<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">Y</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">h</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="rsspace4.html#K5" title="RSSPACE4:func.5">R_VectorSpace_of_BoundedFunctions</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c5">f'</font>, <font color="Maroon" title="c6">h'</font> be  <a href="rsspace4.html#V1" title="RSSPACE4:attr.1">bounded</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</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>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:27"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c5">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font>
 <b>and </b><br/><a NAME="E2:27"/><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon" title="c6">h'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c7">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<a NAME="E3:27"/><i><font color="Green" title="E21">A3</font></i>: 
 <a href="rsspace4.html#K5" title="RSSPACE4:func.5">R_VectorSpace_of_BoundedFunctions</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> is    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of  <a href="lopban_1.html#K5" title="LOPBAN_1:func.5">RealVectSpace</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="rsspace4.html#T8" target="_self" title="RSSPACE4:th.8">Th8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">f1</font> = <font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c9">h1</font> = <font color="Maroon" title="c4">h</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K5" title="LOPBAN_1:func.5">RealVectSpace</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <b>by </b><i><a class="ref" href="rlsub_1.html#T18" title="RLSUB_1:th.18">RLSUB_1:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">f1'</font> = <font color="Maroon" title="c5">f'</font>, <font color="Maroon" title="c11">h1'</font> = <font color="Maroon" title="c6">h'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <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="E22">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_1"/><i><font color="Green" title="E23">A5</font></i>: 
<font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c3">f</font>
 ;<br/><b>let </b><font color="Maroon" title="c12">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><a NAME="E2:27_1"/>
<font color="Maroon" title="c9">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c8">f1</font>
 <b>by </b><i><a class="txt" href="rsspace4.html#E3:27"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="rsspace4.html#E1:27_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="rlsub_1.html#T22" title="RLSUB_1:th.22">RLSUB_1:22</a></i>;<br/><b>hence </b><a NAME="E3:27_1"/>
<font color="Maroon" title="c6">h'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="rsspace4.html#E1:27"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="rsspace4.html#E2:27"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="lopban_1.html#T15" title="LOPBAN_1:th.15">LOPBAN_1:15</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c6">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> <font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f'</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="E2:27_2"/><b>then </b>
<font color="Maroon" title="c9">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c8">f1</font>
 <b>by </b><i><a class="txt" href="rsspace4.html#E1:27"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="rsspace4.html#E2:27"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="lopban_1.html#T15" title="LOPBAN_1:th.15">LOPBAN_1:15</a></i>;<br/><b>hence </b><a NAME="E3:27_2"/>
<font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="rsspace4.html#E3:27"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="rlsub_1.html#T22" title="RLSUB_1:th.22">RLSUB_1:22</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:27"/>
( <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c3">f</font> iff for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c6">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> <font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> )
 <b>by </b><i><a class="txt" href="rsspace4.html#E6:27"><i><font color="Green" title="E22">A4</font></i></a></i>;<br/>


</div>
