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<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Z</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">F</font> be    <a href="seqfunc.html#M1" title="SEQFUNC:mode.1">Functional_Sequence</a> of <font color="Maroon" title="c1">X</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:14"/><i><font color="Green" title="E10">A1</font></i>: 
<font color="Maroon" title="c2">Z</font> <a href="seqfunc.html#R1" title="SEQFUNC:pred.1">common_on_dom</a> <font color="Maroon" title="c3">F</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a> <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E2:14"/><i><font color="Green" title="E11">A2</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 ;<br/>

<b>assume </b><a NAME="E3:14"/><i><font color="Green" title="E12">A3</font></i>: 
for <font color="Olive" title="b1">n</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c2">Z</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <font color="Maroon" title="c4">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Olive" title="b1">n</font></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Z</font>;<br/>

<a NAME="E4:14"/>
<font color="Maroon" title="c3">F</font> <a href="seqfunc.html#R3" title="SEQFUNC:pred.3">is_unif_conv_on</a> <font color="Maroon" title="c2">Z</font>
 
<b>by </b><i><a class="txt" href="tietze.html#E1:14"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="tietze.html#E2:14"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="tietze.html#E3:14"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="tietze.html#T11" target="_self" title="TIETZE:th.11">Th11</a></i>;<br/>
<a NAME="E5:14"/><b>then </b>
<font color="Maroon" title="c3">F</font> <a href="seqfunc.html#R2" title="SEQFUNC:pred.2">is_point_conv_on</a> <font color="Maroon" title="c2">Z</font>
 
<b>by </b><i><a class="ref" href="seqfunc.html#T23" title="SEQFUNC:th.23">SEQFUNC:23</a></i>;<br/>
<a NAME="E6:14"/><b>then </b>
( <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> &amp; <font color="Maroon" title="c2">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> 0</span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Z</font> )
 
<b>by </b><i><a class="txt" href="tietze.html#E1:14"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="seqfunc.html#D10" title="SEQFUNC:def.10">SEQFUNC:def 10</a>, <a class="ref" href="seqfunc.html#D14" title="SEQFUNC:def.14">SEQFUNC:def 14</a></i>;<br/>
<a NAME="E7:14"/><b>then </b>
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> 0</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:14"/><b>then </b><i><font color="Green" title="E13">A4</font></i>: 
( <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> 0</span>)</span></span>)</span> &amp; <font color="Maroon" title="c5">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="power.html#T29" title="POWER:th.29">POWER:29</a>, <a class="ref" href="seq_1.html#D4" title="SEQ_1:def.4">SEQ_1:def 4</a></i>;<br/>
<a NAME="E9:14"/><b>then </b>
( <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> 0</span>)</span>,<font color="Maroon" title="c2">Z</font> <a href="tietze.html#R1" title="TIETZE:pred.1">is_absolutely_bounded_by</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">r</font></span>)</span> &amp; <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Z</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> 0</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="tietze.html#E1:14"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="tietze.html#E2:14"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="tietze.html#E3:14"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="tietze.html#T11" target="_self" title="TIETZE:th.11">Th11</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:14"/><b>then </b>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> 0</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">z</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tietze.html#D1" target="_self" title="TIETZE:def.1">Def1</a></i>;<br/>
<a NAME="E11:14"/><b>then </b>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> 0</span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">z</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="tietze.html#E8:14"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="seq_1.html#D4" title="SEQ_1:def.4">SEQ_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E12:14"/>
( <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">z</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> 0</span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="seqfunc.html#K11" title="SEQFUNC:func.11">lim</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c2">Z</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> 0</span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="tietze.html#T1" target="_self" title="TIETZE:th.1">Th1</a></i>;<br/>


</div>
