscomp_1.html#K2" target="_self">sup</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" target="_self">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K7" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:99_1_1"/><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" target="_self">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K7" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:99_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</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="pscomp_1.html#E3:99_1_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <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> <b>by </b><i><a class="txt" href="pscomp_1.html#E3:99_1_1"><i><font color="Green">E69</font></i></a></i>;<br/><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K7" target="_self">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D17" target="_self">Def17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D16" target="_self">Def16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>

;<br/><b>hence </b><a NAME="E8:99_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pscomp_1.html#E8:99_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="pscomp_1.html#E4:99_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="pscomp_1.html#E5:99_1_1"><i><font color="Green">E71</font></i></a></i>;<br/></div><b>end;</b></div><b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" target="_self">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" target="_self">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K7" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E12:99_1"/>
 <a href="pscomp_1.html#K7" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font> is <a href="pscomp_1.html#V9" target="_self">continuous</a>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E2:99_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="pscomp_1.html#T55" target="_self">Th55</a></i>;<br/><a NAME="E13:99_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" target="_self">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K7" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="pscomp_1.html#V9" target="_self">continuous</a>
 <b>by </b><i><a class="ref" href="pscomp_1.html#T56" target="_self">Th56</a></i>;<br/><a NAME="E14:99_1"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="pscomp_1.html#K9" target="_self">Inv</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" target="_self">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K7" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> is <a href="pscomp_1.html#V9" target="_self">continuous</a>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E11:99_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="pscomp_1.html#T57" target="_self">Th57</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1_2"><b>now </b></a><div class="add"><a NAME="E1:99_1_2"/>
 <a href="pscomp_1.html#K9" target="_self">Inv</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" target="_self">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K7" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> is <a href="seq_2.html#V3">bounded</a>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:99_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="pscomp_1.html#E14:99_1"><i><font color="Green">E69</font></i></a></i>;<br/><a NAME="E2:99_1_2"/><b>then </b>
 <a href="pscomp_1.html#K9" target="_self">Inv</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" target="_self">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K7" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> is <a href="seq_2.html#V1">bounded_above</a>
 <b>by </b><i><a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a></i>;<br/><a NAME="E3:99_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K9" target="_self">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" target="_self">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K7" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V1">bounded_above</a>
 <b>by </b><i><a class="ref" href="pscomp_1.html#D12" target="_self">Def12</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E5:99_1_2"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" target="_self">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" target="_self">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K7" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="seq_4.html#D1">SEQ_4:def 1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:99_1_2_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 or <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_1_2_1_1"><b>suppose </b></a><a NAME="E1:99_1_2_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = 1 as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/><b>take </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font>;<br/><b>thus </b><a NAME="E3:99_1_2_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>let </b><font color="Maroon">c<sub>14</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E4:99_1_2_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" target="_self">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" target="_self">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K7" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>hence </b><a NAME="E5:99_1_2_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:99_1_2_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="pscomp_1.html#E5:99_1_2"><i><font color="Green">E70</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_1_2_1_2"><b>suppose </b></a><a NAME="E1:99_1_2_1_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font>;<br/><b>thus </b><a NAME="E2:99_1_2_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:99_1_2_1_2"><i><font color="Green">E71</font></i></a></i>;<br/><b>thus </b><a NAME="E3:99_1_2_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" target="_self">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" target="_self">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K7" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E5:99_1_2"><i><font color="Green">E70</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E17:99_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E18:99_1"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" target="_self">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" target="_self">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K7" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><a NAME="E19:99_1"/>
1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="pscomp_1.html#E17:99_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E21:99_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E22:99_1"/><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E5:99_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E24:99_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>and </b><br/><a NAME="E25:99_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E21:99_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="pscomp_1.html#E24:99_1"><i><font color="Green">E74</font></i></a></i>;<br/><a NAME="E27:99_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="pscomp_1.html#E5:99_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="pscomp_1.html#E7:99_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="pscomp_1.html#E21:99_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="pscomp_1.html#E25:99_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="pscomp_1.html#D3" target="_self">Def3</a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/><a NAME="E28:99_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E29:99_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><i><font color="Green">E77</font></i>: <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" target="_self">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" target="_self">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K7" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> = 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D18" target="_self">Def18</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K7" target="_self">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D17" target="_self">Def17</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D16" target="_self">Def16</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>

;<br/><a NAME="E31:99_1"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K9" target="_self">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" target="_self">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K7" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" target="_self">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" target="_self">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K7" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</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="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/><a NAME="E32:99_1"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E18:99_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="pscomp_1.html#E30:99_1"><i><font color="Green">E77</font></i></a></i>;<br/><a NAME="E33:99_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E29:99_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="real_2.html#T178">REAL_2:178</a></i>;<br/><a NAME="E34:99_1"/><b>then </b>
1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E17:99_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="real_2.html#T177">REAL_2:177</a></i>;<br/><a NAME="E35:99_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><b>hence </b><a NAME="E36:99_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pscomp_1.html#E22:99_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="pscomp_1.html#E25:99_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:99"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="pscomp_1.html#V9" target="_self">continuous</a> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="pscomp_1.html#V7" target="_self">with_max</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E3:99"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="pscomp_1.html#V9" target="_self">continuous</a>
 ;<br/>

<a NAME="E4:99"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="pscomp_1.html#V7" target="_self">with_max</a>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E2:99"><i><font color="Green">E63</font></i></a></i>;<br/>
<a NAME="E5:99"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="pscomp_1.html#V1" target="_self">with_max</a>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D14" target="_self">Def14</a></i>;<br/>
<a NAME="E6:99"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E7:99"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V1">bounded_above</a>
 <b>by </b><i><a class="ref" href="pscomp_1.html#D12" target="_self">Def12</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_2.html#D5">SEQ_2:def 5</a><br/>

<a NAME="E8:99"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="pscomp_1.html#V8" target="_self">with_min</a>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E2:99"><i><font color="Green">E63</font></i></a>, <a class="txt" href="pscomp_1.html#E3:99"><i><font color="Green">E64</font></i></a>, <a class="ref" href="pscomp_1.html#T60" target="_self">Th60</a></i>;<br/>
<a NAME="E9:99"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="pscomp_1.html#V2" target="_self">with_min</a>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D15" target="_self">Def15</a></i>;<br/>
<a NAME="E10:99"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V2">bounded_below</a>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E11:99"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="seq_2.html#V2">bounded_below</a>
 <b>by </b><i><a class="ref" href="pscomp_1.html#D11" target="_self">Def11</a></i>;<br/>


</div>
