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<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="aofa_000.html#NM1" title="AOFA_000:NM.1">preIfWhileAlgebra</a>;<br/>

<b>assume </b><a NAME="E1:119"/><i><font color="Green" title="E77">A1</font></i>: 
<font color="Maroon" title="c1">A</font> is <a href="freealg.html#V3" title="FREEALG:attr.3">free</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">C</font>, <font color="Maroon" title="c3">I1</font>, <font color="Maroon" title="c4">I2</font>, <font color="Maroon" title="c5">D</font>, <font color="Maroon" title="c6">J1</font>, <font color="Maroon" title="c7">J2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/>

<a NAME="E2:119"/>
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>
 
<b>by </b><i><a class="ref" href="aofa_000.html#D12" target="_self" title="AOFA_000:def.12">Def12</a></i>;<br/>
<a NAME="E3:119"/><b>then </b><i><font color="Green" title="E78">A2</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 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> &amp;  <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 )
 
<b>by </b><i><a class="ref" href="aofa_000.html#T47" target="_self" title="AOFA_000:th.47">Th47</a>, <a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></i>;<br/>
<a NAME="E4:119"/>
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>
 
<b>by </b><i><a class="ref" href="aofa_000.html#D13" target="_self" title="AOFA_000:def.13">Def13</a></i>;<br/>
<a NAME="E5:119"/><b>then </b><i><font color="Green" title="E79">A3</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 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> &amp;  <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 )
 
<b>by </b><i><a class="ref" href="aofa_000.html#T48" target="_self" title="AOFA_000:th.48">Th48</a>, <a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></i>;<br/>
<a NAME="E6:119"/><i><font color="Green" title="E80">A4</font></i>: 
( <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">I1</font>,<font color="Maroon" title="c4">I2</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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> &amp; <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c6">J1</font>,<font color="Maroon" title="c7">J2</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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> )
 
<b>by </b><i><a class="ref" href="catalg_1.html#T12" title="CATALG_1:th.12">CATALG_1:12</a></i>;<br/>
<a NAME="E7:119"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">I1</font>,<font color="Maroon" title="c4">I2</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">I1</font>,<font color="Maroon" title="c4">I2</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T148" title="FINSEQ_2:th.148">FINSEQ_2:148</a></i>;<br/>
<a NAME="E8:119"/><b>then </b>
( <font color="Maroon" title="c2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">I1</font>,<font color="Maroon" title="c4">I2</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> &amp; <font color="Maroon" title="c3">I1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">I1</font>,<font color="Maroon" title="c4">I2</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> &amp; <font color="Maroon" title="c4">I2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">I1</font>,<font color="Maroon" title="c4">I2</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E9:119"/>
(  <a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">I1</font>,<font color="Maroon" title="c4">I2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">C</font> &amp;  <a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">I1</font>,<font color="Maroon" title="c4">I2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">I1</font> &amp;  <a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">I1</font>,<font color="Maroon" title="c4">I2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">I2</font> )
 <b>by </b><i><a class="txt" href="aofa_000.html#E1:119"><i><font color="Green" title="E77">A1</font></i></a>, <a class="txt" href="aofa_000.html#E3:119"><i><font color="Green" title="E78">A2</font></i></a>, <a class="txt" href="aofa_000.html#E6:119"><i><font color="Green" title="E80">A4</font></i></a>, <a class="ref" href="aofa_000.html#T38" target="_self" title="AOFA_000:th.38">Th38</a></i>;<br/>

<a NAME="E10:119"/>
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c6">J1</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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>
 
<b>by </b><i><a class="ref" href="catalg_1.html#T10" title="CATALG_1:th.10">CATALG_1:10</a></i>;<br/>
<b>hence </b><a NAME="E11:119"/>
 <a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">I1</font>,<font color="Maroon" title="c4">I2</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="Maroon" title="c5">D</font>,<font color="Maroon" title="c6">J1</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E1:119"><i><font color="Green" title="E77">A1</font></i></a>, <a class="txt" href="aofa_000.html#E3:119"><i><font color="Green" title="E78">A2</font></i></a>, <a class="txt" href="aofa_000.html#E5:119"><i><font color="Green" title="E79">A3</font></i></a>, <a class="txt" href="aofa_000.html#E6:119"><i><font color="Green" title="E80">A4</font></i></a>, <a class="ref" href="aofa_000.html#T36" target="_self" title="AOFA_000:th.36">Th36</a></i>;<br/>

<b>assume </b><a NAME="E12:119"/>
 <a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">I1</font>,<font color="Maroon" title="c4">I2</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="c5">D</font>,<font color="Maroon" title="c6">J1</font>,<font color="Maroon" title="c7">J2</font>
 ;<br/>

<a NAME="E13:119"/><b>then </b>
<span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">I1</font>,<font color="Maroon" title="c4">I2</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c6">J1</font>,<font color="Maroon" title="c7">J2</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="aofa_000.html#E1:119"><i><font color="Green" title="E77">A1</font></i></a>, <a class="txt" href="aofa_000.html#E3:119"><i><font color="Green" title="E78">A2</font></i></a>, <a class="txt" href="aofa_000.html#E6:119"><i><font color="Green" title="E80">A4</font></i></a>, <a class="ref" href="aofa_000.html#T36" target="_self" title="AOFA_000:th.36">Th36</a></i>;<br/>
<b>hence </b><a NAME="E14:119"/>
( <font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D</font> &amp; <font color="Maroon" title="c3">I1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">J1</font> &amp; <font color="Maroon" title="c4">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">J2</font> )
 <b>by </b><i><a class="ref" href="group_7.html#T3" title="GROUP_7:th.3">GROUP_7:3</a></i>;<br/>


</div>
