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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="clvect_1.html#NM2" title="CLVECT_1:NM.2">ComplexLinearSpace</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="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="clopban1.html#K3" title="CLOPBAN1:func.3">ComplexVectSpace</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span>;<br/>





<b>reconsider </b><font color="Maroon" title="c6">f'</font> = <font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c7">g'</font> = <font color="Maroon" title="c4">g</font>, <font color="Maroon" title="c8">h'</font> = <font color="Maroon" title="c5">h</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL: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> ;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:19_1"/><i><font color="Green" title="E15">A1</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#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c4">g</font>
 ;<br/><b>let </b><font color="Maroon" title="c9">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="E2:19_1"/>
<font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lopban_1.html#K2" title="LOPBAN_1:func.2">FuncAdd</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="funcsdom.html#K1" title="FUNCSDOM:func.1">.</a> <font color="Maroon" title="c6">f'</font>,<font color="Maroon" title="c7">g'</font>
 <b>by </b><i><a class="txt" href="clopban1.html#E1:19_1"><i><font color="Green" title="E15">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E3:19_1"/>
<font color="Maroon" title="c5">h</font> <a href="clopban1.html#K4" title="CLOPBAN1:func.4">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="clopban1.html#K4" title="CLOPBAN1:func.4">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="clopban1.html#K4" title="CLOPBAN1:func.4">.</a> <font color="Maroon" title="c9">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="lopban_1.html#T3" title="LOPBAN_1:th.3">LOPBAN_1:3</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:19"/>
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="c5">h</font> <a href="clopban1.html#K4" title="CLOPBAN1:func.4">.</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="c3">f</font> <a href="clopban1.html#K4" title="CLOPBAN1:func.4">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="clopban1.html#K4" title="CLOPBAN1:func.4">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 ;<br/>

<a NAME="E4:19"/><b>then </b><i><font color="Green" title="E15">A2</font></i>: 
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#K4" title="RLVECT_1:func.4">+</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/>
<b>thus </b><a NAME="E5:19"/>
<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#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="txt" href="clopban1.html#E4:19"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="lopban_1.html#T3" title="LOPBAN_1:th.3">LOPBAN_1:3</a></i>;<br/>


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