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<b>let </b><font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c2">L2</font>, <font color="Maroon" title="c3">T1</font>, <font color="Maroon" title="c4">T2</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="orders_2.html#V2" title="ORDERS_2:attr.2">reflexive</a> <a href="orders_2.html#V4" title="ORDERS_2:attr.4">antisymmetric</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c5">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c3">T1</font>;<br/><b>let </b><font color="Maroon" title="c6">g</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c4">T2</font>;<br/>





<b>assume </b><a NAME="E1:13"/><i><font color="Green" title="E11">A1</font></i>: 
( <font color="Maroon" title="c5">f</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> &amp; <font color="Maroon" title="c6">g</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> )
 ;<br/>

<b>let </b><font color="Maroon" title="c7">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D37" title="WAYBEL_0:def.37">WAYBEL_0:def 37</a><br/>

<b>assume </b><a NAME="E2:13"/>
( not <font color="Maroon" title="c7">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c7">X</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 ;<br/>

<a NAME="E3:13"/><b>then </b>
( not  <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c7">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c7">X</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> &amp; not  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c7">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c7">X</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 
<b>by </b><i><a class="ref" href="yellow_3.html#T21" title="YELLOW_3:th.21">YELLOW_3:21</a>, <a class="ref" href="yellow_3.html#T22" title="YELLOW_3:th.22">YELLOW_3:22</a></i>;<br/>
<a NAME="E4:13"/><b>then </b><i><font color="Green" title="E12">A2</font></i>: 
( <font color="Maroon" title="c5">f</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a>  <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c7">X</font> &amp; <font color="Maroon" title="c6">g</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a>  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c7">X</font> )
 
<b>by </b><i><a class="txt" href="waybel20.html#E1:13"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="waybel_0.html#D37" title="WAYBEL_0:def.37">WAYBEL_0:def 37</a></i>;<br/>
<b>assume </b><a NAME="E5:13"/><i><font color="Green" title="E13">A3</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c7">X</font>,<span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D31" title="WAYBEL_0:def.31">WAYBEL_0:def 31</a><br/>

<a NAME="E6:13"/><b>then </b><i><font color="Green" title="E14">A4</font></i>: 
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c1">L1</font> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c2">L2</font> )
 
<b>by </b><i><a class="ref" href="yellow_3.html#T41" title="YELLOW_3:th.41">YELLOW_3:41</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">iX</font> = <span class="p1"><a href="waybel20.html#K1" title="WAYBEL20:func.1">[:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="waybel20.html#K1" title="WAYBEL20:func.1">:]</a></span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">X</font>;<br/>
<a NAME="E7:13"/><i><font color="Green" title="E15">A5</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:13"/>
<font color="Maroon" title="c7">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>
 
;<br/>
<a NAME="E9:13"/><b>then </b>
<font color="Maroon" title="c7">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#D2" title="YELLOW_3:def.2">YELLOW_3:def 2</a></i>;<br/>
<a NAME="E10:13"/><b>then </b><i><font color="Green" title="E16">A6</font></i>: 
(  <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <span class="p1">(<span class="default"><span class="p2"><a href="waybel20.html#K1" title="WAYBEL20:func.1">[:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="waybel20.html#K1" title="WAYBEL20:func.1">:]</a></span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c7">X</font></span>)</span> &amp;  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <span class="p1">(<span class="default"><span class="p2"><a href="waybel20.html#K1" title="WAYBEL20:func.1">[:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="waybel20.html#K1" title="WAYBEL20:func.1">:]</a></span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c7">X</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="waybel20.html#E7:13"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="waybel20.html#T4" target="_self" title="WAYBEL20:th.4">Th4</a></i>;<br/>
<a NAME="E11:13"/><b>then </b>
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <span class="p1">(<span class="default"><span class="p2"><a href="waybel20.html#K1" title="WAYBEL20:func.1">[:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="waybel20.html#K1" title="WAYBEL20:func.1">:]</a></span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">X</font></span>)</span>,<font color="Maroon" title="c3">T1</font> &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <span class="p1">(<span class="default"><span class="p2"><a href="waybel20.html#K1" title="WAYBEL20:func.1">[:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="waybel20.html#K1" title="WAYBEL20:func.1">:]</a></span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">X</font></span>)</span>,<font color="Maroon" title="c4">T2</font> )
 
<b>by </b><i><a class="txt" href="waybel20.html#E4:13"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="waybel20.html#E6:13"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="waybel_0.html#D31" title="WAYBEL_0:def.31">WAYBEL_0:def 31</a></i>;<br/>
<b>hence </b><a NAME="E12:13"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1"><a href="waybel20.html#K1" title="WAYBEL20:func.1">[:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="waybel20.html#K1" title="WAYBEL20:func.1">:]</a></span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">X</font>,<span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c3">T1</font>,<font color="Maroon" title="c4">T2</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>
 <b>by </b><i><a class="ref" href="yellow_3.html#T41" title="YELLOW_3:th.41">YELLOW_3:41</a></i>;<br/>

<b>hence </b> <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="waybel20.html#K1" title="WAYBEL20:func.1">[:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="waybel20.html#K1" title="WAYBEL20:func.1">:]</a></span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">X</font></span>)</span> = 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p4">(<span class="default"><a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c7">X</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p4">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c7">X</font></span>)</span></span>)</span></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
<b>by </b><i><a class="txt" href="waybel20.html#E10:13"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="waybel20.html#T8" target="_self" title="WAYBEL20:th.8">Th8</a></i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p4">(<span class="default"><a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c7">X</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p4">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c7">X</font></span>)</span></span>)</span></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
<b>by </b><i><a class="txt" href="waybel20.html#E4:13"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="waybel20.html#E6:13"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="waybel_0.html#D31" title="WAYBEL_0:def.31">WAYBEL_0:def 31</a></i>
<br/>.= 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p4">(<span class="default"><a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c7">X</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p4">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c7">X</font></span>)</span></span>)</span></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
<b>by </b><i><a class="txt" href="waybel20.html#E4:13"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="waybel20.html#E6:13"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="waybel_0.html#D31" title="WAYBEL_0:def.31">WAYBEL_0:def 31</a></i>
<br/>.= 
<span class="p1"><a href="waybel20.html#K1" title="WAYBEL20:func.1">[:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="waybel20.html#K1" title="WAYBEL20:func.1">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p2">(<span class="default"><a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c7">X</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p2">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c7">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel20.html#E7:13"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>
<br/>.= 
<span class="p1"><a href="waybel20.html#K1" title="WAYBEL20:func.1">[:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="waybel20.html#K1" title="WAYBEL20:func.1">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c7">X</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel20.html#E5:13"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="waybel20.html#T8" target="_self" title="WAYBEL20:th.8">Th8</a></i>
;<br/><br/>


</div>
