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<b>let </b><font color="Maroon" title="c1">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <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="c1">r</font> &amp; <font color="Maroon" title="c1">r</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>  implies  <a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> )</span><br/>

<b>assume </b><a NAME="E1:92"/>
(  <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="c1">r</font> &amp; <font color="Maroon" title="c1">r</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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font></span><br/>

<a NAME="E2:92"/><b>then </b><i><font color="Green" title="E89">A1</font></i>: 
<font color="Maroon" title="c1">r</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>
 
<b>by </b><i><a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>;<br/>
<a NAME="E3:92"/><i><font color="Green" title="E90">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2"><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>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>
 
<b>by </b><i><a class="txt" href="sin_cos6.html#E8"><i><font color="Green" title="E8">Lm2</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<b>thus </b> <a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <font color="Maroon" title="c1">r</font></span>)</span> = 
<a href="sin_cos6.html#K4" title="SIN_COS6:func.4">arccos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos.html#D23" title="SIN_COS:def.23">SIN_COS:def 23</a></i>
<br/>.= 
<a href="sin_cos6.html#K4" title="SIN_COS6:func.4">arccos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3"><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>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">r</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:92"><i><font color="Green" title="E89">A1</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2"><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>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">r</font>
<b>by </b><i><a class="txt" href="#E2:92"><i><font color="Green" title="E89">A1</font></i></a>, <a class="txt" href="#E3:92"><i><font color="Green" title="E90">A2</font></i></a>, <a class="ref" href="sin_cos6.html#T91" target="_self" title="SIN_COS6:th.91">Th91</a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c1">r</font>
<b>by </b><i><a class="txt" href="#E2:92"><i><font color="Green" title="E89">A1</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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