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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="yellow_3.html#K3">:]</a></span>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="waybel24.html#K1" target="_self">Proj</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/>
<b>assume </b><a NAME="E2:18"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="waybel_0.html#V5">antitone</a>
 ;<br/>

<div><i><font color="Green">E9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E1:18_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><a NAME="E2:18_1"/><i><font color="Green">E11</font></i>: 
<span class="p1">(<span class="default"><a href="waybel24.html#K1" target="_self">Proj</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="waybel24.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E3:18_1"/><i><font color="Green">E12</font></i>: 
<span class="p1">(<span class="default"><a href="waybel24.html#K1" target="_self">Proj</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="waybel24.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E4:18_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E5:18_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="yellow_3.html#K7">]</a></span>
 <b>by </b><i><a class="txt" href="waybel24.html#E1:18_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="yellow_3.html#T11">YELLOW_3:11</a></i>;<br/><b>hence </b><a NAME="E6:18_1"/>
<span class="p1">(<span class="default"><a href="waybel24.html#K1" target="_self">Proj</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="waybel24.html#K1" target="_self">Proj</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="waybel24.html#E2:18"><i><font color="Green">E8</font></i></a>, <a class="txt" href="waybel24.html#E2:18_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="waybel24.html#E3:18_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="waybel_0.html#D5">WAYBEL_0:def 5</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:18_2"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><a NAME="E2:18_2"/><i><font color="Green">E11</font></i>: 
<span class="p1">(<span class="default"><a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="waybel24.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E3:18_2"/><i><font color="Green">E12</font></i>: 
<span class="p1">(<span class="default"><a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="waybel24.html#T8" target="_self">Th8</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/><a NAME="E5:18_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><a NAME="E6:18_2"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7">[</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">]</a></span>
 <b>by </b><i><a class="txt" href="waybel24.html#E1:18_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="yellow_3.html#T11">YELLOW_3:11</a></i>;<br/><b>hence </b><a NAME="E7:18_2"/>
<span class="p1">(<span class="default"><a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="waybel24.html#E2:18"><i><font color="Green">E8</font></i></a>, <a class="txt" href="waybel24.html#E2:18_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="waybel24.html#E3:18_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="waybel_0.html#D5">WAYBEL_0:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:18"/>
(  <a href="waybel24.html#K1" target="_self">Proj</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> is <a href="waybel_0.html#V5">antitone</a> &amp;  <a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> is <a href="waybel_0.html#V5">antitone</a> )
 <b>by </b><i><a class="txt" href="waybel24.html#E3:18"><i><font color="Green">E9</font></i></a>, <a class="ref" href="waybel_9.html#D1">WAYBEL_9:def 1</a></i>;<br/>


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