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

<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   <a href="aofa_000.html#NM1" title="AOFA_000:NM.1">preIfWhileAlgebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">I</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>  holds <br/>(  not <font color="Olive" title="b1">I</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon" title="c1">A</font> or <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon" title="c1">A</font> or  ex <font color="Olive" title="b2">I1</font>, <font color="Olive" title="b3">I2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b3">I2</font> &amp; <font color="Olive" title="b2">I1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b3">I2</font> &amp; <font color="Olive" title="b3">I2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b3">I2</font> ) or  ex <font color="Olive" title="b2">C</font>, <font color="Olive" title="b3">I1</font>, <font color="Olive" title="b4">I2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> st <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Olive" title="b2">C</font>,<font color="Olive" title="b3">I1</font>,<font color="Olive" title="b4">I2</font> or  ex <font color="Olive" title="b2">C</font>, <font color="Olive" title="b3">J</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> st <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Olive" title="b2">C</font>,<font color="Olive" title="b3">J</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">I</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">I</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon" title="c1">A</font> or  ex <font color="Olive" title="b1">I1</font>, <font color="Olive" title="b2">I2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b2">I2</font> &amp; <font color="Olive" title="b1">I1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b2">I2</font> &amp; <font color="Olive" title="b2">I2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b2">I2</font> ) or  ex <font color="Olive" title="b1">C</font>, <font color="Olive" title="b2">I1</font>, <font color="Olive" title="b3">I2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">I1</font>,<font color="Olive" title="b3">I2</font> or  ex <font color="Olive" title="b1">C</font>, <font color="Olive" title="b2">J</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">J</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:91"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
<font color="Maroon" title="c2">I</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E2:91"/><span class="lab"><font color="Green" title="E56">A2</font></span>: 
<font color="Maroon" title="c2">I</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E3:91"/><span class="lab"><font color="Green" title="E57">A3</font></span>: 
for <font color="Olive" title="b1">I1</font>, <font color="Olive" title="b2">I2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b2">I2</font> &amp; <font color="Olive" title="b1">I1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b2">I2</font> holds <br/><font color="Olive" title="b2">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b2">I2</font>
 <span class="kw">and </span><br/><a NAME="E4:91"/><span class="lab"><font color="Green" title="E58">A4</font></span>: 
for <font color="Olive" title="b1">C</font>, <font color="Olive" title="b2">I1</font>, <font color="Olive" title="b3">I2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>  holds <font color="Maroon" title="c2">I</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">I1</font>,<font color="Olive" title="b3">I2</font>
 <span class="kw">and </span><br/><a NAME="E5:91"/><span class="lab"><font color="Green" title="E59">A5</font></span>: 
for <font color="Olive" title="b1">C</font>, <font color="Olive" title="b2">J</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>  holds <font color="Maroon" title="c2">I</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">J</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><span class="lab"><font color="Green" title="E60">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:91_1"/>
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <span class="p2">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> 3,<span class="p3">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">A</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:91_1"/><span class="lab"><font color="Green" title="E61">A7</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <span class="p2">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> 3,<span class="p3">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">A</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:91_1"/><span class="lab"><font color="Green" title="E62">A8</font></span>: 
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <span class="p2">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> 3,<span class="p3">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c4">f</font> = the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V11" title="COMPUT_1:attr.11">ternary</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_000.html#D12" target="_self" title="AOFA_000:def.12">Def12</a></span>;<br/><a NAME="E6:91_1"/>
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_000.html#D12" target="_self" title="AOFA_000:def.12">Def12</a></span>;<br/><a NAME="E7:91_1"/><span class="kw">then </span>
 <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> 3,<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></span>;<br/><span class="kw">then </span> <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <span class="p2">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> 3,<span class="p3">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">A</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="comput_1.html#T25" title="COMPUT_1:th.25">COMPUT_1:25</a></span>
<br/>.= 
3 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="comput_1.html#D27" title="COMPUT_1:def.27">COMPUT_1:def 27</a></span>
;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">C</font>, <font color="Maroon" title="c6">I1</font>, <font color="Maroon" title="c7">I2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:91_1"/><span class="lab"><font color="Green" title="E63">A9</font></span>: 
<font color="Maroon" title="c5">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E11:91_1"/><span class="lab"><font color="Green" title="E64">A10</font></span>: 
<font color="Maroon" title="c6">I1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E12:91_1"/><span class="lab"><font color="Green" title="E65">A11</font></span>: 
<font color="Maroon" title="c7">I2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E13:91_1"/><span class="lab"><font color="Green" title="E66">A12</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c5">C</font>,<font color="Maroon" title="c6">I1</font>,<font color="Maroon" title="c7">I2</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:91_1"><span class="lab"><font color="Green" title="E61">A7</font></span></a>, <a class="ref" href="finseq_2.html#T159" title="FINSEQ_2:th.159">FINSEQ_2:159</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">C</font> = <font color="Maroon" title="c5">C</font>, <font color="Maroon" title="c9">I1</font> = <font color="Maroon" title="c6">I1</font>, <font color="Maroon" title="c10">I2</font> = <font color="Maroon" title="c7">I2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:91_1"><span class="lab"><font color="Green" title="E63">A9</font></span></a>, <a class="txt" href="#E11:91_1"><span class="lab"><font color="Green" title="E64">A10</font></span></a>, <a class="txt" href="#E12:91_1"><span class="lab"><font color="Green" title="E65">A11</font></span></a></span>;<br/><a NAME="E15:91_1"/>
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c8">C</font>,<font color="Maroon" title="c9">I1</font>,<font color="Maroon" title="c10">I2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:91_1"><span class="lab"><font color="Green" title="E62">A8</font></span></a>, <a class="txt" href="#E13:91_1"><span class="lab"><font color="Green" title="E66">A12</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E16:91_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:91"><span class="lab"><font color="Green" title="E58">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E61">A13</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:91_2"/>
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <span class="p2">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> 4,<span class="p3">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">A</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:91_2"/><span class="lab"><font color="Green" title="E62">A14</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <span class="p2">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> 4,<span class="p3">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">A</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:91_2"/><span class="lab"><font color="Green" title="E63">A15</font></span>: 
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <span class="p2">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> 4,<span class="p3">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c4">f</font> = the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 4 as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V10" title="COMPUT_1:attr.10">binary</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_000.html#D13" target="_self" title="AOFA_000:def.13">Def13</a></span>;<br/><a NAME="E6:91_2"/>
4 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_000.html#D13" target="_self" title="AOFA_000:def.13">Def13</a></span>;<br/><a NAME="E7:91_2"/><span class="kw">then </span>
 <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> 4,<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></span>;<br/><span class="kw">then </span> <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <span class="p2">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> 4,<span class="p3">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">A</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="comput_1.html#T25" title="COMPUT_1:th.25">COMPUT_1:25</a></span>
<br/>.= 
2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="comput_1.html#D26" title="COMPUT_1:def.26">COMPUT_1:def 26</a></span>
;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">C</font>, <font color="Maroon" title="c6">J</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:91_2"/><span class="lab"><font color="Green" title="E64">A16</font></span>: 
<font color="Maroon" title="c5">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E11:91_2"/><span class="lab"><font color="Green" title="E65">A17</font></span>: 
<font color="Maroon" title="c6">J</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E12:91_2"/><span class="lab"><font color="Green" title="E66">A18</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">C</font>,<font color="Maroon" title="c6">J</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:91_2"><span class="lab"><font color="Green" title="E62">A14</font></span></a>, <a class="ref" href="finseq_2.html#T157" title="FINSEQ_2:th.157">FINSEQ_2:157</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">C</font> = <font color="Maroon" title="c5">C</font>, <font color="Maroon" title="c8">J</font> = <font color="Maroon" title="c6">J</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:91_2"><span class="lab"><font color="Green" title="E64">A16</font></span></a>, <a class="txt" href="#E11:91_2"><span class="lab"><font color="Green" title="E65">A17</font></span></a></span>;<br/><a NAME="E14:91_2"/>
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> <font color="Maroon" title="c7">C</font>,<font color="Maroon" title="c8">J</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:91_2"><span class="lab"><font color="Green" title="E63">A15</font></span></a>, <a class="txt" href="#E12:91_2"><span class="lab"><font color="Green" title="E66">A18</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E15:91_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:91"><span class="lab"><font color="Green" title="E59">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E8:91"/><span class="lab"><font color="Green" title="E62">A19</font></span>: 
<font color="Maroon" title="c2">I</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b2">I2</font></span>)</span> where <font color="Olive" title="b1">I1</font>, <font color="Olive" title="b2">I2</font> is   <a href="aofa_000.html#NM2" title="AOFA_000:NM.2">Algorithm</a> of <font color="Maroon" title="c1">A</font> : ( <font color="Olive" title="b1">I1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b2">I2</font> &amp; <font color="Olive" title="b2">I2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">I1</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b2">I2</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:91"><span class="lab"><font color="Green" title="E57">A3</font></span></a></span>;<br/>
<a NAME="E9:91"/>
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></span>;<br/>
<a NAME="E10:91"/><span class="kw">then </span>
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <span class="p3">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> 3,<span class="p4">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">A</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:91"><span class="lab"><font color="Green" title="E60">A6</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E11:91"/><span class="kw">then </span>
<font color="Maroon" title="c2">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><a href="aofa_000.html#K12" title="AOFA_000:func.12">EmptyIns</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <span class="p4">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> 3,<span class="p5">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">A</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <span class="p3">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> 4,<span class="p4">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">A</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:91"><span class="lab"><font color="Green" title="E61">A13</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:91"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="#E8:91"><span class="lab"><font color="Green" title="E62">A19</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
