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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">g1</font>, <font color="Maroon" title="c4">M</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">g1</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">M</font> implies  ex <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">M</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">g1</font> ) )</span><br/>

<b>assume </b><a NAME="E1:1"/><i><font color="Green" title="E2">A1</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">g1</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">M</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">M</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">g1</font> )</span><br/>

<a NAME="E2:1"/><i><font color="Green" title="E3">A2</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a>, <a class="ref" href="xreal_1.html#T141" title="XREAL_1:th.141">XREAL_1:141</a>, <a class="txt" href="#E1:1"><i><font color="Green" title="E2">A1</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">r3</font> =  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">M</font></span>)</span></span>)</span>;<br/>
<a NAME="E3:1"/>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">M</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="txt" href="#E2:1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">q</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> <b> such that </b><br/><a NAME="E5:1"/><i><font color="Green" title="E4">A3</font></i>: 
(  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">M</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">q</font> &amp; <font color="Maroon" title="c6">q</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">r</font> = <font color="Maroon" title="c6">q</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">r</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> &amp; <font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">M</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">g1</font> )</span><br/>

<a NAME="E7:1"/><i><font color="Green" title="E5">A4</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">M</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/>
<a NAME="E8:1"/>
<font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">M</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/>
<a NAME="E9:1"/><b>then </b>
<font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">M</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font>
 
<b>by </b><i><a class="txt" href="#E5:1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E10:1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">M</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">M</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">M</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T16" title="XREAL_1:th.16">XREAL_1:16</a></i>;<br/>
<a NAME="E11:1"/><b>then </b><i><font color="Green" title="E6">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">g1</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">M</font>
 
;<br/>
<a NAME="E12:1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">M</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">g1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 1
 
<b>by </b><i><a class="txt" href="#E11:1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="#E1:1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="xreal_1.html#T113" title="XREAL_1:th.113">XREAL_1:113</a></i>;<br/>
<b>hence </b><a NAME="E13:1"/>
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> &amp; <font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">M</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">g1</font> )
 <b>by </b><i><a class="txt" href="#E5:1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="#E7:1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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