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<span class="kw">let </span><font color="Maroon" title="c53">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c53">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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> or <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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:108"/><span class="lab"><font color="Green" title="E93">A1</font></span>: 
<font color="Maroon" title="c53">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c54">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:108"/><span class="lab"><font color="Green" title="E94">A2</font></span>: 
<font color="Maroon" title="c54">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 <span class="kw">and </span><br/><a NAME="E4:108"/><span class="lab"><font color="Green" title="E95">A3</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E5:108"/><span class="lab"><font color="Green" title="E96">A4</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c33">ac</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T99" title="RELAT_1:th.99">RELAT_1:99</a></span>;<br/>
<a NAME="E6:108"/><span class="kw">then </span>
<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#K36" title="SIN_COS:func.36">PI</a> </span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108"><span class="lab"><font color="Green" title="E93">A1</font></span></a>, <a class="ref" href="sin_cos6.html#T87" title="SIN_COS6:th.87">SIN_COS6:87</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c55">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="E8:108"/><span class="lab"><font color="Green" title="E97">A5</font></span>: 
 <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="c55">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108"><span class="lab"><font color="Green" title="E93">A1</font></span></a>, <a class="txt" href="#E5:108"><span class="lab"><font color="Green" title="E96">A4</font></span></a>, <a class="ref" href="sin_cos6.html#T87" title="SIN_COS6:th.87">SIN_COS6:87</a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<a NAME="E9:108"/><span class="lab"><font color="Green" title="E98">A6</font></span>: 
<font color="Maroon" title="c55">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108"><span class="lab"><font color="Green" title="E93">A1</font></span></a>, <a class="txt" href="#E5:108"><span class="lab"><font color="Green" title="E96">A4</font></span></a>, <a class="ref" href="sin_cos6.html#T87" title="SIN_COS6:th.87">SIN_COS6:87</a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c56">x</font> = <font color="Maroon" title="c54">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108"><span class="lab"><font color="Green" title="E94">A2</font></span></a>, <a class="txt" href="toprealb.html#E144"><span class="lab"><font color="Green" title="E91">Lm55</font></span></a></span>;<br/>
<a NAME="E11:108"/><span class="lab"><font color="Green" title="E99">A7</font></span>: 
 <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="c56">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108"><span class="lab"><font color="Green" title="E94">A2</font></span></a>, <a class="txt" href="toprealb.html#E144"><span class="lab"><font color="Green" title="E91">Lm55</font></span></a>, <a class="ref" href="xxreal_1.html#T2" title="XXREAL_1:th.2">XXREAL_1:2</a></span>;<br/>
<a NAME="E12:108"/><span class="lab"><font color="Green" title="E100">A8</font></span>: 
<font color="Maroon" title="c56">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108"><span class="lab"><font color="Green" title="E94">A2</font></span></a>, <a class="txt" href="toprealb.html#E144"><span class="lab"><font color="Green" title="E91">Lm55</font></span></a>, <a class="ref" href="xxreal_1.html#T2" title="XXREAL_1:th.2">XXREAL_1:2</a></span>;<br/>
<font color="Maroon" title="c55">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="c56">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108"><span class="lab"><font color="Green" title="E94">A2</font></span></a>, <a class="txt" href="#E4:108"><span class="lab"><font color="Green" title="E95">A3</font></span></a>, <a class="txt" href="toprealb.html#E144"><span class="lab"><font color="Green" title="E91">Lm55</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
 <a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c56">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos6.html#D4" title="SIN_COS6:def.4">SIN_COS6:def 4</a></span>
;<br/>
<a NAME="E14:108"/><span class="kw">then </span>
<font color="Maroon" title="c55">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:108"><span class="lab"><font color="Green" title="E98">A6</font></span></a>, <a class="txt" href="#E11:108"><span class="lab"><font color="Green" title="E99">A7</font></span></a>, <a class="txt" href="#E12:108"><span class="lab"><font color="Green" title="E100">A8</font></span></a>, <a class="ref" href="sin_cos6.html#T100" title="SIN_COS6:th.100">SIN_COS6:100</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:108"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:108"><span class="lab"><font color="Green" title="E97">A5</font></span></a>, <a class="ref" href="xxreal_1.html#T3" title="XXREAL_1:th.3">XXREAL_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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