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<span class="kw">let </span><font color="Maroon" title="c1">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</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>  st  not  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (1,<font color="Maroon" title="c1">a</font>)</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">A</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (1,<font color="Maroon" title="c1">a</font>)</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">g2</font> =  <a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (1,<font color="Maroon" title="c1">a</font>);<br/>
<span class="kw">assume </span><a NAME="E1:46"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
 not  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (1,<font color="Maroon" title="c1">a</font>)</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a> </span><br/>

<a NAME="E2:46"/>
 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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (1,<font color="Maroon" title="c1">a</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c4">h2</font> = <span class="p1">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (1,<font color="Maroon" title="c1">a</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font>;<br/>
<span class="kw">assume </span><a NAME="E1:46_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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (1,<font color="Maroon" title="c1">a</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:46_1"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
<font color="Maroon" title="c5">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"><span class="p2">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (1,<font color="Maroon" title="c1">a</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:46_1"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (1,<font color="Maroon" title="c1">a</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">d</font> = <font color="Maroon" title="c5">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="#E3:46_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a></span>;<br/>
<a NAME="E6:46_1"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<span class="p1">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (1,<font color="Maroon" title="c1">a</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c6">d</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fcont_1.html#D4" title="FCONT_1:def.4">FCONT_1:def 4</a></span>;<br/>
<a NAME="E7:46_1"/>
<font color="Maroon" title="c6">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="relat_1.html#T57" title="RELAT_1:th.57">RELAT_1:57</a></span>;<br/>
<a NAME="E8:46_1"/><span class="kw">then </span>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (1,<font color="Maroon" title="c1">a</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c1">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c6">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E6:46_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a>, <a class="ref" href="relat_1.html#T58" title="RELAT_1:th.58">RELAT_1:58</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:46_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E3:46_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A5</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (1,<font color="Maroon" title="c1">a</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="seq_4.html#K1" title="SEQ_4: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#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
<a NAME="E4:46"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (1,<font color="Maroon" title="c1">a</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> &amp; <span class="p1">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (1,<font color="Maroon" title="c1">a</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (1,<font color="Maroon" title="c1">a</font>)</span>)</span></span>)</span> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relat_1.html#T69" title="RELAT_1:th.69">RELAT_1:69</a></span>;<br/>
<a NAME="E5:46"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (1,<font color="Maroon" title="c1">a</font>)</span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fcont_1.html#T16" title="FCONT_1:th.16">FCONT_1:16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:46"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (1,<font color="Maroon" title="c1">a</font>)</span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">A</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="fcont_1.html#T23" title="FCONT_1:th.23">FCONT_1:23</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
