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<b>let </b><font color="Maroon">X</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">Y</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="rsspace4.html#K9" target="_self">R_NormSpace_of_BoundedFunctions</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>, <font color="Maroon">g</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="rsspace4.html#K9" target="_self">R_NormSpace_of_BoundedFunctions</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>, <font color="Maroon">h</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="rsspace4.html#K9" target="_self">R_NormSpace_of_BoundedFunctions</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>;<br/><b>let </b><font color="Maroon">f'</font> be  <a href="rsspace4.html#V1" target="_self">bounded</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>, <font color="Maroon">g'</font> be  <a href="rsspace4.html#V1" target="_self">bounded</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>, <font color="Maroon">h'</font> be  <a href="rsspace4.html#V1" target="_self">bounded</a> <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:48"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:48"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">g'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E3:48"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">h'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 ;<br/>

<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1"/>
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">g</font>
 ;<br/><a NAME="E2:48_1"/><b>then </b>
<font color="Maroon">h</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T43">RLVECT_1:43</a></i>;<br/><a NAME="E3:48_1"/><b>then </b>
<font color="Maroon">h</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><a href="rsspace4.html#K9" target="_self">R_NormSpace_of_BoundedFunctions</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>;<br/><a NAME="E4:48_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">h</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><a NAME="E1:48_1_1"/>
<font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, , , , </i>;<br/><a NAME="E2:48_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/><a NAME="E3:48_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">Y</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>;<br/><b>hence </b><a NAME="E4:48_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:48_1"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds  <font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_2"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds  <font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><a NAME="E1:48_2_1"/>
<font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E2:48_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T43">RLVECT_1:43</a></i>;<br/><a NAME="E3:48_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">Y</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>;<br/><b>hence </b><a NAME="E4:48_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T26">RLVECT_1:26</a></i>;<br/></div><b>end;</b></div><a NAME="E3:48_2"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">g</font>
 <b>by </b><i>, , , </i>;<br/><a NAME="E4:48_2"/><b>then </b>
<font color="Maroon">f</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>;<br/><a NAME="E5:48_2"/><b>then </b>
<font color="Maroon">f</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><a href="rsspace4.html#K9" target="_self">R_NormSpace_of_BoundedFunctions</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>;<br/><b>hence </b><a NAME="E6:48_2"/>
<font color="Maroon">f</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:48"/>
( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">g</font> iff for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds  <font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> )
 <b>by </b><i/>;<br/>


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