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<span class="kw">let </span><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">q</font>, <font color="Maroon" title="c3">r</font>, <font color="Maroon" title="c4">s</font> be   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</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"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">s</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> implies <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">s</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c1">p</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">q</font>)</span>)</span></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> )</span><br/>







<span class="kw">assume </span><a NAME="E1:163"/>
<span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">s</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</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"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">s</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c1">p</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">q</font>)</span>)</span></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">u</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a><span class="kw"> such that </span><br/><a NAME="E3:163"/><span class="lab"><font color="Green" title="E69">A1</font></span>: 
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">s</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span>
 <span class="kw">and </span><br/><a NAME="E4:163"/><span class="lab"><font color="Green" title="E70">A2</font></span>: 
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span>
 ;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">t</font> be   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="membered.html#D14" onmouseover="rs('membered/D14')" onmouseout="rh()">MEMBERED:def 14</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="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">s</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> or <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c1">p</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">q</font>)</span>)</span></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> ) &amp; (  not <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c1">p</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">q</font>)</span>)</span></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> or <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">s</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> ) )</span><br/>

<span class="kw">thus </span><a NAME="E5:163"/>
( <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">s</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> implies <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c1">p</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">q</font>)</span>)</span></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</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"> (  not <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c1">p</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">q</font>)</span>)</span></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> or <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">s</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="163_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:163_1"/>
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">s</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</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"> <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c1">p</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">q</font>)</span>)</span></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span></span><br/>

<a NAME="E2:163_1"/><span class="kw">then </span>
( <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">s</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> or <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E3:163_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A3</font></span>: 
( ( <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font> &amp; <font color="Maroon" title="c6">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">s</font> ) or ( <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font> &amp; <font color="Maroon" title="c6">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">q</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T3" target="_self" onmouseover="rs('xxreal_1/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<a NAME="E4:163_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A4</font></span>: 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c1">p</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T23" onmouseover="rs('xxreal_0/T23')" onmouseout="rh()">XXREAL_0:23</a></span>;<br/>
<a NAME="E5:163_1"/>
<font color="Maroon" title="c6">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">q</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:163_1"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T30" onmouseover="rs('xxreal_0/T30')" onmouseout="rh()">XXREAL_0:30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:163_1"/>
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c1">p</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">q</font>)</span>)</span></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:163_1"><span class="lab"><font color="Green" title="E72">A4</font></span></a>, <a class="ref" href="xxreal_1.html#T3" target="_self" onmouseover="rs('xxreal_1/T3')" onmouseout="rh()">Th3</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="E6:163"/><span class="lab"><font color="Green" title="E71">A5</font></span>: 
<font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:163"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="ref" href="xxreal_1.html#T3" target="_self" onmouseover="rs('xxreal_1/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<a NAME="E7:163"/><span class="lab"><font color="Green" title="E72">A6</font></span>: 
<font color="Maroon" title="c5">u</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:163"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="ref" href="xxreal_1.html#T3" target="_self" onmouseover="rs('xxreal_1/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<a NAME="E8:163"/><span class="lab"><font color="Green" title="E73">A7</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:163"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="xxreal_1.html#T3" target="_self" onmouseover="rs('xxreal_1/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<a NAME="E9:163"/><span class="lab"><font color="Green" title="E74">A8</font></span>: 
<font color="Maroon" title="c5">u</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:163"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="xxreal_1.html#T3" target="_self" onmouseover="rs('xxreal_1/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E10:163"/><span class="lab"><font color="Green" title="E75">A9</font></span>: 
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c1">p</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">q</font>)</span>)</span></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</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"> <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">s</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span></span><br/>

<a NAME="E11:163"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A10</font></span>: 
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> (<font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c1">p</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T3" target="_self" onmouseover="rs('xxreal_1/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<a NAME="E12:163"/><span class="lab"><font color="Green" title="E77">A11</font></span>: 
<font color="Maroon" title="c6">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c2">q</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:163"><span class="lab"><font color="Green" title="E75">A9</font></span></a>, <a class="ref" href="xxreal_1.html#T3" target="_self" onmouseover="rs('xxreal_1/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="163_2"><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="E1:163_2"/>
( ( <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font> &amp; <font color="Maroon" title="c6">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">s</font> ) or ( <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font> &amp; <font color="Maroon" title="c6">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">q</font> ) or ( <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font> &amp; <font color="Maroon" title="c6">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">s</font> ) or ( <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font> &amp; <font color="Maroon" title="c6">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">q</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:163"><span class="lab"><font color="Green" title="E76">A10</font></span></a>, <a class="txt" href="#E12:163"><span class="lab"><font color="Green" title="E77">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T21" onmouseover="rs('xxreal_0/T21')" onmouseout="rh()">XXREAL_0:21</a>, <a class="ref" href="xxreal_0.html#T28" onmouseover="rs('xxreal_0/T28')" onmouseout="rh()">XXREAL_0:28</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="163_2_1"><span class="kw">suppose </span></a><a NAME="E1:163_2_1"/>
( ( <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font> &amp; <font color="Maroon" title="c6">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">s</font> ) or ( <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font> &amp; <font color="Maroon" title="c6">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">q</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"> <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">s</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span></span><br/><div class="add"><a NAME="E2:163_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">s</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> or <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T3" target="_self" onmouseover="rs('xxreal_1/T3')" onmouseout="rh()">Th3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:163_2_1"/>
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">s</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</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="163_2_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:163_2_2"/><span class="lab"><font color="Green" title="E78">A12</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font>
 <span class="kw">and </span><br/><a NAME="E2:163_2_2"/><span class="lab"><font color="Green" title="E79">A13</font></span>: 
<font color="Maroon" title="c6">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">s</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"> <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">s</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span></span><br/><div class="add"><a NAME="E3:163_2_2"/>
( <font color="Maroon" title="c5">u</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font> or <font color="Maroon" title="c6">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">u</font> )
 ;<br/><a NAME="E4:163_2_2"/><span class="kw">then </span>
( <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font> or <font color="Maroon" title="c6">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:163"><span class="lab"><font color="Green" title="E71">A5</font></span></a>, <a class="txt" href="#E9:163"><span class="lab"><font color="Green" title="E74">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E5:163_2_2"/><span class="kw">then </span>
( <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">s</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> or <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:163_2_2"><span class="lab"><font color="Green" title="E78">A12</font></span></a>, <a class="txt" href="#E2:163_2_2"><span class="lab"><font color="Green" title="E79">A13</font></span></a>, <a class="ref" href="xxreal_1.html#T3" target="_self" onmouseover="rs('xxreal_1/T3')" onmouseout="rh()">Th3</a></span>;<br/><span class="kw">hence </span><a NAME="E6:163_2_2"/>
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">s</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</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="163_2_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:163_2_3"/><span class="lab"><font color="Green" title="E78">A14</font></span>: 
<font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font>
 <span class="kw">and </span><br/><a NAME="E2:163_2_3"/><span class="lab"><font color="Green" title="E79">A15</font></span>: 
<font color="Maroon" title="c6">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">q</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"> <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">s</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span></span><br/><div class="add"><a NAME="E3:163_2_3"/>
( <font color="Maroon" title="c5">u</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font> or <font color="Maroon" title="c6">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">u</font> )
 ;<br/><a NAME="E4:163_2_3"/><span class="kw">then </span>
( <font color="Maroon" title="c6">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">s</font> or <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">t</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:163"><span class="lab"><font color="Green" title="E72">A6</font></span></a>, <a class="txt" href="#E8:163"><span class="lab"><font color="Green" title="E73">A7</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E5:163_2_3"/><span class="kw">then </span>
( <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">s</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> or <font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:163_2_3"><span class="lab"><font color="Green" title="E78">A14</font></span></a>, <a class="txt" href="#E2:163_2_3"><span class="lab"><font color="Green" title="E79">A15</font></span></a>, <a class="ref" href="xxreal_1.html#T3" target="_self" onmouseover="rs('xxreal_1/T3')" onmouseout="rh()">Th3</a></span>;<br/><span class="kw">hence </span><a NAME="E6:163_2_3"/>
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">s</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</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>
