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<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">g</font>, <font color="Maroon" title="c5">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="c6">f'</font>, <font color="Maroon" title="c7">g'</font>, <font color="Maroon" title="c8">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="E1:25"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c6">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="E2:25"/><i><font color="Green" title="E18">A2</font></i>: 
<font color="Maroon" title="c7">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font>
 <b>and </b><br/><a NAME="E3:25"/><i><font color="Green" title="E19">A3</font></i>: 
<font color="Maroon" title="c8">h'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font>
 ;<br/>

<a NAME="E4:25"/><i><font color="Green" title="E20">A4</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#T7" target="_self" title="RSSPACE4:th.7">Th7</a>, <a class="ref" href="rsspace.html#T13" title="RSSPACE:th.13">RSSPACE:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">f1</font> = <font color="Maroon" title="c3">f</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">g1</font> = <font color="Maroon" title="c4">g</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="txt" href="rsspace4.html#E4:25"><i><font color="Green" title="E20">A4</font></i></a>, <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="c11">h1</font> = <font color="Maroon" title="c5">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="txt" href="rsspace4.html#E4:25"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="rlsub_1.html#T18" title="RLSUB_1:th.18">RLSUB_1:18</a></i>;<br/>
<div><i><font color="Green" title="E21">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1"/><i><font color="Green" title="E22">A6</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="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="E2:25_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="rsspace4.html#E4:25"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="rsspace4.html#E1:25_1"><i><font color="Green" title="E22">A6</font></i></a>, <a class="ref" href="rlsub_1.html#T21" title="RLSUB_1:th.21">RLSUB_1:21</a></i>;<br/><b>hence </b><a NAME="E3:25_1"/>
<font color="Maroon" title="c8">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> <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="c12">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="c12">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="rsspace4.html#E1:25"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="rsspace4.html#E2:25"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="rsspace4.html#E3:25"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="lopban_1.html#T14" title="LOPBAN_1:th.14">LOPBAN_1:14</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_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="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="E2:25_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="txt" href="rsspace4.html#E1:25"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="rsspace4.html#E2:25"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="rsspace4.html#E3:25"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="lopban_1.html#T14" title="LOPBAN_1:th.14">LOPBAN_1:14</a></i>;<br/><b>hence </b><a NAME="E3:25_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="rsspace4.html#E4:25"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="rlsub_1.html#T21" title="RLSUB_1:th.21">RLSUB_1:21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:25"/>
( <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="subset_1.html#M1" title="SUBSET_1:mode.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> )
 <b>by </b><i><a class="txt" href="rsspace4.html#E8:25"><i><font color="Green" title="E21">A5</font></i></a></i>;<br/>


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