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<b>let </b><font color="Maroon" title="c1">f1</font>, <font color="Maroon" title="c2">f2</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c3">x0</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>





<b>assume </b><a NAME="E1:16"/><i><font color="Green" title="E17">A1</font></i>: 
( <font color="Maroon" title="c1">f1</font> <a href="fdiff_3.html#R4" title="FDIFF_3:pred.4">is_left_differentiable_in</a> <font color="Maroon" title="c3">x0</font> &amp; <font color="Maroon" title="c2">f2</font> <a href="fdiff_3.html#R4" title="FDIFF_3:pred.4">is_left_differentiable_in</a> <font color="Maroon" title="c3">x0</font> )
 ;<br/>

<b>given </b><font color="Maroon" title="c4">r0</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><a NAME="E2:16"/><i><font color="Green" title="E18">A2</font></i>: 
( <font color="Maroon" title="c4">r0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; ( for <font color="Olive" title="b1">g</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f2</font> &amp; <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c4">r0</font></span>)</span>,<font color="Maroon" title="c3">x0</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> holds <br/><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 ) )
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:16"/><i><font color="Green" title="E19">A3</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r1</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r1</font></span>)</span>,<font color="Maroon" title="c3">x0</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f1</font> )
 <b>by </b><i><a class="txt" href="fdiff_3.html#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="fdiff_3.html#D4" target="_self" title="FDIFF_3:def.4">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">r2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E6:16"/><i><font color="Green" title="E20">A4</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">r2</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c6">r2</font></span>)</span>,<font color="Maroon" title="c3">x0</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f2</font> )
 <b>by </b><i><a class="txt" href="fdiff_3.html#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="fdiff_3.html#D4" target="_self" title="FDIFF_3:def.4">Def4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">r3</font> =  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font>;<br/>
<a NAME="E7:16"/><i><font color="Green" title="E21">A5</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E2:16"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="fdiff_3.html#E6:16"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<a NAME="E8:16"/><b>then </b><i><font color="Green" title="E22">A6</font></i>: 
<font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x0</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T45" title="XREAL_1:th.45">XREAL_1:45</a></i>;<br/>
<a NAME="E9:16"/>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">r0</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E10:16"/><b>then </b><i><font color="Green" title="E23">A7</font></i>: 
<font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c4">r0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/>
<a NAME="E11:16"/><b>then </b>
<font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c4">r0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">g</font> &amp; <font color="Olive" title="b1">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x0</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E8:16"><i><font color="Green" title="E22">A6</font></i></a></i>;<br/>
<a NAME="E12:16"/><b>then </b><i><font color="Green" title="E24">A8</font></i>: 
<font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c4">r0</font></span>)</span>,<font color="Maroon" title="c3">x0</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E13:16"/>
<font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c4">r0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x0</font>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E8:16"><i><font color="Green" title="E22">A6</font></i></a>, <a class="txt" href="fdiff_3.html#E10:16"><i><font color="Green" title="E23">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E14:16"/><b>then </b><i><font color="Green" title="E25">A9</font></i>: 
<font color="Maroon" title="c3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c4">r0</font></span>)</span>,<font color="Maroon" title="c3">x0</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a></i>;<br/>
<a NAME="E15:16"/><b>then </b><i><font color="Green" title="E26">A10</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">x0</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c4">r0</font></span>)</span>,<font color="Maroon" title="c3">x0</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E12:16"><i><font color="Green" title="E24">A8</font></i></a>, <a class="ref" href="rcomp_1.html#T16" title="RCOMP_1:th.16">RCOMP_1:16</a></i>;<br/>
<a NAME="E16:16"/>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r2</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E17:16"/><b>then </b><i><font color="Green" title="E27">A11</font></i>: 
<font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c6">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/>
<a NAME="E18:16"/><b>then </b>
<font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c6">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">g</font> &amp; <font color="Olive" title="b1">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x0</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E8:16"><i><font color="Green" title="E22">A6</font></i></a></i>;<br/>
<a NAME="E19:16"/><b>then </b><i><font color="Green" title="E28">A12</font></i>: 
<font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c6">r2</font></span>)</span>,<font color="Maroon" title="c3">x0</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E20:16"/>
<font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c6">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x0</font>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E8:16"><i><font color="Green" title="E22">A6</font></i></a>, <a class="txt" href="fdiff_3.html#E17:16"><i><font color="Green" title="E27">A11</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E21:16"/><b>then </b><i><font color="Green" title="E29">A13</font></i>: 
<font color="Maroon" title="c3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c6">r2</font></span>)</span>,<font color="Maroon" title="c3">x0</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a></i>;<br/>
<a NAME="E22:16"/><b>then </b>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">x0</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c6">r2</font></span>)</span>,<font color="Maroon" title="c3">x0</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E19:16"><i><font color="Green" title="E28">A12</font></i></a>, <a class="ref" href="rcomp_1.html#T16" title="RCOMP_1:th.16">RCOMP_1:16</a></i>;<br/>
<a NAME="E23:16"/><b>then </b><i><font color="Green" title="E30">A14</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">x0</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f2</font>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E6:16"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">r</font> =  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c5">r1</font>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span>;<br/>
<a NAME="E24:16"/><i><font color="Green" title="E31">A15</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c5">r1</font>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E4:16"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="fdiff_3.html#E7:16"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<a NAME="E25:16"/><b>then </b><i><font color="Green" title="E32">A16</font></i>: 
<font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c5">r1</font>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x0</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T45" title="XREAL_1:th.45">XREAL_1:45</a></i>;<br/>
<a NAME="E26:16"/>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c5">r1</font>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r1</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E27:16"/><b>then </b><i><font color="Green" title="E33">A17</font></i>: 
<font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c5">r1</font>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/>
<a NAME="E28:16"/><b>then </b>
<font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c5">r1</font>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">g</font> &amp; <font color="Olive" title="b1">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x0</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E25:16"><i><font color="Green" title="E32">A16</font></i></a></i>;<br/>
<a NAME="E29:16"/><b>then </b><i><font color="Green" title="E34">A18</font></i>: 
<font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c5">r1</font>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r1</font></span>)</span>,<font color="Maroon" title="c3">x0</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E30:16"/>
<font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x0</font>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E25:16"><i><font color="Green" title="E32">A16</font></i></a>, <a class="txt" href="fdiff_3.html#E27:16"><i><font color="Green" title="E33">A17</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E31:16"/><b>then </b>
<font color="Maroon" title="c3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r1</font></span>)</span>,<font color="Maroon" title="c3">x0</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a></i>;<br/>
<a NAME="E32:16"/><b>then </b><i><font color="Green" title="E35">A19</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c5">r1</font>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c3">x0</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r1</font></span>)</span>,<font color="Maroon" title="c3">x0</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E29:16"><i><font color="Green" title="E34">A18</font></i></a>, <a class="ref" href="rcomp_1.html#T16" title="RCOMP_1:th.16">RCOMP_1:16</a></i>;<br/>
<a NAME="E33:16"/>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c5">r1</font>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E34:16"/><b>then </b><i><font color="Green" title="E36">A20</font></i>: 
<font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c5">r1</font>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/>
<a NAME="E35:16"/><b>then </b>
<font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c5">r1</font>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">g</font> &amp; <font color="Olive" title="b1">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x0</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E25:16"><i><font color="Green" title="E32">A16</font></i></a></i>;<br/>
<a NAME="E36:16"/><b>then </b><i><font color="Green" title="E37">A21</font></i>: 
<font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c5">r1</font>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">x0</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E37:16"/>
<font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x0</font>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E25:16"><i><font color="Green" title="E32">A16</font></i></a>, <a class="txt" href="fdiff_3.html#E34:16"><i><font color="Green" title="E36">A20</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E38:16"/><b>then </b>
<font color="Maroon" title="c3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">x0</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a></i>;<br/>
<a NAME="E39:16"/><b>then </b><i><font color="Green" title="E38">A22</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c5">r1</font>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c3">x0</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">x0</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E36:16"><i><font color="Green" title="E37">A21</font></i></a>, <a class="ref" href="rcomp_1.html#T16" title="RCOMP_1:th.16">RCOMP_1:16</a></i>;<br/>
<a NAME="E40:16"/><i><font color="Green" title="E39">A23</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c5">r1</font>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c3">x0</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f1</font>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E4:16"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="fdiff_3.html#E32:16"><i><font color="Green" title="E35">A19</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E41:16"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c5">r1</font>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c3">x0</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1"/>
not <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c5">r1</font>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c3">x0</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:16_1"/><i><font color="Green" title="E40">A24</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c5">r1</font>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c3">x0</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">x</font> = <font color="Maroon" title="c9">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="fdiff_3.html#E3:16_1"><i><font color="Green" title="E40">A24</font></i></a></i>;<br/>
<a NAME="E5:16_1"/><i><font color="Green" title="E41">A25</font></i>: 
not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E3:16_1"><i><font color="Green" title="E40">A24</font></i></a>, <a class="ref" href="rfunct_1.html#D8" title="RFUNCT_1:def.8">RFUNCT_1:def 8</a></i>;<br/>
<a NAME="E6:16_1"/><i><font color="Green" title="E42">A26</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">x0</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E39:16"><i><font color="Green" title="E38">A22</font></i></a>, <a class="txt" href="fdiff_3.html#E3:16_1"><i><font color="Green" title="E40">A24</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1_1"/>
(  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f2</font> or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="fdiff_3.html#E5:16_1"><i><font color="Green" title="E41">A25</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_1"><b>suppose </b></a><a NAME="E1:16_1_1_1_1"/>
not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fdiff_3.html#E23:16"><i><font color="Green" title="E30">A14</font></i></a>, <a class="txt" href="fdiff_3.html#E6:16_1"><i><font color="Green" title="E42">A26</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_2"><b>suppose </b></a><a NAME="E1:16_1_1_1_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:16_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f2</font> &amp; <font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/><a NAME="E3:16_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f2</font> &amp; <font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:16_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fdiff_3.html#E2:16"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="fdiff_3.html#E15:16"><i><font color="Green" title="E26">A10</font></i></a>, <a class="txt" href="fdiff_3.html#E6:16_1"><i><font color="Green" title="E42">A26</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:16_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E42:16"/><b>then </b><i><font color="Green" title="E40">A27</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c5">r1</font>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c3">x0</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D8" title="RFUNCT_1:def.8">RFUNCT_1:def 8</a></i>;<br/>
<a NAME="E43:16"/><b>then </b>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c5">r1</font>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c3">x0</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E40:16"><i><font color="Green" title="E39">A23</font></i></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E44:16"/><b>then </b><i><font color="Green" title="E41">A28</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c5">r1</font>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c3">x0</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D4" title="RFUNCT_1:def.4">RFUNCT_1:def 4</a></i>;<br/>
<a NAME="E45:16"/>
for <font color="Olive" title="b1">h</font> being <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><br/> for <font color="Olive" title="b2">c</font> being <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Olive" title="b2">c</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">f2</font></span>)</span> &amp; ( for <font color="Olive" title="b3">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="Olive" title="b1">h</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0 ) holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Olive" title="b2">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Olive" title="b2">c</font></span>)</span></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Olive" title="b2">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Olive" title="b2">c</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">x0</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> <font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">x0</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">h</font> be  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/><b>let </b><font color="Maroon" title="c10">c</font> be  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:16_2"/><i><font color="Green" title="E42">A29</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>and </b><br/><a NAME="E2:16_2"/><i><font color="Green" title="E43">A30</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">f2</font></span>)</span>
 <b>and </b><br/><a NAME="E3:16_2"/><i><font color="Green" title="E44">A31</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>   holds <font color="Maroon" title="c9">h</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0
 ;<br/>

<a NAME="E4:16_2"/><i><font color="Green" title="E45">A32</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E2:16_2"><i><font color="Green" title="E43">A30</font></i></a>, <a class="ref" href="rfunct_1.html#D4" title="RFUNCT_1:def.4">RFUNCT_1:def 4</a></i>;<br/>
<a NAME="E5:16_2"/>
( <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f1</font> &amp; <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E6:16_2"/><b>then </b><i><font color="Green" title="E46">A33</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f1</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> )
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E4:16_2"><i><font color="Green" title="E45">A32</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E7:16_2"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E8:16_2"/><b>then </b><i><font color="Green" title="E47">A34</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f2</font>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E6:16_2"><i><font color="Green" title="E46">A33</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:16_2"/>
 <a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">x0</font>
 
;<br/>
<a NAME="E10:16_2"/><b>then </b><i><font color="Green" title="E48">A35</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">x0</font> )
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="fdiff_3.html#E1:16_2"><i><font color="Green" title="E42">A29</font></i></a>, <a class="txt" href="fdiff_3.html#E3:16_2"><i><font color="Green" title="E44">A31</font></i></a>, <a class="txt" href="fdiff_3.html#E6:16_2"><i><font color="Green" title="E46">A33</font></i></a>, <a class="ref" href="fdiff_3.html#T9" target="_self" title="FDIFF_3:th.9">Th9</a></i>;<br/>
<a NAME="E11:16_2"/>
 <a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> <font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> <font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">x0</font>
 
;<br/>
<a NAME="E12:16_2"/><b>then </b><i><font color="Green" title="E49">A36</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> <font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">x0</font> )
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="fdiff_3.html#E1:16_2"><i><font color="Green" title="E42">A29</font></i></a>, <a class="txt" href="fdiff_3.html#E3:16_2"><i><font color="Green" title="E44">A31</font></i></a>, <a class="txt" href="fdiff_3.html#E8:16_2"><i><font color="Green" title="E47">A34</font></i></a>, <a class="ref" href="fdiff_3.html#T9" target="_self" title="FDIFF_3:th.9">Th9</a></i>;<br/>
<a NAME="E13:16_2"/><i><font color="Green" title="E50">A37</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E6:16_2"><i><font color="Green" title="E46">A33</font></i></a>, <a class="ref" href="rfunct_1.html#D8" title="RFUNCT_1:def.8">RFUNCT_1:def 8</a></i>;<br/>
<a NAME="E14:16_2"/><b>then </b><i><font color="Green" title="E51">A38</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E6:16_2"><i><font color="Green" title="E46">A33</font></i></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E15:16_2"/><i><font color="Green" title="E52">A39</font></i>: 
<font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#NR1" title="SEQ_1:NR.1">is_not_0</a> 
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E13:16_2"><i><font color="Green" title="E50">A37</font></i></a>, <a class="ref" href="rfunct_2.html#T26" title="RFUNCT_2:th.26">RFUNCT_2:26</a></i>;<br/>
<a NAME="E16:16_2"/><i><font color="Green" title="E53">A40</font></i>: 
for <font color="Olive" title="b1">m</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" title="c10">c</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">m</font> be    <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="E1:16_2_1"/>
<font color="Maroon" title="c10">c</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">c</font>
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T10" title="RFUNCT_2:th.10">RFUNCT_2:10</a></i>;<br/>
<b>hence </b><a NAME="E2:16_2_1"/>
<font color="Maroon" title="c10">c</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x0</font>
 <b>by </b><i><a class="txt" href="fdiff_3.html#E1:16_2"><i><font color="Green" title="E42">A29</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:16_2"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c5">r1</font>,<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E4:16"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="fdiff_3.html#E7:16"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<a NAME="E18:16_2"/><b>then </b>
<font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c5">r1</font>,<span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x0</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T45" title="XREAL_1:th.45">XREAL_1:45</a></i>;<br/>
<a NAME="E19:16_2"/><b>then </b><i><font color="Green" title="E54">A41</font></i>: 
<font color="Maroon" title="c3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c5">r1</font>,<span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r0</font>,<font color="Maroon" title="c6">r2</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c3">x0</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a></i>;<br/>
<a NAME="E20:16_2"/><b>then </b><i><font color="Green" title="E55">A42</font></i>: 
<font color="Maroon" title="c3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f1</font>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E40:16"><i><font color="Green" title="E39">A23</font></i></a></i>;<br/>
<a NAME="E21:16_2"/><i><font color="Green" title="E56">A43</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:16_2_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">c</font>
 ;<br/>

<b>hence </b><a NAME="E2:16_2_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f1</font>
 <b>by </b><i><a class="txt" href="fdiff_3.html#E1:16_2"><i><font color="Green" title="E42">A29</font></i></a>, <a class="txt" href="fdiff_3.html#E20:16_2"><i><font color="Green" title="E55">A42</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:16_2"/><i><font color="Green" title="E57">A44</font></i>: 
for <font color="Olive" title="b1">g</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">g</font> holds <br/> ex <font color="Olive" title="b2">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>  st <br/>for <font color="Olive" title="b3">m</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>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">g</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>assume </b><a NAME="E1:16_2_3"/><i><font color="Green" title="E58">A45</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c11">g</font>
 ;<br/>

<b>take </b><font color="Maroon" title="c12">n</font> = 0;<br/>

<b>let </b><font color="Maroon" title="c13">m</font> be    <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/>

<b>assume </b><a NAME="E2:16_2_3"/>
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font>
 ;<br/>

 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span> = 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">c</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_3.html#E21:16_2"><i><font color="Green" title="E56">A43</font></i></a>, <a class="ref" href="rfunct_2.html#T21" title="RFUNCT_2:th.21">RFUNCT_2:21</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_3.html#E16:16_2"><i><font color="Green" title="E53">A40</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E4:16_2_3"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c11">g</font>
 <b>by </b><i><a class="txt" href="fdiff_3.html#E1:16_2_3"><i><font color="Green" title="E58">A45</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:16_2"/><i><font color="Green" title="E58">A46</font></i>: 
( <font color="Maroon" title="c9">h</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c9">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="fdiff_1.html#D1" title="FDIFF_1:def.1">FDIFF_1:def 1</a></i>;<br/>
<a NAME="E24:16_2"/><i><font color="Green" title="E59">A47</font></i>: 
<font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E22:16_2"><i><font color="Green" title="E57">A44</font></i></a>, <a class="ref" href="seq_2.html#D6" title="SEQ_2:def.6">SEQ_2:def 6</a></i>;<br/>
<a NAME="E25:16_2"/><b>then </b><i><font color="Green" title="E60">A48</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E22:16_2"><i><font color="Green" title="E57">A44</font></i></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></i>;<br/>
<a NAME="E26:16_2"/><i><font color="Green" title="E61">A49</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> )
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="fdiff_3.html#E1:16_2"><i><font color="Green" title="E42">A29</font></i></a>, <a class="txt" href="fdiff_3.html#E3:16_2"><i><font color="Green" title="E44">A31</font></i></a>, <a class="txt" href="fdiff_3.html#E6:16_2"><i><font color="Green" title="E46">A33</font></i></a>, <a class="txt" href="fdiff_3.html#E8:16_2"><i><font color="Green" title="E47">A34</font></i></a>, <a class="ref" href="fdiff_3.html#D4" target="_self" title="FDIFF_3:def.4">Def4</a></i>;<br/>
<a NAME="E27:16_2"/><i><font color="Green" title="E62">A50</font></i>: 
<font color="Maroon" title="c3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E42:16"><i><font color="Green" title="E40">A27</font></i></a>, <a class="txt" href="fdiff_3.html#E19:16_2"><i><font color="Green" title="E54">A41</font></i></a></i>;<br/>
<a NAME="E28:16_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:16_2_4"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">c</font>
 ;<br/>

<b>hence </b><a NAME="E2:16_2_4"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="fdiff_3.html#E1:16_2"><i><font color="Green" title="E42">A29</font></i></a>, <a class="txt" href="fdiff_3.html#E27:16_2"><i><font color="Green" title="E62">A50</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E29:16_2"/><b>then </b><i><font color="Green" title="E63">A51</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D8" title="RFUNCT_1:def.8">RFUNCT_1:def 8</a></i>;<br/>
<a NAME="E30:16_2"/><b>then </b><i><font color="Green" title="E64">A52</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E21:16_2"><i><font color="Green" title="E56">A43</font></i></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E31:16_2"/><i><font color="Green" title="E65">A53</font></i>: 
<font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font> <a href="seq_1.html#NR1" title="SEQ_1:NR.1">is_not_0</a> 
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E29:16_2"><i><font color="Green" title="E63">A51</font></i></a>, <a class="ref" href="rfunct_2.html#T26" title="RFUNCT_2:th.26">RFUNCT_2:26</a></i>;<br/>
<a NAME="E32:16_2"/><b>then </b><i><font color="Green" title="E66">A54</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#NR1" title="SEQ_1:NR.1">is_not_0</a> 
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E15:16_2"><i><font color="Green" title="E52">A39</font></i></a>, <a class="ref" href="seq_1.html#T43" title="SEQ_1:th.43">SEQ_1:43</a></i>;<br/>
<a NAME="E33:16_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D8" title="RFUNCT_1:def.8">RFUNCT_1:def 8</a></i>;<br/>
<a NAME="E34:16_2"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E35:16_2"/><b>then </b><i><font color="Green" title="E67">A55</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f2</font>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E29:16_2"><i><font color="Green" title="E63">A51</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E36:16_2"/><i><font color="Green" title="E68">A56</font></i>: 
for <font color="Olive" title="b1">g</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">g</font> holds <br/> ex <font color="Olive" title="b2">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>  st <br/>for <font color="Olive" title="b3">m</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>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">g</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>assume </b><a NAME="E1:16_2_5"/><i><font color="Green" title="E69">A57</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c11">g</font>
 ;<br/>

<b>take </b><font color="Maroon" title="c12">n</font> = 0;<br/>

<b>let </b><font color="Maroon" title="c13">m</font> be    <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/>

<b>assume </b><a NAME="E2:16_2_5"/>
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font>
 ;<br/>

 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span> = 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">c</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_3.html#E35:16_2"><i><font color="Green" title="E67">A55</font></i></a>, <a class="ref" href="rfunct_2.html#T21" title="RFUNCT_2:th.21">RFUNCT_2:21</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_3.html#E16:16_2"><i><font color="Green" title="E53">A40</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E4:16_2_5"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c11">g</font>
 <b>by </b><i><a class="txt" href="fdiff_3.html#E1:16_2_5"><i><font color="Green" title="E69">A57</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E37:16_2"/><b>then </b><i><font color="Green" title="E69">A58</font></i>: 
<font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_2.html#D6" title="SEQ_2:def.6">SEQ_2:def 6</a></i>;<br/>
<a NAME="E38:16_2"/><b>then </b><i><font color="Green" title="E70">A59</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E36:16_2"><i><font color="Green" title="E68">A56</font></i></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></i>;<br/>
<a NAME="E39:16_2"/><i><font color="Green" title="E71">A60</font></i>: 
<font color="Maroon" title="c9">h</font> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E12:16_2"><i><font color="Green" title="E49">A36</font></i></a>, <a class="txt" href="fdiff_3.html#E23:16_2"><i><font color="Green" title="E58">A46</font></i></a>, <a class="ref" href="seq_2.html#T28" title="SEQ_2:th.28">SEQ_2:28</a></i>;<br/>
<a NAME="E40:16_2"/><i><font color="Green" title="E72">A61</font></i>: 
<font color="Maroon" title="c9">h</font> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_6"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">n</font> be    <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="E1:16_2_6_1"/><i><font color="Green" title="E73">A62</font></i>: 
<font color="Maroon" title="c9">h</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="fdiff_3.html#E3:16_2"><i><font color="Green" title="E44">A31</font></i></a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font>
<b>by </b><i><a class="ref" href="seq_1.html#T22" title="SEQ_1:th.22">SEQ_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D8" title="SEQ_1:def.8">SEQ_1:def 8</a></i>
<br/>.= 
1 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_3.html#E1:16_2_6_1"><i><font color="Green" title="E73">A62</font></i></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font>

;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:16_2_6"/>
<font color="Maroon" title="c9">h</font> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E41:16_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_7"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_7_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">n</font> be    <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/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_2.html#T6" title="RFUNCT_2:th.6">RFUNCT_2:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font>

;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:16_2_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E42:16_2"/><b>then </b><i><font color="Green" title="E73">A63</font></i>: 
<font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E37:16_2"><i><font color="Green" title="E69">A58</font></i></a>, <a class="txt" href="fdiff_3.html#E39:16_2"><i><font color="Green" title="E71">A60</font></i></a>, <a class="txt" href="fdiff_3.html#E40:16_2"><i><font color="Green" title="E72">A61</font></i></a>, <a class="ref" href="seq_2.html#T19" title="SEQ_2:th.19">SEQ_2:19</a></i>;<br/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c9">h</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_3.html#E23:16_2"><i><font color="Green" title="E58">A46</font></i></a>, <a class="txt" href="fdiff_3.html#E26:16_2"><i><font color="Green" title="E61">A49</font></i></a>, <a class="ref" href="seq_2.html#T29" title="SEQ_2:th.29">SEQ_2:29</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="fdiff_3.html#E23:16_2"><i><font color="Green" title="E58">A46</font></i></a></i>
;<br/>
<a NAME="E44:16_2"/><b>then </b><i><font color="Green" title="E74">A64</font></i>: 
0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E37:16_2"><i><font color="Green" title="E69">A58</font></i></a>, <a class="txt" href="fdiff_3.html#E38:16_2"><i><font color="Green" title="E70">A59</font></i></a>, <a class="txt" href="fdiff_3.html#E40:16_2"><i><font color="Green" title="E72">A61</font></i></a>, <a class="txt" href="fdiff_3.html#E42:16_2"><i><font color="Green" title="E73">A63</font></i></a>, <a class="ref" href="seq_2.html#T26" title="SEQ_2:th.26">SEQ_2:26</a></i>;<br/>
<a NAME="E45:16_2"/><i><font color="Green" title="E75">A65</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E37:16_2"><i><font color="Green" title="E69">A58</font></i></a>, <a class="txt" href="fdiff_3.html#E42:16_2"><i><font color="Green" title="E73">A63</font></i></a>, <a class="ref" href="seq_2.html#T28" title="SEQ_2:th.28">SEQ_2:28</a></i>;<br/>
<a NAME="E46:16_2"/><i><font color="Green" title="E76">A66</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E37:16_2"><i><font color="Green" title="E69">A58</font></i></a>, <a class="txt" href="fdiff_3.html#E42:16_2"><i><font color="Green" title="E73">A63</font></i></a>, <a class="ref" href="seq_2.html#T29" title="SEQ_2:th.29">SEQ_2:29</a>, <a class="txt" href="fdiff_3.html#E38:16_2"><i><font color="Green" title="E70">A59</font></i></a>, <a class="txt" href="fdiff_3.html#E44:16_2"><i><font color="Green" title="E74">A64</font></i></a></i>;<br/>
<a NAME="E47:16_2"/><i><font color="Green" title="E77">A67</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_9"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_2_9"/>
not  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:16_2_9"/><b>then </b>
<font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E46:16_2"><i><font color="Green" title="E76">A66</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>;<br/>
<b>hence </b><a NAME="E3:16_2_9"/>
contradiction
 <b>by </b><i><a class="txt" href="fdiff_3.html#E2:16"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="fdiff_3.html#E6:16"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="fdiff_3.html#E14:16"><i><font color="Green" title="E25">A9</font></i></a>, <a class="txt" href="fdiff_3.html#E21:16"><i><font color="Green" title="E29">A13</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E48:16_2"/><i><font color="Green" title="E78">A68</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E12:16_2"><i><font color="Green" title="E49">A36</font></i></a>, <a class="txt" href="fdiff_3.html#E24:16_2"><i><font color="Green" title="E59">A47</font></i></a>, <a class="ref" href="seq_2.html#T28" title="SEQ_2:th.28">SEQ_2:28</a></i>;<br/>
<a NAME="E49:16_2"/><i><font color="Green" title="E79">A69</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E10:16_2"><i><font color="Green" title="E48">A35</font></i></a>, <a class="txt" href="fdiff_3.html#E37:16_2"><i><font color="Green" title="E69">A58</font></i></a>, <a class="ref" href="seq_2.html#T28" title="SEQ_2:th.28">SEQ_2:28</a></i>;<br/>
<i><font color="Green" title="E80">A70</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T47" title="RFUNCT_1:th.47">RFUNCT_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T47" title="RFUNCT_1:th.47">RFUNCT_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_3.html#E14:16_2"><i><font color="Green" title="E51">A38</font></i></a>, <a class="ref" href="rfunct_2.html#T23" title="RFUNCT_2:th.23">RFUNCT_2:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_3.html#E13:16_2"><i><font color="Green" title="E50">A37</font></i></a>, <a class="ref" href="rfunct_2.html#T27" title="RFUNCT_2:th.27">RFUNCT_2:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D9" title="SEQ_1:def.9">SEQ_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_3.html#E30:16_2"><i><font color="Green" title="E64">A52</font></i></a>, <a class="ref" href="rfunct_2.html#T23" title="RFUNCT_2:th.23">RFUNCT_2:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_3.html#E29:16_2"><i><font color="Green" title="E63">A51</font></i></a>, <a class="ref" href="rfunct_2.html#T27" title="RFUNCT_2:th.27">RFUNCT_2:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D9" title="SEQ_1:def.9">SEQ_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</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"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_3.html#E15:16_2"><i><font color="Green" title="E52">A39</font></i></a>, <a class="txt" href="fdiff_3.html#E31:16_2"><i><font color="Green" title="E65">A53</font></i></a>, <a class="ref" href="seq_1.html#T58" title="SEQ_1:th.58">SEQ_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T49" title="SEQ_1:th.49">SEQ_1:49</a></i>
;<br/>
<a NAME="E51:16_2"/><i><font color="Green" title="E81">A71</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E48:16_2"><i><font color="Green" title="E78">A68</font></i></a>, <a class="txt" href="fdiff_3.html#E49:16_2"><i><font color="Green" title="E79">A69</font></i></a>, <a class="ref" href="seq_2.html#T25" title="SEQ_2:th.25">SEQ_2:25</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_11"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">n</font> be    <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/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_2.html#T6" title="RFUNCT_2:th.6">RFUNCT_2:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_2.html#T6" title="RFUNCT_2:th.6">RFUNCT_2:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_2.html#T6" title="RFUNCT_2:th.6">RFUNCT_2:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T12" title="SEQ_1:th.12">SEQ_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font>
<b>by </b><i><a class="ref" href="rfunct_2.html#T6" title="RFUNCT_2:th.6">RFUNCT_2:6</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E53:16_2"/><b>then </b><i><font color="Green" title="E82">A72</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K19" title="SEQ_1:func.19">/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E50:16_2"><i><font color="Green" title="E80">A70</font></i></a>, <a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>
<b>then </b> <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</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"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_3.html#E32:16_2"><i><font color="Green" title="E66">A54</font></i></a>, <a class="txt" href="fdiff_3.html#E45:16_2"><i><font color="Green" title="E75">A65</font></i></a>, <a class="txt" href="fdiff_3.html#E47:16_2"><i><font color="Green" title="E77">A67</font></i></a>, <a class="txt" href="fdiff_3.html#E51:16_2"><i><font color="Green" title="E81">A71</font></i></a>, <a class="ref" href="seq_2.html#T38" title="SEQ_2:th.38">SEQ_2:38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_3.html#E48:16_2"><i><font color="Green" title="E78">A68</font></i></a>, <a class="txt" href="fdiff_3.html#E49:16_2"><i><font color="Green" title="E79">A69</font></i></a>, <a class="ref" href="seq_2.html#T26" title="SEQ_2:th.26">SEQ_2:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="fdiff_3.html#E26:16_2"><i><font color="Green" title="E61">A49</font></i></a>, <a class="txt" href="fdiff_3.html#E37:16_2"><i><font color="Green" title="E69">A58</font></i></a>, <a class="ref" href="seq_2.html#T29" title="SEQ_2:th.29">SEQ_2:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">x0</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> <font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">x0</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="fdiff_3.html#E10:16_2"><i><font color="Green" title="E48">A35</font></i></a>, <a class="txt" href="fdiff_3.html#E12:16_2"><i><font color="Green" title="E49">A36</font></i></a>, <a class="txt" href="fdiff_3.html#E24:16_2"><i><font color="Green" title="E59">A47</font></i></a>, <a class="txt" href="fdiff_3.html#E25:16_2"><i><font color="Green" title="E60">A48</font></i></a>, <a class="txt" href="fdiff_3.html#E38:16_2"><i><font color="Green" title="E70">A59</font></i></a>, <a class="txt" href="fdiff_3.html#E46:16_2"><i><font color="Green" title="E76">A66</font></i></a>, <a class="ref" href="seq_2.html#T29" title="SEQ_2:th.29">SEQ_2:29</a></i>
;<br/>
<b>hence </b><a NAME="E55:16_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K18" title="SEQ_1:func.18">"</a> </span>)</span> <a href="seq_1.html#K11" title="SEQ_1:func.11">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">x0</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> <font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">x0</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="fdiff_3.html#E32:16_2"><i><font color="Green" title="E66">A54</font></i></a>, <a class="txt" href="fdiff_3.html#E45:16_2"><i><font color="Green" title="E75">A65</font></i></a>, <a class="txt" href="fdiff_3.html#E47:16_2"><i><font color="Green" title="E77">A67</font></i></a>, <a class="txt" href="fdiff_3.html#E51:16_2"><i><font color="Green" title="E81">A71</font></i></a>, <a class="txt" href="fdiff_3.html#E53:16_2"><i><font color="Green" title="E82">A72</font></i></a>, <a class="ref" href="seq_2.html#T37" title="SEQ_2:th.37">SEQ_2:37</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E46:16"/>
( <font color="Maroon" title="c1">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">f2</font> <a href="fdiff_3.html#R4" title="FDIFF_3:pred.4">is_left_differentiable_in</a> <font color="Maroon" title="c3">x0</font> &amp;  <a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">f2</font></span>)</span>,<font color="Maroon" title="c3">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> <font color="Maroon" title="c1">f1</font>,<font color="Maroon" title="c3">x0</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fdiff_3.html#K1" title="FDIFF_3:func.1">Ldiff</a> <font color="Maroon" title="c2">f2</font>,<font color="Maroon" title="c3">x0</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="fdiff_3.html#E24:16"><i><font color="Green" title="E31">A15</font></i></a>, <a class="txt" href="fdiff_3.html#E44:16"><i><font color="Green" title="E41">A28</font></i></a>, <a class="ref" href="fdiff_3.html#T9" target="_self" title="FDIFF_3:th.9">Th9</a></i>;<br/>


</div>
