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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="supinf_1.html#NM3">R_eal</a>] means ( ex <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>4</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>4</sub></font> holds <br/><span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> ) &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="mesfunc5.html#V7" target="_self">convergent_to_finite_number</a> ) or ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="mesfunc5.html#V8" target="_self">convergent_to_+infty</a> ) or ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="mesfunc5.html#V9" target="_self">convergent_to_-infty</a> ) );<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:77_1_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="mesfunc5.html#V7" target="_self">convergent_to_finite_number</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="mesfunc5.html#V8" target="_self">convergent_to_+infty</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="mesfunc5.html#V9" target="_self">convergent_to_-infty</a> )
 </span><b>by </b><i><a class="txt" href="mesfunc5.html#E1:77"><i><font color="Green">E72</font></i></a>, <a class="ref" href="mesfunc5.html#D11" target="_self">Def11</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_1_1_1"><b>suppose </b></a><a NAME="E1:77_1_1_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="mesfunc5.html#V7" target="_self">convergent_to_finite_number</a>
 ;<br/><div class="add"><a NAME="E2:77_1_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>for <font color="Olive">b<sub>2</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">b<sub>4</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>4</sub></font> holds <br/><span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:77_1_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="mesfunc5.html#D8" target="_self">Def8</a></i>;<br/><b>hence </b><a NAME="E3:77_1_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="mesfunc5.html#E1:77_1_1_1_1"><i><font color="Green">E73</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_1_1_2"><b>suppose </b></a><a NAME="E1:77_1_1_1_2"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="mesfunc5.html#V8" target="_self">convergent_to_+infty</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="mesfunc5.html#V9" target="_self">convergent_to_-infty</a> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:77_1_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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