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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" 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  <a href="measure6.html#K4" title="MEASURE6:func.4">--</a> <font color="Maroon" title="c1">X</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a>  )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c2">s</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><a NAME="E2:52"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<font color="Maroon" title="c2">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" title="XXREAL_2:def.9">XXREAL_2:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="measure6.html#K4" title="MEASURE6:func.4">--</a> <font color="Maroon" title="c1">X</font> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> </span><br/>

<span class="kw">take </span>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">s</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xxreal_2.html#D10" title="XXREAL_2:def.10">XXREAL_2:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">s</font> is    <a href="xxreal_2.html#M1" title="XXREAL_2:mode.1">UpperBound</a> of  <a href="measure6.html#K4" title="MEASURE6:func.4">--</a> <font color="Maroon" title="c1">X</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">t</font> be   <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xxreal_2.html#D1" title="XXREAL_2:def.1">XXREAL_2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure6.html#K4" title="MEASURE6:func.4">--</a> <font color="Maroon" title="c1">X</font> or <font color="Maroon" title="c3">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">s</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:52"/>
<font color="Maroon" title="c3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="measure6.html#K4" title="MEASURE6:func.4">--</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">s</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">r3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <span class="kw"> such that </span><br/><a NAME="E5:52"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c4">r3</font>
 <span class="kw">and </span><br/><a NAME="E6:52"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
<font color="Maroon" title="c4">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">r3</font> = <font color="Maroon" title="c4">r3</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:52"><span class="lab"><font color="Green" title="E24">A3</font></span></a></span>;<br/>
<a NAME="E8:52"/>
<font color="Maroon" title="c5">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E6:52"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="xxreal_2.html#D2" title="XXREAL_2:def.2">XXREAL_2:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:52"/>
<font color="Maroon" title="c3">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:52"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="xreal_1.html#T24" title="XREAL_1:th.24">XREAL_1:24</a></span>; <a class="txt" 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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