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<b>let </b><font color="Maroon" title="c1">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">C</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">V</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="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">C</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V</font>;<br/>







<b>assume </b><a NAME="E1:58"/>
<font color="Maroon" title="c4">f</font> <a href="partfun2.html#R1" title="PARTFUN2:pred.1">is_constant_on</a> <font color="Maroon" title="c1">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">r</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">V</font><b> such that </b><br/><a NAME="E3:58"/><i><font color="Green" title="E25">A1</font></i>: 
for <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</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> <font color="Maroon" title="c4">f</font></span>)</span> holds <br/><font color="Maroon" title="c4">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="ref" href="partfun2.html#D3" title="PARTFUN2:def.3">PARTFUN2:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C</font>;<br/><b>assume </b><a NAME="E1:58_1"/>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</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"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">.||</a></span></span>)</span>
 ;<br/><a NAME="E2:58_1"/><b>then </b><i><font color="Green" title="E26">A2</font></i>: 
( <font color="Maroon" title="c6">c</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"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">.||</a></span> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> )
 <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="E3:58_1"/><b>then </b>
<font color="Maroon" title="c6">c</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> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="vfunct_1.html#D5" target="_self" title="VFUNCT_1:def.5">Def5</a></i>;<br/><a NAME="E4:58_1"/><b>then </b><i><font color="Green" title="E27">A3</font></i>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</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> <font color="Maroon" title="c4">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="vfunct_1.html#E2:58_1"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b><span class="p1"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">.||</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">c</font> = 
<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="c4">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">c</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="vfunct_1.html#E2:58_1"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="vfunct_1.html#D5" target="_self" title="VFUNCT_1:def.5">Def5</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c5">r</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
<b>by </b><i><a class="txt" href="vfunct_1.html#E3:58"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="vfunct_1.html#E4:58_1"><i><font color="Green" title="E27">A3</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:58"/>
<span class="p1"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">||.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">.||</a></span> <a href="partfun2.html#R1" title="PARTFUN2:pred.1">is_constant_on</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="ref" href="partfun2.html#T76" title="PARTFUN2:th.76">PARTFUN2:76</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c6">p</font> =  <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c5">r</font>;<br/><b>let </b><font color="Maroon" title="c7">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C</font>;<br/><b>assume </b><a NAME="E1:58_2"/><i><font color="Green" title="E26">A4</font></i>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</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"><a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">-</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:58_2"/><b>then </b>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</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> <font color="Maroon" title="c4">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="vfunct_1.html#D6" target="_self" title="VFUNCT_1:def.6">Def6</a></i>;<br/><a NAME="E3:58_2"/><b>then </b><i><font color="Green" title="E27">A5</font></i>: 
 <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p1">(<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">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="txt" href="vfunct_1.html#E3:58"><i><font color="Green" title="E25">A1</font></i></a></i>;<br/><a NAME="E4:58_2"/>
<font color="Maroon" title="c7">c</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"><a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">-</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="vfunct_1.html#E1:58_2"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E5:58_2"/>
<span class="p1">(<span class="default"><a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">-</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="txt" href="vfunct_1.html#E3:58_2"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="vfunct_1.html#D6" target="_self" title="VFUNCT_1:def.6">Def6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:58"/>
 <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">-</a> <font color="Maroon" title="c4">f</font> <a href="partfun2.html#R1" title="PARTFUN2:pred.1">is_constant_on</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="ref" href="partfun2.html#D3" title="PARTFUN2:def.3">PARTFUN2:def 3</a></i>;<br/>


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