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<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:82"/><i><font color="Green" title="E63">A1</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 <b>and </b><br/><a NAME="E2:82"/><i><font color="Green" title="E64">A2</font></i>: 
<font color="Maroon" title="c2">Y</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 ;<br/>

<a NAME="E3:82"/><i><font color="Green" title="E65">A3</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#T52" target="_self" title="INTEGRA1:th.52">Th52</a></i>;<br/>
<a NAME="E4:82"/><b>then </b><i><font color="Green" title="E66">A4</font></i>: 
not  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <font color="Maroon" title="c2">Y</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="complsp1.html#T94" title="COMPLSP1:th.94">COMPLSP1:94</a></i>;<br/>
<a NAME="E5:82"/><i><font color="Green" title="E67">A5</font></i>: 
 <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c1">X</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:82"><i><font color="Green" title="E63">A1</font></i></a>, <a class="ref" href="pscomp_1.html#T15" title="PSCOMP_1:th.15">PSCOMP_1:15</a></i>;<br/>
<a NAME="E6:82"/><i><font color="Green" title="E68">A6</font></i>: 
 <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c2">Y</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:82"><i><font color="Green" title="E64">A2</font></i></a>, <a class="ref" href="pscomp_1.html#T15" title="PSCOMP_1:th.15">PSCOMP_1:15</a></i>;<br/>
<a NAME="E7:82"/><b>then </b><i><font color="Green" title="E69">A7</font></i>: 
 <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <font color="Maroon" title="c2">Y</font></span>)</span> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 
<b>by </b><i><a class="txt" href="integra1.html#E3:82"><i><font color="Green" title="E65">A3</font></i></a>, <a class="txt" href="integra1.html#E5:82"><i><font color="Green" title="E67">A5</font></i></a>, <a class="ref" href="complsp1.html#T95" title="COMPLSP1:th.95">COMPLSP1:95</a></i>;<br/>
<a NAME="E8:82"/>
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E5:82"><i><font color="Green" title="E67">A5</font></i></a>, <a class="txt" href="integra1.html#E6:82"><i><font color="Green" title="E68">A6</font></i></a>, <a class="ref" href="complsp1.html#T96" title="COMPLSP1:th.96">COMPLSP1:96</a></i>;<br/>
<b>then </b> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E5:82"><i><font color="Green" title="E67">A5</font></i></a>, <a class="ref" href="pscomp_1.html#T17" title="PSCOMP_1:th.17">PSCOMP_1:17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E6:82"><i><font color="Green" title="E68">A6</font></i></a>, <a class="ref" href="pscomp_1.html#T17" title="PSCOMP_1:th.17">PSCOMP_1:17</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>

;<br/>
<a NAME="E10:82"/><b>then </b>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E3:82"><i><font color="Green" title="E65">A3</font></i></a>, <a class="txt" href="integra1.html#E4:82"><i><font color="Green" title="E66">A4</font></i></a>, <a class="txt" href="integra1.html#E7:82"><i><font color="Green" title="E69">A7</font></i></a>, <a class="ref" href="pscomp_1.html#T17" title="PSCOMP_1:th.17">PSCOMP_1:17</a></i>;<br/>
<a NAME="E11:82"/><b>then </b>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E12:82"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 ;<br/>


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