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<span class="kw">let </span><font color="Maroon" title="c1">x</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <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">seq</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st ( for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Olive" title="b1">seq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> ) holds <br/>( <font color="Olive" title="b1">seq</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Olive" title="b1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">seq</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"> ( ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> ) implies ( <font color="Maroon" title="c2">seq</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:17"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</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">seq</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>  &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> )</span><br/>

<a NAME="E2:17"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<a NAME="E3:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<font color="Maroon" title="c2">seq</font> is  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="valued_0.html#D18" title="VALUED_0:def.18">VALUED_0:def 18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:17"/>
<font color="Maroon" title="c2">seq</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font></span><br/>

<span class="kw">thus </span> <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">seq</font> = 
<font color="Maroon" title="c2">seq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="seq_4.html#T41" title="SEQ_4:th.41">SEQ_4:41</a></span>
<br/>.= 
<font color="Maroon" title="c1">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E14">A1</font></span></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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