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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="rat_1.html#V1">rational</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <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> ;<br/>



<b>assume </b><a NAME="E1:39"/>
not <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font> is <a href="rat_1.html#V1">irrational</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:39"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="rat_1.html#T3">RAT_1:3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E5:39"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="rat_1.html#T3">RAT_1:3</a></i>;<br/>
<a NAME="E6:39"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E3:39"><i><font color="Green">E28</font></i></a>, <a class="txt" href="borsuk_5.html#E5:39"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xcmplx_1.html#T131">XCMPLX_1:131</a></i>;<br/>
<b>hence </b><a NAME="E7:39"/>
contradiction
 ;<br/>


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