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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a> <a href="waybel_0.html#V4">directed-sups-inheriting</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>





<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">c<sub>3</sub></font> as  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">c<sub>1</sub></font> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="yellow16.html#T41" target="_self">Th41</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> , 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> ] means <font color="Maroon">a<sub>1</sub></font> is  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">a<sub>2</sub></font>;<br/>
<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/><b>assume </b><a NAME="E1:61_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">c<sub>3</sub></font>]
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> ;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><a NAME="E3:61_1"/>
(  <a href="yellow_1.html#K5">product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="yellow16.html#K2" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><a NAME="E4:61_1"/><b>then </b>
 <a href="yellow16.html#K1" target="_self">pi</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> is  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="yellow16.html#T37" target="_self">Th37</a></i>;<br/><b>hence </b><a NAME="E5:61_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="yellow16.html#K1" target="_self">pi</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>]
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E1:61_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>]
 ;<br/><a NAME="E2:61_2"/><b>then </b>
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font> implies (  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="waybel_0.html#T7">WAYBEL_0:7</a></i>;<br/><b>hence </b><a NAME="E3:61_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="yellow16.html#R5" target="_self">inherits_sup_of</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="yellow16.html#T42" target="_self">Th42</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:61"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="waybel_0.html#V4">directed-sups-inheriting</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be  <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D4">WAYBEL_0:def 4</a><br/>

<a NAME="E1:61_3"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">c<sub>3</sub></font>] holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">c<sub>3</sub></font> <a href="yellow16.html#R5" target="_self">inherits_sup_of</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>from </b><i><a class="ref" href="yellow16.html#S2" target="_self">YELLOW16:sch 2</a>(<a class="txt" href="yellow16.html#E2:61"><i><font color="Green">E40</font></i></a>, <a class="txt" href="yellow16.html#E3:61"><i><font color="Green">E41</font></i></a>);<br/></i>
<a NAME="E2:61_3"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">c<sub>3</sub></font> <a href="yellow16.html#R5" target="_self">inherits_sup_of</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">c<sub>3</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:61_3"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  not  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">c<sub>3</sub></font> or  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="yellow16.html#D6" target="_self">Def6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:61"/>
<font color="Maroon">c<sub>2</sub></font> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">c<sub>3</sub></font> is  <a href="waybel_0.html#V4">directed-sups-inheriting</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">c<sub>1</sub></font> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


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