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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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> is <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> &amp; <font color="Maroon" title="c2">Y</font> is <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> implies  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">Y</font></span>)</span> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:81"/><i><font color="Green" title="E62">A1</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a>
 <b>and </b><br/><a NAME="E2:81"/><i><font color="Green" title="E63">A2</font></i>: 
<font color="Maroon" title="c2">Y</font> is <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><br/>

<a NAME="E3:81"/><i><font color="Green" title="E64">A3</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">-</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#K2" title="PSCOMP_1:func.2">-</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#K2" title="PSCOMP_1:func.2">-</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="E4:81"/><b>then </b><i><font color="Green" title="E65">A4</font></i>: 
not  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">-</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="E5:81"/><i><font color="Green" title="E66">A5</font></i>: 
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">-</a> <font color="Maroon" title="c1">X</font> is <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a>
 
<b>by </b><i><a class="txt" href="#E1:81"><i><font color="Green" title="E62">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="E6:81"/><i><font color="Green" title="E67">A6</font></i>: 
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">-</a> <font color="Maroon" title="c2">Y</font> is <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a>
 
<b>by </b><i><a class="txt" href="#E2:81"><i><font color="Green" title="E63">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="E7:81"/><b>then </b><i><font color="Green" title="E68">A7</font></i>: 
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">-</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="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a>
 
<b>by </b><i><a class="txt" href="#E3:81"><i><font color="Green" title="E64">A3</font></i></a>, <a class="txt" href="#E5:81"><i><font color="Green" title="E66">A5</font></i></a>, <a class="ref" href="complsp1.html#T95" title="COMPLSP1:th.95">COMPLSP1:95</a></i>;<br/>
<a NAME="E8:81"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">-</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#K2" title="PSCOMP_1:func.2">-</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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">-</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">-</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E5:81"><i><font color="Green" title="E66">A5</font></i></a>, <a class="txt" href="#E6:81"><i><font color="Green" title="E67">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">-</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#K2" title="PSCOMP_1:func.2">-</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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">-</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">-</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E5:81"><i><font color="Green" title="E66">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">-</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E6:81"><i><font color="Green" title="E67">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>

;<br/>
<a NAME="E10:81"/><b>then </b>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">-</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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:81"><i><font color="Green" title="E64">A3</font></i></a>, <a class="txt" href="#E4:81"><i><font color="Green" title="E65">A4</font></i></a>, <a class="txt" href="#E7:81"><i><font color="Green" title="E68">A7</font></i></a>, <a class="ref" href="pscomp_1.html#T17" title="PSCOMP_1:th.17">PSCOMP_1:17</a></i>;<br/>
<b>hence </b><a NAME="E11:81"/>
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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