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<b>let </b><font color="Maroon" title="c53">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:107"/><i><font color="Green" title="E94">A1</font></i>: 
<font color="Maroon" title="c53">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">ac</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <font color="Maroon" title="c44">Q</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c54">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:107"/><i><font color="Green" title="E95">A2</font></i>: 
<font color="Maroon" title="c54">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">ac</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <font color="Maroon" title="c44">Q</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:107"/><i><font color="Green" title="E96">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c33">ac</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <font color="Maroon" title="c44">Q</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c54">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c53">y</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c55">x</font> = <font color="Maroon" title="c54">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="toprealb.html#E3:107"><i><font color="Green" title="E95">A2</font></i></a>, <a class="txt" href="toprealb.html#E145"><i><font color="Green" title="E92">Lm56</font></i></a></i>;<br/>
<a NAME="E6:107"/><i><font color="Green" title="E97">A4</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">ac</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <font color="Maroon" title="c44">Q</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c33">ac</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T99" title="RELAT_1:th.99">RELAT_1:99</a></i>;<br/>
<a NAME="E7:107"/><b>then </b>
<font color="Maroon" title="c53">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E1:107"><i><font color="Green" title="E94">A1</font></i></a>, <a class="ref" href="sin_cos6.html#T87" title="SIN_COS6:th.87">SIN_COS6:87</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c56">y</font> = <font color="Maroon" title="c53">y</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E9:107"/><i><font color="Green" title="E98">A5</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c56">y</font> &amp; <font color="Maroon" title="c56">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a>  )
 
<b>by </b><i><a class="txt" href="toprealb.html#E1:107"><i><font color="Green" title="E94">A1</font></i></a>, <a class="txt" href="toprealb.html#E6:107"><i><font color="Green" title="E97">A4</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a>, <a class="ref" href="sin_cos6.html#T87" title="SIN_COS6:th.87">SIN_COS6:87</a></i>;<br/>
<a NAME="E10:107"/><i><font color="Green" title="E99">A6</font></i>: 
(  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c55">x</font> &amp; <font color="Maroon" title="c55">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="toprealb.html#E3:107"><i><font color="Green" title="E95">A2</font></i></a>, <a class="txt" href="toprealb.html#E145"><i><font color="Green" title="E92">Lm56</font></i></a>, <a class="ref" href="rcomp_2.html#T4" title="RCOMP_2:th.4">RCOMP_2:4</a></i>;<br/>
<font color="Maroon" title="c56">y</font> = 
<a href="sin_cos6.html#K4" title="SIN_COS6:func.4">arccos</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c55">x</font>
<b>by </b><i><a class="txt" href="toprealb.html#E3:107"><i><font color="Green" title="E95">A2</font></i></a>, <a class="txt" href="toprealb.html#E4:107"><i><font color="Green" title="E96">A3</font></i></a>, <a class="txt" href="toprealb.html#E145"><i><font color="Green" title="E92">Lm56</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c55">x</font>
<b>by </b><i><a class="ref" href="sin_cos6.html#D4" title="SIN_COS6:def.4">SIN_COS6:def 4</a></i>
;<br/>
<a NAME="E12:107"/><b>then </b>
<font color="Maroon" title="c56">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> 
 
<b>by </b><i><a class="txt" href="toprealb.html#E10:107"><i><font color="Green" title="E99">A6</font></i></a>, <a class="ref" href="sin_cos6.html#T100" title="SIN_COS6:th.100">SIN_COS6:100</a></i>;<br/>
<a NAME="E13:107"/><b>then </b>
<font color="Maroon" title="c56">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> 
 
<b>by </b><i><a class="txt" href="toprealb.html#E9:107"><i><font color="Green" title="E98">A5</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E14:107"/>
<font color="Maroon" title="c53">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c45">E</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E9:107"><i><font color="Green" title="E98">A5</font></i></a>, <a class="ref" href="rcomp_2.html#T3" title="RCOMP_2:th.3">RCOMP_2:3</a></i>;<br/>


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