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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="E1:13"/><i><font color="Green" title="E16">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>then consider </b><font color="Maroon" title="c4">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:13"/><i><font color="Green" title="E17">A2</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">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="ref" href="fdiff_3.html#D4" target="_self" title="FDIFF_3:def.4">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">r2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:13"/><i><font color="Green" title="E18">A3</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">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#E1:13"><i><font color="Green" title="E16">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="c6">r</font> =  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r1</font>,<font color="Maroon" title="c5">r2</font>;<br/>
<a NAME="E6:13"/><i><font color="Green" title="E19">A4</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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">r1</font>,<font color="Maroon" title="c5">r2</font>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E3:13"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="fdiff_3.html#E5:13"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<a NAME="E7:13"/><b>then </b><i><font color="Green" title="E20">A5</font></i>: 
<font color="Maroon" title="c3">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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">r1</font>,<font color="Maroon" title="c5">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="E8:13"/>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r1</font>,<font color="Maroon" title="c5">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">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="E9:13"/><b>then </b><i><font color="Green" title="E21">A6</font></i>: 
<font color="Maroon" title="c3">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">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#K9" title="REAL_1:func.9">-</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">r1</font>,<font color="Maroon" title="c5">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="E10:13"/><b>then </b>
<font color="Maroon" title="c3">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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">r1</font>,<font color="Maroon" title="c5">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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">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#E7:13"><i><font color="Green" title="E20">A5</font></i></a></i>;<br/>
<a NAME="E11:13"/><b>then </b><i><font color="Green" title="E22">A7</font></i>: 
<font color="Maroon" title="c3">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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">r1</font>,<font color="Maroon" title="c5">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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">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="E12:13"/>
<font color="Maroon" title="c3">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">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#E7:13"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="fdiff_3.html#E9:13"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:13"/><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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">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="E14:13"/><b>then </b><i><font color="Green" title="E23">A8</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#K9" title="REAL_1:func.9">-</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">r1</font>,<font color="Maroon" title="c5">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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">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#E11:13"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="rcomp_1.html#T16" title="RCOMP_1:th.16">RCOMP_1:16</a></i>;<br/>
<a NAME="E15:13"/>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c4">r1</font>,<font color="Maroon" title="c5">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">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="E16:13"/><b>then </b><i><font color="Green" title="E24">A9</font></i>: 
<font color="Maroon" title="c3">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">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#K9" title="REAL_1:func.9">-</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">r1</font>,<font color="Maroon" title="c5">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="E17:13"/><b>then </b>
<font color="Maroon" title="c3">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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">r1</font>,<font color="Maroon" title="c5">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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">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#E7:13"><i><font color="Green" title="E20">A5</font></i></a></i>;<br/>
<a NAME="E18:13"/><b>then </b><i><font color="Green" title="E25">A10</font></i>: 
<font color="Maroon" title="c3">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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">r1</font>,<font color="Maroon" title="c5">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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">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="E19:13"/>
<font color="Maroon" title="c3">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">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#E7:13"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="fdiff_3.html#E16:13"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E20:13"/><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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">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="E21:13"/><b>then </b><i><font color="Green" title="E26">A11</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#K9" title="REAL_1:func.9">-</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">r1</font>,<font color="Maroon" title="c5">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#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">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#E18:13"><i><font color="Green" title="E25">A10</font></i></a>, <a class="ref" href="rcomp_1.html#T16" title="RCOMP_1:th.16">RCOMP_1:16</a></i>;<br/>
<a NAME="E22:13"/><i><font color="Green" title="E27">A12</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#K9" title="REAL_1:func.9">-</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">r1</font>,<font color="Maroon" title="c5">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="c1">f1</font>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E3:13"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="fdiff_3.html#E14:13"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E23:13"/>
<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#K9" title="REAL_1:func.9">-</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">r1</font>,<font color="Maroon" title="c5">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#E5:13"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="fdiff_3.html#E21:13"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E24:13"/><b>then </b><i><font color="Green" title="E28">A13</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#K9" title="REAL_1:func.9">-</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">r1</font>,<font color="Maroon" title="c5">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">(<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> <font color="Maroon" title="c2">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E22:13"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E25:13"/><b>then </b><i><font color="Green" title="E29">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#K9" title="REAL_1:func.9">-</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">r1</font>,<font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c2">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D3" title="SEQ_1:def.3">SEQ_1:def 3</a></i>;<br/>
<a NAME="E26:13"/>
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="seq_1.html#K6" title="SEQ_1:func.6">+</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#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) 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="seq_1.html#K6" title="SEQ_1:func.6">+</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="seq_1.html#K6" title="SEQ_1:func.6">+</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="seq_1.html#K6" title="SEQ_1:func.6">+</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="seq_1.html#K6" title="SEQ_1:func.6">+</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"><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#K7" title="REAL_1:func.7">+</a> <span class="p1">(<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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">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="c8">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="E1:13_1"/><i><font color="Green" title="E30">A15</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">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="E2:13_1"/><i><font color="Green" title="E31">A16</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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">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="seq_1.html#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c2">f2</font></span>)</span>
 <b>and </b><br/><a NAME="E3:13_1"/><i><font color="Green" title="E32">A17</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="c7">h</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<a NAME="E4:13_1"/><i><font color="Green" title="E33">A18</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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">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> <font color="Maroon" title="c2">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="fdiff_3.html#E2:13_1"><i><font color="Green" title="E31">A16</font></i></a>, <a class="ref" href="seq_1.html#D3" title="SEQ_1:def.3">SEQ_1:def 3</a></i>;<br/>
<a NAME="E5:13_1"/>
( <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> <font color="Maroon" title="c2">f2</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; <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> <font color="Maroon" title="c2">f2</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="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E6:13_1"/><b>then </b><i><font color="Green" title="E34">A19</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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">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#E4:13_1"><i><font color="Green" title="E33">A18</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E7:13_1"/>
 <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="E8:13_1"/><b>then </b><i><font color="Green" title="E35">A20</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">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="c8">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="c7">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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">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="c8">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#E1:13"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="fdiff_3.html#E1:13_1"><i><font color="Green" title="E30">A15</font></i></a>, <a class="txt" href="fdiff_3.html#E3:13_1"><i><font color="Green" title="E32">A17</font></i></a>, <a class="txt" href="fdiff_3.html#E6:13_1"><i><font color="Green" title="E34">A19</font></i></a>, <a class="ref" href="fdiff_3.html#T9" target="_self" title="FDIFF_3:th.9">Th9</a></i>;<br/>
<a NAME="E9:13_1"/>
 <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="E10:13_1"/><b>then </b><i><font color="Green" title="E36">A21</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">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="c8">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="c7">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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">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="c8">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#E1:13"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="fdiff_3.html#E1:13_1"><i><font color="Green" title="E30">A15</font></i></a>, <a class="txt" href="fdiff_3.html#E3:13_1"><i><font color="Green" title="E32">A17</font></i></a>, <a class="txt" href="fdiff_3.html#E6:13_1"><i><font color="Green" title="E34">A19</font></i></a>, <a class="ref" href="fdiff_3.html#T9" target="_self" title="FDIFF_3:th.9">Th9</a></i>;<br/>
<a NAME="E11:13_1"/><b>then </b><i><font color="Green" title="E37">A22</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">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="c8">c</font></span>)</span></span>)</span></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="c7">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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">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="c8">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#E8:13_1"><i><font color="Green" title="E35">A20</font></i></a>, <a class="ref" href="seq_2.html#T19" title="SEQ_2:th.19">SEQ_2:19</a></i>;<br/>
<a NAME="E12:13_1"/><i><font color="Green" title="E38">A23</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">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="c8">c</font></span>)</span></span>)</span></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="c7">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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">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="c8">c</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">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="c8">c</font></span>)</span></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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">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="c8">c</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#T24" title="SEQ_1:th.24">SEQ_1:24</a></i>;<br/>
<div><i><font color="Green" title="E39">A24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">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="E1:13_1_1"/><i><font color="Green" title="E40">A25</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">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> <font color="Maroon" title="c2">f2</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">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="E1:13_1_1_1"/>
<font color="Maroon" title="c10">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="c8">c</font>
 ;<br/>

<a NAME="E2:13_1_1_1"/><b>then </b><i><font color="Green" title="E41">A26</font></i>: 
<font color="Maroon" title="c10">x</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#E1:13_1"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:13_1_1_1"/>
<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#K9" title="REAL_1:func.9">-</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">r1</font>,<font color="Maroon" title="c5">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#E7:13"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a></i>;<br/>
<b>hence </b><a NAME="E4:13_1_1_1"/>
<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="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> <font color="Maroon" title="c2">f2</font></span>)</span>
 <b>by </b><i><a class="txt" href="fdiff_3.html#E24:13"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="fdiff_3.html#E2:13_1_1_1"><i><font color="Green" title="E41">A26</font></i></a></i>;<br/>


</div><b>end;</b></div><b>thus </b><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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">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="c8">c</font></span>)</span></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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">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="c8">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font> = 
<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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">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="c8">c</font></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">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="c8">c</font></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">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"><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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">c</font></span>)</span></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p3">(<span class="default"><a href="seq_1.html#K17" title="SEQ_1:func.17">-</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="c8">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">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="c8">c</font></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T15" title="SEQ_1:th.15">SEQ_1:15</a></i>
<br/>.= 
<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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">c</font></span>)</span></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p3">(<span class="default"><a href="seq_1.html#K17" title="SEQ_1:func.17">-</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="c8">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">c</font></span>)</span></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p3">(<span class="default"><a href="seq_1.html#K17" title="SEQ_1:func.17">-</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="c8">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T15" title="SEQ_1:th.15">SEQ_1:15</a></i>
<br/>.= 
<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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">c</font></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_1.html#K17" title="SEQ_1:func.17">-</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="c8">c</font></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">c</font></span>)</span></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p3">(<span class="default"><a href="seq_1.html#K17" title="SEQ_1:func.17">-</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="c8">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">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"><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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">c</font></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_1.html#K17" title="SEQ_1:func.17">-</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="c8">c</font></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">c</font></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_1.html#K17" title="SEQ_1:func.17">-</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="c8">c</font></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span></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"><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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">c</font></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">c</font></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_1.html#K17" title="SEQ_1:func.17">-</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="c8">c</font></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_1.html#K17" title="SEQ_1:func.17">-</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="c8">c</font></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span>

<br/>.= 
<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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">c</font></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">c</font></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</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="c8">c</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_1.html#K17" title="SEQ_1:func.17">-</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="c8">c</font></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T14" title="SEQ_1:th.14">SEQ_1:14</a></i>
<br/>.= 
<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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">c</font></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">c</font></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</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="c8">c</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</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="c8">c</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T14" title="SEQ_1:th.14">SEQ_1:14</a></i>
<br/>.= 
<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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">c</font></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">c</font></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="c8">c</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c8">c</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span>

<br/>.= 
<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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">c</font></span>)</span></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="c8">c</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c8">c</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span></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"><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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">c</font></span>)</span></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="c8">c</font></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</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="c8">c</font></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">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"><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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">c</font></span>)</span></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">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="c8">c</font></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</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="c8">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">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/>.= 
<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#K6" title="SEQ_1:func.6">+</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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">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="c8">c</font></span>)</span> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</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="c8">c</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font>
<b>by </b><i><a class="txt" href="fdiff_3.html#E4:13_1"><i><font color="Green" title="E33">A18</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">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#K6" title="SEQ_1:func.6">+</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="c8">c</font></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c9">n</font>
<b>by </b><i><a class="txt" href="fdiff_3.html#E1:13_1_1"><i><font color="Green" title="E40">A25</font></i></a>, <a class="ref" href="rfunct_2.html#T23" title="RFUNCT_2:th.23">RFUNCT_2:23</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E14:13_1"/><b>then </b><i><font color="Green" title="E40">A27</font></i>: 
<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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">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="c8">c</font></span>)</span></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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">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="c8">c</font></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</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#K6" title="SEQ_1:func.6">+</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">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="seq_1.html#K6" title="SEQ_1:func.6">+</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="c8">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/>
<b>thus </b><a NAME="E15:13_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">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#K6" title="SEQ_1:func.6">+</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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">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#K6" title="SEQ_1:func.6">+</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="c8">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#E11:13_1"><i><font color="Green" title="E37">A22</font></i></a>, <a class="txt" href="fdiff_3.html#E12:13_1"><i><font color="Green" title="E38">A23</font></i></a>, <a class="txt" href="fdiff_3.html#E13:13_1"><i><font color="Green" title="E39">A24</font></i></a>, <a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>

<b>thus </b><a NAME="E16:13_1"/>
 <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="c7">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="seq_1.html#K6" title="SEQ_1:func.6">+</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="c7">h</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <font color="Maroon" title="c8">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="seq_1.html#K6" title="SEQ_1:func.6">+</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="c8">c</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<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#K7" title="REAL_1:func.7">+</a> <span class="p1">(<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>
 <b>by </b><i><a class="txt" href="fdiff_3.html#E8:13_1"><i><font color="Green" title="E35">A20</font></i></a>, <a class="txt" href="fdiff_3.html#E10:13_1"><i><font color="Green" title="E36">A21</font></i></a>, <a class="txt" href="fdiff_3.html#E12:13_1"><i><font color="Green" title="E38">A23</font></i></a>, <a class="txt" href="fdiff_3.html#E14:13_1"><i><font color="Green" title="E40">A27</font></i></a>, <a class="ref" href="seq_2.html#T20" title="SEQ_2:th.20">SEQ_2:20</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E27:13"/>
( <font color="Maroon" title="c1">f1</font> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</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="seq_1.html#K6" title="SEQ_1:func.6">+</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"><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#K7" title="REAL_1:func.7">+</a> <span class="p1">(<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> )
 <b>by </b><i><a class="txt" href="fdiff_3.html#E6:13"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="fdiff_3.html#E25:13"><i><font color="Green" title="E29">A14</font></i></a>, <a class="ref" href="fdiff_3.html#T9" target="_self" title="FDIFF_3:th.9">Th9</a></i>;<br/>


</div>
