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<b>let </b><font color="Maroon">c<sub>53</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:109"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">c<sub>53</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" target="_self">R^1</a> <font color="Maroon">c<sub>44</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>54</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:109"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" target="_self">R^1</a> <font color="Maroon">c<sub>44</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:109"/><i><font color="Green">E99</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" target="_self">R^1</a> <font color="Maroon">c<sub>44</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>54</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>53</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>55</sub></font> = <font color="Maroon">c<sub>54</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="toprealb.html#E3:109"><i><font color="Green">E98</font></i></a>, <a class="txt" href="toprealb.html#E154"><i><font color="Green">Lemma95</font></i></a></i>;<br/>
<a NAME="E6:109"/><i><font color="Green">E100</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" target="_self">R^1</a> <font color="Maroon">c<sub>44</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>33</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/>
<a NAME="E7:109"/>
<font color="Maroon">c<sub>53</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<a href="sin_cos.html#K33">PI</a> </span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E1:109"><i><font color="Green">E97</font></i></a>, <a class="txt" href="toprealb.html#E6:109"><i><font color="Green">E100</font></i></a>, <a class="ref" href="sin_cos6.html#T87">SIN_COS6:87</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>56</sub></font> = <font color="Maroon">c<sub>53</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E9:109"/><i><font color="Green">E101</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>56</sub></font> &amp; <font color="Maroon">c<sub>56</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="sin_cos.html#K33">PI</a>  )
 
<b>by </b><i><a class="txt" href="toprealb.html#E1:109"><i><font color="Green">E97</font></i></a>, <a class="txt" href="toprealb.html#E6:109"><i><font color="Green">E100</font></i></a>, <a class="ref" href="sin_cos6.html#T87">SIN_COS6:87</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E10:109"/><i><font color="Green">E102</font></i>: 
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>55</sub></font> &amp; <font color="Maroon">c<sub>55</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="toprealb.html#E3:109"><i><font color="Green">E98</font></i></a>, <a class="txt" href="toprealb.html#E154"><i><font color="Green">Lemma95</font></i></a>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/>
<font color="Maroon">c<sub>56</sub></font> = 
<a href="sin_cos6.html#K4">arccos</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>55</sub></font>
<b>by </b><i><a class="txt" href="toprealb.html#E3:109"><i><font color="Green">E98</font></i></a>, <a class="txt" href="toprealb.html#E4:109"><i><font color="Green">E99</font></i></a>, <a class="txt" href="toprealb.html#E154"><i><font color="Green">Lemma95</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="sin_cos6.html#K6">arccos</a> <font color="Maroon">c<sub>55</sub></font>
<b>by </b><i><a class="ref" href="sin_cos6.html#D4">SIN_COS6:def 4</a></i>
;<br/>
<a NAME="E12:109"/><b>then </b>
<font color="Maroon">c<sub>56</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="sin_cos.html#K33">PI</a> 
 
<b>by </b><i><a class="txt" href="toprealb.html#E10:109"><i><font color="Green">E102</font></i></a>, <a class="ref" href="sin_cos6.html#T100">SIN_COS6:100</a></i>;<br/>
<a NAME="E13:109"/><b>then </b>
<font color="Maroon">c<sub>56</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="sin_cos.html#K33">PI</a> 
 
<b>by </b><i><a class="txt" href="toprealb.html#E9:109"><i><font color="Green">E101</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E14:109"/>
<font color="Maroon">c<sub>53</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>45</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E9:109"><i><font color="Green">E101</font></i></a>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/>


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