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<b>let </b><font color="Maroon" title="c1">x0</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being  <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>   st <font color="Olive" title="b1">f1</font> <a href="limfunc2.html#R4" title="LIMFUNC2:pred.4">is_right_convergent_in</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Olive" title="b2">f2</font> <a href="limfunc2.html#R4" title="LIMFUNC2:pred.4">is_right_convergent_in</a> <font color="Maroon" title="c1">x0</font> &amp; ( for <font color="Olive" title="b3">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">r</font> holds <br/> ex <font color="Olive" title="b4">g</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b4">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">r</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">g</font> &amp; <font color="Olive" title="b4">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> <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Olive" title="b2">f2</font></span>)</span> ) ) holds <br/>( <font color="Olive" title="b1">f1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Olive" title="b2">f2</font> <a href="limfunc2.html#R4" title="LIMFUNC2:pred.4">is_right_convergent_in</a> <font color="Maroon" title="c1">x0</font> &amp;  <a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Olive" title="b2">f2</font></span>)</span>,<font color="Maroon" title="c1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <font color="Olive" title="b1">f1</font>,<font color="Maroon" title="c1">x0</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <font color="Olive" title="b2">f2</font>,<font color="Maroon" title="c1">x0</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">f1</font> <a href="limfunc2.html#R4" title="LIMFUNC2:pred.4">is_right_convergent_in</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c3">f2</font> <a href="limfunc2.html#R4" title="LIMFUNC2:pred.4">is_right_convergent_in</a> <font color="Maroon" title="c1">x0</font> &amp; ( for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> holds <br/> ex <font color="Olive" title="b2">g</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">g</font> &amp; <font color="Olive" title="b2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c3">f2</font></span>)</span> ) ) implies ( <font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c3">f2</font> <a href="limfunc2.html#R4" title="LIMFUNC2:pred.4">is_right_convergent_in</a> <font color="Maroon" title="c1">x0</font> &amp;  <a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c3">f2</font></span>)</span>,<font color="Maroon" title="c1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">x0</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">x0</font></span>)</span> ) )</span><br/>





<b>assume </b><a NAME="E1:59"/><i><font color="Green" title="E36">A1</font></i>: 
( <font color="Maroon" title="c2">f1</font> <a href="limfunc2.html#R4" title="LIMFUNC2:pred.4">is_right_convergent_in</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c3">f2</font> <a href="limfunc2.html#R4" title="LIMFUNC2:pred.4">is_right_convergent_in</a> <font color="Maroon" title="c1">x0</font> &amp; ( for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> holds <br/> ex <font color="Olive" title="b2">g</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">g</font> &amp; <font color="Olive" title="b2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c3">f2</font></span>)</span> ) ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c3">f2</font> <a href="limfunc2.html#R4" title="LIMFUNC2:pred.4">is_right_convergent_in</a> <font color="Maroon" title="c1">x0</font> &amp;  <a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c3">f2</font></span>)</span>,<font color="Maroon" title="c1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">x0</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">x0</font></span>)</span> )</span><br/>

<a NAME="E2:59"/><b>then </b><i><font color="Green" title="E37">A2</font></i>: 
 <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">f2</font> <a href="limfunc2.html#R4" title="LIMFUNC2:pred.4">is_right_convergent_in</a> <font color="Maroon" title="c1">x0</font>
 
<b>by </b><i><a class="ref" href="limfunc2.html#T61" target="_self" title="LIMFUNC2:th.61">Th61</a></i>;<br/>
<b>hence </b><a NAME="E3:59"/>
<font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c3">f2</font> <a href="limfunc2.html#R4" title="LIMFUNC2:pred.4">is_right_convergent_in</a> <font color="Maroon" title="c1">x0</font>
 <b>by </b><i><a class="txt" href="limfunc2.html#E1:59"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="limfunc2.html#T62" target="_self" title="LIMFUNC2:th.62">Th62</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c3">f2</font></span>)</span>,<font color="Maroon" title="c1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">x0</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">x0</font></span>)</span></span><br/>

<b>thus </b> <a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f1</font> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <font color="Maroon" title="c3">f2</font></span>)</span>,<font color="Maroon" title="c1">x0</font> = 
<span class="p1">(<span class="default"><a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">x0</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <font color="Maroon" title="c3">f2</font></span>)</span>,<font color="Maroon" title="c1">x0</font></span>)</span>
<b>by </b><i><a class="txt" href="limfunc2.html#E1:59"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="limfunc2.html#E2:59"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="limfunc2.html#T62" target="_self" title="LIMFUNC2:th.62">Th62</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">x0</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">x0</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="limfunc2.html#E1:59"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="limfunc2.html#T61" target="_self" title="LIMFUNC2:th.61">Th61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <font color="Maroon" title="c2">f1</font>,<font color="Maroon" title="c1">x0</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <font color="Maroon" title="c3">f2</font>,<font color="Maroon" title="c1">x0</font></span>)</span>

; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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