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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">C</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">V</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">f1</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">f2</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">C</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>;<br/>











<b>assume </b><a NAME="E1:55"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">f1</font> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">X</font> &amp; <font color="Maroon">f2</font> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">Y</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">r1</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:55"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>  st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></span>)</span> holds <br/><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="partfun2.html#D3">PARTFUN2:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">r2</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E5:55"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>  st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font></span>)</span> holds <br/><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font>
 <b>by </b><i>, <a class="ref" href="partfun2.html#D3">PARTFUN2:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>;<br/><b>assume </b><a NAME="E1:55_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="vfunct_1.html#K1" target="_self">+</a> <font color="Maroon">f2</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:55_1"/><b>then </b><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="vfunct_1.html#K1" target="_self">+</a> <font color="Maroon">f2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:55_1"/><b>then </b><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:55_1"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:55_1"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></span>)</span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="vfunct_1.html#K1" target="_self">+</a> <font color="Maroon">f2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">r1</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">r1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">r2</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:55"/>
<font color="Maroon">f1</font> <a href="vfunct_1.html#K1" target="_self">+</a> <font color="Maroon">f2</font> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="partfun2.html#D3">PARTFUN2:def 3</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>;<br/><b>assume </b><a NAME="E1:55_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="vfunct_1.html#K2" target="_self">-</a> <font color="Maroon">f2</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:55_2"/><b>then </b><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="vfunct_1.html#K2" target="_self">-</a> <font color="Maroon">f2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:55_2"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:55_2"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:55_2"/><b>then </b><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font></span>)</span> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="vfunct_1.html#K2" target="_self">-</a> <font color="Maroon">f2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">r1</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">r1</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">r2</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:55"/>
<font color="Maroon">f1</font> <a href="vfunct_1.html#K2" target="_self">-</a> <font color="Maroon">f2</font> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="partfun2.html#D3">PARTFUN2:def 3</a></i>;<br/>


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