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

<span class="kw">let </span><font color="Maroon" title="c1">s</font> be   <a href="scmfsa_1.html#NM1" title="SCMFSA_1:NM.1">SCM+FSA-State</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">s9</font> being  <a href="ami_2.html#NM1" title="AMI_2:NM.1">SCM-State</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Olive" title="b1">s9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="card_3.html#K11" title="CARD_3:func.11">|</a> <a href="scmfsa_1.html#K5" title="SCMFSA_1:func.5">NAT</a> </span>)</span> is   <a href="scmfsa_1.html#NM1" title="SCMFSA_1:NM.1">SCM+FSA-State</a></span><br/><span class="kw">let </span><font color="Maroon" title="c2">s9</font> be   <a href="ami_2.html#NM1" title="AMI_2:NM.1">SCM-State</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c2">s9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="card_3.html#K11" title="CARD_3:func.11">|</a> <a href="scmfsa_1.html#K5" title="SCMFSA_1:func.5">NAT</a> </span>)</span> is   <a href="scmfsa_1.html#NM1" title="SCMFSA_1:NM.1">SCM+FSA-State</a></span><br/>



<a NAME="E1:35"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="scmfsa_1.html#K7" title="SCMFSA_1:func.7">SCM+FSA-OK</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_1.html#K2" title="SCMFSA_1:func.2">SCM+FSA-Memory</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">f</font> =  <a href="scmfsa_1.html#K7" title="SCMFSA_1:func.7">SCM+FSA-OK</a>  as   <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  <a href="scmfsa_1.html#K2" title="SCMFSA_1:func.2">SCM+FSA-Memory</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<span class="lab"><font color="Green" title="E19">A2</font></span>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">s9</font> = 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="ami_2.html#K7" title="AMI_2:func.7">SCM-OK</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></span>
<br/>.= 
 <a href="ami_2.html#K3" title="AMI_2:func.3">SCM-Memory</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">s9</font> implies <font color="Maroon" title="c2">s9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:35_2"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">s9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">s9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><a NAME="E2:35_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A4</font></span>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="scmfsa_1.html#K5" title="SCMFSA_1:func.5">NAT</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a>  or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="scmfsa_1.html#K5" title="SCMFSA_1:func.5">NAT</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:35_2_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="scmfsa_1.html#K5" title="SCMFSA_1:func.5">NAT</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a>  or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="scmfsa_1.html#K5" title="SCMFSA_1:func.5">NAT</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_2"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_1"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="scmfsa_1.html#K5" title="SCMFSA_1:func.5">NAT</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">s9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c4">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K3" title="AMI_2:func.3">SCM-Memory</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E1:35_2"><span class="lab"><font color="Green" title="E20">A3</font></span></a></span>;<br/><a NAME="E3:35_2_1_1"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
<font color="Maroon" title="c2">s9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <a href="ami_2.html#K7" title="AMI_2:func.7">SCM-OK</a> </span>)</span>,<font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D6" title="CARD_3:def.6">CARD_3:def 6</a></span>;<br/><a NAME="E4:35_2_1_1"/><span class="lab"><font color="Green" title="E24">A7</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_1.html#K5" title="SCMFSA_1:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_2_1_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E5:35_2_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="ami_2.html#K7" title="AMI_2:func.7">SCM-OK</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_2.html#K3" title="AMI_2:func.3">SCM-Memory</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><span class="kw">then </span> <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <a href="ami_2.html#K7" title="AMI_2:func.7">SCM-OK</a> </span>)</span>,<font color="Maroon" title="c5">a</font> = 
<a href="ami_2.html#K7" title="AMI_2:func.7">SCM-OK</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#T22" title="CARD_3:th.22">CARD_3:22</a></span>
<br/>.= 
 <a href="scmfsa_1.html#K5" title="SCMFSA_1:func.5">NAT</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_2_1_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="ref" href="ami_2.html#T7" title="AMI_2:th.7">AMI_2:7</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:35_2_1_1"/>
<font color="Maroon" title="c2">s9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_2_1_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E3:35_2_1_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="scmfsa_1.html#T9" target="_self" title="SCMFSA_1:th.9">Th9</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_2"/><span class="lab"><font color="Green" title="E22">A8</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">s9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c4">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K3" title="AMI_2:func.3">SCM-Memory</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E1:35_2"><span class="lab"><font color="Green" title="E20">A3</font></span></a></span>;<br/><a NAME="E3:35_2_1_2"/><span class="lab"><font color="Green" title="E23">A9</font></span>: 
<font color="Maroon" title="c2">s9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <a href="ami_2.html#K7" title="AMI_2:func.7">SCM-OK</a> </span>)</span>,<font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D6" title="CARD_3:def.6">CARD_3:def 6</a></span>;<br/><a NAME="E4:35_2_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="ami_2.html#K7" title="AMI_2:func.7">SCM-OK</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_2.html#K3" title="AMI_2:func.3">SCM-Memory</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E5:35_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A10</font></span>: 
 <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <a href="ami_2.html#K7" title="AMI_2:func.7">SCM-OK</a> </span>)</span>,<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="ami_2.html#K7" title="AMI_2:func.7">SCM-OK</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#T22" title="CARD_3:th.22">CARD_3:22</a></span>;<br/><a NAME="E6:35_2_1_2"/>
<a href="scmfsa_1.html#K7" title="SCMFSA_1:func.7">SCM+FSA-OK</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_2_1_2"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="ref" href="scmfsa_1.html#T10" target="_self" title="SCMFSA_1:th.10">Th10</a></span>;<br/><span class="kw">hence </span><a NAME="E7:35_2_1_2"/>
<font color="Maroon" title="c2">s9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_2_1_2"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E5:35_2_1_2"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="txt" href="#E3:35_2_1_2"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="ref" href="ami_2.html#T10" title="AMI_2:th.10">AMI_2:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:35_2_1_3"/><span class="lab"><font color="Green" title="E22">A11</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="scmfsa_1.html#K5" title="SCMFSA_1:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">s9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:35_2_1_3"/>
 <a href="scmfsa_1.html#K6" title="SCMFSA_1:func.6">SCM+FSA-Instr</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="ami_2.html#K6" title="AMI_2:func.6">SCM-Instr</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">{<span class="default"> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">J</font>,<span class="p4"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Olive" title="b2">c</font>,<font color="Olive" title="b4">f2</font>,<font color="Olive" title="b3">b</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">J</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> 13, <font color="Olive" title="b2">c</font>, <font color="Olive" title="b3">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="scmfsa_1.html#K3" title="SCMFSA_1:func.3">SCM+FSA-Data-Loc</a> , <font color="Olive" title="b4">f2</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="scmfsa_1.html#K4" title="SCMFSA_1:func.4">SCM+FSA-Data*-Loc</a>  : <font color="Olive" title="b1">J</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">9,10</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">{<span class="default"> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">K</font>,<span class="p4"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Olive" title="b2">c1</font>,<font color="Olive" title="b3">f1</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">K</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> 13, <font color="Olive" title="b2">c1</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="scmfsa_1.html#K3" title="SCMFSA_1:func.3">SCM+FSA-Data-Loc</a> , <font color="Olive" title="b3">f1</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="scmfsa_1.html#K4" title="SCMFSA_1:func.4">SCM+FSA-Data*-Loc</a>  : <font color="Olive" title="b1">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">11,12</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> </span>}</span> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></span>;<br/><a NAME="E3:35_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A12</font></span>: 
 <a href="ami_2.html#K6" title="AMI_2:func.6">SCM-Instr</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="scmfsa_1.html#K6" title="SCMFSA_1:func.6">SCM+FSA-Instr</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c4">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K3" title="AMI_2:func.3">SCM-Memory</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E1:35_2"><span class="lab"><font color="Green" title="E20">A3</font></span></a></span>;<br/><a NAME="E5:35_2_1_3"/><span class="lab"><font color="Green" title="E24">A13</font></span>: 
<font color="Maroon" title="c2">s9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <a href="ami_2.html#K7" title="AMI_2:func.7">SCM-OK</a> </span>)</span>,<font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D6" title="CARD_3:def.6">CARD_3:def 6</a></span>;<br/><a NAME="E6:35_2_1_3"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="ami_2.html#K7" title="AMI_2:func.7">SCM-OK</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_2.html#K3" title="AMI_2:func.3">SCM-Memory</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E7:35_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A14</font></span>: 
 <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <a href="ami_2.html#K7" title="AMI_2:func.7">SCM-OK</a> </span>)</span>,<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="ami_2.html#K7" title="AMI_2:func.7">SCM-OK</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#T22" title="CARD_3:th.22">CARD_3:22</a></span>;<br/><a NAME="E8:35_2_1_3"/>
( <a href="ami_2.html#K7" title="AMI_2:func.7">SCM-OK</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_2.html#K6" title="AMI_2:func.6">SCM-Instr</a>  &amp; <a href="scmfsa_1.html#K7" title="SCMFSA_1:func.7">SCM+FSA-OK</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_1.html#K6" title="SCMFSA_1:func.6">SCM+FSA-Instr</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_2_1_3"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="ref" href="scmfsa_1.html#T11" target="_self" title="SCMFSA_1:th.11">Th11</a>, <a class="ref" href="ami_2.html#T11" title="AMI_2:th.11">AMI_2:11</a></span>;<br/><span class="kw">hence </span><a NAME="E9:35_2_1_3"/>
<font color="Maroon" title="c2">s9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_2_1_3"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="txt" href="#E7:35_2_1_3"><span class="lab"><font color="Green" title="E25">A14</font></span></a>, <a class="txt" href="#E5:35_2_1_3"><span class="lab"><font color="Green" title="E24">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E5:35"/><span class="kw">then </span>
<font color="Maroon" title="c1">s</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c2">s9</font> is   <a href="scmfsa_1.html#NM1" title="SCMFSA_1:NM.1">SCM+FSA-State</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="pre_circ.html#T9" title="PRE_CIRC:th.9">PRE_CIRC:9</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:35"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c2">s9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">s</font> <a href="card_3.html#K11" title="CARD_3:func.11">|</a> <a href="scmfsa_1.html#K5" title="SCMFSA_1:func.5">NAT</a> </span>)</span> is   <a href="scmfsa_1.html#NM1" title="SCMFSA_1:NM.1">SCM+FSA-State</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#T69" title="CARD_3:th.69">CARD_3:69</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
