<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon" title="c3">n</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c4">p</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c5">F</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon" title="c6">y</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:45"/><i><font color="Green" title="E34">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> )
 <b>and </b><br/><a NAME="E2:45"/><i><font color="Green" title="E35">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font>  holds <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <b>and </b><br/><a NAME="E3:45"/><i><font color="Green" title="E36">A3</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font> holds <br/>( <font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> )
 ;<br/>

<a NAME="E4:45"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">F</font> holds <br/><font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:45_1"/><i><font color="Green" title="E37">A4</font></i>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">F</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">i</font> = <font color="Maroon" title="c7">i</font> as    <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="E3:45_1"/>
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E1:45"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="convfun1.html#E1:45_1"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:45_1"/><b>then </b><i><font color="Green" title="E38">A5</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="convfun1.html#E3:45"><i><font color="Green" title="E36">A3</font></i></a></i>;<br/>
<a NAME="E5:45_1"/><i><font color="Green" title="E39">A6</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 
<b>by </b><i><a class="txt" href="convfun1.html#E2:45"><i><font color="Green" title="E35">A2</font></i></a></i>;<br/>
<a NAME="E6:45_1"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> 0
 
<b>by </b><i><a class="ref" href="measure6.html#D1" title="MEASURE6:def.1">MEASURE6:def 1</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">SUPINF_2:def 1</a></i>;<br/>
<a NAME="E7:45_1"/><b>then </b><i><font color="Green" title="E40">A7</font></i>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E4:45_1"><i><font color="Green" title="E38">A5</font></i></a>, <a class="ref" href="measure6.html#T13" title="MEASURE6:th.13">MEASURE6:13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_1"/>
(  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  or <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or (  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp; <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1"><b>suppose </b></a><a NAME="E1:45_1_1_1"/>
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 ;<br/><div class="add"><a NAME="E2:45_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="convfun1.html#E4:45_1"><i><font color="Green" title="E38">A5</font></i></a>, <a class="ref" href="extreal1.html#D1" title="EXTREAL1:def.1">EXTREAL1:def 1</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">SUPINF_2:def 1</a></i>;<br/><b>hence </b><a NAME="E3:45_1_1_1"/>
<font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_2"><b>suppose </b></a><a NAME="E1:45_1_1_2"/>
<font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c9">w</font> = <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="convfun1.html#E5:45_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="supinf_2.html#T2" title="SUPINF_2:th.2">SUPINF_2:2</a></i>;<br/><a NAME="E3:45_1_1_2"/>
<font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c9">w</font>
 <b>by </b><i><a class="txt" href="convfun1.html#E4:45_1"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="convfun1.html#E32"><i><font color="Green" title="E29">Lm17</font></i></a></i>;<br/><b>hence </b><a NAME="E4:45_1_1_2"/>
<font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_3"><b>suppose </b></a><a NAME="E1:45_1_1_3"/><i><font color="Green" title="E41">A8</font></i>: 
(  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp; <font color="Maroon" title="c2">f</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 ;<br/><div class="add"><a NAME="E2:45_1_1_3"/><b>then </b>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="convfun1.html#E7:45_1"><i><font color="Green" title="E40">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E3:45_1_1_3"/>
<font color="Maroon" title="c5">F</font> <a href="supinf_2.html#K13" title="SUPINF_2:func.13">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <b>by </b><i><a class="txt" href="convfun1.html#E4:45_1"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="convfun1.html#E1:45_1_1_3"><i><font color="Green" title="E41">A8</font></i></a>, <a class="ref" href="extreal1.html#D1" title="EXTREAL1:def.1">EXTREAL1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:45"/>
not  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  <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="c5">F</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div>
