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<b>consider </b><font color="Maroon">p0</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E2:7_1_1"/><i><font color="Green">E42</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E3:7_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">p0</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="finseq_4.html#T73">FINSEQ_4:73</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p0</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <b>by </b><i>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">ll</font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="bhsp_5.html#K3" target="_self">PO</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">p</font>,<font color="Maroon">x</font>;<br/>
<a NAME="E5:7_1_1"/>
 ex <font color="Olive">q0</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">q0</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">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="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> holds <br/><font color="Olive">q0</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>) ) )
 
<b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<b>then consider </b><font color="Maroon">q0</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E7:7_1_1"/><i><font color="Green">E44</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q0</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">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="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> holds <br/><font color="Maroon">q0</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="bhsp_5.html#K3" target="_self">PO</a> <font color="Olive">i</font>,<font color="Maroon">p</font>,<font color="Maroon">x</font> ) )
 ;<br/>
<a NAME="E8:7_1_1"/><i><font color="Green">E45</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">q0</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:7_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">q0</font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><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="E1:7_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">q0</font>
 ;<br/><b>then </b><i><font color="Green">E48</font></i>: <font color="Maroon">q0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
 <a href="bhsp_5.html#K3" target="_self">PO</a> <font color="Maroon">i</font>,<font color="Maroon">p</font>,<font color="Maroon">x</font>
<b>by </b><i>, </i>
<br/>.= 
the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span>

;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> <b>by </b><i>, , <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/><a NAME="E4:7_1_1_1"/>
the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">X</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="bhsp_1.html#D1">BHSP_1:def 1</a></i>;<br/><b>hence </b><a NAME="E5:7_1_1_1"/>
<font color="Maroon">q0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q0</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T14">FINSEQ_2:14</a></i>;<br/>
<b>take </b>
<a href="binop_2.html#K33">addreal</a>  <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">q</font>
;<br/>

<b>thus </b><a NAME="E12:7_1_1"/>
 ex <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">p</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp;  ex <font color="Olive">q</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>(  <a href="finsop_1.html#K5">dom</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Olive">p</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Olive">q</font> holds <br/><font color="Olive">q</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="bhsp_5.html#K3" target="_self">PO</a> <font color="Olive">i</font>,<font color="Olive">p</font>,<font color="Maroon">x</font> ) &amp; <a href="binop_2.html#K33">addreal</a>  <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K33">addreal</a>  <a href="finsop_1.html#K2">"**"</a> <font color="Olive">q</font> ) )
 <b>by </b><i>, , , , </i>;<br/>


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