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<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be   <a href="clvect_1.html#NM2" title="CLVECT_1:NM.2">ComplexLinearSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g</font>, <font color="Maroon" title="c5">h</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K6" title="CLOPBAN1:func.6">C_VectorSpace_of_LinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span>;<br/>



<b>reconsider </b><font color="Maroon" title="c6">f'</font> = <font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c7">g'</font> = <font color="Maroon" title="c4">g</font>, <font color="Maroon" title="c8">h'</font> = <font color="Maroon" title="c5">h</font> as   <a href="clopban1.html#NM1" title="CLOPBAN1:NM.1">LinearOperator</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> <b>by </b><i><a class="ref" href="clopban1.html#D5" target="_self" title="CLOPBAN1:def.5">Def5</a></i>;<br/>
<a NAME="E2:34"/><i><font color="Green" title="E22">A1</font></i>: 
 <a href="clopban1.html#K6" title="CLOPBAN1:func.6">C_VectorSpace_of_LinearOperators</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> is    <a href="clvect_1.html#M1" title="CLVECT_1:mode.1">Subspace</a> of  <a href="clopban1.html#K3" title="CLOPBAN1:func.3">ComplexVectSpace</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="clopban1.html#T16" target="_self" title="CLOPBAN1:th.16">Th16</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">f1</font> = <font color="Maroon" title="c3">f</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K3" title="CLOPBAN1:func.3">ComplexVectSpace</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <b>by </b><i><a class="ref" href="clvect_1.html#T30" title="CLVECT_1:th.30">CLVECT_1:30</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">g1</font> = <font color="Maroon" title="c4">g</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K3" title="CLOPBAN1:func.3">ComplexVectSpace</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <b>by </b><i><a class="txt" href="clopban1.html#E2:34"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="clvect_1.html#T30" title="CLVECT_1:th.30">CLVECT_1:30</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">h1</font> = <font color="Maroon" title="c5">h</font> as   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K3" title="CLOPBAN1:func.3">ComplexVectSpace</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <b>by </b><i><a class="txt" href="clopban1.html#E2:34"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="clvect_1.html#T30" title="CLVECT_1:th.30">CLVECT_1:30</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">f1'</font> = <font color="Maroon" title="c6">f'</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font> <b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">g1'</font> = <font color="Maroon" title="c7">g'</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font> <b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">h1'</font> = <font color="Maroon" title="c8">h'</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font> <b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<div><i><font color="Green" title="E23">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:34_1"/><i><font color="Green" title="E24">A3</font></i>: 
<font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">g</font>
 ;<br/><b>let </b><font color="Maroon" title="c15">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><a NAME="E2:34_1"/>
<font color="Maroon" title="c11">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">g1</font>
 <b>by </b><i><a class="txt" href="clopban1.html#E2:34"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="clopban1.html#E1:34_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="clvect_1.html#T33" title="CLVECT_1:th.33">CLVECT_1:33</a></i>;<br/><b>hence </b><a NAME="E3:34_1"/>
<font color="Maroon" title="c8">h'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="clopban1.html#T12" target="_self" title="CLOPBAN1:th.12">Th12</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:34_2"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c8">h'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 ;<br/><a NAME="E2:34_2"/><b>then </b>
<font color="Maroon" title="c11">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">g1</font>
 <b>by </b><i><a class="ref" href="clopban1.html#T12" target="_self" title="CLOPBAN1:th.12">Th12</a></i>;<br/><b>hence </b><a NAME="E3:34_2"/>
<font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="txt" href="clopban1.html#E2:34"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="clvect_1.html#T33" title="CLVECT_1:th.33">CLVECT_1:33</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:34"/>
( <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">g</font> iff for <font color="Olive" title="b1">x</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c5">h</font> <a href="clopban1.html#K7" title="CLOPBAN1:func.7">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="clopban1.html#K7" title="CLOPBAN1:func.7">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="clopban1.html#K7" title="CLOPBAN1:func.7">.</a> <font color="Olive" title="b1">x</font></span>)</span> )
 <b>by </b><i><a class="txt" href="clopban1.html#E9:34"><i><font color="Green" title="E23">A2</font></i></a></i>;<br/>


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