<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font> iff <span class="p1">(<span class="default"><a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <span class="p2">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <span class="p2">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <span class="p1">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E1:45"/>
( <span class="p1">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font> implies <span class="p1">(<span class="default"><a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <span class="p2">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <span class="p2">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <span class="p1">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="roughs_5.html#T5" target="_self" onmouseover="rs('roughs_5/T5')" onmouseout="rh()">FlipCompose</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <span class="p2">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <span class="p2">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <span class="p1">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span> implies <span class="p1">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:45"/>
<span class="p1">(<span class="default"><a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <span class="p2">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <span class="p2">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <span class="p1">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span><br/>

<a NAME="E3:45"/><span class="kw">then </span>
 <a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <span class="p3">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <span class="p3">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="roughs_2.html#T23" onmouseover="rs('roughs_2/T23')" onmouseout="rh()">ROUGHS_2:23</a></span>;<br/>
<span class="kw">then </span> <a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font> = 
<span class="p1">(<span class="default"><a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <span class="p2">(<span class="default"><a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <span class="p3">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <span class="p2">(<span class="default"><a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <span class="p3">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="roughs_5.html#T5" target="_self" onmouseover="rs('roughs_5/T5')" onmouseout="rh()">FlipCompose</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <span class="p2">(<span class="default"><a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <span class="p3">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="roughs_2.html#T23" onmouseover="rs('roughs_2/T23')" onmouseout="rh()">ROUGHS_2:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="roughs_2.html#T23" onmouseover="rs('roughs_2/T23')" onmouseout="rh()">ROUGHS_2:23</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:45"/>
<span class="p1">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="roughs_5.html#K5" title="ROUGHS_5:func.5">f_0</a> <font color="Maroon" title="c1">R</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
