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<b>let </b><font color="Maroon" title="c2">r</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <span class="p1">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) implies for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/>1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> )</span><br/>

<b>set </b><font color="Maroon" title="c3">s</font> =  <a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font>;<br/>
<b>set </b><font color="Maroon" title="c4">s2</font> =  <a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font>;<br/>
<b>assume </b><a NAME="E1:72"/><i><font color="Green" title="E51">A1</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <span class="p1">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/>1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span><br/>

<a NAME="E2:72"/><b>then </b><i><font color="Green" title="E52">A2</font></i>: 
<span class="p1">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>;<br/>
<a NAME="E3:72"/><i><font color="Green" title="E53">A3</font></i>: 
( <span class="p1">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="#E1:72"><i><font color="Green" title="E51">A1</font></i></a></i>;<br/>
<a NAME="E4:72"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="square_1.html#T74" title="SQUARE_1:th.74">SQUARE_1:74</a></i>;<br/>
<a NAME="E5:72"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E3:72"><i><font color="Green" title="E53">A3</font></i></a>, <a class="ref" href="xreal_1.html#T131" title="XREAL_1:th.131">XREAL_1:131</a></i>;<br/>
<a NAME="E6:72"/><b>then </b><i><font color="Green" title="E54">A4</font></i>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:72"><i><font color="Green" title="E53">A3</font></i></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a>, <a class="ref" href="xreal_1.html#T78" title="XREAL_1:th.78">XREAL_1:78</a></i>;<br/>
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> = 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p5">(<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> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p5">(<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></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="real_3.html#D7" target="_self" title="REAL_3:def.7">Def7</a></i>
<br/>.= 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="real_3.html#D7" target="_self" title="REAL_3:def.7">Def7</a></i>
<br/>.= 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="real_3.html#D7" target="_self" title="REAL_3:def.7">Def7</a></i>
<br/>.= 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="real_3.html#D7" target="_self" title="REAL_3:def.7">Def7</a></i>
<br/>.= 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1</span>)</span></span>)</span>

;<br/>
<a NAME="E8:72"/><b>then </b><i><font color="Green" title="E55">A5</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[1]
 
<b>by </b><i><a class="txt" href="#E6:72"><i><font color="Green" title="E54">A4</font></i></a>, <a class="ref" href="real_3.html#D7" target="_self" title="REAL_3:def.7">Def7</a></i>;<br/>
<a NAME="E9:72"/><i><font color="Green" title="E56">A6</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">n</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:72_2"/>
( <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:72_2"/><i><font color="Green" title="E57">A7</font></i>: 
<span class="p1">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E1:72"><i><font color="Green" title="E51">A1</font></i></a></i>;<br/>
<a NAME="E3:72_2"/><i><font color="Green" title="E58">A8</font></i>: 
( <span class="p1">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="#E2:72"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="real_3.html#T52" target="_self" title="REAL_3:th.52">Th52</a></i>;<br/>
<a NAME="E4:72_2"/><b>then </b><i><font color="Green" title="E59">A9</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T131" title="XREAL_1:th.131">XREAL_1:131</a></i>;<br/>
<a NAME="E5:72_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E3:72_2"><i><font color="Green" title="E58">A8</font></i></a>, <a class="ref" href="square_1.html#T74" title="SQUARE_1:th.74">SQUARE_1:74</a></i>;<br/>
<a NAME="E6:72_2"/><b>then </b><i><font color="Green" title="E60">A10</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E2:72_2"><i><font color="Green" title="E57">A7</font></i></a>, <a class="ref" href="xreal_1.html#T131" title="XREAL_1:th.131">XREAL_1:131</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="real_3.html#D7" target="_self" title="REAL_3:def.7">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E8:72_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="#E4:72_2"><i><font color="Green" title="E59">A9</font></i></a>, <a class="txt" href="#E6:72_2"><i><font color="Green" title="E60">A10</font></i></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a>, <a class="ref" href="xreal_1.html#T78" title="XREAL_1:th.78">XREAL_1:78</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E10:72"/><i><font color="Green" title="E57">A11</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S8" title="NAT_1:sch.8">NAT_1:sch 8</a>(<a class="txt" href="#E8:72"><i><font color="Green" title="E55">A5</font></i></a>, <a class="txt" href="#E9:72"><i><font color="Green" title="E56">A6</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c5">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 implies 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> )</span><br/>

<b>thus </b><a NAME="E11:72"/>
( <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 implies 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK6" title="REAL_3:NK.6">scf</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#NK10" title="REAL_3:NK.10">c_d</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E10:72"><i><font color="Green" title="E57">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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