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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="clvect_1.html#NM5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="csspace4.html#K5" target="_self">C_VectorSpace_of_BoundedFunctions</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>, <font color="Maroon">g</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="csspace4.html#K5" target="_self">C_VectorSpace_of_BoundedFunctions</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>, <font color="Maroon">h</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="csspace4.html#K5" target="_self">C_VectorSpace_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="csspace4.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="csspace4.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="csspace4.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="E1:31"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:31"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">g'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E3:31"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">h'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 ;<br/>

<a NAME="E4:31"/><i><font color="Green">E47</font></i>: 
 <a href="csspace4.html#K5" target="_self">C_VectorSpace_of_BoundedFunctions</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> is    <a href="clvect_1.html#M1">Subspace</a> of  <a href="clopban1.html#K3">ComplexVectSpace</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K3">ComplexVectSpace</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <b>by </b><i><a class="ref" href="clvect_1.html#T30">CLVECT_1:30</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">g</font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K3">ComplexVectSpace</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <b>by </b><i><a class="txt" href="csspace4.html#E1"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="clvect_1.html#T30">CLVECT_1:30</a></i>;<br/>
<b>reconsider </b><font color="Maroon">h1</font> = <font color="Maroon">h</font> as   <a href="clvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K3">ComplexVectSpace</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <b>by </b><i><a class="txt" href="csspace4.html#E1"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="clvect_1.html#T30">CLVECT_1:30</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f1'</font> = <font color="Maroon">f'</font>, <font color="Maroon">g1'</font> = <font color="Maroon">g'</font>, <font color="Maroon">h1'</font> = <font color="Maroon">h'</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">g</font>
 ;<br/><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="E2:31_1"/>
<font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="txt" href="csspace4.html#E1"><i><font color="Green">Lemma27</font></i></a>, , <a class="ref" href="clvect_1.html#T33">CLVECT_1:33</a></i>;<br/><b>hence </b><a NAME="E3:31_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#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>, , , <a class="ref" href="clopban1.html#T12">CLOPBAN1:12</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_2"/>
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#K4">+</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/><a NAME="E2:31_2"/><b>then </b>
<font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">g1</font>
 <b>by </b><i>, , , <a class="ref" href="clopban1.html#T12">CLOPBAN1:12</a></i>;<br/><b>hence </b><a NAME="E3:31_2"/>
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="csspace4.html#E1"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="clvect_1.html#T33">CLVECT_1:33</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:31"/>
( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="rlvect_1.html#K4">+</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#K4">+</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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