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<b>let </b><font color="Maroon" title="c1">CNS</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">RNS</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="nfcont_1.html#NM1" title="NFCONT_1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">RNS</font>,<font color="Maroon" title="c1">CNS</font>;<br/>







<b>assume </b><a NAME="E1:171"/><i><font color="Green" title="E123">A1</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="ncfcont1.html#R15" title="NCFCONT1:pred.15">is_Lipschitzian_on</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:171"/><i><font color="Green" title="E124">A2</font></i>: 
( 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</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">RNS</font>  st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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="c3">X</font> holds <br/><span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</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#K6" title="RLVECT_1:func.6">-</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="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</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"><font color="Olive" title="b1">x1</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Olive" title="b2">x2</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> ) )
 <b>by </b><i><a class="ref" href="ncfcont1.html#D29" target="_self" title="NCFCONT1:def.29">Def29</a></i>;<br/>
<a NAME="E4:171"/><i><font color="Green" title="E125">A3</font></i>: 
<font color="Maroon" title="c3">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="txt" href="ncfcont1.html#E1:171"><i><font color="Green" title="E123">A1</font></i></a>, <a class="ref" href="ncfcont1.html#D29" target="_self" title="NCFCONT1:def.29">Def29</a></i>;<br/>
<a NAME="E5:171"/><b>then </b><i><font color="Green" title="E126">A4</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="171_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x0</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">RNS</font>;<br/><b>assume </b><a NAME="E1:171_1"/><i><font color="Green" title="E127">A5</font></i>: 
<font color="Maroon" title="c6">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="171_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">g</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:171_1_1"/><i><font color="Green" title="E128">A6</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">g</font>
 ;<br/><b>set </b><font color="Maroon" title="c8">s</font> = <font color="Maroon" title="c7">g</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font>;<br/><b>take </b><font color="Maroon" title="c9">s'</font> = <font color="Maroon" title="c7">g</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font>;<br/><a NAME="E2:171_1_1"/><i><font color="Green" title="E129">A7</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> 
 <b>by </b><i><a class="txt" href="ncfcont1.html#E3:171"><i><font color="Green" title="E124">A2</font></i></a>, <a class="ref" href="xreal_1.html#T124" title="XREAL_1:th.124">XREAL_1:124</a></i>;<br/><a NAME="E3:171_1_1"/><i><font color="Green" title="E130">A8</font></i>: 
<font color="Maroon" title="c9">s'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="171_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">x1</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">RNS</font>;<br/><b>assume </b><a NAME="E1:171_1_1_1"/><i><font color="Green" title="E131">A9</font></i>: 
( <font color="Maroon" title="c10">x1</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"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> &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="c10">x1</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c6">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">g</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font> )
 ;<br/><a NAME="E2:171_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</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"><font color="Maroon" title="c10">x1</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c6">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E3:171"><i><font color="Green" title="E124">A2</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><a NAME="E3:171_1_1_1"/><b>then </b><i><font color="Green" title="E132">A10</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</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"><font color="Maroon" title="c10">x1</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c6">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">g</font>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E3:171"><i><font color="Green" title="E124">A2</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>;<br/><a NAME="E4:171_1_1_1"/>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</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="c10">x1</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="c6">x0</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</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"><font color="Maroon" title="c10">x1</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c6">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E3:171"><i><font color="Green" title="E124">A2</font></i></a>, <a class="txt" href="ncfcont1.html#E5:171"><i><font color="Green" title="E126">A4</font></i></a>, <a class="txt" href="ncfcont1.html#E1:171_1"><i><font color="Green" title="E127">A5</font></i></a>, <a class="txt" href="ncfcont1.html#E1:171_1_1_1"><i><font color="Green" title="E131">A9</font></i></a></i>;<br/><a NAME="E5:171_1_1_1"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</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="c10">x1</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="c6">x0</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">g</font>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E3:171_1_1_1"><i><font color="Green" title="E132">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:171_1_1_1"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c10">x1</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</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="c6">x0</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">g</font>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E1:171_1_1_1"><i><font color="Green" title="E131">A9</font></i></a>, <a class="ref" href="partfun2.html#T32" title="PARTFUN2:th.32">PARTFUN2:32</a></i>;<br/><b>hence </b><a NAME="E7:171_1_1_1"/>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c10">x1</font></span>)</span> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">x0</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">g</font>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E5:171"><i><font color="Green" title="E126">A4</font></i></a>, <a class="txt" href="ncfcont1.html#E1:171_1"><i><font color="Green" title="E127">A5</font></i></a>, <a class="ref" href="partfun2.html#T32" title="PARTFUN2:th.32">PARTFUN2:32</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:171_1_1"/>
( 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">s'</font> &amp; ( for <font color="Olive" title="b1">x1</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">RNS</font>  st <font color="Olive" title="b1">x1</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"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> &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#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c6">x0</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">s'</font> holds <br/><span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <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#K6" title="RLVECT_1:func.6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">x0</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">g</font> ) )
 <b>by </b><i><a class="txt" href="ncfcont1.html#E1:171_1_1"><i><font color="Green" title="E128">A6</font></i></a>, <a class="txt" href="ncfcont1.html#E2:171_1_1"><i><font color="Green" title="E129">A7</font></i></a>, <a class="txt" href="ncfcont1.html#E3:171_1_1"><i><font color="Green" title="E130">A8</font></i></a>, <a class="ref" href="xreal_1.html#T131" title="XREAL_1:th.131">XREAL_1:131</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:171_1"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font> <a href="ncfcont1.html#R3" title="NCFCONT1:pred.3">is_continuous_in</a> <font color="Maroon" title="c6">x0</font>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E5:171"><i><font color="Green" title="E126">A4</font></i></a>, <a class="txt" href="ncfcont1.html#E1:171_1"><i><font color="Green" title="E127">A5</font></i></a>, <a class="ref" href="ncfcont1.html#T31" target="_self" title="NCFCONT1:th.31">Th31</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:171"/>
<font color="Maroon" title="c4">f</font> <a href="ncfcont1.html#R9" title="NCFCONT1:pred.9">is_continuous_on</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="ncfcont1.html#E4:171"><i><font color="Green" title="E125">A3</font></i></a>, <a class="ref" href="ncfcont1.html#D23" target="_self" title="NCFCONT1:def.23">Def23</a></i>;<br/>


</div>
