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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</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">k</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon" title="c1">x</font> <a href="mesfunc7.html#K4" title="MESFUNC7:func.4">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mesfunc7.html#K4" title="MESFUNC7:func.4">|^</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <font color="Maroon" title="c1">x</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span><font color="Maroon" title="c1">x</font> <a href="mesfunc7.html#K4" title="MESFUNC7:func.4">|^</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mesfunc7.html#K4" title="MESFUNC7:func.4">|^</a> $1</span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <font color="Maroon" title="c1">x</font>;<br/>
<a NAME="E1:24"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">k</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_1"/>
<font color="Maroon" title="c1">x</font> <a href="mesfunc7.html#K4" title="MESFUNC7:func.4">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mesfunc7.html#K4" title="MESFUNC7:func.4">|^</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <font color="Maroon" title="c1">x</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">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">k1</font> = <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E3:24_1"/>
 <a href="mesfunc7.html#K2" title="MESFUNC7:func.2">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc7.html#K2" title="MESFUNC7:func.2">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k1</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T60" title="FINSEQ_2:th.60">FINSEQ_2:60</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:24_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc7.html#T8" target="_self" title="MESFUNC7:th.8">Th8</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="E2:24"/>
<font color="Maroon" title="c1">x</font> <a href="mesfunc7.html#K4" title="MESFUNC7:func.4">|^</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <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>  <a href="mesfunc7.html#K2" title="MESFUNC7:func.2">Product</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T59" title="FINSEQ_2:th.59">FINSEQ_2:59</a></span>;<br/>
<a NAME="E3:24"/><span class="kw">then </span>
<font color="Maroon" title="c1">x</font> <a href="mesfunc7.html#K4" title="MESFUNC7:func.4">|^</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <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> <font color="Maroon" title="c1">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc7.html#T7" target="_self" title="MESFUNC7:th.7">Th7</a></span>;<br/>
<a NAME="E4:24"/><span class="kw">then </span>
<font color="Maroon" title="c1">x</font> <a href="mesfunc7.html#K4" title="MESFUNC7:func.4">|^</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <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> <a href="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <font color="Maroon" title="c1">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T81" title="XXREAL_3:th.81">XXREAL_3:81</a></span>;<br/>
<a NAME="E5:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc7.html#T6" target="_self" title="MESFUNC7:th.6">Th6</a>, <a class="ref" href="finseq_2.html#T58" title="FINSEQ_2:th.58">FINSEQ_2:58</a></span>;<br/>
<a NAME="E6:24"/>
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E5:24"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E13">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E7:24"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon" title="c1">x</font> <a href="mesfunc7.html#K4" title="MESFUNC7:func.4">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mesfunc7.html#K4" title="MESFUNC7:func.4">|^</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="extreal1.html#K1" title="EXTREAL1:func.1">*</a> <font color="Maroon" title="c1">x</font>
 ; <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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