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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font>, <font color="Maroon" title="c3">T</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="struct_0.html#NM10" title="STRUCT_0:NM.10">PartFunc</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font>;<br/>







<b>assume </b><a NAME="E1:8"/><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="nfcont_2.html#R1" title="NFCONT_2:pred.1">is_uniformly_continuous_on</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<a NAME="E2:8"/><b>then </b>
<font color="Maroon" title="c1">X</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> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="ref" href="nfcont_2.html#D1" target="_self" title="NFCONT_2:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E3:8"/><i><font color="Green" title="E6">A2</font></i>: 
<font color="Maroon" title="c1">X</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"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span>
 <b>by </b><i><a class="ref" href="nfcont_1.html#D2" title="NFCONT_1:def.2">NFCONT_1:def 2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="nfcont_2.html#D2" target="_self" title="NFCONT_2:def.2">NFCONT_2:def 2</a><br/>

<b>let </b><font color="Maroon" title="c5">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E4:8"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E6:8"/><i><font color="Green" title="E7">A3</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">s</font> &amp; ( for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Olive" title="b2">x2</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">s</font> holds <br/><span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">x1</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">x2</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font> ) )
 <b>by </b><i><a class="txt" href="nfcont_2.html#E1:8"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="nfcont_2.html#D1" target="_self" title="NFCONT_2:def.1">Def1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">s</font>
;<br/>

<b>thus </b><a NAME="E7:8"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">s</font>
 <b>by </b><i><a class="txt" href="nfcont_2.html#E6:8"><i><font color="Green" title="E7">A3</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c7">x1</font>, <font color="Maroon" title="c8">x2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font>;<br/>



<b>assume </b><a NAME="E8:8"/><i><font color="Green" title="E8">A4</font></i>: 
( <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c8">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c8">x2</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">s</font> )
 ;<br/>

<b>then </b> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">x2</font></span>)</span></span>)</span> = 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">x2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nfcont_2.html#E3:8"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">x2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nfcont_2.html#E3:8"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="nfcont_2.html#E8:8"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">x1</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">x2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nfcont_2.html#E3:8"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="nfcont_2.html#E8:8"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="nfcont_1.html#D2" title="NFCONT_1:def.2">NFCONT_1:def 2</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">x1</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">x2</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span></span>)</span>
<b>by </b><i><a class="txt" href="nfcont_2.html#E3:8"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="nfcont_2.html#E8:8"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="nfcont_1.html#D2" title="NFCONT_1:def.2">NFCONT_1:def 2</a></i>
;<br/>
<a NAME="E10:8"/><b>then </b><i><font color="Green" title="E9">A5</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">x2</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 
<b>by </b><i><a class="ref" href="normsp_1.html#T13" title="NORMSP_1:th.13">NORMSP_1:13</a></i>;<br/>
<a NAME="E11:8"/>
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">x2</font></span>)</span></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font>
 
<b>by </b><i><a class="txt" href="nfcont_2.html#E6:8"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="nfcont_2.html#E8:8"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:8"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="nfcont_1.html#K1" title="NFCONT_1:func.1">.||</a></span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">x2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="txt" href="nfcont_2.html#E10:8"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


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