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<b>let </b><font color="Maroon">X</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>, <font color="Maroon">Y</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</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#K8" target="_self">R_VectorSpace_of_LinearOperators</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#K8" target="_self">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font>, <font color="Maroon">h'</font> = <font color="Maroon">h</font> as   <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="E2:38"/><i><font color="Green">E33</font></i>: 
 <a href="lopban_1.html#K8" target="_self">R_VectorSpace_of_LinearOperators</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#K5" target="_self">RealVectSpace</a> the <a href="struct_0.html#U1">carrier</a> of <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#K5" target="_self">RealVectSpace</a> the <a href="struct_0.html#U1">carrier</a> of <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#K5" target="_self">RealVectSpace</a> the <a href="struct_0.html#U1">carrier</a> of <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">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><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">f</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="E2:38_1"/>
<font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">f1</font>
 <b>by </b><i>, , <a class="ref" href="rlsub_1.html#T22">RLSUB_1:22</a></i>;<br/><b>hence </b><a NAME="E3:38_1"/>
<font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</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>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_2"/>
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="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</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>
 ;<br/><a NAME="E2:38_2"/><b>then </b>
<font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">f1</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:38_2"/>
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="rlsub_1.html#T22">RLSUB_1:22</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:38"/>
( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">f</font> iff for <font color="Olive">x</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> holds  <font color="Maroon">h</font> <a href="lopban_1.html#K9" target="_self">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="lopban_1.html#K9" target="_self">.</a> <font color="Olive">x</font></span>)</span> )
 <b>by </b><i/>;<br/>


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