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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">g</font>, <font color="Olive" title="b3">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> holds <br/> ( <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">g</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="b3">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> <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> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</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">g</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"><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"> ( <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">g</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="c4">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> <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 href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</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">consider </span><font color="Maroon" title="c5">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="E2:13"/><span class="lab"><font color="Green" title="E5">AS1</font></span>: 
( <font color="Maroon" title="c5">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="c5">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="c6">f1</font> = <font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c7">g1</font> = <font color="Maroon" title="c3">g</font>, <font color="Maroon" title="c8">h1</font> = <font color="Maroon" title="c4">h</font> as   <a href="hahnban1.html#NM2" title="HAHNBAN1:NM.2">linear-Functional</a> of <font color="Maroon" title="c5">Y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13"><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="lab"><font color="Green" title="E6">A2</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="13_1"><span class="kw">now </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">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">g</font> implies  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="c4">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> <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 href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="dualsp01.html#K6" title="DUALSP01:func.6">.</a> <font color="Olive" title="b1">x</font></span>)</span> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:13_1"/><span class="lab"><font color="Green" title="E7">A3</font></span>: 
<font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">g</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="c4">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> <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 href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</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">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="c4">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> <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 href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</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="c5">Y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13"><span class="lab"><font color="Green" title="E5">AS1</font></span></a></span>;<br/><a NAME="E3:13_1"/>
<font color="Maroon" title="c8">h1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">f1</font> <a href="hahnban1.html#K3" title="HAHNBAN1:func.3">+</a> <font color="Maroon" title="c7">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="txt" href="#E2:13"><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="E4:13_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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x1</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">g1</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#D3" onmouseover="rs('hahnban1/D3')" onmouseout="rh()">HAHNBAN1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E5:13_1"/>
<font color="Maroon" title="c4">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> <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 href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</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><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="13_2"><span class="kw">now </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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> holds  <font color="Maroon" title="c4">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> <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 href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</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="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">g</font> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:13_2"/>
 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="c4">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> <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 href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</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="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">g</font></span><br/><a NAME="E2:13_2"/><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="c5">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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</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">g1</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="#E2:13"><span class="lab"><font color="Green" title="E5">AS1</font></span></a></span>;<br/><a NAME="E3:13_2"/><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="c6">f1</font> <a href="hahnban1.html#K3" title="HAHNBAN1:func.3">+</a> <font color="Maroon" title="c7">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hahnban1.html#D3" onmouseover="rs('hahnban1/D3')" onmouseout="rh()">HAHNBAN1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:13_2"/>
<font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13"><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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:13"/>
( <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">g</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="c4">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> <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 href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="dualsp01.html#K6" title="DUALSP01:func.6">.</a> <font color="Olive" title="b1">x</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:13"><span class="lab"><font color="Green" title="E6">A2</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/>


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