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<span class="kw">let </span><font color="Maroon" title="c1">f</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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> <font color="Maroon" title="c1">f</font> holds <br/><font color="Maroon" title="c1">f</font> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">f</font> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <font color="Olive" title="b1">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font> implies <font color="Maroon" title="c1">f</font> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">f</font> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <font color="Maroon" title="c2">x</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:23"/><span class="lab"><font color="Green" title="E12">A1</font></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> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">f</font> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">f</font> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <font color="Maroon" title="c2">x</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c1">f</font> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span> or  not <font color="Maroon" title="c3">a</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c1">f</font> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <font color="Maroon" title="c2">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:23"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">f</font> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c3">a</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c1">f</font> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <font color="Maroon" title="c2">x</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:23"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:23"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>
 ;<br/>
<a NAME="E6:23"/><span class="lab"><font color="Green" title="E15">A4</font></span>: 
 <a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> (<font color="Maroon" title="c1">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T69" title="FUNCT_7:th.69">FUNCT_7:69</a></span>;<br/>
<a NAME="E7:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A5</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> (<font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>;<br/>
<a NAME="E8:23"/>
<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="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> (<font color="Maroon" title="c1">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E6:23"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:23"/>
 not <font color="Maroon" title="c3">a</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c1">f</font> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <font color="Maroon" title="c2">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:23"><span class="lab"><font color="Green" title="E16">A5</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/>


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