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<span class="kw">let </span><font color="Maroon" title="c1">P</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="poset_1.html#K6" title="POSET_1:func.6">ConPoset</a> (<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c1">P</font>)</span>)</span>  st <font color="Olive" title="b1">p1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">p2</font> holds <br/>( <font color="Olive" title="b1">p1</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="c1">P</font>) &amp; <font color="Olive" title="b2">p2</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="c1">P</font>) &amp;  ex <font color="Olive" title="b3">g1</font>, <font color="Olive" title="b4">g2</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="c1">P</font> st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g1</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">g2</font> &amp; <font color="Olive" title="b3">g1</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Olive" title="b4">g2</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="poset_1.html#K6" title="POSET_1:func.6">ConPoset</a> (<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c1">P</font>)</span>)</span>; <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="c2">p1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">p2</font> implies ( <font color="Maroon" title="c2">p1</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="c1">P</font>) &amp; <font color="Maroon" title="c3">p2</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="c1">P</font>) &amp;  ex <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</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="c1">P</font> st <br/>( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g2</font> &amp; <font color="Olive" title="b1">g1</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Olive" title="b2">g2</font> ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:64"/>
<font color="Maroon" title="c2">p1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">p2</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="c2">p1</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="c1">P</font>) &amp; <font color="Maroon" title="c3">p2</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="c1">P</font>) &amp;  ex <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</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="c1">P</font> st <br/>( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g2</font> &amp; <font color="Olive" title="b1">g1</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Olive" title="b2">g2</font> ) )</span><br/>

<a NAME="E2:64"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A1</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</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="c1">P</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a></span>;<br/>
<a NAME="E3:64"/>
 ex <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</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="c1">P</font> st <br/>( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g2</font> &amp; <font color="Olive" title="b1">g1</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Olive" title="b2">g2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c4">g1</font>, <font color="Maroon" title="c5">g2</font> being   <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="c1">P</font><span class="kw"> such that </span><br/><a NAME="E2:64_1"/><span class="lab"><font color="Green" title="E51">A2</font></span>: 
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g1</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> &amp; <font color="Maroon" title="c4">g1</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">g2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="ref" href="poset_1.html#D7" target="_self" title="POSET_1:def.7">Def7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">h1</font> = <font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c7">h2</font> = <font color="Maroon" title="c3">p2</font> as   <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="c1">P</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="poset_1.html#E52"><span class="lab"><font color="Green" title="E46">Lm29</font></span></a></span>;<br/>
<a NAME="E4:64_1"/>
<font color="Maroon" title="c4">g1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">h1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_1"><span class="lab"><font color="Green" title="E51">A2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">g1</font> = <font color="Maroon" title="c4">g1</font> as   <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="c1">P</font> ;<br/>
<a NAME="E6:64_1"/>
<font color="Maroon" title="c5">g2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">h2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_1"><span class="lab"><font color="Green" title="E51">A2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">g2</font> = <font color="Maroon" title="c5">g2</font> as   <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="c1">P</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">g1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">g2</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="c1">P</font> st <br/>( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g1</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g2</font> &amp; <font color="Maroon" title="c8">g1</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Olive" title="b1">g2</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c9">g2</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="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g1</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g2</font> &amp; <font color="Maroon" title="c8">g1</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">g2</font> )</span><br/>

<span class="kw">thus </span><a NAME="E8:64_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g1</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g2</font> &amp; <font color="Maroon" title="c8">g1</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">g2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_1"><span class="lab"><font color="Green" title="E51">A2</font></span></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">hence </span><a NAME="E4:64"/>
( <font color="Maroon" title="c2">p1</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="c1">P</font>) &amp; <font color="Maroon" title="c3">p2</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="c1">P</font>) &amp;  ex <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</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="c1">P</font> st <br/>( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g2</font> &amp; <font color="Olive" title="b1">g1</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Olive" title="b2">g2</font> ) )
 ; <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>
