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<span class="kw">let </span><font color="Maroon" title="c1">P</font>, <font color="Maroon" title="c2">Q</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>   <a href="poset_1.html#V1" title="POSET_1:attr.1">chain-complete</a>  <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a>  <a href="poset_1.html#K4" title="POSET_1:func.4">ConFuncs</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">R</font> =  <a href="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font>;<br/>
<a NAME="E1:41"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="poset_1.html#K4" title="POSET_1:func.4">ConFuncs</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="poset_1.html#K4" title="POSET_1:func.4">ConFuncs</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c4">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:41_1"/><span class="lab"><font color="Green" title="E34">B1</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="poset_1.html#K4" title="POSET_1:func.4">ConFuncs</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c4">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">x1</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">P</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Q</font><span class="kw"> such that </span><br/><a NAME="E3:41_1"/><span class="lab"><font color="Green" title="E35">B3</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x1</font>
 <span class="kw">and </span><br/><a NAME="E4:41_1"/><span class="lab"><font color="Green" title="E36">B4</font></span>: 
 ex <font color="Olive" title="b1">f</font> being  <a href="poset_1.html#V2" title="POSET_1:attr.2">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font> st <font color="Olive" title="b1">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">x1</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">f</font> being   <a href="poset_1.html#V2" title="POSET_1:attr.2">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font><span class="kw"> such that </span><br/><a NAME="E6:41_1"/><span class="lab"><font color="Green" title="E37">B5</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_1"><span class="lab"><font color="Green" title="E36">B4</font></span></a></span>;<br/>
<a NAME="E7:41_1"/>
<font color="Maroon" title="c6">f</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="poset_1.html#E31"><span class="lab"><font color="Green" title="E28">Lemlessthan01</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:41_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c4">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1"><span class="lab"><font color="Green" title="E34">B1</font></span></a>, <a class="txt" href="#E3:41_1"><span class="lab"><font color="Green" title="E35">B3</font></span></a>, <a class="txt" href="#E6:41_1"><span class="lab"><font color="Green" title="E37">B5</font></span></a>, <a class="ref" href="poset_1.html#D7" target="_self" title="POSET_1:def.7">DefConRelat</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:41"/>
 <a href="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a>  <a href="poset_1.html#K4" title="POSET_1:func.4">ConFuncs</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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