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<b>let </b><font color="Maroon">L1</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> , <font color="Maroon">L2</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> , <font color="Maroon">T1</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> , <font color="Maroon">T2</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">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L1</font>,<font color="Maroon">T1</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L2</font>,<font color="Maroon">T2</font>;<br/>





<b>assume </b><a NAME="E1:10"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">f</font> is <a href="waybel_0.html#V17">infs-preserving</a> &amp; <font color="Maroon">g</font> is <a href="waybel_0.html#V17">infs-preserving</a> )
 ;<br/>

<b>let </b><font color="Maroon">X</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">L1</font>,<font color="Maroon">L2</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#D32">WAYBEL_0:def 32</a><br/>

<a NAME="E2:10"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">f</font> <a href="waybel_0.html#R3">preserves_inf_of</a>  <a href="yellow_3.html#K4">proj1</a> <font color="Maroon">X</font> &amp; <font color="Maroon">g</font> <a href="waybel_0.html#R3">preserves_inf_of</a>  <a href="yellow_3.html#K5">proj2</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#D32">WAYBEL_0:def 32</a></i>;<br/>
<b>assume </b><a NAME="E3:10"/><i><font color="Green">E37</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">X</font>,<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</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#D30">WAYBEL_0:def 30</a><br/>

<a NAME="E4:10"/><b>then </b><i><font color="Green">E39</font></i>: 
(  <a href="yellow_0.html#R2">ex_inf_of</a>  <a href="yellow_3.html#K4">proj1</a> <font color="Maroon">X</font>,<font color="Maroon">L1</font> &amp;  <a href="yellow_0.html#R2">ex_inf_of</a>  <a href="yellow_3.html#K5">proj2</a> <font color="Maroon">X</font>,<font color="Maroon">L2</font> )
 
<b>by </b><i><a class="ref" href="yellow_3.html#T42">YELLOW_3:42</a></i>;<br/>
<b>set </b><font color="Maroon">iX</font> = <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>;<br/>
<a NAME="E5:10"/><i><font color="Green">E40</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L2</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:10"/>
<font color="Maroon">X</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">L1</font>,<font color="Maroon">L2</font></span><a href="yellow_3.html#K3">:]</a></span>
 
;<br/>
<a NAME="E7:10"/><b>then </b>
<font color="Maroon">X</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">L1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L2</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="E8:10"/><b>then </b><i><font color="Green">E41</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">f</font>,<font color="Maroon">g</font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</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">X</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">f</font>,<font color="Maroon">g</font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</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">X</font></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:10"/><b>then </b>
(  <a href="yellow_0.html#R2">ex_inf_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">f</font>,<font color="Maroon">g</font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">T1</font> &amp;  <a href="yellow_0.html#R2">ex_inf_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">f</font>,<font color="Maroon">g</font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">T2</font> )
 
<b>by </b><i>, , <a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a></i>;<br/>
<b>hence </b><a NAME="E10:10"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>,<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">T1</font>,<font color="Maroon">T2</font></span><a href="yellow_3.html#K3">:]</a></span>
 <b>by </b><i><a class="ref" href="yellow_3.html#T42">YELLOW_3:42</a></i>;<br/>

<b>hence </b> <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</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#NK4" target="_self">inf</a> <span class="p3">(<span class="default"><font color="Maroon">f</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">X</font></span>)</span></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"><font color="Maroon">g</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">X</font></span>)</span></span>)</span></span>)</span></span><a href="yellow_3.html#K7">]</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p4">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">X</font></span>)</span></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"><font color="Maroon">g</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">X</font></span>)</span></span>)</span></span>)</span></span><a href="yellow_3.html#K7">]</a></span>
<b>by </b><i>, , <a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</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">f</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p4">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p4">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span></span><a href="yellow_3.html#K7">]</a></span>
<b>by </b><i>, , <a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a></i>
<br/>.= 
<span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">X</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p2">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i>, <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">f</font>,<font color="Maroon">g</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#NK4" target="_self">inf</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i>, </i>
;<br/><br/>


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