<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">s</font> be   <a href="scmfsa_1.html#NM1">SCM+FSA-State</a>;<br/><b>let </b><font color="Maroon">s'</font> be   <a href="ami_2.html#NM1">SCM-State</a>;<br/>



<a NAME="E1:26"/>
 <a href="relat_1.html#K2">rng</a> <a href="scmfsa_1.html#K6" target="_self">SCM+FSA-OK</a>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="numbers.html#K4">INT</a> ,<span class="p2">(<span class="default"><a href="numbers.html#K4">INT</a>  <a href="finseq_1.html#K13">*</a> </span>)</span></span><a href="tarski.html#K2">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="scmfsa_1.html#K4" target="_self">SCM+FSA-Instr</a> ,<a href="scmfsa_1.html#K3" target="_self">SCM+FSA-Instr-Loc</a> </span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E2:26"/><b>then </b><i><font color="Green">E34</font></i>: 
not  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <a href="scmfsa_1.html#K6" target="_self">SCM+FSA-OK</a> 
 
<b>by </b><i><a class="ref" href="setfam_1.html#D9">SETFAM_1:def 9</a></i>;<br/>
<a NAME="E3:26"/><i><font color="Green">E35</font></i>: 
 <a href="relat_1.html#K1">dom</a> <a href="scmfsa_1.html#K6" target="_self">SCM+FSA-OK</a>  <a href="hidden.html#R1">=</a>  <a href="numbers.html#K4">INT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> =  <a href="scmfsa_1.html#K6" target="_self">SCM+FSA-OK</a>  as  <a href="relat_1.html#V2">non-empty</a>  <a href="pboole.html#M1">ManySortedSet</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a>, <a class="ref" href="relat_1.html#D9">RELAT_1:def 9</a></i>;<br/>
<i><font color="Green">E36</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">s'</font> = 
 <a href="relat_1.html#K1">dom</a> <a href="ami_2.html#K5">SCM-OK</a> 
<b>by </b><i><a class="ref" href="card_3.html#T18">CARD_3:18</a></i>
<br/>.= 
 <a href="numbers.html#K5">NAT</a> 
<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="26_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="E1:26_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">s'</font>
 ;<br/><a NAME="E2:26_2"/><b>then </b><i><font color="Green">E38</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="ami_3.html#K1">SCM</a> </span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <a href="ami_2.html#K2">SCM-Data-Loc</a>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ami_2.html#K3">SCM-Instr-Loc</a>  )
 <b>by </b><i>, <a class="ref" href="ami_3.html#D1">AMI_3:def 1</a>, <a class="ref" href="ami_5.html#T23">AMI_5:23</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="ami_3.html#K1">SCM</a> </span>)</span></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ami_2.html#K2">SCM-Data-Loc</a>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ami_2.html#K3">SCM-Instr-Loc</a>  )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1_1"><b>suppose </b></a><a NAME="E1:26_2_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2">IC</a> <a href="ami_3.html#K1">SCM</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:26_2_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="ami_1.html#K2">IC</a> <a href="ami_3.html#K1">SCM</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, </i>;<br/><a NAME="E4:26_2_1_1"/>
 <a href="relat_1.html#K1">dom</a> <a href="ami_2.html#K5">SCM-OK</a>  <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>then </b><i><font color="Green">E44</font></i>:  <a href="card_3.html#K5">pi</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <a href="ami_2.html#K5">SCM-OK</a> </span>)</span>,<font color="Maroon">a</font> = 
<a href="ami_2.html#K5">SCM-OK</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="card_3.html#T22">CARD_3:22</a></i>
<br/>.= 
 <a href="ami_2.html#K3">SCM-Instr-Loc</a> 
<b>by </b><i>, <a class="ref" href="ami_2.html#T7">AMI_2:7</a>, <a class="ref" href="ami_3.html#T4">AMI_3:4</a></i>
;<br/><a NAME="E6:26_2_1_1"/>
<font color="Maroon">s'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5">pi</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <a href="ami_2.html#K5">SCM-OK</a> </span>)</span>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/><b>hence </b><a NAME="E7:26_2_1_1"/>
<font color="Maroon">s'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , <a class="ref" href="ami_3.html#T4">AMI_3:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1_2"><b>suppose </b></a><a NAME="E1:26_2_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ami_2.html#K2">SCM-Data-Loc</a> 
 ;<br/><div class="add"><a NAME="E2:26_2_1_2"/><b>then </b><i><font color="Green">E46</font></i>: 
<a href="scmfsa_1.html#K6" target="_self">SCM+FSA-OK</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K4">INT</a> 
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, </i>;<br/><a NAME="E4:26_2_1_2"/>
 <a href="relat_1.html#K1">dom</a> <a href="ami_2.html#K5">SCM-OK</a>  <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:26_2_1_2"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="card_3.html#K5">pi</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <a href="ami_2.html#K5">SCM-OK</a> </span>)</span>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <a href="ami_2.html#K5">SCM-OK</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="card_3.html#T22">CARD_3:22</a></i>;<br/><a NAME="E6:26_2_1_2"/>
<font color="Maroon">s'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5">pi</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <a href="ami_2.html#K5">SCM-OK</a> </span>)</span>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/><b>hence </b><a NAME="E7:26_2_1_2"/>
<font color="Maroon">s'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , <a class="ref" href="ami_2.html#T10">AMI_2:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1_3"><b>suppose </b></a><a NAME="E1:26_2_1_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ami_2.html#K3">SCM-Instr-Loc</a> 
 ;<br/><div class="add"><a NAME="E2:26_2_1_3"/><b>then </b><i><font color="Green">E49</font></i>: 
<a href="ami_2.html#K5">SCM-OK</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="ami_2.html#K4">SCM-Instr</a> 
 <b>by </b><i><a class="ref" href="ami_2.html#T11">AMI_2:11</a></i>;<br/><a NAME="E3:26_2_1_3"/><i><font color="Green">E50</font></i>: 
<a href="scmfsa_1.html#K6" target="_self">SCM+FSA-OK</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_1.html#K4" target="_self">SCM+FSA-Instr</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E4:26_2_1_3"/>
 <a href="scmfsa_1.html#K4" target="_self">SCM+FSA-Instr</a>  <a href="hidden.html#R1">=</a> <a href="ami_2.html#K4">SCM-Instr</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">J</font>,<span class="p4"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Olive">c</font>,<font color="Olive">f2</font>,<font color="Olive">b</font></span><a href="finseq_4.html#K3">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">J</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> 13, <font color="Olive">c</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="scmfsa_1.html#K1" target="_self">SCM+FSA-Data-Loc</a> , <font color="Olive">b</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="scmfsa_1.html#K1" target="_self">SCM+FSA-Data-Loc</a> , <font color="Olive">f2</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="scmfsa_1.html#K2" target="_self">SCM+FSA-Data*-Loc</a>  : <font color="Olive">J</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">9,10</span><a href="tarski.html#K2">}</a></span> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p2">{<span class="default"> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">K</font>,<span class="p4"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Olive">c1</font>,<font color="Olive">f1</font></span><a href="finseq_4.html#K2">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">K</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> 13, <font color="Olive">c1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="scmfsa_1.html#K1" target="_self">SCM+FSA-Data-Loc</a> , <font color="Olive">f1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="scmfsa_1.html#K2" target="_self">SCM+FSA-Data*-Loc</a>  : <font color="Olive">K</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">11,12</span><a href="tarski.html#K2">}</a></span> </span>}</span> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/><a NAME="E5:26_2_1_3"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="ami_2.html#K4">SCM-Instr</a>  <a href="tarski.html#R1">c=</a>  <a href="scmfsa_1.html#K4" target="_self">SCM+FSA-Instr</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, </i>;<br/><a NAME="E7:26_2_1_3"/>
 <a href="relat_1.html#K1">dom</a> <a href="ami_2.html#K5">SCM-OK</a>  <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E8:26_2_1_3"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="card_3.html#K5">pi</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <a href="ami_2.html#K5">SCM-OK</a> </span>)</span>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <a href="ami_2.html#K5">SCM-OK</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="card_3.html#T22">CARD_3:22</a></i>;<br/><a NAME="E9:26_2_1_3"/>
<font color="Maroon">s'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5">pi</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <a href="ami_2.html#K5">SCM-OK</a> </span>)</span>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/><b>hence </b><a NAME="E10:26_2_1_3"/>
<font color="Maroon">s'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E7:26"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">s</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">s'</font> is   <a href="scmfsa_1.html#NM1">SCM+FSA-State</a>
 
<b>by </b><i>, , <a class="ref" href="numbers.html#T17">NUMBERS:17</a>, <a class="ref" href="pre_circ.html#T9">PRE_CIRC:9</a></i>;<br/>
<a NAME="E8:26"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <a href="scmfsa_1.html#K6" target="_self">SCM+FSA-OK</a> 
 
;<br/>
<a NAME="E9:26"/>
 <a href="card_3.html#K4">product</a> <a href="scmfsa_1.html#K6" target="_self">SCM+FSA-OK</a>  <a href="tarski.html#R1">c=</a>  <a href="ami_1.html#K7">sproduct</a> <a href="scmfsa_1.html#K6" target="_self">SCM+FSA-OK</a> 
 
<b>by </b><i><a class="ref" href="ami_1.html#T27">AMI_1:27</a></i>;<br/>
<a NAME="E10:26"/><b>then </b>
<font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <a href="scmfsa_1.html#K3" target="_self">SCM+FSA-Instr-Loc</a>  <a href="hidden.html#R2">in</a>  <a href="ami_1.html#K7">sproduct</a> <a href="scmfsa_1.html#K6" target="_self">SCM+FSA-OK</a> 
 
<b>by </b><i>, <a class="ref" href="ami_1.html#T41">AMI_1:41</a></i>;<br/>
<b>hence </b><a NAME="E11:26"/>
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">s'</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> <a href="scmfsa_1.html#K3" target="_self">SCM+FSA-Instr-Loc</a> </span>)</span> is   <a href="scmfsa_1.html#NM1">SCM+FSA-State</a>
 <b>by </b><i>, <a class="ref" href="ami_1.html#T29">AMI_1:29</a></i>;<br/>


</div>
