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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</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> ;<br/>

<b>assume </b><a NAME="E1:50"/>
<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>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">p2</font> being  <a href="rat_1.html#V1" title="RAT_1:attr.1">rational</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:50"/><i><font color="Green" title="E28">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="c3">p1</font> &amp; <font color="Maroon" title="c3">p1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">p2</font> &amp; <font color="Maroon" title="c4">p2</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="borsuk_5.html#T47" target="_self" title="BORSUK_5:th.47">Th47</a></i>;<br/>
<a NAME="E4:50"/><i><font color="Green" title="E29">A2</font></i>: 
<font color="Maroon" title="c4">p2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E3:50"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">x</font> =  <a href="int_1.html#K4" title="INT_1:func.4">frac</a> <a href="power.html#K7" title="POWER:func.7">number_e</a> ;<br/>
<b>set </b><font color="Maroon" title="c6">y</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="int_1.html#K4" title="INT_1:func.4">frac</a> <a href="power.html#K7" title="POWER:func.7">number_e</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_1.html#K4" title="INT_1:func.4">frac</a> <a href="power.html#K7" title="POWER:func.7">number_e</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">p2</font></span>)</span>;<br/>
<a NAME="E5:50"/><i><font color="Green" title="E30">A3</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="int_1.html#K4" title="INT_1:func.4">frac</a> <a href="power.html#K7" title="POWER:func.7">number_e</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_1.html#K4" title="INT_1:func.4">frac</a> <a href="power.html#K7" title="POWER:func.7">number_e</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_1.html#K4" title="INT_1:func.4">frac</a> <a href="power.html#K7" title="POWER:func.7">number_e</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">p1</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E6:50"/>
<span class="p1">(<span class="default"><a href="int_1.html#K4" title="INT_1:func.4">frac</a> <a href="power.html#K7" title="POWER:func.7">number_e</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">p1</font></span>)</span> is <a href="rat_1.html#V1" title="RAT_1:attr.1">irrational</a>
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E4:50"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="borsuk_5.html#T42" target="_self" title="BORSUK_5:th.42">Th42</a></i>;<br/>
<a NAME="E7:50"/><b>then </b><i><font color="Green" title="E31">A4</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="int_1.html#K4" title="INT_1:func.4">frac</a> <a href="power.html#K7" title="POWER:func.7">number_e</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_1.html#K4" title="INT_1:func.4">frac</a> <a href="power.html#K7" title="POWER:func.7">number_e</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">p2</font></span>)</span> is <a href="rat_1.html#V1" title="RAT_1:attr.1">irrational</a>
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E5:50"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="borsuk_5.html#T38" target="_self" title="BORSUK_5:th.38">Th38</a></i>;<br/>
<a NAME="E8:50"/><i><font color="Green" title="E32">A5</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="int_1.html#K4" title="INT_1:func.4">frac</a> <a href="power.html#K7" title="POWER:func.7">number_e</a>  &amp;  <a href="int_1.html#K4" title="INT_1:func.4">frac</a> <a href="power.html#K7" title="POWER:func.7">number_e</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 
<b>by </b><i><a class="ref" href="int_1.html#T69" title="INT_1:th.69">INT_1:69</a></i>;<br/>
<a NAME="E9:50"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="int_1.html#K4" title="INT_1:func.4">frac</a> <a href="power.html#K7" title="POWER:func.7">number_e</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_1.html#K4" title="INT_1:func.4">frac</a> <a href="power.html#K7" title="POWER:func.7">number_e</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">p2</font>
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E3:50"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="xreal_1.html#T180" title="XREAL_1:th.180">XREAL_1:180</a></i>;<br/>
<a NAME="E10:50"/><b>then </b><i><font color="Green" title="E33">A6</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="int_1.html#K4" title="INT_1:func.4">frac</a> <a href="power.html#K7" title="POWER:func.7">number_e</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_1.html#K4" title="INT_1:func.4">frac</a> <a href="power.html#K7" title="POWER:func.7">number_e</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">p2</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="txt" href="borsuk_5.html#E3:50"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:50"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="int_1.html#K4" title="INT_1:func.4">frac</a> <a href="power.html#K7" title="POWER:func.7">number_e</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_1.html#K4" title="INT_1:func.4">frac</a> <a href="power.html#K7" title="POWER:func.7">number_e</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">p1</font>
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E3:50"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="borsuk_5.html#E8:50"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="xreal_1.html#T179" title="XREAL_1:th.179">XREAL_1:179</a></i>;<br/>
<a NAME="E12:50"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><a href="int_1.html#K4" title="INT_1:func.4">frac</a> <a href="power.html#K7" title="POWER:func.7">number_e</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_1.html#K4" title="INT_1:func.4">frac</a> <a href="power.html#K7" title="POWER:func.7">number_e</a> </span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">a</font>
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E3:50"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E13:50"/>
 ex <font color="Olive" title="b1">p</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a> <a href="irrat_1.html#NV1" title="IRRAT_1:NV.1">irrational</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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">p</font> &amp; <font color="Olive" title="b1">p</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="txt" href="borsuk_5.html#E7:50"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="borsuk_5.html#E10:50"><i><font color="Green" title="E33">A6</font></i></a></i>;<br/>


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