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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">f</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  ex <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <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="Olive" title="b2">r</font> &amp; <font color="Olive" title="b1">f</font> <a href="rfunct_2.html#R3" title="RFUNCT_2:pred.3">is_non_decreasing_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b2">r</font></span>)</span>,<font color="Maroon" title="c1">x0</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <font color="Olive" title="b1">f</font> <a href="rfunct_2.html#R4" title="RFUNCT_2:pred.4">is_non_increasing_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b2">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; not <font color="Olive" title="b1">f</font> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b2">r</font></span>)</span>,<font color="Maroon" title="c1">x0</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; not <font color="Olive" title="b1">f</font> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b2">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> ) &amp; ( for <font color="Olive" title="b2">r1</font>, <font color="Olive" title="b3">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b2">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b3">r2</font> holds <br/> ex <font color="Olive" title="b4">g1</font>, <font color="Olive" title="b5">g2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b4">g1</font> &amp; <font color="Olive" title="b4">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Olive" title="b4">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b5">g2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b3">r2</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b5">g2</font> &amp; <font color="Olive" title="b5">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font> ) ) holds <br/><font color="Olive" title="b1">f</font> <a href="limfunc3.html#R2" title="LIMFUNC3:pred.2">is_divergent_to+infty_in</a> <font color="Maroon" title="c1">x0</font></span><br/><b>let </b><font color="Maroon" title="c2">f</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"> (  ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <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="Olive" title="b1">r</font> &amp; <font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#R3" title="RFUNCT_2:pred.3">is_non_decreasing_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b1">r</font></span>)</span>,<font color="Maroon" title="c1">x0</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#R4" title="RFUNCT_2:pred.4">is_non_increasing_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b1">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; not <font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b1">r</font></span>)</span>,<font color="Maroon" title="c1">x0</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; not <font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b1">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> ) &amp; ( for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r2</font> holds <br/> ex <font color="Olive" title="b3">g1</font>, <font color="Olive" title="b4">g2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b3">g1</font> &amp; <font color="Olive" title="b3">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Olive" title="b3">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Olive" title="b4">g2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r2</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b4">g2</font> &amp; <font color="Olive" title="b4">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> ) ) implies <font color="Maroon" title="c2">f</font> <a href="limfunc3.html#R2" title="LIMFUNC3:pred.2">is_divergent_to+infty_in</a> <font color="Maroon" title="c1">x0</font> )</span><br/>



<b>given </b><font color="Maroon" title="c3">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><a NAME="E2:25"/><i><font color="Green" title="E19">A1</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="c3">r</font> &amp; <font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#R3" title="RFUNCT_2:pred.3">is_non_decreasing_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c1">x0</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <font color="Maroon" title="c2">f</font> <a href="rfunct_2.html#R4" title="RFUNCT_2:pred.4">is_non_increasing_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; not <font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c1">x0</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; not <font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c1">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r2</font> &amp; ( for <font color="Olive" title="b3">g1</font>, <font color="Olive" title="b4">g2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  holds <br/>(  not <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b3">g1</font> or  not <font color="Olive" title="b3">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> or  not <font color="Olive" title="b3">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> or  not <font color="Olive" title="b4">g2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r2</font> or  not <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b4">g2</font> or  not <font color="Olive" title="b4">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> ) ) ) or <font color="Maroon" title="c2">f</font> <a href="limfunc3.html#R2" title="LIMFUNC3:pred.2">is_divergent_to+infty_in</a> <font color="Maroon" title="c1">x0</font> )</span><br/>

<b>assume </b><a NAME="E3:25"/>
for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r2</font> holds <br/> ex <font color="Olive" title="b3">g1</font>, <font color="Olive" title="b4">g2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b3">g1</font> &amp; <font color="Olive" title="b3">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Olive" title="b3">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Olive" title="b4">g2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r2</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b4">g2</font> &amp; <font color="Olive" title="b4">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="limfunc3.html#R2" title="LIMFUNC3:pred.2">is_divergent_to+infty_in</a> <font color="Maroon" title="c1">x0</font></span><br/>

<a NAME="E4:25"/><b>then </b><i><font color="Green" title="E20">A2</font></i>: 
( ( 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="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</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="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">g</font> &amp; <font color="Olive" title="b2">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">x0</font> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</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#R1" title="XXREAL_0:pred.1">&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#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Maroon" title="c1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> ) ) )
 
<b>by </b><i><a class="ref" href="limfunc3.html#T8" target="_self" title="LIMFUNC3:th.8">Th8</a></i>;<br/>
<a NAME="E5:25"/><b>then </b><i><font color="Green" title="E21">A3</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="limfunc2.html#R2" title="LIMFUNC2:pred.2">is_left_divergent_to+infty_in</a> <font color="Maroon" title="c1">x0</font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E2:25"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="limfunc2.html#T31" title="LIMFUNC2:th.31">LIMFUNC2:31</a></i>;<br/>
<a NAME="E6:25"/>
<font color="Maroon" title="c2">f</font> <a href="limfunc2.html#R5" title="LIMFUNC2:pred.5">is_right_divergent_to+infty_in</a> <font color="Maroon" title="c1">x0</font>
 
<b>by </b><i><a class="txt" href="limfunc3.html#E2:25"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="limfunc3.html#E4:25"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="limfunc2.html#T35" title="LIMFUNC2:th.35">LIMFUNC2:35</a></i>;<br/>
<b>hence </b><a NAME="E7:25"/>
<font color="Maroon" title="c2">f</font> <a href="limfunc3.html#R2" title="LIMFUNC3:pred.2">is_divergent_to+infty_in</a> <font color="Maroon" title="c1">x0</font>
 <b>by </b><i><a class="txt" href="limfunc3.html#E5:25"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="limfunc3.html#T15" target="_self" title="LIMFUNC3:th.15">Th15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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