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<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</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>  &amp; <font color="Maroon" title="c2">Y</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a>  implies <font color="Maroon" title="c1">X</font> <a href="member_1.html#K9" title="MEMBER_1:func.9">++</a> <font color="Maroon" title="c2">Y</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:144"/>
<font color="Maroon" title="c1">X</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</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="c2">Y</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a>  or <font color="Maroon" title="c1">X</font> <a href="member_1.html#K9" title="MEMBER_1:func.9">++</a> <font color="Maroon" title="c2">Y</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a>  )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">r1</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><br/><a NAME="E3:144"/><span class="lab"><font color="Green" title="E78">A1</font></span>: 
<font color="Maroon" title="c3">r1</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="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D9" title="XXREAL_2:def.9">XXREAL_2:def 9</a></span>;<br/>
<a NAME="E4:144"/><span class="lab"><font color="Green" title="E79">A2</font></span>: 
 for <font color="Olive" title="b1">r</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>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:144"><span class="lab"><font color="Green" title="E78">A1</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">assume </span><a NAME="E5:144"/>
<font color="Maroon" title="c2">Y</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</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> <a href="member_1.html#K9" title="MEMBER_1:func.9">++</a> <font color="Maroon" title="c2">Y</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">r2</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><br/><a NAME="E7:144"/><span class="lab"><font color="Green" title="E80">A3</font></span>: 
<font color="Maroon" title="c4">r2</font> is    <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of <font color="Maroon" title="c2">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D9" title="XXREAL_2:def.9">XXREAL_2:def 9</a></span>;<br/>
<a NAME="E8:144"/><span class="lab"><font color="Green" title="E81">A4</font></span>: 
 for <font color="Olive" title="b1">r</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>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> holds <br/><font color="Maroon" title="c4">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:144"><span class="lab"><font color="Green" title="E80">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">take </span>
<font color="Maroon" title="c3">r1</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c4">r2</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"> <font color="Maroon" title="c3">r1</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c4">r2</font> is    <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of <font color="Maroon" title="c1">X</font> <a href="member_1.html#K9" title="MEMBER_1:func.9">++</a> <font color="Maroon" title="c2">Y</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">r</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#D2" title="XXREAL_2:def.2">XXREAL_2:def 2</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="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="member_1.html#K9" title="MEMBER_1:func.9">++</a> <font color="Maroon" title="c2">Y</font> or <font color="Maroon" title="c3">r1</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c4">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E9:144"/>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="member_1.html#K9" title="MEMBER_1:func.9">++</a> <font color="Maroon" title="c2">Y</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">r1</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c4">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font></span><br/>

<a NAME="E10:144"/><span class="kw">then </span>
<font color="Maroon" title="c5">r</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="Olive" title="b1">r22</font> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <font color="Olive" title="b2">r12</font></span>)</span> where <font color="Olive" title="b1">r22</font>, <font color="Olive" title="b2">r12</font> is    <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>  : ( <font color="Olive" title="b1">r22</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">r12</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="member_1.html#D6" title="MEMBER_1:def.6">MEMBER_1:def 6</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">r22</font>, <font color="Maroon" title="c7">r12</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="E12:144"/><span class="lab"><font color="Green" title="E82">A5</font></span>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r22</font> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <font color="Maroon" title="c7">r12</font>
 <span class="kw">and </span><br/><a NAME="E13:144"/><span class="lab"><font color="Green" title="E83">A6</font></span>: 
<font color="Maroon" title="c6">r22</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">and </span><br/><a NAME="E14:144"/><span class="lab"><font color="Green" title="E84">A7</font></span>: 
<font color="Maroon" title="c7">r12</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">r9</font> = <font color="Maroon" title="c6">r22</font>, <font color="Maroon" title="c9">r99</font> = <font color="Maroon" title="c7">r12</font> as   <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">by</span> <span class="lab"><a class="txt" href="#E13:144"><span class="lab"><font color="Green" title="E83">A6</font></span></a>, <a class="txt" href="#E14:144"><span class="lab"><font color="Green" title="E84">A7</font></span></a></span>;<br/>
<a NAME="E16:144"/><span class="lab"><font color="Green" title="E85">A8</font></span>: 
<font color="Maroon" title="c4">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">r99</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:144"><span class="lab"><font color="Green" title="E81">A4</font></span></a>, <a class="txt" href="#E14:144"><span class="lab"><font color="Green" title="E84">A7</font></span></a></span>;<br/>
<a NAME="E17:144"/>
<font color="Maroon" title="c3">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">r9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:144"><span class="lab"><font color="Green" title="E79">A2</font></span></a>, <a class="txt" href="#E13:144"><span class="lab"><font color="Green" title="E83">A6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:144"/>
<font color="Maroon" title="c3">r1</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c4">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:144"><span class="lab"><font color="Green" title="E82">A5</font></span></a>, <a class="txt" href="#E16:144"><span class="lab"><font color="Green" title="E85">A8</font></span></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</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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