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<b>let </b><font color="Maroon">V</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">W</font> be    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">L</font> be    <a href="rlvect_2.html#M1">Linear_Combination</a> of <font color="Maroon">V</font>;<br/>





<b>assume </b><a NAME="E1:25"/><i><font color="Green">E24</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>
 ;<br/>

<b>let </b><font color="Maroon">K</font> be    <a href="rlvect_2.html#M1">Linear_Combination</a> of <font color="Maroon">W</font>;<br/>

<b>assume </b><a NAME="E2:25"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>
 ;<br/>

<a NAME="E3:25"/><i><font color="Green">E29</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="rusub_1.html#D1">RUSUB_1:def 1</a></i>;<br/>
<a NAME="E4:25"/><i><font color="Green">E31</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:25_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">K</font>
 ;<br/><b>then consider </b><font color="Maroon">w</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">W</font><b> such that </b><br/><a NAME="E3:25_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font>
 <b>and </b><br/><a NAME="E4:25_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">K</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E5:25_1"/>
( <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">w</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> )
 <b>by </b><i>, , <a class="ref" href="rusub_3.html#D1" target="_self">Def1</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="ref" href="rusub_1.html#T3">RUSUB_1:3</a></i>;<br/><b>hence </b><a NAME="E6:25_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E6:25"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:25_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font>
 ;<br/><b>then consider </b><font color="Maroon">v</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:25_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E4:25_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E5:25_2"/>
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, , , <a class="ref" href="rusub_3.html#D1" target="_self">Def1</a>, , <a class="txt" href="rusub_3.html#E1:25"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E6:25_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">K</font>
 <b>by </b><i>, <a class="ref" href="rusub_3.html#D1" target="_self">Def1</a>, </i>;<br/></div><b>end;</b></div>
<a NAME="E8:25"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">K</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E9:25"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon">F</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E11:25"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">F</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E12:25"/><i><font color="Green">E69</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E13:25"/><i><font color="Green">E70</font></i>: 
 <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_2.html#D10">RLVECT_2:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">W</font><b> such that </b><br/><a NAME="E15:25"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">G</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E16:25"/><i><font color="Green">E72</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E17:25"/><i><font color="Green">E73</font></i>: 
 <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_2.html#D10">RLVECT_2:def 10</a></i>;<br/>
<a NAME="E18:25"/>
<font color="Maroon">F</font>,<font color="Maroon">G</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i>, <a class="txt" href="rusub_3.html#E1:25"><i><font color="Green">E24</font></i></a>, , , <a class="txt" href="rusub_3.html#E2:25"><i><font color="Green">E28</font></i></a>, <a class="ref" href="rfinseq.html#T39">RFINSEQ:39</a></i>;<br/>
<b>then consider </b><font color="Maroon">P</font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font><b> such that </b><br/><a NAME="E20:25"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="rfinseq.html#T17">RFINSEQ:17</a></i>;<br/>
<b>set </b><font color="Maroon">p</font> = <font color="Maroon">L</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">F</font>;<br/>
<a NAME="E21:25"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T48">FINSEQ_2:48</a></i>;<br/>
<a NAME="E22:25"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">F</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a>, <a class="ref" href="rlvect_2.html#D9">RLVECT_2:def 9</a></i>;<br/>
<a NAME="E23:25"/><b>then </b><i><font color="Green">E75</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E24:25"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#D9">RLVECT_2:def 9</a></i>;<br/>
<a NAME="E25:25"/><b>then </b><i><font color="Green">E83</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">q</font> = <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font> as   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">W</font> <b>by </b><i><a class="ref" href="finseq_2.html#T51">FINSEQ_2:51</a></i>;<br/>
<a NAME="E27:25"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T48">FINSEQ_2:48</a></i>;<br/>
<a NAME="E28:25"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">G</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a>, <a class="ref" href="rlvect_2.html#D9">RLVECT_2:def 9</a></i>;<br/>
<a NAME="E29:25"/><b>then </b><i><font color="Green">E84</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">G</font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:25_3"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">F</font></span>)</span>
 ;<br/><b>set </b><font color="Maroon">v</font> = <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>;<br/><a NAME="E2:25_3"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="rusub_3.html#E4:25"><i><font color="Green">E31</font></i></a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>set </b><font color="Maroon">j</font> = <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>;<br/><a NAME="E3:25_3"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T67">FUNCT_2:67</a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/><a NAME="E4:25_3"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="rusub_3.html#E4:25"><i><font color="Green">E31</font></i></a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:25_3"/><b>then </b>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><i><font color="Green">E88</font></i>: <font color="Maroon">G</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> = 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="txt" href="rusub_3.html#E4:25"><i><font color="Green">E31</font></i></a>, , , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
;<br/><a NAME="E8:25_3"/>
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="rusub_3.html#E4:25"><i><font color="Green">E31</font></i></a>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">w</font> = <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">W</font> <b>by </b><i>, </i>;<br/><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="rusub_3.html#E4:25"><i><font color="Green">E31</font></i></a>, , , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font>
<b>by </b><i>, , <a class="ref" href="rusub_3.html#T1" target="_self">Th1</a>, <a class="ref" href="rlvect_2.html#D9">RLVECT_2:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="rusub_1.html#T7">RUSUB_1:7</a></i>
;<br/><b>hence </b><a NAME="E11:25_3"/>
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="rlvect_2.html#D9">RLVECT_2:def 9</a></i>;<br/></div><b>end;</b></div>
<a NAME="E31:25"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">L</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="rusub_3.html#E4:25"><i><font color="Green">E31</font></i></a>, , <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>
<a NAME="E32:25"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#D9">RLVECT_2:def 9</a></i>;<br/>
<a NAME="E33:25"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">G</font></span>)</span> ;<br/>
<a NAME="E35:25"/>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">P</font>
 
;<br/>
<a NAME="E36:25"/><b>then </b><i><font color="Green">E90</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="rlvect_2.html#K5">(#)</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="rlvect_2.html#T9">RLVECT_2:9</a></i>;<br/>
<a NAME="E37:25"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E38:25"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">q'</font> = <font color="Maroon">q</font> as   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font><b> such that </b><br/><a NAME="E41:25"/><i><font color="Green">E92</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span>
 <b>and </b><br/><a NAME="E42:25"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E43:25"/><i><font color="Green">E94</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">w</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">W</font>  st <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> &amp; <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Olive">w</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D12">RLVECT_1:def 12</a></i>;<br/>
<a NAME="E44:25"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E45:25"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E47:25"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="rusub_3.html#T1" target="_self">Th1</a>, <a class="ref" href="rusub_1.html#T4">RUSUB_1:4</a></i>;<br/>
<a NAME="E48:25"/><i><font color="Green">E96</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q'</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/>



<b>assume </b><a NAME="E1:25_4"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q'</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/>

<a NAME="E2:25_4"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:25_4"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">v'</font> = <font color="Maroon">v</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">W</font> <b>by </b><i>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/>
<a NAME="E5:25_4"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v'</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:25_4"/>
<font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="rusub_1.html#T6">RUSUB_1:6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E49:25"/>
<font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q'</font></span>)</span> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>
 
;<br/>
<b>hence </b><a NAME="E50:25"/>
(  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K3">Carrier</a> <font color="Maroon">K</font> &amp;  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_2.html#K6">Sum</a> <font color="Maroon">K</font> )
 <b>by </b><i>, <a class="ref" href="rusub_3.html#D1" target="_self">Def1</a>, , <a class="txt" href="rusub_3.html#E3:25"><i><font color="Green">E29</font></i></a>, , , , <a class="ref" href="rusub_3.html#T2" target="_self">Th2</a>, , <a class="ref" href="rlvect_1.html#D12">RLVECT_1:def 12</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
