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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <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="kw">means </span>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <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="kw">means </span>$1 is    <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of <font color="Maroon" title="c1">X</font>;<br/>
<a NAME="E1:3_2_1"/><span class="lab"><font color="Green" title="E3">A9</font></span>: 
for <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being  <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>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">r</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">s</font>] holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D2" target="_self" title="XXREAL_2:def.2">Def2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c2">s</font> being   <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="kw"> such that </span><br/><a NAME="E3:3_2_1"/><span class="lab"><font color="Green" title="E4">A10</font></span>: 
for <font color="Olive" title="b1">r</font> being  <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>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">r</font>] holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font>
 <span class="kw">and </span><br/><a NAME="E4:3_2_1"/><span class="lab"><font color="Green" title="E5">A11</font></span>: 
for <font color="Olive" title="b1">r</font> being  <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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] holds <br/><font color="Maroon" title="c2">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font>
 <span class="kw">from</span> <span class="lab"><a class="ref" href="xxreal_1.html#S1" title="XXREAL_1:sch.1">XXREAL_1:sch 1</a>(<a class="txt" href="#E1:3_2_1"><span class="lab"><font color="Green" title="E3">A9</font></span></a>);<br/></span>
<span class="kw">take </span>
<font color="Maroon" title="c2">s</font>
; <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="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> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E5:3_2_1"/>
<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="kw">by</span> <span class="lab"><a class="txt" href="#E4:3_2_1"><span class="lab"><font color="Green" title="E5">A11</font></span></a>, <a class="ref" href="xxreal_2.html#D2" target="_self" title="XXREAL_2:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font></span><br/>

<span class="kw">thus </span><a NAME="E6:3_2_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_2_1"><span class="lab"><font color="Green" title="E4">A10</font></span></a></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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