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<span class="kw">let </span><font color="Maroon" title="c2">seq1</font>, <font color="Maroon" title="c3">seq2</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</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="c2">seq1</font> is  <a href="limfunc1.html#V2" title="LIMFUNC1:attr.2">divergent_to-infty</a>  &amp;  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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; ( for <font color="Olive" title="b2">n</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>  holds  <font color="Maroon" title="c3">seq2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">r</font> ) ) implies <font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq2</font> is  <a href="limfunc1.html#V2" title="LIMFUNC1:attr.2">divergent_to-infty</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:57"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<font color="Maroon" title="c2">seq1</font> is  <a href="limfunc1.html#V2" title="LIMFUNC1:attr.2">divergent_to-infty</a> 
 <span class="kw">and </span><br/><a NAME="E2:57"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
 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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; ( for <font color="Olive" title="b2">n</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>  holds  <font color="Maroon" title="c3">seq2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">r</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="c2">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq2</font> is  <a href="limfunc1.html#V2" title="LIMFUNC1:attr.2">divergent_to-infty</a> </span><br/>

<a NAME="E3:57"/>
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c2">seq1</font> is  <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="limfunc1.html#T41" target="_self" title="LIMFUNC1:th.41">Th41</a></span>;<br/>
<a NAME="E4:57"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A3</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq2</font> is  <a href="limfunc1.html#V1" title="LIMFUNC1:attr.1">divergent_to+infty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="limfunc1.html#T49" target="_self" title="LIMFUNC1:th.49">Th49</a></span>;<br/>
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <font color="Maroon" title="c2">seq1</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T26" title="SEQ_1:th.26">SEQ_1:26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T31" title="SEQ_1:th.31">SEQ_1:31</a></span>
<br/>.= 
<font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_1.html#T35" title="SEQ_1:th.35">SEQ_1:35</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:57"/>
<font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Maroon" title="c3">seq2</font> is  <a href="limfunc1.html#V2" title="LIMFUNC1:attr.2">divergent_to-infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:57"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="limfunc1.html#T40" target="_self" title="LIMFUNC1:th.40">Th40</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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