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<b>let </b><font color="Maroon">X</font> be   <a href="normsp_1.html#NM1">RealNormSpace</a>, <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="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K11" target="_self">R_VectorSpace_of_BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>, <font color="Maroon">g</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K11" target="_self">R_VectorSpace_of_BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>, <font color="Maroon">h</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K11" target="_self">R_VectorSpace_of_BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>;<br/>



<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font>, <font color="Maroon">g'</font> = <font color="Maroon">g</font>, <font color="Maroon">h'</font> = <font color="Maroon">h</font> as  <a href="lopban_1.html#V3" target="_self">bounded</a> <a href="lopban_1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<a NAME="E2:54"/><i><font color="Green">E33</font></i>: 
 <a href="lopban_1.html#K11" target="_self">R_VectorSpace_of_BoundedLinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font> is    <a href="rlsub_1.html#M1">Subspace</a> of  <a href="lopban_1.html#K8" target="_self">R_VectorSpace_of_LinearOperators</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="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K8" target="_self">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <b>by </b><i><a class="ref" href="rlsub_1.html#T18">RLSUB_1:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">g</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K8" target="_self">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <b>by </b><i>, <a class="ref" href="rlsub_1.html#T18">RLSUB_1:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">h1</font> = <font color="Maroon">h</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K8" target="_self">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <b>by </b><i>, <a class="ref" href="rlsub_1.html#T18">RLSUB_1:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f1'</font> = <font color="Maroon">f'</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <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/>
<b>reconsider </b><font color="Maroon">g1'</font> = <font color="Maroon">g'</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <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/>
<b>reconsider </b><font color="Maroon">h1'</font> = <font color="Maroon">h'</font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <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><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:54_1"/><i><font color="Green">E42</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="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><a NAME="E2:54_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="ref" href="lopban_1.html#T2" target="_self">Th2</a>, <a class="ref" href="rlsub_1.html#T21">RLSUB_1:21</a></i>;<br/><b>hence </b><a NAME="E3:54_1"/>
<font color="Maroon">h</font> <a href="lopban_1.html#K12" target="_self">.</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="lopban_1.html#K12" target="_self">.</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="lopban_1.html#K12" target="_self">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E10:54"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> holds  <font color="Maroon">h</font> <a href="lopban_1.html#K12" target="_self">.</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="lopban_1.html#K12" target="_self">.</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="lopban_1.html#K12" target="_self">.</a> <font color="Olive">x</font></span>)</span>
 ;<br/>

<a NAME="E11:54"/><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/>;<br/>
<b>hence </b><a NAME="E12:54"/>
<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="ref" href="rlsub_1.html#T21">RLSUB_1:21</a></i>;<br/>


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