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<b>let </b><font color="Maroon" title="c3">g1</font>, <font color="Maroon" title="c4">g2</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:44_1_2"/><i><font color="Green" title="E31">A4</font></i>: 
<font color="Maroon" title="c3">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font>
 <b>and </b><br/><a NAME="E2:44_1_2"/><i><font color="Green" title="E32">A5</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  <font color="Maroon" title="c3">g1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><a href="int_1.html#K4" title="INT_1:func.4">frac</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E3:44_1_2"/><i><font color="Green" title="E33">A6</font></i>: 
<font color="Maroon" title="c4">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font>
 <b>and </b><br/><a NAME="E4:44_1_2"/><i><font color="Green" title="E34">A7</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  <font color="Maroon" title="c4">g2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><a href="int_1.html#K4" title="INT_1:func.4">frac</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">g2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">r1</font> = <font color="Maroon" title="c2">r</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">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/>
<a NAME="E6:44_1_2"/><i><font color="Green" title="E35">A8</font></i>: 
for <font color="Olive" title="b1">n</font> being   <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/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y1</font>, <font color="Olive" title="b4">y2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b4">y2</font>] holds <br/><font color="Olive" title="b3">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">y2</font>
 
;<br/>
<a NAME="E7:44_1_2"/><i><font color="Green" title="E36">A9</font></i>: 
( <font color="Maroon" title="c3">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r1</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c3">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c3">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<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>] ) )
 
<b>by </b><i><a class="txt" href="real_3.html#E1:44_1_2"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="real_3.html#E2:44_1_2"><i><font color="Green" title="E32">A5</font></i></a></i>;<br/>
<a NAME="E8:44_1_2"/><i><font color="Green" title="E37">A10</font></i>: 
( <font color="Maroon" title="c4">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r1</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c4">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c4">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<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>] ) )
 
<b>by </b><i><a class="txt" href="real_3.html#E3:44_1_2"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="real_3.html#E4:44_1_2"><i><font color="Green" title="E34">A7</font></i></a></i>;<br/>
<b>thus </b><a NAME="E9:44_1_2"/>
<font color="Maroon" title="c3">g1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">g2</font>
 <b>from </b><i><a class="ref" href="recdef_1.html#S10" title="RECDEF_1:sch.10">RECDEF_1:sch 10</a>(<a class="txt" href="real_3.html#E7:44_1_2"><i><font color="Green" title="E36">A9</font></i></a>, <a class="txt" href="real_3.html#E8:44_1_2"><i><font color="Green" title="E37">A10</font></i></a>, <a class="txt" href="real_3.html#E6:44_1_2"><i><font color="Green" title="E35">A8</font></i></a>);</i><br/>


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