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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:70_1_1_1"/>
( <font color="Maroon" title="c1">seq</font> is  <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a>  or <font color="Maroon" title="c1">seq</font> is  <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a>  or <font color="Maroon" title="c1">seq</font> is  <a href="mesfunc5.html#V9" title="MESFUNC5:attr.9">convergent_to_-infty</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="ref" href="mesfunc5.html#D11" target="_self" title="MESFUNC5:def.11">Def11</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:70_1_1_1_1"/><span class="lab"><font color="Green" title="E67">A2</font></span>: 
<font color="Maroon" title="c1">seq</font> is  <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> st <br/>(  ex <font color="Olive" title="b2">g</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> &amp; (  for <font color="Olive" title="b3">p</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">p</font> holds <br/> ex <font color="Olive" title="b4">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/> for <font color="Olive" title="b5">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">m</font> holds <br/><span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b5">m</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">p</font> ) &amp; <font color="Maroon" title="c1">seq</font> is  <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a>  ) or ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c1">seq</font> is  <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a>  ) or ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c1">seq</font> is  <a href="mesfunc5.html#V9" title="MESFUNC5:attr.9">convergent_to_-infty</a>  ) )</span><br/><div class="add"><a NAME="E2:70_1_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">g</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/> for <font color="Olive" title="b2">p</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">p</font> holds <br/> ex <font color="Olive" title="b3">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/> for <font color="Olive" title="b4">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">m</font> holds <br/><span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b4">m</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Olive" title="b1">g</font></span>)</span></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mesfunc5.html#D8" target="_self" title="MESFUNC5:def.8">Def8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:70_1_1_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> st <br/>(  ex <font color="Olive" title="b2">g</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> &amp; (  for <font color="Olive" title="b3">p</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">p</font> holds <br/> ex <font color="Olive" title="b4">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/> for <font color="Olive" title="b5">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">m</font> holds <br/><span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b5">m</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">p</font> ) &amp; <font color="Maroon" title="c1">seq</font> is  <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a>  ) or ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c1">seq</font> is  <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a>  ) or ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c1">seq</font> is  <a href="mesfunc5.html#V9" title="MESFUNC5:attr.9">convergent_to_-infty</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1_1_1"><span class="lab"><font color="Green" title="E67">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:70_1_1_1_2"/>
( <font color="Maroon" title="c1">seq</font> is  <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a>  or <font color="Maroon" title="c1">seq</font> is  <a href="mesfunc5.html#V9" title="MESFUNC5:attr.9">convergent_to_-infty</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> st <br/>(  ex <font color="Olive" title="b2">g</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> &amp; (  for <font color="Olive" title="b3">p</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">p</font> holds <br/> ex <font color="Olive" title="b4">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/> for <font color="Olive" title="b5">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">m</font> holds <br/><span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b5">m</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">p</font> ) &amp; <font color="Maroon" title="c1">seq</font> is  <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a>  ) or ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c1">seq</font> is  <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a>  ) or ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c1">seq</font> is  <a href="mesfunc5.html#V9" title="MESFUNC5:attr.9">convergent_to_-infty</a>  ) )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:70_1_1_1_2"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a> st <br/>(  ex <font color="Olive" title="b2">g</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> &amp; (  for <font color="Olive" title="b3">p</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">p</font> holds <br/> ex <font color="Olive" title="b4">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/> for <font color="Olive" title="b5">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b5">m</font> holds <br/><span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b5">m</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">p</font> ) &amp; <font color="Maroon" title="c1">seq</font> is  <a href="mesfunc5.html#V7" title="MESFUNC5:attr.7">convergent_to_finite_number</a>  ) or ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c1">seq</font> is  <a href="mesfunc5.html#V8" title="MESFUNC5:attr.8">convergent_to_+infty</a>  ) or ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c1">seq</font> is  <a href="mesfunc5.html#V9" title="MESFUNC5:attr.9">convergent_to_-infty</a>  ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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