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<span class="kw">let </span><font color="Maroon" title="c1">T</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>   <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>   non  <a href="abcmiz_0.html#V4" title="ABCMIZ_0:attr.4">void</a>   <a href="abcmiz_0.html#L3" title="ABCMIZ_0:struct.3">TAS-structure</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">t</font> being   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font><br/>  for <font color="Olive" title="b2">v</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">v</font> <a href="abcmiz_0.html#R5" title="ABCMIZ_0:pred.5">is_properly_applicable_to</a> <font color="Olive" title="b1">t</font> holds <br/><font color="Olive" title="b2">v</font> <a href="abcmiz_0.html#R3" title="ABCMIZ_0:pred.3">is_applicable_to</a> <font color="Olive" title="b1">t</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">t</font> be   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">v</font> <a href="abcmiz_0.html#R5" title="ABCMIZ_0:pred.5">is_properly_applicable_to</a> <font color="Maroon" title="c2">t</font> holds <br/><font color="Olive" title="b1">v</font> <a href="abcmiz_0.html#R3" title="ABCMIZ_0:pred.3">is_applicable_to</a> <font color="Maroon" title="c2">t</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">v</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">v</font> <a href="abcmiz_0.html#R5" title="ABCMIZ_0:pred.5">is_properly_applicable_to</a> <font color="Maroon" title="c2">t</font> implies <font color="Maroon" title="c3">v</font> <a href="abcmiz_0.html#R3" title="ABCMIZ_0:pred.3">is_applicable_to</a> <font color="Maroon" title="c2">t</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:98"/><span class="lab"><font color="Green" title="E53">A1</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">a</font> being   <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c1">T</font><br/>  for <font color="Olive" title="b3">s</font> being   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> holds <br/><font color="Olive" title="b2">a</font> <a href="abcmiz_0.html#R4" title="ABCMIZ_0:pred.4">is_properly_applicable_to</a> <font color="Olive" title="b3">s</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="abcmiz_0.html#D28" target="_self" onmouseover="rs('abcmiz_0/D28')" onmouseout="rh()">ABCMIZ_0:def 28</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">v</font> <a href="abcmiz_0.html#R3" title="ABCMIZ_0:pred.3">is_applicable_to</a> <font color="Maroon" title="c2">t</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="abcmiz_0.html#D21" target="_self" onmouseover="rs('abcmiz_0/D21')" onmouseout="rh()">ABCMIZ_0:def 21</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c1">T</font><br/>  for <font color="Olive" title="b2">s</font> being   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font>  st <font color="Maroon" title="c4">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> holds <br/><font color="Olive" title="b1">a</font> <a href="abcmiz_0.html#R1" title="ABCMIZ_0:pred.1">is_applicable_to</a> <font color="Olive" title="b2">s</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">a</font> be   <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s</font> being   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font>  st <font color="Maroon" title="c4">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> holds <br/><font color="Maroon" title="c5">a</font> <a href="abcmiz_0.html#R1" title="ABCMIZ_0:pred.1">is_applicable_to</a> <font color="Olive" title="b1">s</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">s</font> be   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> implies <font color="Maroon" title="c5">a</font> <a href="abcmiz_0.html#R1" title="ABCMIZ_0:pred.1">is_applicable_to</a> <font color="Maroon" title="c6">s</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:98"/><span class="lab"><font color="Green" title="E54">A2</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">v</font>
 <span class="kw">and </span><br/><a NAME="E3:98"/><span class="lab"><font color="Green" title="E55">A3</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">and </span><br/><a NAME="E4:98"/><span class="lab"><font color="Green" title="E56">A4</font></span>: 
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> (<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c2">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="abcmiz_0.html#R1" title="ABCMIZ_0:pred.1">is_applicable_to</a> <font color="Maroon" title="c6">s</font></span><br/>

<a NAME="E5:98"/>
<font color="Maroon" title="c5">a</font> <a href="abcmiz_0.html#R4" title="ABCMIZ_0:pred.4">is_properly_applicable_to</a> <font color="Maroon" title="c6">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:98"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="txt" href="#E2:98"><span class="lab"><font color="Green" title="E54">A2</font></span></a>, <a class="txt" href="#E3:98"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="txt" href="#E4:98"><span class="lab"><font color="Green" title="E56">A4</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:98"/>
<font color="Maroon" title="c5">a</font> <a href="abcmiz_0.html#R1" title="ABCMIZ_0:pred.1">is_applicable_to</a> <font color="Maroon" title="c6">s</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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