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<b>let </b><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> ;<br/><b>let </b><font color="Maroon" title="c2">y</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>let </b><font color="Maroon" title="c3">k</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>





<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ] <b>means </b><font color="Maroon" title="c1">x</font> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> $1;<br/>
<a NAME="E2:27"/>
<font color="Maroon" title="c1">x</font> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K9" title="MESFUNC1:func.9">1.</a> 
 
<b>by </b><i><a class="ref" href="mesfunc7.html#T6" target="_self" title="MESFUNC7:th.6">Th6</a>, <a class="ref" href="finseq_2.html#T72" title="FINSEQ_2:th.72">FINSEQ_2:72</a></i>;<br/>
<a NAME="E3:27"/><b>then </b><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>;<br/>
<a NAME="E4:27"/><i><font color="Green" title="E17">A3</font></i>: 
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>   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="27_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">k</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">y1</font> = <font color="Maroon" title="c2">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>assume </b><a NAME="E2:27_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>]
 ;<br/>

<a NAME="E3:27_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y1</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">y1</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E1:27"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="extreal1.html#T13" title="EXTREAL1:th.13">EXTREAL1:13</a></i>;<br/>
<a NAME="E4:27_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>;<br/>
<b>hence </b><a NAME="E5:27_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="ref" href="mesfunc7.html#T10" target="_self" title="MESFUNC7:th.10">Th10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:27"/>
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">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="mesfunc7.html#E3:27"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="mesfunc7.html#E4:27"><i><font color="Green" title="E17">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:27"/>
<font color="Maroon" title="c1">x</font> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">k</font>
 ;<br/>


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