<?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="struct_0.html#V3">empty</a> <a href="orders_2.html#V4">antisymmetric</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3" target="_self">:]</a></span>;<br/>



<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>5</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E2:81"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3" target="_self">:]</a></span>
 
;<br/>
<a NAME="E3:81"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</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>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#D2" target="_self">Def2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/>
<a NAME="E5:81"/>
not  <a href="funct_5.html#K1">proj1</a> <font color="Maroon">c<sub>6</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="yellow_3.html#K4" target="_self">proj1</a> <font color="Maroon">c<sub>3</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E7:81"/>
not  <a href="funct_5.html#K2">proj2</a> <font color="Maroon">c<sub>6</sub></font> is <a href="xboole_0.html#V1">empty</a>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> =  <a href="yellow_3.html#K5" target="_self">proj2</a> <font color="Maroon">c<sub>3</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>assume </b><a NAME="E9:81"/>
( <span class="p1"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3" target="_self">:]</a></span> is <a href="lattice3.html#V3">complete</a> or  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3" target="_self">:]</a></span> )
 ;<br/>

<a NAME="E10:81"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3" target="_self">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E11:81"/><b>then </b><i><font color="Green">E25</font></i>: 
(  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="yellow_3.html#T42" target="_self">Th42</a></i>;<br/>
<a NAME="E12:81"/><b>then </b><i><font color="Green">E26</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1"><a href="yellow_3.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="yellow_3.html#K6" target="_self">:]</a></span>,<span class="p1"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3" target="_self">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#T40" target="_self">Th40</a></i>;<br/>
<a NAME="E13:81"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3" target="_self">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#D2" target="_self">Def2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:81"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</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="yellow_3.html#E15:81"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="yellow_3.html#E15:81"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E18:81"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>11</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><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>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E1:81_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:81_1"/><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="funct_5.html#D1">FUNCT_5:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font> <b>by </b><i><a class="txt" href="yellow_3.html#E3:81_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_5.html#T4">FUNCT_5:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E6:81_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="yellow_3.html#K7" target="_self">]</a></span>
 
<b>by </b><i><a class="txt" href="yellow_3.html#E10:81"><i><font color="Green">E24</font></i></a>, <a class="txt" href="yellow_3.html#E15:81"><i><font color="Green">E27</font></i></a>, <a class="ref" href="yellow_0.html#D10">YELLOW_0:def 10</a></i>;<br/>
<a NAME="E7:81_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="yellow_3.html#K7" target="_self">]</a></span>
 
<b>by </b><i><a class="txt" href="yellow_3.html#E3:81_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<b>hence </b><a NAME="E8:81_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="yellow_3.html#T11" target="_self">Th11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:81"/>
<font color="Maroon">c<sub>8</sub></font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>12</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2"><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>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E1:81_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:81_2"/><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="funct_5.html#D2">FUNCT_5:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="yellow_3.html#E3:81_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_5.html#T4">FUNCT_5:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E6:81_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="yellow_3.html#K7" target="_self">]</a></span>
 
<b>by </b><i><a class="txt" href="yellow_3.html#E10:81"><i><font color="Green">E24</font></i></a>, <a class="txt" href="yellow_3.html#E15:81"><i><font color="Green">E27</font></i></a>, <a class="ref" href="yellow_0.html#D10">YELLOW_0:def 10</a></i>;<br/>
<a NAME="E7:81_2"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="yellow_3.html#K7" target="_self">]</a></span>
 
<b>by </b><i><a class="txt" href="yellow_3.html#E3:81_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<b>hence </b><a NAME="E8:81_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="yellow_3.html#T11" target="_self">Th11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:81"/><b>then </b>
(  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>11</sub></font> &amp;  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow_3.html#E11:81"><i><font color="Green">E25</font></i></a>, <a class="txt" href="yellow_3.html#E18:81"><i><font color="Green">E28</font></i></a>, <a class="ref" href="yellow_0.html#D10">YELLOW_0:def 10</a></i>;<br/>
<a NAME="E21:81"/><b>then </b><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_3.html#E15:81"><i><font color="Green">E27</font></i></a>, <a class="ref" href="yellow_3.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E22:81"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="yellow_3.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="yellow_3.html#K6" target="_self">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E23:81"/><b>then </b>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="yellow_3.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="yellow_3.html#K6" target="_self">:]</a></span>
 
<b>by </b><i><a class="txt" href="yellow_3.html#E10:81"><i><font color="Green">E24</font></i></a>, <a class="txt" href="yellow_3.html#E12:81"><i><font color="Green">E26</font></i></a>, <a class="ref" href="yellow_0.html#T35">YELLOW_0:35</a></i>;<br/>
<a NAME="E24:81"/><b>then </b>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K4" target="_self">proj1</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K5" target="_self">proj2</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="yellow_3.html#K7" target="_self">]</a></span>
 
<b>by </b><i><a class="txt" href="yellow_3.html#E11:81"><i><font color="Green">E25</font></i></a>, <a class="ref" href="yellow_3.html#T44" target="_self">Th44</a></i>;<br/>
<b>hence </b><a NAME="E25:81"/>
<span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K4" target="_self">proj1</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K5" target="_self">proj2</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="yellow_3.html#E21:81"><i><font color="Green">E29</font></i></a>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>


</div>
