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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="dualsp01.html#K7" title="DUALSP01:func.7">LinearFunctionals</a> <font color="Maroon" title="c1">X</font> is  <a href="rlsub_1.html#V1" title="RLSUB_1:attr.1">linearly-closed</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">W</font> =  <a href="dualsp01.html#K7" title="DUALSP01:func.7">LinearFunctionals</a> <font color="Maroon" title="c1">X</font>;<br/>
<a NAME="E1:20"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
 for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span>)</span>  st <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="dualsp01.html#K7" title="DUALSP01:func.7">LinearFunctionals</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="dualsp01.html#K7" title="DUALSP01:func.7">LinearFunctionals</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="dualsp01.html#K7" title="DUALSP01:func.7">LinearFunctionals</a> <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="20_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">u</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="dualsp01.html#K7" title="DUALSP01:func.7">LinearFunctionals</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="dualsp01.html#K7" title="DUALSP01:func.7">LinearFunctionals</a> <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="dualsp01.html#K7" title="DUALSP01:func.7">LinearFunctionals</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:20_1"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
( <font color="Maroon" title="c3">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="dualsp01.html#K7" title="DUALSP01:func.7">LinearFunctionals</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="dualsp01.html#K7" title="DUALSP01:func.7">LinearFunctionals</a> <font color="Maroon" title="c1">X</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="dualsp01.html#K7" title="DUALSP01:func.7">LinearFunctionals</a> <font color="Maroon" title="c1">X</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">u</font> as   <a href="hahnban.html#NM1" title="HAHNBAN:NM.1">Functional</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T66" onmouseover="rs('funct_2/T66')" onmouseout="rh()">FUNCT_2:66</a></span>;<br/>
<a NAME="E3:20_1"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
<font color="Maroon" title="c5">f</font> is  <a href="hahnban.html#V2" title="HAHNBAN:attr.2">additive</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="20_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="hahnban.html#D2" onmouseover="rs('hahnban/D2')" onmouseout="rh()">HAHNBAN:def 2</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="real_1.html#K3" title="REAL_1:func.3">K55</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="c6">x</font></span>)</span>,<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="c7">y</font></span>)</span>)</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">vZ1</font> = <font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c9">uZ1</font> = <font color="Maroon" title="c4">u</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) ;<br/>
<a NAME="E2:20_1_1"/><span class="lab"><font color="Green" title="E11">A4</font></span>: 
<font color="Maroon" title="c9">uZ1</font> is   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="dualsp01.html#D6" target="_self" onmouseover="rs('dualsp01/D6')" onmouseout="rh()">Def7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">uZ11</font> = <font color="Maroon" title="c9">uZ1</font> as   <a href="hahnban.html#V2" title="HAHNBAN:attr.2">additive</a>   <a href="hahnban.html#V3" title="HAHNBAN:attr.3">homogeneous</a>  <a href="hahnban.html#NM1" title="HAHNBAN:NM.1">Functional</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="dualsp01.html#D6" target="_self" onmouseover="rs('dualsp01/D6')" onmouseout="rh()">Def7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">x1</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c12">y1</font> = <font color="Maroon" title="c7">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E5:20_1_1"/><span class="lab"><font color="Green" title="E12">A5</font></span>: 
<font color="Maroon" title="c8">vZ1</font> is   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="dualsp01.html#D6" target="_self" onmouseover="rs('dualsp01/D6')" onmouseout="rh()">Def7</a></span>;<br/>
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">uZ1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">vZ1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcsdom.html#T1" onmouseover="rs('funcsdom/T1')" onmouseout="rh()">FUNCSDOM:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">uZ1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">uZ1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">vZ1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1_1"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="ref" href="hahnban.html#D2" onmouseover="rs('hahnban/D2')" onmouseout="rh()">HAHNBAN:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">uZ1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">uZ1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">vZ1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vZ1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_1_1"><span class="lab"><font color="Green" title="E12">A5</font></span></a>, <a class="ref" href="hahnban.html#D2" onmouseover="rs('hahnban/D2')" onmouseout="rh()">HAHNBAN:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">uZ1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">vZ1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">uZ1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">vZ1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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="c6">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">uZ1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">vZ1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcsdom.html#T1" onmouseover="rs('funcsdom/T1')" onmouseout="rh()">FUNCSDOM:1</a></span>
<br/>.= 
<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="c6">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">uZ1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vZ1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span>

;<br/>
<span class="kw">hence </span><a NAME="E7:20_1_1"/>
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c6">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0: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="c7">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcsdom.html#T1" onmouseover="rs('funcsdom/T1')" onmouseout="rh()">FUNCSDOM:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:20_1"/>
<font color="Maroon" title="c5">f</font> is  <a href="hahnban.html#V3" title="HAHNBAN:attr.3">homogeneous</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="20_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="hahnban.html#D3" onmouseover="rs('hahnban/D3')" onmouseout="rh()">HAHNBAN:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds  <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c6">x</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">s</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c6">x</font></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c8">v1</font> = <font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c9">u1</font> = <font color="Maroon" title="c4">u</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) ;<br/>
<a NAME="E2:20_1_2"/><span class="lab"><font color="Green" title="E11">A6</font></span>: 
( <font color="Maroon" title="c9">u1</font> is   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c8">v1</font> is   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c1">X</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="dualsp01.html#D6" target="_self" onmouseover="rs('dualsp01/D6')" onmouseout="rh()">Def7</a></span>;<br/>
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">u1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcsdom.html#T1" onmouseover="rs('funcsdom/T1')" onmouseout="rh()">FUNCSDOM:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">u1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1_2"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="ref" href="hahnban.html#D3" onmouseover="rs('hahnban/D3')" onmouseout="rh()">HAHNBAN:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">u1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1_2"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="ref" href="hahnban.html#D3" onmouseover="rs('hahnban/D3')" onmouseout="rh()">HAHNBAN:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c7">s</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">u1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>

;<br/>
<span class="kw">hence </span><a NAME="E4:20_1_2"/>
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c6">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcsdom.html#T1" onmouseover="rs('funcsdom/T1')" onmouseout="rh()">FUNCSDOM:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:20_1"/>
<font color="Maroon" title="c3">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c4">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="dualsp01.html#K7" title="DUALSP01:func.7">LinearFunctionals</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_1"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="dualsp01.html#D6" target="_self" onmouseover="rs('dualsp01/D6')" onmouseout="rh()">Def7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:20"/>
 for <font color="Olive" title="b1">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b2">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span>)</span>  st <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="dualsp01.html#K7" title="DUALSP01:func.7">LinearFunctionals</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="dualsp01.html#K7" title="DUALSP01:func.7">LinearFunctionals</a> <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="20_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">a</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span>)</span>  st <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="dualsp01.html#K7" title="DUALSP01:func.7">LinearFunctionals</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="dualsp01.html#K7" title="DUALSP01:func.7">LinearFunctionals</a> <font color="Maroon" title="c1">X</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="dualsp01.html#K7" title="DUALSP01:func.7">LinearFunctionals</a> <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="dualsp01.html#K7" title="DUALSP01:func.7">LinearFunctionals</a> <font color="Maroon" title="c1">X</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:20_2"/><span class="lab"><font color="Green" title="E9">A8</font></span>: 
<font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="dualsp01.html#K7" title="DUALSP01:func.7">LinearFunctionals</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="dualsp01.html#K7" title="DUALSP01:func.7">LinearFunctionals</a> <font color="Maroon" title="c1">X</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v</font> as   <a href="hahnban.html#NM1" title="HAHNBAN:NM.1">Functional</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T66" onmouseover="rs('funct_2/T66')" onmouseout="rh()">FUNCT_2:66</a></span>;<br/>
<a NAME="E3:20_2"/><span class="lab"><font color="Green" title="E10">A9</font></span>: 
<font color="Maroon" title="c5">f</font> is  <a href="hahnban.html#V2" title="HAHNBAN:attr.2">additive</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="20_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="hahnban.html#D2" onmouseover="rs('hahnban/D2')" onmouseout="rh()">HAHNBAN:def 2</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="real_1.html#K3" title="REAL_1:func.3">K55</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="c6">x</font></span>)</span>,<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="c7">y</font></span>)</span>)</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">vZ1</font> = <font color="Maroon" title="c4">v</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) ;<br/>
<a NAME="E2:20_2_1"/><span class="lab"><font color="Green" title="E11">A10</font></span>: 
<font color="Maroon" title="c8">vZ1</font> is   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp01.html#D6" target="_self" onmouseover="rs('dualsp01/D6')" onmouseout="rh()">Def7</a>, <a class="txt" href="#E1:20_2"><span class="lab"><font color="Green" title="E9">A8</font></span></a></span>;<br/>
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">y</font></span>)</span> = 
<font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">vZ1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcsdom.html#T4" onmouseover="rs('funcsdom/T4')" onmouseout="rh()">FUNCSDOM:4</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">vZ1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vZ1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_2_1"><span class="lab"><font color="Green" title="E11">A10</font></span></a>, <a class="ref" href="hahnban.html#D2" onmouseover="rs('hahnban/D2')" onmouseout="rh()">HAHNBAN:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vZ1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vZ1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span>

<br/>.= 
<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="c6">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vZ1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcsdom.html#T4" onmouseover="rs('funcsdom/T4')" onmouseout="rh()">FUNCSDOM:4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:20_2_1"/>
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c6">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0: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="c7">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcsdom.html#T4" onmouseover="rs('funcsdom/T4')" onmouseout="rh()">FUNCSDOM:4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:20_2"/>
<font color="Maroon" title="c5">f</font> is  <a href="hahnban.html#V3" title="HAHNBAN:attr.3">homogeneous</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="20_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">X</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="hahnban.html#D3" onmouseover="rs('hahnban/D3')" onmouseout="rh()">HAHNBAN:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds  <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c6">x</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">s</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c6">x</font></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c8">vZ1</font> = <font color="Maroon" title="c4">v</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>) ;<br/>
<a NAME="E2:20_2_2"/><span class="lab"><font color="Green" title="E11">A11</font></span>: 
<font color="Maroon" title="c8">vZ1</font> is   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp01.html#D6" target="_self" onmouseover="rs('dualsp01/D6')" onmouseout="rh()">Def7</a>, <a class="txt" href="#E1:20_2"><span class="lab"><font color="Green" title="E9">A8</font></span></a></span>;<br/>
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x</font></span>)</span> = 
<font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">vZ1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcsdom.html#T4" onmouseover="rs('funcsdom/T4')" onmouseout="rh()">FUNCSDOM:4</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vZ1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_2_2"><span class="lab"><font color="Green" title="E11">A11</font></span></a>, <a class="ref" href="hahnban.html#D3" onmouseover="rs('hahnban/D3')" onmouseout="rh()">HAHNBAN:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c7">s</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vZ1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>

;<br/>
<span class="kw">hence </span><a NAME="E4:20_2_2"/>
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c6">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcsdom.html#T4" onmouseover="rs('funcsdom/T4')" onmouseout="rh()">FUNCSDOM:4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:20_2"/>
<font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="dualsp01.html#K7" title="DUALSP01:func.7">LinearFunctionals</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_2"><span class="lab"><font color="Green" title="E10">A9</font></span></a>, <a class="ref" href="dualsp01.html#D6" target="_self" onmouseover="rs('dualsp01/D6')" onmouseout="rh()">Def7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:20"/>
 <a href="dualsp01.html#K7" title="DUALSP01:func.7">LinearFunctionals</a> <font color="Maroon" title="c1">X</font> is  <a href="rlsub_1.html#V1" title="RLSUB_1:attr.1">linearly-closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E8">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
