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<span class="kw">let </span><font color="Maroon" title="c5">F1</font>, <font color="Maroon" title="c6">F2</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a>  the <a href="memstr_0.html#U1" title="MEMSTR_0:sel.1">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c1">s</font> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c5">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">I</font>)</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="extpro_1.html#K8" title="EXTPRO_1:func.8">LifeSpan</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">I</font>)</span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c2">I</font></span>)</span>,<span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span>) ) &amp; <font color="Maroon" title="c6">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c1">s</font> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c6">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">I</font>)</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="extpro_1.html#K8" title="EXTPRO_1:func.8">LifeSpan</a> (<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">I</font>)</span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c2">I</font></span>)</span>,<span class="p3">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3</span>)</span>) ) implies <font color="Maroon" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:17_1_2"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<font color="Maroon" title="c5">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c1">s</font>
 <span class="kw">and </span><br/><a NAME="E2:17_1_2"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c5">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c5">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font>)
 <span class="kw">and </span><br/><a NAME="E3:17_1_2"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
<font color="Maroon" title="c6">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c1">s</font>
 <span class="kw">and </span><br/><a NAME="E4:17_1_2"/><span class="lab"><font color="Green" title="E12">A6</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c6">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c6">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">s</font> = <font color="Maroon" title="c1">s</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a>  the <a href="memstr_0.html#U1" title="MEMSTR_0:sel.1">Object-Kind</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#T107" title="CARD_3:th.107">CARD_3:107</a></span>;<br/>
<a NAME="E6:17_1_2"/><span class="lab"><font color="Green" title="E13">A7</font></span>: 
<font color="Maroon" title="c5">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c7">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1_2"><span class="lab"><font color="Green" title="E9">A3</font></span></a></span>;<br/>
<a NAME="E7:17_1_2"/><span class="lab"><font color="Green" title="E14">A8</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c5">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c5">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1_2"><span class="lab"><font color="Green" title="E10">A4</font></span></a></span>;<br/>
<a NAME="E8:17_1_2"/><span class="lab"><font color="Green" title="E15">A9</font></span>: 
<font color="Maroon" title="c6">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c7">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1_2"><span class="lab"><font color="Green" title="E11">A5</font></span></a></span>;<br/>
<a NAME="E9:17_1_2"/><span class="lab"><font color="Green" title="E16">A10</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c6">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c6">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1_2"><span class="lab"><font color="Green" title="E12">A6</font></span></a></span>;<br/>
<a NAME="E10:17_1_2"/>
<font color="Maroon" title="c5">F1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c6">F2</font>
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S16" title="NAT_1:sch.16">NAT_1:sch 16</a>(<a class="txt" href="#E6:17_1_2"><span class="lab"><font color="Green" title="E13">A7</font></span></a>, <a class="txt" href="#E7:17_1_2"><span class="lab"><font color="Green" title="E14">A8</font></span></a>, <a class="txt" href="#E8:17_1_2"><span class="lab"><font color="Green" title="E15">A9</font></span></a>, <a class="txt" href="#E9:17_1_2"><span class="lab"><font color="Green" title="E16">A10</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E11:17_1_2"/>
<font color="Maroon" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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