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<span class="kw">let </span><font color="Maroon" title="c1">V</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">  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">h</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">V</font> <a href="dualsp01.html#K5" title="DUALSP01:func.5">*'</a></span>)</span><br/>  for <font color="Olive" title="b3">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds <br/> ( <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">f</font> iff  for <font color="Olive" title="b4">x</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> holds  <font color="Olive" title="b2">h</font> <a href="dualsp01.html#K6" title="DUALSP01:func.6">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="dualsp01.html#K6" title="DUALSP01:func.6">.</a> <font color="Olive" title="b4">x</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">h</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">V</font> <a href="dualsp01.html#K5" title="DUALSP01:func.5">*'</a></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">  for <font color="Olive" title="b1">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds <br/> ( <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">f</font> iff  for <font color="Olive" title="b2">x</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> holds  <font color="Maroon" title="c3">h</font> <a href="dualsp01.html#K6" title="DUALSP01:func.6">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="dualsp01.html#K6" title="DUALSP01:func.6">.</a> <font color="Olive" title="b2">x</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">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"> ( <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">f</font> iff  for <font color="Olive" title="b1">x</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> holds  <font color="Maroon" title="c3">h</font> <a href="dualsp01.html#K6" title="DUALSP01:func.6">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="dualsp01.html#K6" title="DUALSP01:func.6">.</a> <font color="Olive" title="b1">x</font></span>)</span> )</span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c5">a1</font> = <font color="Maroon" title="c4">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">Y</font> being   <a href="vectsp_1.html#NM9" title="VECTSP_1:NM.9">VectSp</a> of  <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> <span class="kw"> such that </span><br/><a NAME="E3:14"/><span class="lab"><font color="Green" title="E5">AS1</font></span>: 
( <font color="Maroon" title="c6">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dualsp01.html#K2" title="DUALSP01:func.2">RLSp2RVSp</a> <font color="Maroon" title="c1">V</font> &amp; <font color="Maroon" title="c1">V</font> <a href="dualsp01.html#K5" title="DUALSP01:func.5">*'</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dualsp01.html#K4" title="DUALSP01:func.4">RVSp2RLSp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Y</font> <a href="hahnban1.html#K8" title="HAHNBAN1:func.8">*'</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dualsp01.html#D5" target="_self" onmouseover="rs('dualsp01/D5')" onmouseout="rh()">def2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">f1</font> = <font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c8">h1</font> = <font color="Maroon" title="c3">h</font> as   <a href="hahnban1.html#NM2" title="HAHNBAN1:NM.2">linear-Functional</a> of <font color="Maroon" title="c6">Y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E5">AS1</font></span></a>, <a class="ref" href="hahnban1.html#D10" onmouseover="rs('hahnban1/D10')" onmouseout="rh()">HAHNBAN1:def 10</a></span>;<br/>
<div><span class="kw">hereby </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"> ( (  for <font color="Olive" title="b1">x</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> holds  <font color="Maroon" title="c3">h</font> <a href="dualsp01.html#K6" title="DUALSP01:func.6">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="dualsp01.html#K6" title="DUALSP01:func.6">.</a> <font color="Olive" title="b1">x</font></span>)</span> ) implies <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">f</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:14_1"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
<font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">f</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">  for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> holds  <font color="Maroon" title="c3">h</font> <a href="dualsp01.html#K6" title="DUALSP01:func.6">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="dualsp01.html#K6" title="DUALSP01:func.6">.</a> <font color="Olive" title="b1">x</font></span>)</span></span><br/><div><span class="kw">hereby </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>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c9">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</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">h</font> <a href="dualsp01.html#K6" title="DUALSP01:func.6">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="dualsp01.html#K6" title="DUALSP01:func.6">.</a> <font color="Maroon" title="c9">x</font></span>)</span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c10">x1</font> = <font color="Maroon" title="c9">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">Y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E5">AS1</font></span></a></span>;<br/><a NAME="E2:14_1_1"/>
<font color="Maroon" title="c8">h1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">a1</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <font color="Maroon" title="c7">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E5">AS1</font></span></a>, <a class="ref" href="hahnban1.html#D10" onmouseover="rs('hahnban1/D10')" onmouseout="rh()">HAHNBAN1:def 10</a></span>;<br/><a NAME="E3:14_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#D6" onmouseover="rs('hahnban1/D6')" onmouseout="rh()">HAHNBAN1:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E4:14_1_1"/>
<font color="Maroon" title="c3">h</font> <a href="dualsp01.html#K6" title="DUALSP01:func.6">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="dualsp01.html#K6" title="DUALSP01:func.6">.</a> <font color="Maroon" title="c9">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"> verum</span><br/></div>
<span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E6:14"/>
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> holds  <font color="Maroon" title="c3">h</font> <a href="dualsp01.html#K6" title="DUALSP01:func.6">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="dualsp01.html#K6" title="DUALSP01:func.6">.</a> <font color="Olive" title="b1">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">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">f</font></span><br/>

<a NAME="E7:14"/><span class="kw">then </span>
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">Y</font> holds  <font color="Maroon" title="c8">h1</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="c5">a1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E5">AS1</font></span></a></span>;<br/>
<a NAME="E8:14"/><span class="kw">then </span>
<font color="Maroon" title="c8">h1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">a1</font> <a href="hahnban1.html#K6" title="HAHNBAN1:func.6">*</a> <font color="Maroon" title="c7">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#D6" onmouseover="rs('hahnban1/D6')" onmouseout="rh()">HAHNBAN1:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:14"/>
<font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E5">AS1</font></span></a>, <a class="ref" href="hahnban1.html#D10" onmouseover="rs('hahnban1/D10')" onmouseout="rh()">HAHNBAN1:def 10</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/>


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