<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">f</font>, <font color="Maroon" title="c2">g</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f</font> holds <br/><font color="Olive" title="b1">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K3" title="GRAPH_2:func.3">^'</a> <font color="Maroon" title="c2">g</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f</font> implies <font color="Maroon" title="c3">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K3" title="GRAPH_2:func.3">^'</a> <font color="Maroon" title="c2">g</font></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:10"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f</font>
 ; <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="c3">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K3" title="GRAPH_2:func.3">^'</a> <font color="Maroon" title="c2">g</font></span>)</span></span><br/>

<a NAME="E2:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T29" title="FINSEQ_4:th.29">FINSEQ_4:29</a></span>;<br/>
<a NAME="E3:10"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="finseq_4.html#T30" title="FINSEQ_4:th.30">FINSEQ_4:30</a></span>;<br/>
<a NAME="E4:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A4</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E5:10"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">f</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K3" title="GRAPH_2:func.3">^'</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">f</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K3" title="GRAPH_2:func.3">^'</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:10_1"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">and </span><br/><a NAME="E2:10_1"/><span class="lab"><font color="Green" title="E15">A7</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K3" title="GRAPH_2:func.3">^'</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font></span><br/>

<a NAME="E3:10_1"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E2:10_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E4:10_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A9</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E5:10_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E6:10_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K3" title="GRAPH_2:func.3">^'</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E3:10_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="ref" href="graph_2.html#T14" title="GRAPH_2:th.14">GRAPH_2:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:10_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K3" title="GRAPH_2:func.3">^'</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E4:10_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="ref" href="finseq_4.html#T34" title="FINSEQ_4:th.34">FINSEQ_4:34</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><span class="kw">end;</span></div>
<a NAME="E6:10"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K3" title="GRAPH_2:func.3">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal8.html#T7" target="_self" title="TOPREAL8:th.7">Th7</a></span>;<br/>
<a NAME="E7:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A10</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K3" title="GRAPH_2:func.3">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T32" title="FINSEQ_3:th.32">FINSEQ_3:32</a></span>;<br/>
<a NAME="E8:10"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E9:10"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K3" title="GRAPH_2:func.3">^'</a> <font color="Maroon" title="c2">g</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="c3">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E4:10"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="graph_2.html#T14" title="GRAPH_2:th.14">GRAPH_2:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:10"/>
<font color="Maroon" title="c3">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K3" title="GRAPH_2:func.3">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E7:10"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="txt" href="#E5:10"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="finseq_6.html#T4" title="FINSEQ_6:th.4">FINSEQ_6:4</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>
