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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>, <font color="Maroon">c<sub>4</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>5</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>





<b>assume </b><a NAME="E1:12"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> is <a href="waybel_0.html#V18">sups-preserving</a> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="waybel_0.html#V18">sups-preserving</a> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM2">Subset</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D33">WAYBEL_0:def 33</a><br/>

<a NAME="E2:12"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="waybel_0.html#R4">preserves_sup_of</a>  <a href="yellow_3.html#K4">proj1</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="waybel_0.html#R4">preserves_sup_of</a>  <a href="yellow_3.html#K5">proj2</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel20.html#E1:12"><i><font color="Green">E10</font></i></a>, <a class="ref" href="waybel_0.html#D33">WAYBEL_0:def 33</a></i>;<br/>
<b>assume </b><a NAME="E3:12"/><i><font color="Green">E12</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>7</sub></font>,<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>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a><br/>

<a NAME="E4:12"/><b>then </b><i><font color="Green">E13</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="yellow_3.html#K4">proj1</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="yellow_3.html#K5">proj2</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="yellow_3.html#T41">YELLOW_3:41</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font>;<br/>
<a NAME="E5:12"/><i><font color="Green">E14</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:12"/>
<font color="Maroon">c<sub>7</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">[:</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>
 
;<br/>
<a NAME="E7:12"/><b>then </b>
<font color="Maroon">c<sub>7</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">YELLOW_3:def 2</a></i>;<br/>
<a NAME="E8:12"/><b>then </b><i><font color="Green">E15</font></i>: 
(  <a href="yellow_3.html#K4">proj1</a> <span class="p1">(<span class="default"><span class="p2"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> &amp;  <a href="yellow_3.html#K5">proj2</a> <span class="p1">(<span class="default"><span class="p2"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="waybel20.html#E5:12"><i><font color="Green">E14</font></i></a>, <a class="ref" href="waybel20.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E9:12"/><b>then </b>
(  <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="yellow_3.html#K4">proj1</a> <span class="p1">(<span class="default"><span class="p2"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="yellow_3.html#K5">proj2</a> <span class="p1">(<span class="default"><span class="p2"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel20.html#E2:12"><i><font color="Green">E11</font></i></a>, <a class="txt" href="waybel20.html#E4:12"><i><font color="Green">E13</font></i></a>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>;<br/>
<b>hence </b><a NAME="E10:12"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="yellow_3.html#K3">:]</a></span>
 <b>by </b><i><a class="ref" href="yellow_3.html#T41">YELLOW_3:41</a></i>;<br/>

<b>hence </b> <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> = 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p4">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p4">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="yellow_3.html#K7">]</a></span>
<b>by </b><i><a class="txt" href="waybel20.html#E8:12"><i><font color="Green">E15</font></i></a>, <a class="ref" href="waybel20.html#T8" target="_self">Th8</a></i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p4">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p4">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="yellow_3.html#K7">]</a></span>
<b>by </b><i><a class="txt" href="waybel20.html#E2:12"><i><font color="Green">E11</font></i></a>, <a class="txt" href="waybel20.html#E4:12"><i><font color="Green">E13</font></i></a>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p4">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p4">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span></span><a href="yellow_3.html#K7">]</a></span>
<b>by </b><i><a class="txt" href="waybel20.html#E2:12"><i><font color="Green">E11</font></i></a>, <a class="txt" href="waybel20.html#E4:12"><i><font color="Green">E13</font></i></a>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>
<br/>.= 
<span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span><a href="yellow_3.html#K7">]</a></span>
<b>by </b><i><a class="txt" href="waybel20.html#E5:12"><i><font color="Green">E14</font></i></a>, <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>
<br/>.= 
<span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel20.html#E3:12"><i><font color="Green">E12</font></i></a>, <a class="ref" href="waybel20.html#T8" target="_self">Th8</a></i>
;<br/><br/>


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