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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="seq_4.html#K27" title="SEQ_4:func.27">+</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/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:81"/><span class="lab"><font color="Green" title="E61">A1</font></span>: 
<font color="Maroon" title="c1">X</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> 
 <span class="kw">and </span><br/><a NAME="E2:81"/><span class="lab"><font color="Green" title="E62">A2</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="seq_4.html#K27" title="SEQ_4:func.27">+</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"/><span class="lab"><font color="Green" title="E63">A3</font></span>: 
 <a href="measure6.html#K4" title="MEASURE6:func.4">-</a> <font color="Maroon" title="c2">Y</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81"><span class="lab"><font color="Green" title="E62">A2</font></span></a>, <a class="ref" href="measure6.html#T77" title="MEASURE6:th.77">MEASURE6:77</a></span>;<br/>
<a NAME="E4:81"/><span class="lab"><font color="Green" title="E64">A4</font></span>: 
 <a href="measure6.html#K4" title="MEASURE6:func.4">-</a> <font color="Maroon" title="c1">X</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="ref" href="measure6.html#T77" title="MEASURE6:th.77">MEASURE6:77</a></span>;<br/>
<a NAME="E5:81"/><span class="kw">then </span>
 <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="measure6.html#K4" title="MEASURE6:func.4">-</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_4.html#K27" title="SEQ_4:func.27">+</a> <span class="p2">(<span class="default"><a href="measure6.html#K4" title="MEASURE6: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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p2">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">-</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="measure6.html#K4" title="MEASURE6:func.4">-</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:81"><span class="lab"><font color="Green" title="E63">A3</font></span></a>, <a class="ref" href="seq_4.html#T142" title="SEQ_4:th.142">SEQ_4:142</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E65">A5</font></span>:  <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="measure6.html#K4" title="MEASURE6:func.4">-</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_4.html#K27" title="SEQ_4:func.27">+</a> <span class="p2">(<span class="default"><a href="measure6.html#K4" title="MEASURE6: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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">-</a> <span class="p4">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">-</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="measure6.html#K4" title="MEASURE6:func.4">-</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:81"><span class="lab"><font color="Green" title="E64">A4</font></span></a>, <a class="ref" href="measure6.html#T79" title="MEASURE6:th.79">MEASURE6:79</a></span>
<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="measure6.html#K4" title="MEASURE6:func.4">-</a> <span class="p4">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">-</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:81"><span class="lab"><font color="Green" title="E63">A3</font></span></a>, <a class="ref" href="measure6.html#T79" title="MEASURE6:th.79">MEASURE6:79</a></span>
<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="E7:81"/><span class="lab"><font color="Green" title="E66">A6</font></span>: 
<span class="p1">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">-</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="seq_4.html#K27" title="SEQ_4:func.27">+</a> <span class="p1">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">-</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#K4" title="MEASURE6:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="seq_4.html#K27" title="SEQ_4:func.27">+</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T52" target="_self" title="INTEGRA1:th.52">Th52</a></span>;<br/>
<a NAME="E8:81"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A7</font></span>: 
not  <a href="measure6.html#K4" title="MEASURE6:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="seq_4.html#K27" title="SEQ_4:func.27">+</a> <font color="Maroon" title="c2">Y</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T140" title="SEQ_4:th.140">SEQ_4:140</a></span>;<br/>
<a NAME="E9:81"/>
 <a href="measure6.html#K4" title="MEASURE6:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="seq_4.html#K27" title="SEQ_4:func.27">+</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:81"><span class="lab"><font color="Green" title="E66">A6</font></span></a>, <a class="txt" href="#E4:81"><span class="lab"><font color="Green" title="E64">A4</font></span></a>, <a class="txt" href="#E3:81"><span class="lab"><font color="Green" title="E63">A3</font></span></a>, <a class="ref" href="seq_4.html#T141" title="SEQ_4:th.141">SEQ_4:141</a></span>;<br/>
<a NAME="E10:81"/><span class="kw">then </span>
 <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="measure6.html#K4" title="MEASURE6:func.4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K4" title="MEASURE6:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="seq_4.html#K27" title="SEQ_4:func.27">+</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:81"><span class="lab"><font color="Green" title="E66">A6</font></span></a>, <a class="txt" href="#E8:81"><span class="lab"><font color="Green" title="E67">A7</font></span></a>, <a class="txt" href="#E6:81"><span class="lab"><font color="Green" title="E65">A5</font></span></a>, <a class="ref" href="measure6.html#T79" title="MEASURE6:th.79">MEASURE6:79</a></span>;<br/>
<span class="kw">hence </span><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="seq_4.html#K27" title="SEQ_4:func.27">+</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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