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<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,  <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <font color="Maroon" title="c1">T</font>) <span class="kw">-&gt; </span>  <a href="turing_1.html#NM4" title="TURING_1:NM.4">All-State</a> of <font color="Maroon" title="c1">T</font> =  <a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> $2;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">F1</font>, <font color="Maroon" title="c4">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"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default">the <a href="turing_1.html#U2" title="TURING_1:sel.2">FStates</a> of <font color="Maroon" title="c1">T</font>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ,<span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ,the <a href="turing_1.html#U1" title="TURING_1:sel.1">Symbols</a> of <font color="Maroon" title="c1">T</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</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="c3">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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">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="c3">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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) &amp; <font color="Maroon" title="c4">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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">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="c4">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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) implies <font color="Maroon" title="c3">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:23_1_2"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
<font color="Maroon" title="c3">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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font>
 <span class="kw">and </span><br/><a NAME="E2:23_1_2"/><span class="lab"><font color="Green" title="E8">A3</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="c3">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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:23_1_2"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
<font color="Maroon" title="c4">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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font>
 <span class="kw">and </span><br/><a NAME="E4:23_1_2"/><span class="lab"><font color="Green" title="E10">A5</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="c4">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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="turing_1.html#K9" title="TURING_1:func.9">Following</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font></span>)</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="c3">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F2</font></span><br/>

<a NAME="E5:23_1_2"/><span class="lab"><font color="Green" title="E11">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="c3">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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c3">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="#E2:23_1_2"><span class="lab"><font color="Green" title="E8">A3</font></span></a></span>;<br/>
<a NAME="E6:23_1_2"/><span class="lab"><font color="Green" title="E12">A7</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="c4">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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">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="#E4:23_1_2"><span class="lab"><font color="Green" title="E10">A5</font></span></a></span>;<br/>
<a NAME="E7:23_1_2"/><span class="lab"><font color="Green" title="E13">A8</font></span>: 
<font color="Maroon" title="c4">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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_2"><span class="lab"><font color="Green" title="E9">A4</font></span></a></span>;<br/>
<a NAME="E8:23_1_2"/><span class="lab"><font color="Green" title="E14">A9</font></span>: 
<font color="Maroon" title="c3">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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_2"><span class="lab"><font color="Green" title="E7">A2</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E9:23_1_2"/>
<font color="Maroon" title="c3">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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="#E8:23_1_2"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="txt" href="#E5:23_1_2"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="txt" href="#E7:23_1_2"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="txt" href="#E6:23_1_2"><span class="lab"><font color="Green" title="E12">A7</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/>


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