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<b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b</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="E1:7"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span> )
 ;<br/>

<a NAME="E2:7"/><i><font color="Green">E26</font></i>: 
(  <a href="sin_cos.html#K20">sin</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> &amp;  <a href="sin_cos.html#K20">sin</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <a href="sin_cos.html#K19">sin</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="sin_cos.html#D21">SIN_COS:def 21</a></i>;<br/>
<a NAME="E3:7"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  &amp;  <a href="relat_1.html#K1">dom</a> <a href="sin_cos.html#K19">sin</a>  <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a>  )
 
<b>by </b><i><a class="ref" href="sin_cos.html#D20">SIN_COS:def 20</a>, <a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E4:7"/><b>then </b><i><font color="Green">E29</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <a href="sin_cos.html#K19">sin</a> </span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K35">PI</a>  <a href="real_1.html#K6">/</a> 2</span>)</span>,<a href="sin_cos.html#K35">PI</a> </span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <a href="sin_cos.html#K19">sin</a> </span>)</span> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:7"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> implies  <a href="sin_cos.html#K20">sin</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="sin_cos.html#K20">sin</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="rfunct_2.html#D3">RFUNCT_2:def 3</a>, <a class="ref" href="sin_cos2.html#T3">SIN_COS2:3</a></i>;<br/>

<b>assume </b><a NAME="E6:7"/><i><font color="Green">E30</font></i>: 
 <a href="sin_cos.html#K20">sin</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="sin_cos.html#K20">sin</a> <font color="Maroon">b</font>
 ;<br/>

<b>assume </b><a NAME="E7:7"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E8:7"/><b>then </b>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:7"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="rfunct_2.html#D3">RFUNCT_2:def 3</a>, <a class="ref" href="sin_cos2.html#T3">SIN_COS2:3</a></i>;<br/>


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