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<div class="add">

<a NAME="E1:61"/>
not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:61_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:61_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">p</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> <a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">p</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:61_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos2.html#T15" target="_self" title="SIN_COS2:th.15">Th15</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/>


</div><span class="kw">end;</span></div>
<a NAME="E2:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A1</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a>  <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">/</a> <a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> </span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> &amp; <a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a>  <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T142" title="FUNCT_1:th.142">FUNCT_1:142</a>, <a class="ref" href="rfunct_1.html#D4" title="RFUNCT_1:def.4">RFUNCT_1:def 4</a></span>;<br/>
<a NAME="E3:61"/>
for <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds <br/><a href="sin_cos2.html#K7" title="SIN_COS2:func.7">tanh</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a>  <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">/</a> <a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">p</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  implies <a href="sin_cos2.html#K7" title="SIN_COS2:func.7">tanh</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a>  <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">/</a> <a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">p</font> )</span><br/>

<a NAME="E1:61_2"/>
<font color="Maroon" title="c1">p</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a>  <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">/</a> <a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">p</font> = 
<span class="p1">(<span class="default"><a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="sin_cos2.html#T30" target="_self" title="SIN_COS2:th.30">Th30</a>, <a class="ref" href="rfunct_1.html#D4" title="RFUNCT_1:def.4">RFUNCT_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></span>
<br/>.= 
<a href="sin_cos2.html#K7" title="SIN_COS2:func.7">tanh</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos2.html#T17" target="_self" title="SIN_COS2:th.17">Th17</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:61_2"/>
( <font color="Maroon" title="c1">p</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  implies <a href="sin_cos2.html#K7" title="SIN_COS2:func.7">tanh</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a>  <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">/</a> <a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">p</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:61"/>
 <a href="sin_cos2.html#K7" title="SIN_COS2:func.7">tanh</a>  <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <a href="sin_cos2.html#K1" title="SIN_COS2:func.1">sinh</a>  <a href="rfunct_1.html#K3" title="RFUNCT_1:func.3">/</a> <a href="sin_cos2.html#K4" title="SIN_COS2:func.4">cosh</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="sin_cos2.html#T30" target="_self" title="SIN_COS2:th.30">Th30</a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</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/>


</div>
