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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">Z</font> be   <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font> implies  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Z</font> )</span><br/>

<a NAME="E1:8"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
 <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:8"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Z</font></span><br/>

<a NAME="E3:8"/>
<span class="p1">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</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="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">x</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="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</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> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:8_1"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E2:8_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c1">Z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></span>;<br/>
<a NAME="E3:8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A4</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T34" title="FUNCT_1:th.34">FUNCT_1:34</a></span>;<br/>
<a NAME="E4:8_1"/>
<span class="p1">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_1"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></span>;<br/>
<a NAME="E5:8_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8_1"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:8_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="txt" href="#E3:8_1"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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="E4:8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A5</font></span>: 
<span class="p1">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</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="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></span>;<br/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c1">Z</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p2"><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 class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#D8" title="RFUNCT_1:def.8">RFUNCT_1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c1">Z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="ref" href="funct_1.html#T34" title="FUNCT_1:th.34">FUNCT_1:34</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:8"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun2.html#K1" title="PARTFUN2:func.1">id</a> <font color="Maroon" title="c1">Z</font></span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a>, <a class="ref" href="sin_cos.html#T27" title="SIN_COS:th.27">SIN_COS:27</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>
