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<span class="kw">let </span><font color="Maroon" title="c3">L1</font>, <font color="Maroon" title="c4">L2</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">Intf</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 <br/>(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">Intf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c2">a</font> &amp; ( for <font color="Olive" title="b2">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">Intf</font> holds <br/><font color="Olive" title="b1">Intf</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">a</font>,<font color="Olive" title="b2">x</font> ) &amp; <font color="Olive" title="b1">Intf</font> is  <a href="limfunc1.html#V3" title="LIMFUNC1:attr.3">convergent_in+infty</a>  &amp; <font color="Maroon" title="c3">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</a> <font color="Olive" title="b1">Intf</font> ) &amp;  ex <font color="Olive" title="b1">Intf</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 <br/>(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">Intf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c2">a</font> &amp; ( for <font color="Olive" title="b2">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">Intf</font> holds <br/><font color="Olive" title="b1">Intf</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">a</font>,<font color="Olive" title="b2">x</font> ) &amp; <font color="Olive" title="b1">Intf</font> is  <a href="limfunc1.html#V3" title="LIMFUNC1:attr.3">convergent_in+infty</a>  &amp; <font color="Maroon" title="c4">L2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</a> <font color="Olive" title="b1">Intf</font> ) implies <font color="Maroon" title="c3">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">L2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:10_1_2"/>
 ex <font color="Olive" title="b1">Intf1</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 <br/>(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">Intf1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c2">a</font> &amp; ( for <font color="Olive" title="b2">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">Intf1</font> holds <br/><font color="Olive" title="b1">Intf1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">a</font>,<font color="Olive" title="b2">x</font> ) &amp; <font color="Olive" title="b1">Intf1</font> is  <a href="limfunc1.html#V3" title="LIMFUNC1:attr.3">convergent_in+infty</a>  &amp; <font color="Maroon" title="c3">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</a> <font color="Olive" title="b1">Intf1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( for <font color="Olive" title="b1">Intf</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>   holds <br/>(  not  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">Intf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c2">a</font> or  ex <font color="Olive" title="b2">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">Intf</font> &amp; not <font color="Olive" title="b1">Intf</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">a</font>,<font color="Olive" title="b2">x</font> ) or  not <font color="Olive" title="b1">Intf</font> is  <a href="limfunc1.html#V3" title="LIMFUNC1:attr.3">convergent_in+infty</a>  or  not <font color="Maroon" title="c4">L2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</a> <font color="Olive" title="b1">Intf</font> ) or <font color="Maroon" title="c3">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">L2</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">Intf1</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> <span class="kw"> such that </span><br/><a NAME="E3:10_1_2"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">Intf1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">and </span><br/><a NAME="E4:10_1_2"/><span class="lab"><font color="Green" title="E11">A4</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">Intf1</font> holds <br/><font color="Maroon" title="c5">Intf1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">x</font>
 <span class="kw">and </span><br/><a NAME="E5:10_1_2"/>
<font color="Maroon" title="c5">Intf1</font> is  <a href="limfunc1.html#V3" title="LIMFUNC1:attr.3">convergent_in+infty</a> 
 <span class="kw">and </span><br/><a NAME="E6:10_1_2"/><span class="lab"><font color="Green" title="E12">A5</font></span>: 
<font color="Maroon" title="c3">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</a> <font color="Maroon" title="c5">Intf1</font>
 ;<br/>
<span class="kw">assume </span><a NAME="E7:10_1_2"/>
 ex <font color="Olive" title="b1">Intf2</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 <br/>(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">Intf2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c2">a</font> &amp; ( for <font color="Olive" title="b2">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">Intf2</font> holds <br/><font color="Olive" title="b1">Intf2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">a</font>,<font color="Olive" title="b2">x</font> ) &amp; <font color="Olive" title="b1">Intf2</font> is  <a href="limfunc1.html#V3" title="LIMFUNC1:attr.3">convergent_in+infty</a>  &amp; <font color="Maroon" title="c4">L2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</a> <font color="Olive" title="b1">Intf2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">L2</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">Intf2</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> <span class="kw"> such that </span><br/><a NAME="E9:10_1_2"/><span class="lab"><font color="Green" title="E13">A6</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">Intf2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K3" title="LIMFUNC1:func.3">right_closed_halfline</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">and </span><br/><a NAME="E10:10_1_2"/><span class="lab"><font color="Green" title="E14">A7</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">Intf2</font> holds <br/><font color="Maroon" title="c6">Intf2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">x</font>
 <span class="kw">and </span><br/><a NAME="E11:10_1_2"/>
<font color="Maroon" title="c6">Intf2</font> is  <a href="limfunc1.html#V3" title="LIMFUNC1:attr.3">convergent_in+infty</a> 
 <span class="kw">and </span><br/><a NAME="E12:10_1_2"/><span class="lab"><font color="Green" title="E15">A8</font></span>: 
<font color="Maroon" title="c4">L2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K5" title="LIMFUNC1:func.5">lim_in+infty</a> <font color="Maroon" title="c6">Intf2</font>
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">Intf1</font> implies <font color="Maroon" title="c5">Intf1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Intf2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:10_1_2_1"/><span class="lab"><font color="Green" title="E16">A9</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">Intf1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">Intf1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Intf2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span><br/><span class="kw">hence </span><font color="Maroon" title="c5">Intf1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font> = 
 <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_1_2"><span class="lab"><font color="Green" title="E11">A4</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c6">Intf2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_1_2"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="#E9:10_1_2"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="txt" href="#E10:10_1_2"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="txt" href="#E1:10_1_2_1"><span class="lab"><font color="Green" title="E16">A9</font></span></a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E14:10_1_2"/>
<font color="Maroon" title="c3">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">L2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_1_2"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="#E6:10_1_2"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="txt" href="#E9:10_1_2"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="txt" href="#E12:10_1_2"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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