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<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">  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="roughs_4.html#NM1" title="ROUGHS_4:NM.1">map</a> of <font color="Maroon" title="c1">R</font> holds   <a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <font color="Olive" title="b1">f</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> <font color="Olive" title="b2">g</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</font> be   <a href="roughs_4.html#NM1" title="ROUGHS_4:NM.1">map</a> of <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">  <a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <font color="Maroon" title="c2">f</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> <font color="Maroon" title="c3">g</font></span>)</span></span><br/>



<span class="kw">set </span><font color="Maroon" title="c4">fg</font> =  <a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">ff</font> =  <a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <font color="Maroon" title="c2">f</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">gg</font> =  <a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <font color="Maroon" title="c3">g</font>;<br/>
<a NAME="E1:7"/>
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> holds  <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"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <font color="Maroon" title="c2">f</font></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> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <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"> <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"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <font color="Maroon" title="c2">f</font></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> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font></span><br/>

<a NAME="E1:7_1"/>
<font color="Maroon" title="c7">x</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 
;<br/>
<a NAME="E2:7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E3:7_1"/><span class="lab"><font color="Green" title="E5">a2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<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"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="roughs_2.html#D14" onmouseover="rs('roughs_2/D14')" onmouseout="rh()">ROUGHS_2:def 14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a>, <a class="txt" href="#E2:7_1"><span class="lab"><font color="Green" title="E4">A1</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="roughs_2.html#D14" onmouseover="rs('roughs_2/D14')" onmouseout="rh()">ROUGHS_2:def 14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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> <font color="Maroon" title="c3">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="roughs_2.html#D14" onmouseover="rs('roughs_2/D14')" onmouseout="rh()">ROUGHS_2:def 14</a></span>
<br/>.= 
<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> <font color="Maroon" title="c2">f</font></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> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_1"><span class="lab"><font color="Green" title="E5">a2</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:7_1"/>
<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"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <font color="Maroon" title="c2">f</font></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> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:7"/>
 <a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="roughs_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <font color="Maroon" title="c2">f</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> <font color="Maroon" title="c3">g</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"> verum</span><br/>


</div>
