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<b>let </b><font color="Maroon">X1</font> be   <a href="struct_0.html#NM2">Subset</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">X2</font> be   <a href="struct_0.html#NM2">Subset</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>assume </b><b>that </b><br/><a NAME="E1:45_1_2"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> iff <font color="Olive">x</font> is  <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>and </b><br/><a NAME="E2:45_1_2"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> iff <font color="Olive">x</font> is  <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> )
 ;<br/>

<a NAME="E3:45_1_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:45_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 ;<br/>

<a NAME="E2:45_1_2_1"/><b>then </b>
<font color="Maroon">x</font> is  <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/>
<b>hence </b><a NAME="E3:45_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:45_1_2"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E5:45_1_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:45_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font>
 ;<br/>

<a NAME="E2:45_1_2_2"/><b>then </b>
<font color="Maroon">x</font> is  <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/>
<b>hence </b><a NAME="E3:45_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E6:45_1_2"/><b>then </b>
<font color="Maroon">X2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X1</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:45_1_2"/>
<font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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