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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <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" 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">p</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Maroon" title="c1">X</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a>  holds <br/><font color="Olive" title="b1">p</font> <a href="measure6.html#K1" title="MEASURE6:func.1">++</a> <font color="Maroon" title="c1">X</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</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"> ( <font color="Maroon" title="c1">X</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a>  implies <font color="Maroon" title="c2">p</font> <a href="measure6.html#K1" title="MEASURE6:func.1">++</a> <font color="Maroon" title="c1">X</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a>  )</span><br/>



<span class="kw">given </span><font color="Maroon" title="c3">s</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><a NAME="E2:62"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<font color="Maroon" title="c3">s</font> is    <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of <font color="Maroon" title="c1">X</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xxreal_2.html#D9" onmouseover="rs('xxreal_2/D9')" onmouseout="rh()">XXREAL_2:def 9</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"> <font color="Maroon" title="c2">p</font> <a href="measure6.html#K1" title="MEASURE6:func.1">++</a> <font color="Maroon" title="c1">X</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> </span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">s</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xxreal_2.html#D9" onmouseover="rs('xxreal_2/D9')" onmouseout="rh()">XXREAL_2:def 9</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"> <font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">s</font> is    <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of <font color="Maroon" title="c2">p</font> <a href="measure6.html#K1" title="MEASURE6:func.1">++</a> <font color="Maroon" title="c1">X</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">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="xxreal_2.html#D2" onmouseover="rs('xxreal_2/D2')" onmouseout="rh()">XXREAL_2:def 2</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="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">p</font> <a href="measure6.html#K1" title="MEASURE6:func.1">++</a> <font color="Maroon" title="c1">X</font> or <font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:62"/>
<font color="Maroon" title="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">p</font> <a href="measure6.html#K1" title="MEASURE6:func.1">++</a> <font color="Maroon" title="c1">X</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="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t</font></span><br/>

<a NAME="E4:62"/><span class="kw">then </span>
<font color="Maroon" title="c4">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"><font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">r3</font></span>)</span> where <font color="Olive" title="b1">r3</font> is   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> : <font color="Olive" title="b1">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="measure6.html#E57"><span class="lab"><font color="Green" title="E25">Lm5</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">r3</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E6:62"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">r3</font>
 <span class="kw">and </span><br/><a NAME="E7:62"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c5">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/>
<a NAME="E8:62"/>
<font color="Maroon" title="c5">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:62"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E7:62"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="xxreal_2.html#D2" onmouseover="rs('xxreal_2/D2')" onmouseout="rh()">XXREAL_2:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:62"/>
<font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:62"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</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/>


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