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<b>set </b><font color="Maroon" title="c1">S</font> =  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ;<br/>
<b>set </b><font color="Maroon" title="c2">T</font> = <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon">F<sub>3</sub></font>(),<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<b>set </b><font color="Maroon" title="c3">C</font> = <font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>();<br/>
<b>set </b><font color="Maroon" title="c4">II</font> = <font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span>;<br/>
<a NAME="E1:118_1"/><i><font color="Green" title="E69">Z0</font></i>: 
( <font color="Maroon">F<sub>6</sub></font>() is <a href="aofa_000.html#V14" title="AOFA_000:attr.14">complying_with_catenation</a> &amp; <font color="Maroon">F<sub>6</sub></font>() <a href="aofa_000.html#R3" title="AOFA_000:pred.3">complies_with_while_wrt</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon">F<sub>3</sub></font>(),<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="aofa_000.html#D32" title="AOFA_000:def.32">AOFA_000:def 32</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">s1</font> = <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon">F<sub>9</sub></font>(),<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K47" title="AOFA_I00:func.47">:=</a> <font color="Maroon">F<sub>10</sub></font>()</span>)</span> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c6">s2</font> = <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">s1</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  ;<br/>
<a NAME="E4:118_1"/><i><font color="Green" title="E70">S1</font></i>: 
( <font color="Maroon" title="c5">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>10</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">F<sub>9</sub></font>() &amp; <font color="Maroon" title="c5">s1</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() )
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E5:118"><i><font color="Green" title="E68">D</font></i></a>, <a class="ref" href="aofa_i00.html#T26" target="_self" title="AOFA_I00:th.26">Th111</a></i>;<br/>
<a NAME="E5:118_1"/><b>then </b><i><font color="Green" title="E71">S2</font></i>: 
( <font color="Maroon" title="c6">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>10</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">F<sub>9</sub></font>() &amp; <font color="Maroon" title="c6">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() )
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E5:118"><i><font color="Green" title="E68">D</font></i></a>, <a class="ref" href="aofa_i00.html#T35" target="_self" title="AOFA_I00:th.35">Th114</a></i>;<br/>
<a NAME="E6:118_1"/><i><font color="Green" title="E72">Sn</font></i>: 
<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon">F<sub>9</sub></font>(),<font color="Maroon">F<sub>5</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">s1</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E1:118"><i><font color="Green" title="E64">F</font></i></a>, <a class="ref" href="aofa_000.html#D29" title="AOFA_000:def.29">AOFA_000:def 29</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ] <b>means </b>$1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>();<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ] <b>means </b>( <font color="Maroon">P<sub>1</sub></font>[$1] &amp; ( <font color="Maroon">F<sub>10</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">F<sub>9</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() implies <span class="p1">(<span class="default">$1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>()</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() ) &amp; $1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() );<br/>
<a NAME="E7:118_1"/>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>10</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">F<sub>9</sub></font>()</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">F<sub>10</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">F<sub>9</sub></font>()
 
<b>by </b><i><a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></i>;<br/>
<a NAME="E8:118_1"/><b>then </b><i><font color="Green" title="E73">A1</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">s1</font>]
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E2:118"><i><font color="Green" title="E65">A</font></i></a>, <a class="txt" href="aofa_i00.html#E4:118_1"><i><font color="Green" title="E70">S1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:118_1"/><i><font color="Green" title="E74">A2</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">s2</font>]
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E3:118"><i><font color="Green" title="E66">B</font></i></a>, <a class="txt" href="aofa_i00.html#E2:118"><i><font color="Green" title="E65">A</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:118_1_1"/>
( <font color="Maroon">F<sub>10</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">F<sub>9</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() or <font color="Maroon">F<sub>10</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">F<sub>9</sub></font>() <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_1_1_1"><b>suppose </b></a><a NAME="E1:118_1_1_1"/><i><font color="Green" title="E75">C1</font></i>: 
<font color="Maroon">F<sub>10</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">F<sub>9</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>()
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon">F<sub>9</sub></font>(),<font color="Maroon">F<sub>5</sub></font>()] &amp; ( <font color="Maroon">F<sub>10</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">F<sub>9</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() implies <span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon">F<sub>9</sub></font>(),<font color="Maroon">F<sub>5</sub></font>()</span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 ) &amp; ( <font color="Maroon">F<sub>10</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">F<sub>9</sub></font>() <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() implies <span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon">F<sub>9</sub></font>(),<font color="Maroon">F<sub>5</sub></font>()</span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>10</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">F<sub>9</sub></font>() ) &amp; <span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon">F<sub>9</sub></font>(),<font color="Maroon">F<sub>5</sub></font>()</span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() )</span><br/><div class="add"><a NAME="E2:118_1_1_1"/><b>then </b><i><font color="Green" title="E76">B'</font></i>: 
<font color="Maroon" title="c6">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="aofa_i00.html#E4:118_1"><i><font color="Green" title="E70">S1</font></i></a>, <a class="ref" href="aofa_i00.html#T35" target="_self" title="AOFA_I00:th.35">Th114</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ] <b>means </b>( <font color="Maroon">P<sub>1</sub></font>[$1] &amp; <font color="Maroon">S<sub>1</sub></font>[$1] );<br/><a NAME="E3:118_1_1_1"/><i><font color="Green" title="E77">BB</font></i>: 
( <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">s1</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon">F<sub>3</sub></font>(),<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">s1</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span>] )
 <b>by </b><i><a class="txt" href="aofa_i00.html#E2:118"><i><font color="Green" title="E65">A</font></i></a>, <a class="txt" href="aofa_i00.html#E3:118"><i><font color="Green" title="E66">B</font></i></a>, <a class="txt" href="aofa_i00.html#E1:118_1_1_1"><i><font color="Green" title="E75">C1</font></i></a>, <a class="txt" href="aofa_i00.html#E5:118_1"><i><font color="Green" title="E71">S2</font></i></a>, <a class="txt" href="aofa_i00.html#E2:118_1_1_1"><i><font color="Green" title="E76">B'</font></i></a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  =  <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">$1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default">$1 <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>()</span>)</span></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E4:118_1_1_1"/><i><font color="Green" title="E78">Ct</font></i>: 
for <font color="Olive" title="b1">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">s</font>] holds <br/>( ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span></span>)</span>] implies <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon">F<sub>3</sub></font>(),<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon">F<sub>3</sub></font>(),<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span></span>)</span>] ) &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">s</font>) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">s</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c7">s</font>] implies ( ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span></span>)</span>] implies <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon">F<sub>3</sub></font>(),<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon">F<sub>3</sub></font>(),<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span></span>)</span>] ) &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">s</font>) ) )</span><br/>

<b>assume </b><a NAME="E1:118_1_1_1_1"/><i><font color="Green" title="E79">00</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c7">s</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span></span>)</span>] implies <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon">F<sub>3</sub></font>(),<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon">F<sub>3</sub></font>(),<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span></span>)</span>] ) &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">s</font>) )</span><br/>

<b>reconsider </b><font color="Maroon" title="c8">s'</font> = <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<font color="Maroon">F<sub>4</sub></font>() as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c9">s''</font> = <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c8">s'</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">s'''</font> = <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">s''</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  ;<br/>
<a NAME="E5:118_1_1_1_1"/><i><font color="Green" title="E80">01</font></i>: 
( <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span> &amp; <font color="Maroon" title="c9">s''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="aofa_000.html#D29" title="AOFA_000:def.29">AOFA_000:def 29</a></i>;<br/>
<a NAME="E6:118_1_1_1_1"/><i><font color="Green" title="E81">02</font></i>: 
( <font color="Maroon" title="c10">s'''</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">s''</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>() &amp; <font color="Maroon" title="c10">s'''</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">s''</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() &amp; ( <font color="Maroon" title="c9">s''</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>() <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">s''</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() implies <font color="Maroon" title="c10">s'''</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon" title="c9">s''</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">s''</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() implies <font color="Maroon" title="c10">s'''</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E5:118"><i><font color="Green" title="E68">D</font></i></a>, <a class="ref" href="aofa_i00.html#T35" target="_self" title="AOFA_I00:th.35">Th114</a></i>;<br/>
<a NAME="E7:118_1_1_1_1"/><i><font color="Green" title="E82">03</font></i>: 
( <font color="Maroon" title="c8">s'</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>() &amp; <font color="Maroon" title="c8">s'</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() )
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E4:118"><i><font color="Green" title="E67">C</font></i></a>, <a class="txt" href="aofa_i00.html#E1:118_1_1_1_1"><i><font color="Green" title="E79">00</font></i></a></i>;<br/>
<a NAME="E8:118_1_1_1_1"/><i><font color="Green" title="E83">04</font></i>: 
( <font color="Maroon" title="c9">s''</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s'</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>()</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">s''</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s'</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c9">s''</font>] )
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E1:118_1_1_1_1"><i><font color="Green" title="E79">00</font></i></a>, <a class="txt" href="aofa_i00.html#E5:118_1_1_1_1"><i><font color="Green" title="E80">01</font></i></a>, <a class="txt" href="aofa_i00.html#E3:118"><i><font color="Green" title="E66">B</font></i></a>, <a class="txt" href="aofa_i00.html#E5:118"><i><font color="Green" title="E68">D</font></i></a>, <a class="ref" href="aofa_i00.html#T28" target="_self" title="AOFA_I00:th.28">Th012</a></i>;<br/>
<b>hence </b><a NAME="E9:118_1_1_1_1"/>
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span></span>)</span>] iff <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon">F<sub>3</sub></font>(),<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="aofa_i00.html#E5:118_1_1_1_1"><i><font color="Green" title="E80">01</font></i></a>, <a class="txt" href="aofa_i00.html#E6:118_1_1_1_1"><i><font color="Green" title="E81">02</font></i></a>, <a class="txt" href="aofa_i00.html#E3:118"><i><font color="Green" title="E66">B</font></i></a>, <a class="ref" href="aofa_i00.html#T2" target="_self" title="AOFA_I00:th.2">LemTS</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">s</font>)</span><br/>

<b>reconsider </b><font color="Maroon" title="c11">j</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>()</span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="aofa_i00.html#E1:118_1_1_1_1"><i><font color="Green" title="E79">00</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/>
<a NAME="E11:118_1_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">s'''</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j</font>
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E6:118_1_1_1_1"><i><font color="Green" title="E81">02</font></i></a>, <a class="txt" href="aofa_i00.html#E7:118_1_1_1_1"><i><font color="Green" title="E82">03</font></i></a>, <a class="txt" href="aofa_i00.html#E8:118_1_1_1_1"><i><font color="Green" title="E83">04</font></i></a>, <a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></i>;<br/>
<a NAME="E12:118_1_1_1_1"/><b>then </b>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">s'''</font>) <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">s</font>)
 
<b>by </b><i><a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></i>;<br/>
<b>hence </b><a NAME="E13:118_1_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span></span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">s</font>)
 <b>by </b><i><a class="txt" href="aofa_i00.html#E5:118_1_1_1_1"><i><font color="Green" title="E80">01</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E5:118_1_1_1"/><i><font color="Green" title="E79">C'</font></i>: 
<font color="Maroon">F<sub>6</sub></font>() <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span>,<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">s1</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span>
 <b>from </b><i><a class="ref" href="aofa_000.html#S3" title="AOFA_000:sch.3">AOFA_000:sch 3</a>(<a class="txt" href="aofa_i00.html#E3:118_1_1_1"><i><font color="Green" title="E77">BB</font></i></a>, <a class="txt" href="aofa_i00.html#E4:118_1_1_1"><i><font color="Green" title="E78">Ct</font></i></a>);<br/></i><a NAME="E6:118_1_1_1"/><i><font color="Green" title="E80">D'</font></i>: 
for <font color="Olive" title="b1">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s</font>] &amp; <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon">F<sub>3</sub></font>(),<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">s</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">s</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">s</font>] &amp; <font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon">F<sub>3</sub></font>(),<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">s</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span>] )</span><br/>

<b>assume </b><a NAME="E1:118_1_1_1_2"/><i><font color="Green" title="E81">01</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">s</font>] &amp; <font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon">F<sub>3</sub></font>(),<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">s</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span>]</span><br/>

<b>thus </b><a NAME="E2:118_1_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span>]
 <b>by </b><i><a class="txt" href="aofa_i00.html#E3:118"><i><font color="Green" title="E66">B</font></i></a>, <a class="txt" href="aofa_i00.html#E1:118_1_1_1_2"><i><font color="Green" title="E81">01</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon">F<sub>10</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">F<sub>9</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p3">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>()</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() ) &amp; <span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() )</span><br/>

<b>reconsider </b><font color="Maroon" title="c8">s'</font> = <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<font color="Maroon">F<sub>4</sub></font>() as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c9">s''</font> = <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c8">s'</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>  ;<br/>
<a NAME="E5:118_1_1_1_2"/><i><font color="Green" title="E82">03</font></i>: 
( <font color="Maroon" title="c8">s'</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>() &amp; <font color="Maroon" title="c8">s'</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() &amp; <font color="Maroon" title="c9">s''</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">s'</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>()</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">s''</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s'</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() )
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E1:118_1_1_1_2"><i><font color="Green" title="E81">01</font></i></a>, <a class="txt" href="aofa_i00.html#E4:118"><i><font color="Green" title="E67">C</font></i></a>, <a class="txt" href="aofa_i00.html#E5:118"><i><font color="Green" title="E68">D</font></i></a>, <a class="ref" href="aofa_i00.html#T28" target="_self" title="AOFA_I00:th.28">Th012</a></i>;<br/>
<a NAME="E6:118_1_1_1_2"/>
<font color="Maroon" title="c9">s''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="aofa_000.html#D29" title="AOFA_000:def.29">AOFA_000:def 29</a></i>;<br/>
<b>hence </b><a NAME="E7:118_1_1_1_2"/>
( ( <font color="Maroon">F<sub>10</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">F<sub>9</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p3">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p4">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>()</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() ) &amp; <span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() )
 <b>by </b><i><a class="txt" href="aofa_i00.html#E1:118_1_1_1_2"><i><font color="Green" title="E81">01</font></i></a>, <a class="txt" href="aofa_i00.html#E5:118_1_1_1_2"><i><font color="Green" title="E82">03</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E7:118_1_1_1"/><i><font color="Green" title="E81">E'</font></i>: 
for <font color="Olive" title="b1">s</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s</font>] holds <br/>( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span>] &amp; ( <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon">F<sub>3</sub></font>(),<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span>] ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span>] implies <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon">F<sub>3</sub></font>(),<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">s</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">s</font>] implies ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span>] &amp; ( <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon">F<sub>3</sub></font>(),<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span>] ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span>] implies <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon">F<sub>3</sub></font>(),<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) ) )</span><br/>

<b>assume </b><a NAME="E1:118_1_1_1_3"/><i><font color="Green" title="E82">01</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">s</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span>] &amp; ( <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon">F<sub>3</sub></font>(),<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span>] ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span>] implies <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon">F<sub>3</sub></font>(),<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/>

<b>hence </b><a NAME="E2:118_1_1_1_3"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span>]
 <b>by </b><i><a class="txt" href="aofa_i00.html#E3:118"><i><font color="Green" title="E66">B</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon">F<sub>10</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">F<sub>9</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>()</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() ) &amp; <span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() &amp; ( <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon">F<sub>3</sub></font>(),<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span>] ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span>] implies <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon">F<sub>3</sub></font>(),<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/>

<a NAME="E3:118_1_1_1_3"/>
( <span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>() &amp; <span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() &amp; ( <font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>() <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() implies <span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">s</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() implies <span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 
<b>by </b><i><a class="txt" href="aofa_i00.html#E5:118"><i><font color="Green" title="E68">D</font></i></a>, <a class="ref" href="aofa_i00.html#T35" target="_self" title="AOFA_I00:th.35">Th114</a></i>;<br/>
<b>hence </b><a NAME="E4:118_1_1_1_3"/>
( ( <font color="Maroon">F<sub>10</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">F<sub>9</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>()</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() ) &amp; <span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span></span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() &amp; ( <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon">F<sub>3</sub></font>(),<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span>] ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span>] implies <font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon">F<sub>3</sub></font>(),<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 <b>by </b><i><a class="txt" href="aofa_i00.html#E1:118_1_1_1_3"><i><font color="Green" title="E82">01</font></i></a>, <a class="ref" href="aofa_i00.html#T2" target="_self" title="AOFA_I00:th.2">LemTS</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E8:118_1_1_1"/><i><font color="Green" title="E82">F'</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">s1</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span>] &amp;  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">s1</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K74" title="AOFA_I00:func.74">leq</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <span class="p3">(<span class="default"><font color="Maroon">F<sub>7</sub></font>() <a href="aofa_i00.html#K57" title="AOFA_I00:func.57">+=</a> 1</span>)</span></span>)</span></span>)</span>] )
 <b>from </b><i><a class="ref" href="aofa_000.html#S5" title="AOFA_000:sch.5">AOFA_000:sch 5</a>(<a class="txt" href="aofa_i00.html#E8:118_1"><i><font color="Green" title="E73">A1</font></i></a>, <a class="txt" href="aofa_i00.html#E5:118_1_1_1"><i><font color="Green" title="E79">C'</font></i></a>, <a class="txt" href="aofa_i00.html#E6:118_1_1_1"><i><font color="Green" title="E80">D'</font></i></a>, <a class="txt" href="aofa_i00.html#E7:118_1_1_1"><i><font color="Green" title="E81">E'</font></i></a>);<br/></i><a NAME="E9:118_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon">F<sub>9</sub></font>(),<font color="Maroon">F<sub>5</sub></font>()</span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>() <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="aofa_i00.html#E6:118_1"><i><font color="Green" title="E72">Sn</font></i></a>, <a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/><a NAME="E10:118_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon">F<sub>9</sub></font>(),<font color="Maroon">F<sub>5</sub></font>()</span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>()</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></i>;<br/><a NAME="E11:118_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon">F<sub>9</sub></font>(),<font color="Maroon">F<sub>5</sub></font>()</span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>()</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>()
 <b>by </b><i><a class="txt" href="aofa_i00.html#E1:118_1_1_1"><i><font color="Green" title="E75">C1</font></i></a>, <a class="txt" href="aofa_i00.html#E6:118_1"><i><font color="Green" title="E72">Sn</font></i></a>, <a class="txt" href="aofa_i00.html#E8:118_1_1_1"><i><font color="Green" title="E82">F'</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E12:118_1_1_1"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon">F<sub>9</sub></font>(),<font color="Maroon">F<sub>5</sub></font>()] &amp; ( <font color="Maroon">F<sub>10</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">F<sub>9</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() implies <span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon">F<sub>9</sub></font>(),<font color="Maroon">F<sub>5</sub></font>()</span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 ) &amp; ( <font color="Maroon">F<sub>10</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">F<sub>9</sub></font>() <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() implies <span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon">F<sub>9</sub></font>(),<font color="Maroon">F<sub>5</sub></font>()</span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>10</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">F<sub>9</sub></font>() ) &amp; <span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon">F<sub>9</sub></font>(),<font color="Maroon">F<sub>5</sub></font>()</span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() )
 <b>by </b><i><a class="txt" href="aofa_i00.html#E1:118_1_1_1"><i><font color="Green" title="E75">C1</font></i></a>, <a class="txt" href="aofa_i00.html#E1:118"><i><font color="Green" title="E64">F</font></i></a>, <a class="txt" href="aofa_i00.html#E8:118_1_1_1"><i><font color="Green" title="E82">F'</font></i></a>, <a class="ref" href="aofa_000.html#D29" title="AOFA_000:def.29">AOFA_000:def 29</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="118_1_1_2"><b>suppose </b></a><a NAME="E1:118_1_1_2"/><i><font color="Green" title="E75">C2</font></i>: 
<font color="Maroon">F<sub>10</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">F<sub>9</sub></font>() <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>()
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon">F<sub>9</sub></font>(),<font color="Maroon">F<sub>5</sub></font>()] &amp; ( <font color="Maroon">F<sub>10</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">F<sub>9</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() implies <span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon">F<sub>9</sub></font>(),<font color="Maroon">F<sub>5</sub></font>()</span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 ) &amp; ( <font color="Maroon">F<sub>10</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">F<sub>9</sub></font>() <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() implies <span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon">F<sub>9</sub></font>(),<font color="Maroon">F<sub>5</sub></font>()</span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>10</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">F<sub>9</sub></font>() ) &amp; <span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon">F<sub>9</sub></font>(),<font color="Maroon">F<sub>5</sub></font>()</span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() )</span><br/><div class="add"><a NAME="E2:118_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">s2</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="aofa_i00.html#E4:118_1"><i><font color="Green" title="E70">S1</font></i></a>, <a class="ref" href="aofa_i00.html#T35" target="_self" title="AOFA_I00:th.35">Th114</a></i>;<br/><a NAME="E3:118_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">s2</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon">F<sub>2</sub></font>(),<a href="mesfunc1.html#K2" title="MESFUNC1:func.2">INT</a> </span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> <font color="Maroon">F<sub>3</sub></font>(),<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="aofa_i00.html#T2" target="_self" title="AOFA_I00:th.2">LemTS</a></i>;<br/><b>hence </b><a NAME="E4:118_1_1_2"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon">F<sub>9</sub></font>(),<font color="Maroon">F<sub>5</sub></font>()] &amp; ( <font color="Maroon">F<sub>10</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">F<sub>9</sub></font>() <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() implies <span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon">F<sub>9</sub></font>(),<font color="Maroon">F<sub>5</sub></font>()</span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>()</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 ) &amp; ( <font color="Maroon">F<sub>10</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">F<sub>9</sub></font>() <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() implies <span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon">F<sub>9</sub></font>(),<font color="Maroon">F<sub>5</sub></font>()</span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>7</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>10</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">F<sub>9</sub></font>() ) &amp; <span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon">F<sub>9</sub></font>(),<font color="Maroon">F<sub>5</sub></font>()</span>)</span> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>9</sub></font>() <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon">F<sub>8</sub></font>() )
 <b>by </b><i><a class="txt" href="aofa_i00.html#E9:118_1"><i><font color="Green" title="E74">A2</font></i></a>, <a class="txt" href="aofa_i00.html#E1:118_1_1_2"><i><font color="Green" title="E75">C2</font></i></a>, <a class="txt" href="aofa_i00.html#E5:118_1"><i><font color="Green" title="E71">S2</font></i></a>, <a class="txt" href="aofa_i00.html#E6:118_1"><i><font color="Green" title="E72">Sn</font></i></a>, <a class="txt" href="aofa_i00.html#E1:118_1"><i><font color="Green" title="E69">Z0</font></i></a>, <a class="ref" href="aofa_000.html#D31" title="AOFA_000:def.31">AOFA_000:def 31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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