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<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">  for <font color="Olive" title="b1">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<span class="p1">(<span class="default"><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>)</span><br/>  for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</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"><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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b3">y</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<span class="p1">(<span class="default"><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>)</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">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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"><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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">y</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</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"><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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font></span>)</span></span><br/>





<a NAME="E1:54"/><span class="lab"><font color="Green" title="E27">AA</font></span>: 
<span class="p1">(<span class="default"><a href="roughs_5.html#K4" title="ROUGHS_5:func.4">ff_0</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">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font></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_5.html#D5" target="_self" onmouseover="rs('roughs_5/D5')" onmouseout="rh()">Defff</a></span>;<br/>
<a NAME="E2:54"/><span class="lab"><font color="Green" title="E28">AB</font></span>: 
<span class="p1">(<span class="default"><a href="roughs_5.html#K4" title="ROUGHS_5:func.4">ff_0</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"><font color="Maroon" title="c3">x</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">x</font> <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_5.html#D5" target="_self" onmouseover="rs('roughs_5/D5')" onmouseout="rh()">Defff</a></span>;<br/>
<a NAME="E3:54"/><span class="lab"><font color="Green" title="E29">AC</font></span>: 
<span class="p1">(<span class="default"><a href="roughs_5.html#K4" title="ROUGHS_5:func.4">ff_0</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"><font color="Maroon" title="c4">y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">y</font> <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_5.html#D5" target="_self" onmouseover="rs('roughs_5/D5')" onmouseout="rh()">Defff</a></span>;<br/>
<span class="kw">thus </span><a NAME="E4:54"/>
<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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font></span>)</span>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</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_2.html#K5" title="ROUGHS_2:func.5">Flip</a> <span class="p2">(<span class="default"><a href="roughs_5.html#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="54_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">t</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span> or <font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:54_1"/><span class="lab"><font color="Green" title="E30">zz</font></span>: 
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</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"> <font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font></span>)</span></span><br/>

<a NAME="E2:54_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">ZZ</font></span>: 
( <font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:54_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="roughs_5.html#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</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="c3">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/>
<a NAME="E4:54_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">Z1</font></span>: 
 not <font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="roughs_5.html#K4" title="ROUGHS_5:func.4">ff_0</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"><font color="Maroon" title="c3">x</font> <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="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E5:54_1"/>
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="roughs_5.html#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</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="c4">y</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="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E31">ZZ</font></span></a>, <a class="ref" href="roughs_2.html#D14" onmouseover="rs('roughs_2/D14')" onmouseout="rh()">ROUGHS_2:def 14</a></span>;<br/>
<a NAME="E6:54_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">V1</font></span>: 
 not <font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="roughs_5.html#K4" title="ROUGHS_5:func.4">ff_0</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"><font color="Maroon" title="c4">y</font> <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="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E7:54_1"/>
 not <font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="roughs_5.html#K4" title="ROUGHS_5:func.4">ff_0</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">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="54_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:54_1_1"/>
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="roughs_5.html#K4" title="ROUGHS_5:func.4">ff_0</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">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></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"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E3:54_1_1"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
( <font color="Maroon" title="c5">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">w</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E27">AA</font></span></a></span>;<br/>
<a NAME="E4:54_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">w</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <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="xboole_1.html#T54" onmouseover="rs('xboole_1/T54')" onmouseout="rh()">XBOOLE_1:54</a>, <a class="txt" href="#E3:54_1_1"><span class="lab"><font color="Green" title="E34">A1</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:54_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">w</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">x</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">w</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T70" onmouseover="rs('xboole_1/T70')" onmouseout="rh()">XBOOLE_1:70</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:54_1_1_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">w</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">x</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</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"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:54_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54_1"><span class="lab"><font color="Green" title="E32">Z1</font></span></a>, <a class="txt" href="#E3:54_1_1"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E28">AB</font></span></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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:54_1_1_1_2"/>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">w</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</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"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:54_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54_1"><span class="lab"><font color="Green" title="E33">V1</font></span></a>, <a class="txt" href="#E3:54"><span class="lab"><font color="Green" title="E29">AC</font></span></a>, <a class="txt" href="#E3:54_1_1"><span class="lab"><font color="Green" title="E34">A1</font></span></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"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E8:54_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="roughs_5.html#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font></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="txt" href="#E1:54_1"><span class="lab"><font color="Green" title="E30">zz</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:54_1"/>
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</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>; <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">let </span><font color="Maroon" title="c5">t</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font></span>)</span> or <font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E5:54"/><span class="lab"><font color="Green" title="E30">v0</font></span>: 
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</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"> <font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span></span><br/>

<a NAME="E6:54"/><span class="kw">then </span>
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="roughs_5.html#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font></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="roughs_2.html#D14" onmouseover="rs('roughs_2/D14')" onmouseout="rh()">ROUGHS_2:def 14</a></span>;<br/>
<a NAME="E7:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">ww</font></span>: 
 not <font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font></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="txt" href="#E1:54"><span class="lab"><font color="Green" title="E27">AA</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E8:54"/><span class="lab"><font color="Green" title="E32">vc</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <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="xboole_1.html#T54" onmouseover="rs('xboole_1/T54')" onmouseout="rh()">XBOOLE_1:54</a></span>;<br/>
<a NAME="E9:54"/><span class="lab"><font color="Green" title="E33">w1</font></span>: 
 not <font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="roughs_5.html#K4" title="ROUGHS_5:func.4">ff_0</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"><font color="Maroon" title="c3">x</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="54_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:54_2"/>
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="roughs_5.html#K4" title="ROUGHS_5:func.4">ff_0</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"><font color="Maroon" title="c3">x</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></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"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E3:54_2"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
( <font color="Maroon" title="c5">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">w</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">x</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E28">AB</font></span></a></span>;<br/>
<a NAME="E4:54_2"/>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">w</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:54"><span class="lab"><font color="Green" title="E32">vc</font></span></a>, <a class="txt" href="#E3:54_2"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="xboole_1.html#T63" onmouseover="rs('xboole_1/T63')" onmouseout="rh()">XBOOLE_1:63</a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:54_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:54"><span class="lab"><font color="Green" title="E31">ww</font></span></a>, <a class="txt" href="#E3:54_2"><span class="lab"><font color="Green" title="E34">A1</font></span></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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:54"/><span class="lab"><font color="Green" title="E34">z1</font></span>: 
 not <font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="roughs_5.html#K4" title="ROUGHS_5:func.4">ff_0</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"><font color="Maroon" title="c4">y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="54_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:54_3"/>
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="roughs_5.html#K4" title="ROUGHS_5:func.4">ff_0</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"><font color="Maroon" title="c4">y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></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"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E3:54_3"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
( <font color="Maroon" title="c5">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">w</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54"><span class="lab"><font color="Green" title="E29">AC</font></span></a></span>;<br/>
<a NAME="E4:54_3"/>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">w</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:54"><span class="lab"><font color="Green" title="E32">vc</font></span></a>, <a class="txt" href="#E3:54_3"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="xboole_1.html#T63" onmouseover="rs('xboole_1/T63')" onmouseout="rh()">XBOOLE_1:63</a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:54_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:54"><span class="lab"><font color="Green" title="E31">ww</font></span></a>, <a class="txt" href="#E3:54_3"><span class="lab"><font color="Green" title="E35">A1</font></span></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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E11:54"/>
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="roughs_5.html#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</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="c3">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="txt" href="#E9:54"><span class="lab"><font color="Green" title="E33">w1</font></span></a>, <a class="txt" href="#E5:54"><span class="lab"><font color="Green" title="E30">v0</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E12:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">W1</font></span>: 
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font>
 
<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/>
<a NAME="E13:54"/>
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="roughs_5.html#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</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="c4">y</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="txt" href="#E10:54"><span class="lab"><font color="Green" title="E34">z1</font></span></a>, <a class="txt" href="#E5:54"><span class="lab"><font color="Green" title="E30">v0</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E14:54"/><span class="kw">then </span>
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font>
 
<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="kw">hence </span><a NAME="E15:54"/>
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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#K4" title="ROUGHS_5:func.4">ff_0</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:54"><span class="lab"><font color="Green" title="E35">W1</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</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"> verum</span><br/>


</div>
