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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:54"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> being  <a href="rat_1.html#V1">rational</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:54"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="borsuk_5.html#T47" target="_self">Th47</a></i>;<br/>
<a NAME="E4:54"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E3:54"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="int_1.html#K4">frac</a> <a href="power.html#K7">number_e</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="int_1.html#K4">frac</a> <a href="power.html#K7">number_e</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_1.html#K4">frac</a> <a href="power.html#K7">number_e</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>
<a NAME="E5:54"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="int_1.html#K4">frac</a> <a href="power.html#K7">number_e</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_1.html#K4">frac</a> <a href="power.html#K7">number_e</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_1.html#K4">frac</a> <a href="power.html#K7">number_e</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E6:54"/>
<span class="p1">(<span class="default"><a href="int_1.html#K4">frac</a> <a href="power.html#K7">number_e</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="rat_1.html#V1">irrational</a>
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E4:54"><i><font color="Green">E40</font></i></a>, <a class="ref" href="borsuk_5.html#T42" target="_self">Th42</a></i>;<br/>
<a NAME="E7:54"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="int_1.html#K4">frac</a> <a href="power.html#K7">number_e</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_1.html#K4">frac</a> <a href="power.html#K7">number_e</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is <a href="rat_1.html#V1">irrational</a>
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E5:54"><i><font color="Green">E41</font></i></a>, <a class="ref" href="borsuk_5.html#T38" target="_self">Th38</a></i>;<br/>
<a NAME="E8:54"/><i><font color="Green">E43</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="int_1.html#K4">frac</a> <a href="power.html#K7">number_e</a>  &amp;  <a href="int_1.html#K4">frac</a> <a href="power.html#K7">number_e</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 
<b>by </b><i><a class="ref" href="int_1.html#T69">INT_1:69</a></i>;<br/>
<a NAME="E9:54"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="int_1.html#K4">frac</a> <a href="power.html#K7">number_e</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_1.html#K4">frac</a> <a href="power.html#K7">number_e</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E3:54"><i><font color="Green">E39</font></i></a>, <a class="txt" href="borsuk_5.html#E58"><i><font color="Green">Lemma37</font></i></a></i>;<br/>
<a NAME="E10:54"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="int_1.html#K4">frac</a> <a href="power.html#K7">number_e</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_1.html#K4">frac</a> <a href="power.html#K7">number_e</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E3:54"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:54"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="int_1.html#K4">frac</a> <a href="power.html#K7">number_e</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_1.html#K4">frac</a> <a href="power.html#K7">number_e</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E3:54"><i><font color="Green">E39</font></i></a>, <a class="txt" href="borsuk_5.html#E8:54"><i><font color="Green">E43</font></i></a>, <a class="txt" href="borsuk_5.html#E57"><i><font color="Green">Lemma36</font></i></a></i>;<br/>
<a NAME="E12:54"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="int_1.html#K4">frac</a> <a href="power.html#K7">number_e</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_1.html#K4">frac</a> <a href="power.html#K7">number_e</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E3:54"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E13:54"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a> <a href="rat_1.html#V1">irrational</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="borsuk_5.html#E7:54"><i><font color="Green">E42</font></i></a>, <a class="txt" href="borsuk_5.html#E10:54"><i><font color="Green">E44</font></i></a></i>;<br/>


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