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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="freealg.html#V1" title="FREEALG:attr.1">disjoint_with_NAT</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">C1</font>, <font color="Olive" title="b2">C2</font>, <font color="Olive" title="b3">I1</font>, <font color="Olive" title="b4">I2</font>, <font color="Olive" title="b5">J1</font>, <font color="Olive" title="b6">J2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<font color="Maroon" title="c1">X</font></span>)</span>  st  <a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Olive" title="b1">C1</font>,<font color="Olive" title="b3">I1</font>,<font color="Olive" title="b4">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="Olive" title="b2">C2</font>,<font color="Olive" title="b5">J1</font>,<font color="Olive" title="b6">J2</font> holds <br/>( <font color="Olive" title="b1">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">C2</font> &amp; <font color="Olive" title="b3">I1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">J1</font> &amp; <font color="Olive" title="b4">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">J2</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">S</font> =  <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">A</font> =  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">C1</font>, <font color="Maroon" title="c5">C2</font>, <font color="Maroon" title="c6">I1</font>, <font color="Maroon" title="c7">I2</font>, <font color="Maroon" title="c8">J1</font>, <font color="Maroon" title="c9">J2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ,<font color="Maroon" title="c1">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c4">C1</font>,<font color="Maroon" title="c6">I1</font>,<font color="Maroon" title="c7">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">C2</font>,<font color="Maroon" title="c8">J1</font>,<font color="Maroon" title="c9">J2</font> implies ( <font color="Maroon" title="c4">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">C2</font> &amp; <font color="Maroon" title="c6">I1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">J1</font> &amp; <font color="Maroon" title="c7">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">J2</font> ) )</span><br/>

<a NAME="E1:107"/><span class="lab"><font color="Green" title="E69">A1</font></span>: 
 <a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c4">C1</font>,<font color="Maroon" title="c6">I1</font>,<font color="Maroon" title="c7">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">C1</font>,<font color="Maroon" title="c6">I1</font>,<font color="Maroon" title="c7">I2</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_000.html#T63" target="_self" title="AOFA_000:th.63">Th63</a></span>;<br/>
<a NAME="E2:107"/>
 <a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c5">C2</font>,<font color="Maroon" title="c8">J1</font>,<font color="Maroon" title="c9">J2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</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">C2</font>,<font color="Maroon" title="c8">J1</font>,<font color="Maroon" title="c9">J2</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_000.html#T63" target="_self" title="AOFA_000:th.63">Th63</a></span>;<br/>
<a NAME="E3:107"/><span class="kw">then </span>
(  <a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c4">C1</font>,<font color="Maroon" title="c6">I1</font>,<font color="Maroon" title="c7">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">C2</font>,<font color="Maroon" title="c8">J1</font>,<font color="Maroon" title="c9">J2</font> implies <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">C1</font>,<font color="Maroon" title="c6">I1</font>,<font color="Maroon" title="c7">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="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c5">C2</font>,<font color="Maroon" title="c8">J1</font>,<font color="Maroon" title="c9">J2</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:107"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="ref" href="trees_4.html#T15" title="TREES_4:th.15">TREES_4:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:107"/>
(  <a href="aofa_000.html#K14" title="AOFA_000:func.14">if-then-else</a> <font color="Maroon" title="c4">C1</font>,<font color="Maroon" title="c6">I1</font>,<font color="Maroon" title="c7">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">C2</font>,<font color="Maroon" title="c8">J1</font>,<font color="Maroon" title="c9">J2</font> implies ( <font color="Maroon" title="c4">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">C2</font> &amp; <font color="Maroon" title="c6">I1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">J1</font> &amp; <font color="Maroon" title="c7">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">J2</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T99" title="FINSEQ_1:th.99">FINSEQ_1:99</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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