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<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="ami_1.html#U4">Instructions</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> ;<br/>

<a NAME="E1:2"/><i><font color="Green">E3</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K1">Load</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="scmfsa_7.html#D1">SCMFSA_7:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K1">Load</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:2_1"/><i><font color="Green">E4</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K1">Load</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K1">Load</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:2_1"/><i><font color="Green">E5</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K5">insloc</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="scmfsa7b.html#E1:2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="scmfsa7b.html#E3:2_1"><i><font color="Green">E4</font></i></a></i>;<br/><a NAME="E6:2_1"/><i><font color="Green">E6</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="scmfsa7b.html#E3:2_1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="scmfsa7b.html#E5:2_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="scmfsa_7.html#T26">SCMFSA_7:26</a></i>;<br/><b>then </b><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="scmfsa7b.html#E3:2_1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="scmfsa7b.html#E5:2_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="scmfsa_7.html#D1">SCMFSA_7:def 1</a></i>
;<br/><b>hence </b><a NAME="E8:2_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="scmfsa7b.html#E6:2_1"><i><font color="Green">E6</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:2"/><b>then </b><i><font color="Green">E4</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K1">Load</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T19">FUNCT_1:19</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:2_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:2_2"/><i><font color="Green">E5</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:2_2"/><i><font color="Green">E6</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="scmfsa7b.html#E3:2_2"><i><font color="Green">E5</font></i></a>, <a class="ref" href="finseq_3.html#T25">FINSEQ_3:25</a></i>;<br/><a NAME="E6:2_2"/><i><font color="Green">E7</font></i>: 
 <a href="scmfsa_2.html#K5">insloc</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K1">Load</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa7b.html#E1:2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="scmfsa7b.html#E3:2_2"><i><font color="Green">E5</font></i></a></i>;<br/><a NAME="E7:2_2"/>
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="scmfsa7b.html#E4:2_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:2_2"/><i><font color="Green">E8</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; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="scmfsa7b.html#E3:2_2"><i><font color="Green">E5</font></i></a></i>;<br/><i><font color="Green">E9</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i><a class="txt" href="scmfsa7b.html#E9:2_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>

;<br/><span class="p1">(<span class="default"><a href="scmfsa_7.html#K1">Load</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> = 
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="scmfsa7b.html#E6:2_2"><i><font color="Green">E7</font></i></a>, <a class="ref" href="scmfsa_7.html#D1">SCMFSA_7:def 1</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="scmfsa7b.html#E3:2_2"><i><font color="Green">E5</font></i></a>, <a class="txt" href="scmfsa7b.html#E10:2_2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E12:2_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K1">Load</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K1">Load</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="scmfsa7b.html#E6:2_2"><i><font color="Green">E7</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K1">Load</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T19">FUNCT_1:19</a></i>;<br/>
<b>hence </b><a NAME="E6:2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K1">Load</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="scmfsa7b.html#E3:2"><i><font color="Green">E4</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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