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<span class="kw">set </span><font color="Maroon" title="c2">R</font> =  the   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">J</font> = <font color="Maroon" title="c1">I</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a>  the   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/>
<a NAME="E1:47_1_1"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a>  the   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">J</font> = <font color="Maroon" title="c1">I</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a>  the   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  as   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">J</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">J</font> is  <a href="yellow_1.html#V1" title="YELLOW_1:attr.1">RelStr-yielding</a>  &amp; <font color="Maroon" title="c4">J</font> is  <a href="waybel_3.html#V4" title="WAYBEL_3:attr.4">non-Empty</a>  &amp; <font color="Maroon" title="c4">J</font> is  <a href="waybel_3.html#V5" title="WAYBEL_3:attr.5">reflexive-yielding</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E3:47_1_1"/>
<font color="Maroon" title="c4">J</font> is  <a href="yellow_1.html#V1" title="YELLOW_1:attr.1">RelStr-yielding</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">J</font> is  <a href="waybel_3.html#V4" title="WAYBEL_3:attr.4">non-Empty</a>  &amp; <font color="Maroon" title="c4">J</font> is  <a href="waybel_3.html#V5" title="WAYBEL_3:attr.5">reflexive-yielding</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E4:47_1_1"/>
<font color="Maroon" title="c4">J</font> is  <a href="waybel_3.html#V4" title="WAYBEL_3:attr.4">non-Empty</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">J</font> is  <a href="waybel_3.html#V5" title="WAYBEL_3:attr.5">reflexive-yielding</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">S</font> be    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel_3.html#D7" target="_self" title="WAYBEL_3:def.7">WAYBEL_3:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <font color="Maroon" title="c4">J</font> implies  not <font color="Maroon" title="c5">S</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:47_1_1_1"/>
<font color="Maroon" title="c5">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">J</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c5">S</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> </span><br/>

<span class="kw">hence </span><a NAME="E2:47_1_1_1"/>
 not <font color="Maroon" title="c5">S</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c5">S</font> be    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel_3.html#D8" target="_self" title="WAYBEL_3:def.8">WAYBEL_3:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <font color="Maroon" title="c4">J</font> implies <font color="Maroon" title="c5">S</font> is  <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E5:47_1_1"/>
<font color="Maroon" title="c5">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">J</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">S</font> is  <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> </span><br/>

<span class="kw">hence </span><a NAME="E6:47_1_1"/>
<font color="Maroon" title="c5">S</font> is  <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_1_1"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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