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<div class="add">

<b>set </b><font color="Maroon">c<sub>1</sub></font> =  <a href="complfld.html#K1">F_Complex</a> ;<br/>
<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_1.html#L4">VectSpStr</a> of  <a href="complfld.html#K1">F_Complex</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="bilinear.html#NM1">Form</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>



<b>thus </b><a NAME="E1:108"/>
 <a href="bilinear.html#K12">leftker</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="bilinear.html#K12">leftker</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:108_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12">leftker</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:108_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>3</sub></font> holds <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  ) )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><a NAME="E1:108_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="complfld.html#K2">*'</a>  <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 <b>by </b><i><a class="txt" href="hermitan.html#E3:108_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="complfld.html#T83">COMPLFLD:83</a></i>;<br/><b>hence </b><a NAME="E2:108_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 <b>by </b><i><a class="ref" href="hermitan.html#D8" target="_self">Def8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:108_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12">leftker</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span>
 <b>by </b><i><a class="txt" href="hermitan.html#E3:108_1"><i><font color="Green">E55</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:108"/>
 <a href="bilinear.html#K12">leftker</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="bilinear.html#K12">leftker</a> <font color="Maroon">c<sub>4</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:108_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12">leftker</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:108_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>3</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  ) )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><a NAME="E1:108_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="complfld.html#K2">*'</a>  <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 <b>by </b><i><a class="txt" href="hermitan.html#E3:108_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="complfld.html#T83">COMPLFLD:83</a></i>;<br/><a NAME="E2:108_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="complfld.html#K2">*'</a>  <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 <b>by </b><i><a class="ref" href="hermitan.html#D8" target="_self">Def8</a></i>;<br/><b>hence </b><a NAME="E3:108_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:108_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12">leftker</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="hermitan.html#E3:108_2"><i><font color="Green">E55</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:108"/>
 <a href="bilinear.html#K13">rightker</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="bilinear.html#K13">rightker</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:108_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K13">rightker</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E3:108_3"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  ) )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:108_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="complfld.html#K2">*'</a>  <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 <b>by </b><i><a class="txt" href="hermitan.html#E3:108_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="complfld.html#T83">COMPLFLD:83</a></i>;<br/><b>hence </b><a NAME="E2:108_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 <b>by </b><i><a class="ref" href="hermitan.html#D8" target="_self">Def8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:108_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K13">rightker</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span>
 <b>by </b><i><a class="txt" href="hermitan.html#E3:108_3"><i><font color="Green">E55</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:108"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K13">rightker</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E6:108"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  ) )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:108_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="complfld.html#K2">*'</a>  <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 <b>by </b><i><a class="txt" href="hermitan.html#E6:108"><i><font color="Green">E55</font></i></a>, <a class="ref" href="complfld.html#T83">COMPLFLD:83</a></i>;<br/><a NAME="E2:108_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="complfld.html#K2">*'</a>  <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 <b>by </b><i><a class="ref" href="hermitan.html#D8" target="_self">Def8</a></i>;<br/><b>hence </b><a NAME="E3:108_4"/>
<font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:108"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K13">rightker</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="hermitan.html#E6:108"><i><font color="Green">E55</font></i></a></i>;<br/>


</div>
