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<span class="kw">set </span><font color="Maroon" title="c5">L</font> =  <a href="bilinear.html#K14" title="BILINEAR:func.14">RKer</a> <font color="Maroon" title="c4">f</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">vq</font> =  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c3">W</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K14" title="BILINEAR:func.14">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span>);<br/>
<span class="kw">let </span><font color="Maroon" title="c7">f1</font>, <font color="Maroon" title="c8">f2</font> be   <a href="bilinear.html#V1" title="BILINEAR:attr.1">additiveFAF</a>   <a href="bilinear.html#V3" title="BILINEAR:attr.3">homogeneousFAF</a>  <a href="bilinear.html#NM1" title="BILINEAR:NM.1">Form</a> of <font color="Maroon" title="c2">V</font>,<span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K14" title="BILINEAR:func.14">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span>)</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="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K14" title="BILINEAR:func.14">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span>)</span>)</span><br/>  for <font color="Olive" title="b2">v</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V</font><br/>  for <font color="Olive" title="b3">w</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c3">W</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">w</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K14" title="BILINEAR:func.14">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span> holds <br/><font color="Maroon" title="c7">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b2">v</font>,<font color="Olive" title="b1">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">w</font>) ) &amp; (  for <font color="Olive" title="b1">A</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K14" title="BILINEAR:func.14">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span>)</span>)</span><br/>  for <font color="Olive" title="b2">v</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V</font><br/>  for <font color="Olive" title="b3">w</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c3">W</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">w</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K14" title="BILINEAR:func.14">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span> holds <br/><font color="Maroon" title="c8">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b2">v</font>,<font color="Olive" title="b1">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">w</font>) ) implies <font color="Maroon" title="c7">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:105_1_2"/><span class="lab"><font color="Green" title="E42">A13</font></span>: 
 for <font color="Olive" title="b1">A</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K14" title="BILINEAR:func.14">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span>)</span>)</span><br/>  for <font color="Olive" title="b2">v</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V</font><br/>  for <font color="Olive" title="b3">a</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c3">W</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K14" title="BILINEAR:func.14">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span> holds <br/><font color="Maroon" title="c7">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b2">v</font>,<font color="Olive" title="b1">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">a</font>)
 <span class="kw">and </span><br/><a NAME="E2:105_1_2"/><span class="lab"><font color="Green" title="E43">A14</font></span>: 
 for <font color="Olive" title="b1">A</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K14" title="BILINEAR:func.14">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span>)</span>)</span><br/>  for <font color="Olive" title="b2">v</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V</font><br/>  for <font color="Olive" title="b3">a</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c3">W</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K14" title="BILINEAR:func.14">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span> holds <br/><font color="Maroon" title="c8">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b2">v</font>,<font color="Olive" title="b1">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">a</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="c7">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f2</font></span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="105_1_2_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">  for <font color="Olive" title="b1">v</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">V</font><br/>  for <font color="Olive" title="b2">A</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K14" title="BILINEAR:func.14">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span>)</span>)</span> holds  <font color="Maroon" title="c7">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">A</font>)</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">v</font> be   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c2">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">  for <font color="Olive" title="b1">A</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K14" title="BILINEAR:func.14">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span>)</span>)</span> holds  <font color="Maroon" title="c7">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c9">v</font>,<font color="Olive" title="b1">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c9">v</font>,<font color="Olive" title="b1">A</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c10">A</font> be   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Maroon" title="c3">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K14" title="BILINEAR:func.14">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span>)</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="c7">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">A</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">A</font>)</span><br/><span class="kw">consider </span><font color="Maroon" title="c11">a</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Maroon" title="c3">W</font><span class="kw"> such that </span><br/><a NAME="E2:105_1_2_1"/><span class="lab"><font color="Green" title="E44">A15</font></span>: 
<font color="Maroon" title="c10">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K14" title="BILINEAR:func.14">RKer</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp10.html#T22" onmouseover="rs('vectsp10/T22')" onmouseout="rh()">VECTSP10:22</a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c7">f1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">A</font>) = 
<font color="Maroon" title="c4">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c11">a</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105_1_2"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="txt" href="#E2:105_1_2_1"><span class="lab"><font color="Green" title="E44">A15</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c8">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">A</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:105_1_2"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="txt" href="#E2:105_1_2_1"><span class="lab"><font color="Green" title="E44">A15</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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:105_1_2"/>
<font color="Maroon" title="c7">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f2</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"> verum</span><br/>


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