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<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be  <a href="funct_1.html#V3">V7</a> non <a href="realset1.html#V1">trivial</a> <a href="hahnban1.html#NM2">linear-Functional</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:55_1_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K8">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="hahnban1.html#K7">0Functional</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>and </b><br/><a NAME="E2:55_1_2"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K8">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="hahnban1.html#K7">0Functional</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E3:55_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="vectsp_5.html#R1">is_the_direct_sum_of</a> <font color="Maroon">c<sub>4</sub></font>, <a href="vectsp_7.html#K1">Lin</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="vectsp_5.html#D5">VECTSP_5:def 5</a></i>;<br/>
<a NAME="E4:55_1_2"/><b>then </b><i><font color="Green">E31</font></i>: 
(  <a href="vectsp_1.html#G4">VectSpStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="vectsp_1.html#U2">lmult</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="vectsp_5.html#K1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="vectsp_5.html#K2">/\</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp_4.html#K1">(0).</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="vectsp_5.html#D4">VECTSP_5:def 4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_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>2</sub></font>;<br/><a NAME="E1:55_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="vectsp_5.html#K1">+</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="vectsp10.html#E4:55_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</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="E3:55_1_2_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="vectsp_5.html#T5">VECTSP_5:5</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:55_1_2_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="vectsp10.html#E3:55_1_2_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="vectsp10.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E6:55_1_2_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="vectsp10.html#E3:55_1_2_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><b>then </b><i><font color="Green">E35</font></i>: <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K8">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="vectsp10.html#E1:55_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="vectsp10.html#E2:55_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="vectsp10.html#E6:55_1_2_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><b>thus </b><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="vectsp10.html#E3:55_1_2_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="vectsp10.html#E5:55_1_2_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="hahnban1.html#D11">HAHNBAN1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D12">HAHNBAN1:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="vectsp10.html#E1:55_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="vectsp10.html#E2:55_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="hahnban1.html#D12">HAHNBAN1:def 12</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="vectsp10.html#E3:55_1_2_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="vectsp10.html#E5:55_1_2_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="vectsp10.html#E7:55_1_2_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="hahnban1.html#D11">HAHNBAN1:def 11</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:55_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


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