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<span class="kw">let </span><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">N</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p2">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>  st (  for <font color="Olive" title="b2">p</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>)<br/>  for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st (  for <font color="Olive" title="b5">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  holds <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b4">b</font> ) holds <br/><font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font>,<font color="Olive" title="b4">b</font> ) &amp; (  for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">p9</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>)<br/>  for <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st (  for <font color="Olive" title="b6">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds <br/> ( ( <font color="Olive" title="b4">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">i</font>,<font color="Olive" title="b5">b</font> implies <font color="Olive" title="b4">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b3">p9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">i</font>,<font color="Olive" title="b5">b</font> ) &amp; ( <font color="Olive" title="b4">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b3">p9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">i</font>,<font color="Olive" title="b5">b</font> implies <font color="Olive" title="b4">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">i</font>,<font color="Olive" title="b5">b</font> ) &amp; ( <font color="Olive" title="b5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">i</font>,<font color="Olive" title="b4">a</font> implies <font color="Olive" title="b5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b3">p9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">i</font>,<font color="Olive" title="b4">a</font> ) &amp; ( <font color="Olive" title="b5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b3">p9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">i</font>,<font color="Olive" title="b4">a</font> implies <font color="Olive" title="b5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">i</font>,<font color="Olive" title="b4">a</font> ) ) ) holds <br/>( <font color="Olive" title="b4">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font>,<font color="Olive" title="b5">b</font> iff <font color="Olive" title="b4">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">p9</font>,<font color="Olive" title="b5">b</font> ) ) &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 holds <br/> ex <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> st <br/> for <font color="Olive" title="b3">p</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>)<br/>  for <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b4">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b3">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font>,<font color="Olive" title="b5">b</font> holds <br/><font color="Olive" title="b4">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">p</font>,<font color="Olive" title="b5">b</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p2">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">p</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>)<br/>  for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st (  for <font color="Olive" title="b4">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  holds <font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">i</font>,<font color="Olive" title="b3">b</font> ) holds <br/><font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">b</font> ) &amp; (  for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">p9</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>)<br/>  for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st (  for <font color="Olive" title="b5">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds <br/> ( ( <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b4">b</font> implies <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b2">p9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b4">b</font> ) &amp; ( <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b2">p9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b4">b</font> implies <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b4">b</font> ) &amp; ( <font color="Olive" title="b4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b3">a</font> implies <font color="Olive" title="b4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b2">p9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b3">a</font> ) &amp; ( <font color="Olive" title="b4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b2">p9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b3">a</font> implies <font color="Olive" title="b4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b3">a</font> ) ) ) holds <br/>( <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b4">b</font> iff <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p9</font>,<font color="Olive" title="b4">b</font> ) ) &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3 implies  ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> st <br/> for <font color="Olive" title="b2">p</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>)<br/>  for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>,<font color="Olive" title="b4">b</font> holds <br/><font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font>,<font color="Olive" title="b4">b</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:24"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
 for <font color="Olive" title="b1">p</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>)<br/>  for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st (  for <font color="Olive" title="b4">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  holds <font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">i</font>,<font color="Olive" title="b3">b</font> ) holds <br/><font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">b</font>
 <span class="kw">and </span><br/><a NAME="E2:24"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
 for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">p9</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>)<br/>  for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st (  for <font color="Olive" title="b5">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds <br/> ( ( <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b4">b</font> implies <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b2">p9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b4">b</font> ) &amp; ( <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b2">p9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b4">b</font> implies <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b4">b</font> ) &amp; ( <font color="Olive" title="b4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b3">a</font> implies <font color="Olive" title="b4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b2">p9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b3">a</font> ) &amp; ( <font color="Olive" title="b4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b2">p9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b3">a</font> implies <font color="Olive" title="b4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b3">a</font> ) ) ) holds <br/>( <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b4">b</font> iff <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p9</font>,<font color="Olive" title="b4">b</font> )
 <span class="kw">and </span><br/><a NAME="E3:24"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 3
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> st <br/> for <font color="Olive" title="b2">p</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>)<br/>  for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>,<font color="Olive" title="b4">b</font> holds <br/><font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font>,<font color="Olive" title="b4">b</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>] <span class="kw">means </span>(  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> $2 holds <br/>$2 <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $1,<font color="Olive" title="b1">a</font> or  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> $2 holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $1,$2 );<br/>
<a NAME="E4:24"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
 for <font color="Olive" title="b1">p</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>)<br/>  for <font color="Olive" title="b2">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st (  for <font color="Olive" title="b3">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">i</font>,<font color="Olive" title="b2">b</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">b</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:24_1"/>
 ex <font color="Olive" title="b1">p</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>) ex <font color="Olive" title="b2">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> st <br/>( (  for <font color="Olive" title="b3">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">i</font>,<font color="Olive" title="b2">b</font>] ) &amp;  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">b</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">p</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>), <font color="Maroon" title="c5">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E3:24_1"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
 ex <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Olive" title="b1">a</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font> )
 <span class="kw">and </span><br/><a NAME="E4:24_1"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
 ex <font color="Olive" title="b1">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Olive" title="b1">c</font> )
 <span class="kw">and </span><br/><a NAME="E5:24_1"/><span class="lab"><font color="Green" title="E24">A7</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c5">b</font>]
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">a9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E7:24_1"/><span class="lab"><font color="Green" title="E25">A8</font></span>: 
( <font color="Maroon" title="c6">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c6">a9</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">c9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E9:24_1"/><span class="lab"><font color="Green" title="E26">A9</font></span>: 
( <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c9</font> &amp; <font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c7">c9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a></span>;<br/>
<a NAME="E10:24_1"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> &amp; <font color="Olive" title="b1">a</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Olive" title="b2">c</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:24_1_1_1"/>
( <font color="Maroon" title="c6">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c9</font> or <font color="Maroon" title="c6">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c9</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:24_1_1_1_1"/><span class="lab"><font color="Green" title="E27">A10</font></span>: 
<font color="Maroon" title="c6">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> &amp; <font color="Olive" title="b1">a</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Olive" title="b2">c</font> )</span><br/><div class="add"><span class="kw">take </span>
<font color="Maroon" title="c6">a9</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Maroon" title="c6">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> &amp; <font color="Maroon" title="c6">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> &amp; <font color="Maroon" title="c6">a9</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Olive" title="b1">c</font> )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c7">c9</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c9</font> &amp; <font color="Maroon" title="c6">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c9</font> &amp; <font color="Maroon" title="c6">a9</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c7">c9</font> )</span><br/><span class="kw">thus </span><a NAME="E2:24_1_1_1_1"/>
( <font color="Maroon" title="c6">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c9</font> &amp; <font color="Maroon" title="c6">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c9</font> &amp; <font color="Maroon" title="c6">a9</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c7">c9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:24_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E9:24_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E1:24_1_1_1_1"><span class="lab"><font color="Green" title="E27">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:24_1_1_1_2"/><span class="lab"><font color="Green" title="E27">A11</font></span>: 
<font color="Maroon" title="c6">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> &amp; <font color="Olive" title="b1">a</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Olive" title="b2">c</font> )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c8">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E3:24_1_1_1_2"/><span class="lab"><font color="Green" title="E28">A12</font></span>: 
( <font color="Maroon" title="c8">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c8">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="arrow.html#T2" target="_self" title="ARROW:th.2">Th2</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:24_1_1_1_2_1"/>
( <font color="Maroon" title="c8">d</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font> or <font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c8">d</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:24_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E29">A13</font></span>: 
<font color="Maroon" title="c8">d</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> &amp; <font color="Olive" title="b1">a</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Olive" title="b2">c</font> )</span><br/><div class="add"><span class="kw">take </span>
<font color="Maroon" title="c8">d</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Maroon" title="c8">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> &amp; <font color="Maroon" title="c8">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> &amp; <font color="Maroon" title="c8">d</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Olive" title="b1">c</font> )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c7">c9</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c9</font> &amp; <font color="Maroon" title="c8">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c9</font> &amp; <font color="Maroon" title="c8">d</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c7">c9</font> )</span><br/><span class="kw">thus </span><a NAME="E2:24_1_1_1_2_1_1"/>
( <font color="Maroon" title="c8">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c9</font> &amp; <font color="Maroon" title="c8">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c9</font> &amp; <font color="Maroon" title="c8">d</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c7">c9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24_1"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="txt" href="#E1:24_1_1_1_2"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E3:24_1_1_1_2"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="txt" href="#E1:24_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E29">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:24_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E29">A14</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c8">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> &amp; <font color="Olive" title="b1">a</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Olive" title="b2">c</font> )</span><br/><div class="add"><span class="kw">take </span>
<font color="Maroon" title="c6">a9</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Maroon" title="c6">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> &amp; <font color="Maroon" title="c6">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> &amp; <font color="Maroon" title="c6">a9</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Olive" title="b1">c</font> )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c8">d</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c6">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c6">a9</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c8">d</font> )</span><br/><span class="kw">thus </span><a NAME="E2:24_1_1_1_2_1_2"/>
( <font color="Maroon" title="c6">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c6">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c6">a9</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c8">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:24_1"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E3:24_1_1_1_2"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="txt" href="#E1:24_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E29">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E12:24_1"/><span class="lab"><font color="Green" title="E27">A15</font></span>: 
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c</font> )
 <span class="kw">and </span><br/><a NAME="E13:24_1"/><span class="lab"><font color="Green" title="E28">A16</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c</font>
 <span class="kw">and </span><br/><a NAME="E14:24_1"/><span class="lab"><font color="Green" title="E29">A17</font></span>: 
( <font color="Maroon" title="c8">a</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c9">c</font> )
 ;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font>] <span class="kw">means </span>( ( <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1,<font color="Maroon" title="c5">b</font> implies <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c5">b</font> ) &amp; ( <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c5">b</font> implies <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1,<font color="Maroon" title="c5">b</font> ) &amp; ( <font color="Maroon" title="c5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1,<font color="Maroon" title="c8">a</font> implies <font color="Maroon" title="c5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c8">a</font> ) &amp; ( <font color="Maroon" title="c5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c8">a</font> implies <font color="Maroon" title="c5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1,<font color="Maroon" title="c8">a</font> ) &amp; ( <font color="Maroon" title="c5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1,<font color="Maroon" title="c9">c</font> implies <font color="Maroon" title="c5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c9">c</font> ) &amp; ( <font color="Maroon" title="c5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c9">c</font> implies <font color="Maroon" title="c5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1,<font color="Maroon" title="c9">c</font> ) &amp; ( <font color="Maroon" title="c9">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1,<font color="Maroon" title="c5">b</font> implies <font color="Maroon" title="c9">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c5">b</font> ) &amp; ( <font color="Maroon" title="c9">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c5">b</font> implies <font color="Maroon" title="c9">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1,<font color="Maroon" title="c5">b</font> ) &amp; <font color="Maroon" title="c9">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c8">a</font> );<br/>
<a NAME="E15:24_1"/><span class="lab"><font color="Green" title="E30">A18</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  ex <font color="Olive" title="b2">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">o</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">i</font>,<font color="Olive" title="b1">o</font>]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:24_1_2_1"/>
(  for <font color="Olive" title="b1">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> holds <br/><font color="Maroon" title="c5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">i</font>,<font color="Olive" title="b1">c</font> or  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c5">b</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_1"><span class="lab"><font color="Green" title="E24">A7</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:24_1_2_1_1"/>
 for <font color="Olive" title="b1">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> holds <br/><font color="Maroon" title="c5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">i</font>,<font color="Olive" title="b1">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">i</font>,<font color="Olive" title="b1">o</font>]</span><br/><div class="add"><a NAME="E2:24_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A19</font></span>: 
( <font color="Maroon" title="c5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c9">c</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:24_1"><span class="lab"><font color="Green" title="E27">A15</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E4:24_1_2_1_1"/><span class="lab"><font color="Green" title="E32">A20</font></span>: 
( <font color="Maroon" title="c5">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c9">c</font> &amp; <font color="Maroon" title="c9">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c8">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:24_1"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="txt" href="#E13:24_1"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="ref" href="arrow.html#T7" target="_self" title="ARROW:th.7">Th7</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c11">o</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">o</font>]</span><br/><span class="kw">thus </span><a NAME="E5:24_1_2_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">o</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_2_1_1"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="txt" href="#E4:24_1_2_1_1"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a>, <a class="ref" href="arrow.html#T5" target="_self" title="ARROW:th.5">Th5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:24_1_2_1_2"/>
 for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c5">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">i</font>,<font color="Olive" title="b1">o</font>]</span><br/><div class="add"><a NAME="E2:24_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A21</font></span>: 
( <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c9">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c5">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:24_1"><span class="lab"><font color="Green" title="E27">A15</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E4:24_1_2_1_2"/><span class="lab"><font color="Green" title="E32">A22</font></span>: 
( <font color="Maroon" title="c9">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c8">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c5">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:24_1"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="txt" href="#E13:24_1"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="ref" href="arrow.html#T7" target="_self" title="ARROW:th.7">Th7</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c11">o</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">o</font>]</span><br/><span class="kw">thus </span><a NAME="E5:24_1_2_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">o</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_2_1_2"><span class="lab"><font color="Green" title="E31">A21</font></span></a>, <a class="txt" href="#E4:24_1_2_1_2"><span class="lab"><font color="Green" title="E32">A22</font></span></a>, <a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a>, <a class="ref" href="arrow.html#T5" target="_self" title="ARROW:th.5">Th5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c10">p9</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E17:24_1"/><span class="lab"><font color="Green" title="E31">A23</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c10">p9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E15:24_1"><span class="lab"><font color="Green" title="E30">A18</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c11">p9</font> = <font color="Maroon" title="c10">p9</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>) <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T8" title="FUNCT_2:th.8">FUNCT_2:8</a></span>;<br/>
<a NAME="E19:24_1"/>
( <font color="Maroon" title="c8">a</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">p9</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">p9</font>,<font color="Maroon" title="c9">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E14:24_1"><span class="lab"><font color="Green" title="E29">A17</font></span></a>, <a class="txt" href="#E17:24_1"><span class="lab"><font color="Green" title="E31">A23</font></span></a></span>;<br/>
<a NAME="E20:24_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">a</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">p9</font>,<font color="Maroon" title="c9">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arrow.html#T5" target="_self" title="ARROW:th.5">Th5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:24_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E17:24_1"><span class="lab"><font color="Green" title="E31">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:24"/><span class="lab"><font color="Green" title="E22">A24</font></span>: 
 for <font color="Olive" title="b1">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  ex <font color="Olive" title="b2">nb</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> ex <font color="Olive" title="b3">pI</font>, <font color="Olive" title="b4">pII</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>) st <br/>( (  for <font color="Olive" title="b5">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Olive" title="b5">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">nb</font> holds <br/><font color="Olive" title="b3">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font> ) &amp; (  for <font color="Olive" title="b5">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b1">b</font>] ) &amp; (  for <font color="Olive" title="b5">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b4">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b1">b</font>] ) &amp; (  for <font color="Olive" title="b5">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b1">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b3">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">nb</font>,<font color="Olive" title="b5">a</font> ) &amp; (  for <font color="Olive" title="b5">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b5">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b4">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">nb</font>,<font color="Olive" title="b1">b</font> ) &amp; (  for <font color="Olive" title="b5">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b1">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">pI</font>,<font color="Olive" title="b5">a</font> ) &amp; (  for <font color="Olive" title="b5">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b5">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">pII</font>,<font color="Olive" title="b1">b</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c4">t</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E2:24_2"/><span class="lab"><font color="Green" title="E23">A25</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">N</font>
 <span class="kw">and </span><br/><a NAME="E3:24_2"/><span class="lab"><font color="Green" title="E24">A26</font></span>: 
<font color="Maroon" title="c4">t</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T58" title="FINSEQ_4:th.58">FINSEQ_4:58</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">t</font> = <font color="Maroon" title="c4">t</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">N</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_2"><span class="lab"><font color="Green" title="E23">A25</font></span></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">nb</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> ex <font color="Olive" title="b2">pI</font>, <font color="Olive" title="b3">pII</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>) st <br/>( (  for <font color="Olive" title="b4">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Olive" title="b4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">nb</font> holds <br/><font color="Olive" title="b2">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">i</font> ) &amp; (  for <font color="Olive" title="b4">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">i</font>,<font color="Maroon" title="c6">b</font>] ) &amp; (  for <font color="Olive" title="b4">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">i</font>,<font color="Maroon" title="c6">b</font>] ) &amp; (  for <font color="Olive" title="b4">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Maroon" title="c6">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b2">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">nb</font>,<font color="Olive" title="b4">a</font> ) &amp; (  for <font color="Olive" title="b4">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Olive" title="b4">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b3">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">nb</font>,<font color="Maroon" title="c6">b</font> ) &amp; (  for <font color="Olive" title="b4">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Maroon" title="c6">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">pI</font>,<font color="Olive" title="b4">a</font> ) &amp; (  for <font color="Olive" title="b4">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Olive" title="b4">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">pII</font>,<font color="Maroon" title="c6">b</font> ) )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">oI</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E6:24_2"/><span class="lab"><font color="Green" title="E25">A27</font></span>: 
 for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Maroon" title="c6">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">oI</font>,<font color="Olive" title="b1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arrow.html#T8" target="_self" title="ARROW:th.8">Th8</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">oII</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E8:24_2"/><span class="lab"><font color="Green" title="E26">A28</font></span>: 
 for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">oII</font>,<font color="Maroon" title="c6">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arrow.html#T9" target="_self" title="ARROW:th.9">Th9</a></span>;<br/>
<a NAME="E9:24_2"/><span class="lab"><font color="Green" title="E27">A29</font></span>: 
 for <font color="Olive" title="b1">k0</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  ex <font color="Olive" title="b2">p</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>) st <br/>( (  for <font color="Olive" title="b3">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Olive" title="b3">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k0</font> holds <br/><font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font> ) &amp; (  for <font color="Olive" title="b3">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Olive" title="b3">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">k0</font> holds <br/><font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">k0</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>) st <br/>( (  for <font color="Olive" title="b2">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k0</font> holds <br/><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font> ) &amp; (  for <font color="Olive" title="b2">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">k0</font> holds <br/><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font> ) )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font>] <span class="kw">means </span> ex <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> &amp; ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k0</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font> ) &amp; ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">k0</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font> ) );<br/>
<a NAME="E1:24_2_1"/><span class="lab"><font color="Green" title="E28">A30</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  ex <font color="Olive" title="b2">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">o</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">i</font>,<font color="Olive" title="b1">o</font>]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c11">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:24_2_1_1"/><span class="lab"><font color="Green" title="E29">A31</font></span>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font>
 <span class="kw">and </span><br/><a NAME="E3:24_2_1_1"/><span class="lab"><font color="Green" title="E30">A32</font></span>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_2"><span class="lab"><font color="Green" title="E23">A25</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">k</font> = <font color="Maroon" title="c11">k</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_2_1_1"><span class="lab"><font color="Green" title="E29">A31</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:24_2_1_1_1"/>
( <font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k0</font> or <font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">k0</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:24_2_1_1_1_1"/><span class="lab"><font color="Green" title="E31">A33</font></span>: 
<font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">i</font>,<font color="Olive" title="b1">o</font>]</span><br/><div class="add"><span class="kw">take </span>
<font color="Maroon" title="c8">oII</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c8">oII</font>]</span><br/><span class="kw">thus </span><a NAME="E2:24_2_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c8">oII</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_2_1_1"><span class="lab"><font color="Green" title="E29">A31</font></span></a>, <a class="txt" href="#E3:24_2_1_1"><span class="lab"><font color="Green" title="E30">A32</font></span></a>, <a class="txt" href="#E1:24_2_1_1_1_1"><span class="lab"><font color="Green" title="E31">A33</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:24_2_1_1_1_2"/><span class="lab"><font color="Green" title="E31">A34</font></span>: 
<font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">k0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">i</font>,<font color="Olive" title="b1">o</font>]</span><br/><div class="add"><span class="kw">take </span>
<font color="Maroon" title="c7">oI</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c7">oI</font>]</span><br/><span class="kw">thus </span><a NAME="E2:24_2_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c7">oI</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_2_1_1"><span class="lab"><font color="Green" title="E29">A31</font></span></a>, <a class="txt" href="#E3:24_2_1_1"><span class="lab"><font color="Green" title="E30">A32</font></span></a>, <a class="txt" href="#E1:24_2_1_1_1_2"><span class="lab"><font color="Green" title="E31">A34</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c10">p</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:24_2_1"/><span class="lab"><font color="Green" title="E29">A35</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c10">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E1:24_2_1"><span class="lab"><font color="Green" title="E28">A30</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c11">p</font> = <font color="Maroon" title="c10">p</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>) <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T8" title="FUNCT_2:th.8">FUNCT_2:8</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c11">p</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k0</font> holds <br/><font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font> ) &amp; (  for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">k0</font> holds <br/><font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E5:24_2_1"/>
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k0</font> holds <br/><font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">k0</font> holds <br/><font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k0</font> implies <font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:24_2_1_2"/><span class="lab"><font color="Green" title="E30">A36</font></span>: 
<font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font>
 <span class="kw">and </span><br/><a NAME="E2:24_2_1_2"/><span class="lab"><font color="Green" title="E31">A37</font></span>: 
<font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c13">i</font> = <font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_2_1_2"><span class="lab"><font color="Green" title="E30">A36</font></span></a>, <a class="ref" href="partfun1.html#T4" title="PARTFUN1:th.4">PARTFUN1:4</a></span>;<br/>
<a NAME="E4:24_2_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c11">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_2_1"><span class="lab"><font color="Green" title="E29">A35</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:24_2_1_2"/>
<font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_2"><span class="lab"><font color="Green" title="E24">A26</font></span></a>, <a class="txt" href="#E1:24_2_1_2"><span class="lab"><font color="Green" title="E30">A36</font></span></a>, <a class="txt" href="#E2:24_2_1_2"><span class="lab"><font color="Green" title="E31">A37</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c12">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">k0</font> implies <font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:24_2_1"/><span class="lab"><font color="Green" title="E30">A38</font></span>: 
<font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font>
 <span class="kw">and </span><br/><a NAME="E7:24_2_1"/><span class="lab"><font color="Green" title="E31">A39</font></span>: 
<font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">k0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c13">i</font> = <font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24_2_1"><span class="lab"><font color="Green" title="E30">A38</font></span></a>, <a class="ref" href="partfun1.html#T4" title="PARTFUN1:th.4">PARTFUN1:4</a></span>;<br/>
<a NAME="E9:24_2_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c11">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_2_1"><span class="lab"><font color="Green" title="E29">A35</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:24_2_1"/>
<font color="Maroon" title="c11">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_2"><span class="lab"><font color="Green" title="E24">A26</font></span></a>, <a class="txt" href="#E6:24_2_1"><span class="lab"><font color="Green" title="E30">A38</font></span></a>, <a class="txt" href="#E7:24_2_1"><span class="lab"><font color="Green" title="E31">A39</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">p</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>)  st (  for <font color="Olive" title="b2">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $1 holds <br/><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font> ) &amp; (  for <font color="Olive" title="b2">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> $1 holds <br/><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font> ) holds <br/> for <font color="Olive" title="b2">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c6">b</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">kII9</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">t</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E11:24_2"/><span class="lab"><font color="Green" title="E28">A40</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">kII9</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">p</font> be    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">kII9</font> holds <br/><font color="Maroon" title="c10">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font> ) &amp; (  for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">kII9</font> holds <br/><font color="Maroon" title="c10">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font> ) implies  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c6">b</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:24_2_2"/><span class="lab"><font color="Green" title="E29">A41</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">kII9</font> holds <br/><font color="Maroon" title="c10">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font>
 <span class="kw">and </span><br/><a NAME="E2:24_2_2"/>
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">kII9</font> holds <br/><font color="Maroon" title="c10">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font>
 ; <a class="txt" 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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c6">b</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c11">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c11">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c6">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:24_2_2"/><span class="lab"><font color="Green" title="E30">A42</font></span>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c6">b</font></span><br/>

<a NAME="E4:24_2_2"/>
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  holds <font color="Maroon" title="c11">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c10">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c6">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c10">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c6">b</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c13">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:24_2_2_1"/><span class="lab"><font color="Green" title="E31">A43</font></span>: 
<font color="Maroon" title="c13">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font>
 <span class="kw">and </span><br/><a NAME="E3:24_2_2_1"/><span class="lab"><font color="Green" title="E32">A44</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_2"><span class="lab"><font color="Green" title="E23">A25</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">k</font> = <font color="Maroon" title="c13">k</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_2_2_1"><span class="lab"><font color="Green" title="E31">A43</font></span></a></span>;<br/>
<a NAME="E5:24_2_2_1"/>
<font color="Maroon" title="c14">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_2_2_1"><span class="lab"><font color="Green" title="E31">A43</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E6:24_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">kII9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E7:24_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_2_2"><span class="lab"><font color="Green" title="E29">A41</font></span></a>, <a class="txt" href="#E2:24_2_2_1"><span class="lab"><font color="Green" title="E31">A43</font></span></a>, <a class="txt" href="#E3:24_2_2_1"><span class="lab"><font color="Green" title="E32">A44</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:24_2_2_1"/>
<font color="Maroon" title="c11">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c10">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c6">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24_2"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="txt" href="#E3:24_2_2"><span class="lab"><font color="Green" title="E30">A42</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:24_2_2"/>
<font color="Maroon" title="c11">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">p</font>,<font color="Maroon" title="c6">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E18">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:24_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A45</font></span>: 
 ex <font color="Olive" title="b1">kII9</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">kII9</font>]
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">kII</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E14:24_2"/><span class="lab"><font color="Green" title="E30">A46</font></span>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">kII</font>] &amp; (  for <font color="Olive" title="b1">k0</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k0</font>] holds <br/><font color="Olive" title="b1">k0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">kII</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E12:24_2"><span class="lab"><font color="Green" title="E29">A45</font></span></a>);<br/></span>
<span class="kw">consider </span><font color="Maroon" title="c11">pII</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>)<span class="kw"> such that </span><br/><a NAME="E16:24_2"/><span class="lab"><font color="Green" title="E31">A47</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">kII</font> holds <br/><font color="Maroon" title="c11">pII</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font>
 <span class="kw">and </span><br/><a NAME="E17:24_2"/><span class="lab"><font color="Green" title="E32">A48</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">kII</font> holds <br/><font color="Maroon" title="c11">pII</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a></span>;<br/>
<a NAME="E18:24_2"/><span class="lab"><font color="Green" title="E33">A49</font></span>: 
<font color="Maroon" title="c10">kII</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:24_2_3"/><span class="lab"><font color="Green" title="E34">A50</font></span>: 
<font color="Maroon" title="c10">kII</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c12">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E3:24_2_3"/><span class="lab"><font color="Green" title="E35">A51</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="arrow.html#T1" target="_self" title="ARROW:th.1">Th1</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E4:24_2_3"/><span class="lab"><font color="Green" title="E36">A52</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  holds <font color="Maroon" title="c6">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c12">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c12">a</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c14">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:24_2_3_1"/><span class="lab"><font color="Green" title="E37">A53</font></span>: 
<font color="Maroon" title="c14">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font>
 <span class="kw">and </span><br/><a NAME="E3:24_2_3_1"/><span class="lab"><font color="Green" title="E38">A54</font></span>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_2"><span class="lab"><font color="Green" title="E23">A25</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">k</font> = <font color="Maroon" title="c14">k</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_2_3_1"><span class="lab"><font color="Green" title="E37">A53</font></span></a></span>;<br/>
<a NAME="E5:24_2_3_1"/>
<font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_2_3_1"><span class="lab"><font color="Green" title="E37">A53</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E6:24_2_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:24_2"><span class="lab"><font color="Green" title="E32">A48</font></span></a>, <a class="txt" href="#E1:24_2_3"><span class="lab"><font color="Green" title="E34">A50</font></span></a>, <a class="txt" href="#E2:24_2_3_1"><span class="lab"><font color="Green" title="E37">A53</font></span></a>, <a class="txt" href="#E3:24_2_3_1"><span class="lab"><font color="Green" title="E38">A54</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:24_2_3_1"/>
<font color="Maroon" title="c6">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c12">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E3:24_2_3"><span class="lab"><font color="Green" title="E35">A51</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:24_2_3"/><span class="lab"><font color="Green" title="E37">A55</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">pII</font>,<font color="Maroon" title="c6">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:24_2"><span class="lab"><font color="Green" title="E30">A46</font></span></a>, <a class="txt" href="#E16:24_2"><span class="lab"><font color="Green" title="E31">A47</font></span></a>, <a class="txt" href="#E17:24_2"><span class="lab"><font color="Green" title="E32">A48</font></span></a>, <a class="txt" href="#E3:24_2_3"><span class="lab"><font color="Green" title="E35">A51</font></span></a></span>;<br/>
<a NAME="E6:24_2_3"/>
<font color="Maroon" title="c6">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">pII</font>,<font color="Maroon" title="c12">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E4:24_2_3"><span class="lab"><font color="Green" title="E36">A52</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:24_2_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_2_3"><span class="lab"><font color="Green" title="E37">A55</font></span></a>, <a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">kI</font> = <font color="Maroon" title="c10">kII</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">kI</font> = <font color="Maroon" title="c12">kI</font> as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E21:24_2"/><span class="lab"><font color="Green" title="E34">A56</font></span>: 
<font color="Maroon" title="c10">kII</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">kI</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
;<br/>
<a NAME="E22:24_2"/>
<font color="Maroon" title="c13">kI</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:24_2"><span class="lab"><font color="Green" title="E33">A49</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E23:24_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A57</font></span>: 
<font color="Maroon" title="c13">kI</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T7" title="INT_1:th.7">INT_1:7</a></span>;<br/>
<a NAME="E24:24_2"/>
<font color="Maroon" title="c10">kII</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">kII9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:24_2"><span class="lab"><font color="Green" title="E28">A40</font></span></a>, <a class="txt" href="#E14:24_2"><span class="lab"><font color="Green" title="E30">A46</font></span></a></span>;<br/>
<a NAME="E25:24_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">kI</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">kII9</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E26:24_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A58</font></span>: 
<font color="Maroon" title="c13">kI</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:24_2"><span class="lab"><font color="Green" title="E35">A57</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">nb</font> = <font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">kI</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T4" title="PARTFUN1:th.4">PARTFUN1:4</a></span>;<br/>
<a NAME="E28:24_2"/><span class="lab"><font color="Green" title="E37">A59</font></span>: 
<font color="Maroon" title="c13">kI</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">kII</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:24_2"><span class="lab"><font color="Green" title="E34">A56</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">pI</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>)<span class="kw"> such that </span><br/><a NAME="E30:24_2"/><span class="lab"><font color="Green" title="E38">A60</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">kI</font> holds <br/><font color="Maroon" title="c15">pI</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font>
 <span class="kw">and </span><br/><a NAME="E31:24_2"/><span class="lab"><font color="Green" title="E39">A61</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c13">kI</font> holds <br/><font color="Maroon" title="c15">pI</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font>
 <span class="kw">and </span><br/><a NAME="E32:24_2"/><span class="lab"><font color="Green" title="E40">A62</font></span>: 
 ex <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> &amp;  not <font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">pI</font>,<font color="Maroon" title="c6">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:24_2"><span class="lab"><font color="Green" title="E30">A46</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c14">nb</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">pI</font>, <font color="Olive" title="b2">pII</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>) st <br/>( (  for <font color="Olive" title="b3">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Olive" title="b3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">nb</font> holds <br/><font color="Olive" title="b1">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">i</font> ) &amp; (  for <font color="Olive" title="b3">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">i</font>,<font color="Maroon" title="c6">b</font>] ) &amp; (  for <font color="Olive" title="b3">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">i</font>,<font color="Maroon" title="c6">b</font>] ) &amp; (  for <font color="Olive" title="b3">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Maroon" title="c6">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">nb</font>,<font color="Olive" title="b3">a</font> ) &amp; (  for <font color="Olive" title="b3">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b2">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">nb</font>,<font color="Maroon" title="c6">b</font> ) &amp; (  for <font color="Olive" title="b3">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Maroon" title="c6">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">pI</font>,<font color="Olive" title="b3">a</font> ) &amp; (  for <font color="Olive" title="b3">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">pII</font>,<font color="Maroon" title="c6">b</font> ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c15">pI</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">pII</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>) st <br/>( (  for <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Olive" title="b2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">nb</font> holds <br/><font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">i</font> ) &amp; (  for <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">i</font>,<font color="Maroon" title="c6">b</font>] ) &amp; (  for <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">i</font>,<font color="Maroon" title="c6">b</font>] ) &amp; (  for <font color="Olive" title="b2">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Maroon" title="c6">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">nb</font>,<font color="Olive" title="b2">a</font> ) &amp; (  for <font color="Olive" title="b2">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">nb</font>,<font color="Maroon" title="c6">b</font> ) &amp; (  for <font color="Olive" title="b2">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Maroon" title="c6">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">pI</font>,<font color="Olive" title="b2">a</font> ) &amp; (  for <font color="Olive" title="b2">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">pII</font>,<font color="Maroon" title="c6">b</font> ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c11">pII</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">nb</font> holds <br/><font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> ) &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c6">b</font>] ) &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c6">b</font>] ) &amp; (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Maroon" title="c6">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">nb</font>,<font color="Olive" title="b1">a</font> ) &amp; (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">nb</font>,<font color="Maroon" title="c6">b</font> ) &amp; (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Maroon" title="c6">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">pI</font>,<font color="Olive" title="b1">a</font> ) &amp; (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">pII</font>,<font color="Maroon" title="c6">b</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E33:24_2"/>
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">nb</font> holds <br/><font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c6">b</font>] ) &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c6">b</font>] ) &amp; (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Maroon" title="c6">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">nb</font>,<font color="Olive" title="b1">a</font> ) &amp; (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">nb</font>,<font color="Maroon" title="c6">b</font> ) &amp; (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Maroon" title="c6">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">pI</font>,<font color="Olive" title="b1">a</font> ) &amp; (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">pII</font>,<font color="Maroon" title="c6">b</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">nb</font> implies <font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_2_4"/><span class="lab"><font color="Green" title="E41">A63</font></span>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">nb</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c17">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:24_2_4"/><span class="lab"><font color="Green" title="E42">A64</font></span>: 
<font color="Maroon" title="c17">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font>
 <span class="kw">and </span><br/><a NAME="E4:24_2_4"/><span class="lab"><font color="Green" title="E43">A65</font></span>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_2"><span class="lab"><font color="Green" title="E23">A25</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">k</font> = <font color="Maroon" title="c17">k</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_2_4"><span class="lab"><font color="Green" title="E42">A64</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_4_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:24_2_4_1"/>
( <font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">kI</font> or <font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">kI</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_2_4"><span class="lab"><font color="Green" title="E41">A63</font></span></a>, <a class="txt" href="#E4:24_2_4"><span class="lab"><font color="Green" title="E43">A65</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:24_2_4_1_1"/>
<font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">kI</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span><br/><div class="add"><a NAME="E2:24_2_4_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c18">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">kII</font> &amp; <font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:24_2"><span class="lab"><font color="Green" title="E34">A56</font></span></a>, <a class="txt" href="#E30:24_2"><span class="lab"><font color="Green" title="E38">A60</font></span></a>, <a class="txt" href="#E3:24_2_4"><span class="lab"><font color="Green" title="E42">A64</font></span></a>, <a class="txt" href="#E4:24_2_4"><span class="lab"><font color="Green" title="E43">A65</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E3:24_2_4_1_1"/>
<font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:24_2"><span class="lab"><font color="Green" title="E31">A47</font></span></a>, <a class="txt" href="#E3:24_2_4"><span class="lab"><font color="Green" title="E42">A64</font></span></a>, <a class="txt" href="#E4:24_2_4"><span class="lab"><font color="Green" title="E43">A65</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:24_2_4_1_2"/>
<font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c13">kI</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font></span><br/><div class="add"><a NAME="E2:24_2_4_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c18">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c10">kII</font> &amp; <font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:24_2"><span class="lab"><font color="Green" title="E34">A56</font></span></a>, <a class="txt" href="#E31:24_2"><span class="lab"><font color="Green" title="E39">A61</font></span></a>, <a class="txt" href="#E3:24_2_4"><span class="lab"><font color="Green" title="E42">A64</font></span></a>, <a class="txt" href="#E4:24_2_4"><span class="lab"><font color="Green" title="E43">A65</font></span></a>, <a class="ref" href="int_1.html#T7" title="INT_1:th.7">INT_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:24_2_4_1_2"/>
<font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:24_2"><span class="lab"><font color="Green" title="E32">A48</font></span></a>, <a class="txt" href="#E3:24_2_4"><span class="lab"><font color="Green" title="E42">A64</font></span></a>, <a class="txt" href="#E4:24_2_4"><span class="lab"><font color="Green" title="E43">A65</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E34:24_2"/><span class="lab"><font color="Green" title="E41">A66</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c6">b</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c6">b</font>] ) &amp; (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Maroon" title="c6">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">nb</font>,<font color="Olive" title="b1">a</font> ) &amp; (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">nb</font>,<font color="Maroon" title="c6">b</font> ) &amp; (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Maroon" title="c6">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">pI</font>,<font color="Olive" title="b1">a</font> ) &amp; (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">pII</font>,<font color="Maroon" title="c6">b</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c6">b</font>]</span><br/>

<a NAME="E1:24_2_5"/>
 ex <font color="Olive" title="b1">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_2"><span class="lab"><font color="Green" title="E23">A25</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E2:24_2_5"/><span class="kw">then </span>
( <font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font> or <font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:24_2"><span class="lab"><font color="Green" title="E38">A60</font></span></a>, <a class="txt" href="#E31:24_2"><span class="lab"><font color="Green" title="E39">A61</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:24_2_5"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c6">b</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E8:24_2"><span class="lab"><font color="Green" title="E26">A28</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E35:24_2"/>
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c6">b</font>]
  <a class="txt" 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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Maroon" title="c6">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">nb</font>,<font color="Olive" title="b1">a</font> ) &amp; (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">nb</font>,<font color="Maroon" title="c6">b</font> ) &amp; (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Maroon" title="c6">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">pI</font>,<font color="Olive" title="b1">a</font> ) &amp; (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">pII</font>,<font color="Maroon" title="c6">b</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c6">b</font>]</span><br/>

<a NAME="E1:24_2_6"/>
 ex <font color="Olive" title="b1">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_2"><span class="lab"><font color="Green" title="E23">A25</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E2:24_2_6"/><span class="kw">then </span>
( <font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font> or <font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:24_2"><span class="lab"><font color="Green" title="E31">A47</font></span></a>, <a class="txt" href="#E17:24_2"><span class="lab"><font color="Green" title="E32">A48</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:24_2_6"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c6">b</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E8:24_2"><span class="lab"><font color="Green" title="E26">A28</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<a NAME="E36:24_2"/>
<font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">nb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oI</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:24_2"><span class="lab"><font color="Green" title="E36">A58</font></span></a>, <a class="txt" href="#E31:24_2"><span class="lab"><font color="Green" title="E39">A61</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E37:24_2"/>
 for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Maroon" title="c6">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c15">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">nb</font>,<font color="Olive" title="b1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a></span>; <a class="txt" 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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">nb</font>,<font color="Maroon" title="c6">b</font> ) &amp; (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Maroon" title="c6">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">pI</font>,<font color="Olive" title="b1">a</font> ) &amp; (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">pII</font>,<font color="Maroon" title="c6">b</font> ) )</span><br/>

<a NAME="E38:24_2"/>
<font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">nb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">oII</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:24_2"><span class="lab"><font color="Green" title="E31">A47</font></span></a>, <a class="txt" href="#E26:24_2"><span class="lab"><font color="Green" title="E36">A58</font></span></a>, <a class="txt" href="#E28:24_2"><span class="lab"><font color="Green" title="E37">A59</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E39:24_2"/>
 for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c11">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">nb</font>,<font color="Maroon" title="c6">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24_2"><span class="lab"><font color="Green" title="E26">A28</font></span></a></span>; <a class="txt" 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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Maroon" title="c6">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">pI</font>,<font color="Olive" title="b1">a</font> ) &amp; (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">pII</font>,<font color="Maroon" title="c6">b</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E40:24_2"/>
 for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Maroon" title="c6">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">pI</font>,<font color="Olive" title="b1">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E32:24_2"><span class="lab"><font color="Green" title="E40">A62</font></span></a>, <a class="txt" href="#E34:24_2"><span class="lab"><font color="Green" title="E41">A66</font></span></a></span>; <a class="txt" 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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">pII</font>,<font color="Maroon" title="c6">b</font></span><br/>

<span class="kw">thus </span><a NAME="E41:24_2"/>
 for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">pII</font>,<font color="Maroon" title="c6">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:24_2"><span class="lab"><font color="Green" title="E30">A46</font></span></a>, <a class="txt" href="#E16:24_2"><span class="lab"><font color="Green" title="E31">A47</font></span></a>, <a class="txt" href="#E17:24_2"><span class="lab"><font color="Green" title="E32">A48</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:24"/><span class="lab"><font color="Green" title="E23">A67</font></span>: 
 for <font color="Olive" title="b1">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  ex <font color="Olive" title="b2">nb</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> ex <font color="Olive" title="b3">pI</font>, <font color="Olive" title="b4">pII</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>) st <br/>( (  for <font color="Olive" title="b5">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Olive" title="b5">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">nb</font> holds <br/><font color="Olive" title="b3">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font> ) &amp; (  for <font color="Olive" title="b5">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font>,<font color="Olive" title="b1">b</font>] ) &amp; (  for <font color="Olive" title="b5">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b1">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">pI</font>,<font color="Olive" title="b5">a</font> ) &amp; (  for <font color="Olive" title="b5">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b5">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">pII</font>,<font color="Olive" title="b1">b</font> ) &amp; (  for <font color="Olive" title="b5">p</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>)<br/>  for <font color="Olive" title="b6">a</font>, <font color="Olive" title="b7">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> &amp; <font color="Olive" title="b7">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> &amp; <font color="Olive" title="b6">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b5">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">nb</font>,<font color="Olive" title="b7">c</font> holds <br/><font color="Olive" title="b6">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">p</font>,<font color="Olive" title="b7">c</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">nb</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> ex <font color="Olive" title="b2">pI</font>, <font color="Olive" title="b3">pII</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>) st <br/>( (  for <font color="Olive" title="b4">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Olive" title="b4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">nb</font> holds <br/><font color="Olive" title="b2">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">i</font> ) &amp; (  for <font color="Olive" title="b4">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">i</font>,<font color="Maroon" title="c4">b</font>] ) &amp; (  for <font color="Olive" title="b4">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">pI</font>,<font color="Olive" title="b4">a</font> ) &amp; (  for <font color="Olive" title="b4">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Olive" title="b4">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">pII</font>,<font color="Maroon" title="c4">b</font> ) &amp; (  for <font color="Olive" title="b4">p</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>)<br/>  for <font color="Olive" title="b5">a</font>, <font color="Olive" title="b6">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Olive" title="b6">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Olive" title="b5">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b4">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">nb</font>,<font color="Olive" title="b6">c</font> holds <br/><font color="Olive" title="b5">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">p</font>,<font color="Olive" title="b6">c</font> ) )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">nb</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>, <font color="Maroon" title="c6">pI</font>, <font color="Maroon" title="c7">pII</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>)<span class="kw"> such that </span><br/><a NAME="E2:24_3"/><span class="lab"><font color="Green" title="E24">A68</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">nb</font> holds <br/><font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>
 <span class="kw">and </span><br/><a NAME="E3:24_3"/><span class="lab"><font color="Green" title="E25">A69</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">b</font>]
 <span class="kw">and </span><br/><a NAME="E4:24_3"/><span class="lab"><font color="Green" title="E26">A70</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">b</font>]
 <span class="kw">and </span><br/><a NAME="E5:24_3"/><span class="lab"><font color="Green" title="E27">A71</font></span>: 
 for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">nb</font>,<font color="Olive" title="b1">a</font>
 <span class="kw">and </span><br/><a NAME="E6:24_3"/><span class="lab"><font color="Green" title="E28">A72</font></span>: 
 for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">nb</font>,<font color="Maroon" title="c4">b</font>
 <span class="kw">and </span><br/><a NAME="E7:24_3"/><span class="lab"><font color="Green" title="E29">A73</font></span>: 
( (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">pI</font>,<font color="Olive" title="b1">a</font> ) &amp; (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">pII</font>,<font color="Maroon" title="c4">b</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24"><span class="lab"><font color="Green" title="E22">A24</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">nb</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">pI</font>, <font color="Olive" title="b2">pII</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>) st <br/>( (  for <font color="Olive" title="b3">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Olive" title="b3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">nb</font> holds <br/><font color="Olive" title="b1">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">i</font> ) &amp; (  for <font color="Olive" title="b3">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">i</font>,<font color="Maroon" title="c4">b</font>] ) &amp; (  for <font color="Olive" title="b3">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">pI</font>,<font color="Olive" title="b3">a</font> ) &amp; (  for <font color="Olive" title="b3">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">pII</font>,<font color="Maroon" title="c4">b</font> ) &amp; (  for <font color="Olive" title="b3">p</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>)<br/>  for <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Olive" title="b5">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Olive" title="b4">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b3">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">nb</font>,<font color="Olive" title="b5">c</font> holds <br/><font color="Olive" title="b4">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">p</font>,<font color="Olive" title="b5">c</font> ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c6">pI</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">pII</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>) st <br/>( (  for <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Olive" title="b2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">nb</font> holds <br/><font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">i</font> ) &amp; (  for <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">i</font>,<font color="Maroon" title="c4">b</font>] ) &amp; (  for <font color="Olive" title="b2">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">pI</font>,<font color="Olive" title="b2">a</font> ) &amp; (  for <font color="Olive" title="b2">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">pII</font>,<font color="Maroon" title="c4">b</font> ) &amp; (  for <font color="Olive" title="b2">p</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>)<br/>  for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Olive" title="b4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">nb</font>,<font color="Olive" title="b4">c</font> holds <br/><font color="Olive" title="b3">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font>,<font color="Olive" title="b4">c</font> ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c7">pII</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">nb</font> holds <br/><font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> ) &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">b</font>] ) &amp; (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">pI</font>,<font color="Olive" title="b1">a</font> ) &amp; (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">pII</font>,<font color="Maroon" title="c4">b</font> ) &amp; (  for <font color="Olive" title="b1">p</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>)<br/>  for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Olive" title="b3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">nb</font>,<font color="Olive" title="b3">c</font> holds <br/><font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">c</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E8:24_3"/>
( (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">nb</font> holds <br/><font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> ) &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">b</font>] ) &amp; (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">pI</font>,<font color="Olive" title="b1">a</font> ) &amp; (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">pII</font>,<font color="Maroon" title="c4">b</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_3"><span class="lab"><font color="Green" title="E24">A68</font></span></a>, <a class="txt" href="#E3:24_3"><span class="lab"><font color="Green" title="E25">A69</font></span></a>, <a class="txt" href="#E7:24_3"><span class="lab"><font color="Green" title="E29">A73</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>)<br/>  for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Olive" title="b3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">nb</font>,<font color="Olive" title="b3">c</font> holds <br/><font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">c</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">p</font> be    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>); <a class="txt" 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>, <font color="Olive" title="b2">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Olive" title="b2">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">nb</font>,<font color="Olive" title="b2">c</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font>,<font color="Olive" title="b2">c</font></span><br/><span class="kw">let </span><font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c10">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c10">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">nb</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c10">c</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E9:24_3"/><span class="lab"><font color="Green" title="E30">A74</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">and </span><br/><a NAME="E10:24_3"/><span class="lab"><font color="Green" title="E31">A75</font></span>: 
<font color="Maroon" title="c10">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">and </span><br/><a NAME="E11:24_3"/><span class="lab"><font color="Green" title="E32">A76</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">nb</font>,<font color="Maroon" title="c10">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c10">c</font></span><br/>

<a NAME="E12:24_3"/><span class="lab"><font color="Green" title="E33">A77</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:24_3"><span class="lab"><font color="Green" title="E32">A76</font></span></a>, <a class="ref" href="arrow.html#T3" target="_self" title="ARROW:th.3">Th3</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font>] <span class="kw">means </span>( ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c9">a</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1,<font color="Maroon" title="c9">a</font> ) &amp; ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">nb</font> implies ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c10">c</font> ) ) &amp; ( $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">nb</font> implies ( ( (  for <font color="Olive" title="b1">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1,<font color="Olive" title="b1">d</font> ) implies ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c9">a</font> &amp; <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c10">c</font> ) ) &amp; ( (  for <font color="Olive" title="b1">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Olive" title="b1">d</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1,<font color="Maroon" title="c4">b</font> ) implies ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> $2,<font color="Maroon" title="c4">b</font> ) ) ) ) );<br/>
<a NAME="E13:24_3"/><span class="lab"><font color="Green" title="E34">A78</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  ex <font color="Olive" title="b2">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">o</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">i</font>,<font color="Olive" title="b1">o</font>]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:24_3_1_1"/>
( <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">nb</font> or <font color="Maroon" title="c11">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">nb</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:24_3_1_1_1"/><span class="lab"><font color="Green" title="E35">A79</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">nb</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">i</font>,<font color="Olive" title="b1">o</font>]</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c12">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E3:24_3_1_1_1"/><span class="lab"><font color="Green" title="E36">A80</font></span>: 
( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c10">c</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24_3"><span class="lab"><font color="Green" title="E30">A74</font></span></a>, <a class="txt" href="#E10:24_3"><span class="lab"><font color="Green" title="E31">A75</font></span></a>, <a class="txt" href="#E12:24_3"><span class="lab"><font color="Green" title="E33">A77</font></span></a>, <a class="ref" href="arrow.html#T7" target="_self" title="ARROW:th.7">Th7</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c12">o</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">o</font>]</span><br/><span class="kw">thus </span><a NAME="E4:24_3_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">o</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:24_3"><span class="lab"><font color="Green" title="E32">A76</font></span></a>, <a class="txt" href="#E1:24_3_1_1_1"><span class="lab"><font color="Green" title="E35">A79</font></span></a>, <a class="txt" href="#E3:24_3_1_1_1"><span class="lab"><font color="Green" title="E36">A80</font></span></a>, <a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a>, <a class="ref" href="arrow.html#T5" target="_self" title="ARROW:th.5">Th5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:24_3_1_1_2"/><span class="lab"><font color="Green" title="E35">A81</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">nb</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">i</font>,<font color="Olive" title="b1">o</font>]</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_3_1_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:24_3_1_1_2_1"/>
(  for <font color="Olive" title="b1">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Olive" title="b1">d</font> or  for <font color="Olive" title="b1">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Olive" title="b1">d</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c4">b</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_3"><span class="lab"><font color="Green" title="E26">A70</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_3_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:24_3_1_1_2_1_1"/>
 for <font color="Olive" title="b1">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Olive" title="b1">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">i</font>,<font color="Olive" title="b1">o</font>]</span><br/><div class="add"><a NAME="E2:24_3_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24_3"><span class="lab"><font color="Green" title="E30">A74</font></span></a></span>;<br/><a NAME="E3:24_3_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A82</font></span>: 
 not <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E5:24_3_1_1_2_1_1"/><span class="lab"><font color="Green" title="E37">A83</font></span>: 
( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c9">a</font> &amp; <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c10">c</font> &amp; ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c9">a</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c9">a</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24_3"><span class="lab"><font color="Green" title="E30">A74</font></span></a>, <a class="txt" href="#E10:24_3"><span class="lab"><font color="Green" title="E31">A75</font></span></a>, <a class="ref" href="arrow.html#T10" target="_self" title="ARROW:th.10">Th10</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c12">o</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">o</font>]</span><br/><span class="kw">thus </span><a NAME="E6:24_3_1_1_2_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">o</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24_3"><span class="lab"><font color="Green" title="E30">A74</font></span></a>, <a class="txt" href="#E1:24_3_1_1_2"><span class="lab"><font color="Green" title="E35">A81</font></span></a>, <a class="txt" href="#E3:24_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E36">A82</font></span></a>, <a class="txt" href="#E5:24_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E37">A83</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_3_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:24_3_1_1_2_1_2"/>
 for <font color="Olive" title="b1">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Olive" title="b1">d</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">i</font>,<font color="Olive" title="b1">o</font>]</span><br/><div class="add"><a NAME="E2:24_3_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24_3"><span class="lab"><font color="Green" title="E30">A74</font></span></a></span>;<br/><a NAME="E3:24_3_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A84</font></span>: 
 not <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c9">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E5:24_3_1_1_2_1_2"/><span class="lab"><font color="Green" title="E37">A85</font></span>: 
( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c4">b</font> &amp; ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c9">a</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c9">a</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24_3"><span class="lab"><font color="Green" title="E30">A74</font></span></a>, <a class="txt" href="#E10:24_3"><span class="lab"><font color="Green" title="E31">A75</font></span></a>, <a class="ref" href="arrow.html#T11" target="_self" title="ARROW:th.11">Th11</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c12">o</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">o</font>]</span><br/><span class="kw">thus </span><a NAME="E6:24_3_1_1_2_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">o</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24_3"><span class="lab"><font color="Green" title="E30">A74</font></span></a>, <a class="txt" href="#E1:24_3_1_1_2"><span class="lab"><font color="Green" title="E35">A81</font></span></a>, <a class="txt" href="#E3:24_3_1_1_2_1_2"><span class="lab"><font color="Green" title="E36">A84</font></span></a>, <a class="txt" href="#E5:24_3_1_1_2_1_2"><span class="lab"><font color="Green" title="E37">A85</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c11">pIII</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E15:24_3"/><span class="lab"><font color="Green" title="E35">A86</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c11">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E13:24_3"><span class="lab"><font color="Green" title="E34">A78</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c12">pIII</font> = <font color="Maroon" title="c11">pIII</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>) <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T8" title="FUNCT_2:th.8">FUNCT_2:8</a></span>;<br/>
<a NAME="E17:24_3"/>
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds <br/> ( ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c9">a</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c9">a</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> ) )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_3_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:24_3_2_1"/>
( <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">nb</font> or <font color="Maroon" title="c13">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">nb</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_3_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:24_3_2_1_1"/>
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">nb</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> ) )</span><br/><div class="add"><a NAME="E2:24_3_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24_3"><span class="lab"><font color="Green" title="E28">A72</font></span></a>, <a class="txt" href="#E9:24_3"><span class="lab"><font color="Green" title="E30">A74</font></span></a>, <a class="txt" href="#E15:24_3"><span class="lab"><font color="Green" title="E35">A86</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:24_3_2_1_1"/>
( ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_3_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:24_3_2_1_2"/><span class="lab"><font color="Green" title="E36">A87</font></span>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">nb</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> ) )</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_3_2_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:24_3_2_1_2_1"/>
(  for <font color="Olive" title="b1">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Olive" title="b1">d</font> or  for <font color="Olive" title="b1">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Olive" title="b1">d</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_3"><span class="lab"><font color="Green" title="E26">A70</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_3_2_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:24_3_2_1_2_1_1"/>
 for <font color="Olive" title="b1">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Olive" title="b1">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> ) )</span><br/><div class="add"><a NAME="E2:24_3_2_1_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> &amp; <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24_3"><span class="lab"><font color="Green" title="E30">A74</font></span></a>, <a class="txt" href="#E15:24_3"><span class="lab"><font color="Green" title="E35">A86</font></span></a>, <a class="txt" href="#E1:24_3_2_1_2"><span class="lab"><font color="Green" title="E36">A87</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:24_3_2_1_2_1_1"/>
( ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_3_2_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:24_3_2_1_2_1_2"/>
 for <font color="Olive" title="b1">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Olive" title="b1">d</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> ) )</span><br/><div class="add"><a NAME="E2:24_3_2_1_2_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24_3"><span class="lab"><font color="Green" title="E30">A74</font></span></a>, <a class="txt" href="#E15:24_3"><span class="lab"><font color="Green" title="E35">A86</font></span></a>, <a class="txt" href="#E1:24_3_2_1_2"><span class="lab"><font color="Green" title="E36">A87</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:24_3_2_1_2_1_2"/>
( ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c9">a</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E18:24_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A88</font></span>: 
( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">pII</font>,<font color="Maroon" title="c4">b</font> iff <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">pIII</font>,<font color="Maroon" title="c4">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E19">A2</font></span></a></span>;<br/>
<a NAME="E19:24_3"/>
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds <br/> ( ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">b</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_3_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:24_3_3_1"/>
( <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">nb</font> or <font color="Maroon" title="c13">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">nb</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_3_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:24_3_3_1_1"/>
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">nb</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) )</span><br/><div class="add"><a NAME="E2:24_3_3_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> &amp; <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_3"><span class="lab"><font color="Green" title="E27">A71</font></span></a>, <a class="txt" href="#E10:24_3"><span class="lab"><font color="Green" title="E31">A75</font></span></a>, <a class="txt" href="#E15:24_3"><span class="lab"><font color="Green" title="E35">A86</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:24_3_3_1_1"/>
( ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_3_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:24_3_3_1_2"/><span class="lab"><font color="Green" title="E37">A89</font></span>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">nb</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) )</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_3_3_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:24_3_3_1_2_1"/>
(  for <font color="Olive" title="b1">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Olive" title="b1">d</font> or  for <font color="Olive" title="b1">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Olive" title="b1">d</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_3"><span class="lab"><font color="Green" title="E26">A70</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_3_3_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:24_3_3_1_2_1_1"/><span class="lab"><font color="Green" title="E38">A90</font></span>: 
 for <font color="Olive" title="b1">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Olive" title="b1">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) )</span><br/><div class="add"><a NAME="E2:24_3_3_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:24_3"><span class="lab"><font color="Green" title="E31">A75</font></span></a></span>;<br/><a NAME="E3:24_3_3_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A91</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_3"><span class="lab"><font color="Green" title="E24">A68</font></span></a>, <a class="txt" href="#E1:24_3_3_1_2"><span class="lab"><font color="Green" title="E37">A89</font></span></a></span>;<br/><a NAME="E4:24_3_3_1_2_1_1"/>
<font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:24_3"><span class="lab"><font color="Green" title="E35">A86</font></span></a>, <a class="txt" href="#E1:24_3_3_1_2"><span class="lab"><font color="Green" title="E37">A89</font></span></a>, <a class="txt" href="#E1:24_3_3_1_2_1_1"><span class="lab"><font color="Green" title="E38">A90</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:24_3_3_1_2_1_1"/>
( ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_3_3_1_2_1_1"><span class="lab"><font color="Green" title="E39">A91</font></span></a>, <a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_3_3_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:24_3_3_1_2_1_2"/><span class="lab"><font color="Green" title="E38">A92</font></span>: 
 for <font color="Olive" title="b1">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Olive" title="b1">d</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) )</span><br/><div class="add"><a NAME="E2:24_3_3_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:24_3"><span class="lab"><font color="Green" title="E31">A75</font></span></a></span>;<br/><a NAME="E3:24_3_3_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A93</font></span>: 
<font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_3"><span class="lab"><font color="Green" title="E24">A68</font></span></a>, <a class="txt" href="#E1:24_3_3_1_2"><span class="lab"><font color="Green" title="E37">A89</font></span></a></span>;<br/><a NAME="E4:24_3_3_1_2_1_2"/>
<font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:24_3"><span class="lab"><font color="Green" title="E35">A86</font></span></a>, <a class="txt" href="#E1:24_3_3_1_2"><span class="lab"><font color="Green" title="E37">A89</font></span></a>, <a class="txt" href="#E1:24_3_3_1_2_1_2"><span class="lab"><font color="Green" title="E38">A92</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:24_3_3_1_2_1_2"/>
( ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> implies <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c10">c</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) &amp; ( <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c12">pIII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c10">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c4">b</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_3_3_1_2_1_2"><span class="lab"><font color="Green" title="E39">A93</font></span></a>, <a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E20:24_3"/><span class="kw">then </span>
( <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">pI</font>,<font color="Maroon" title="c10">c</font> iff <font color="Maroon" title="c4">b</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">pIII</font>,<font color="Maroon" title="c10">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E19">A2</font></span></a></span>;<br/>
<a NAME="E21:24_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">pIII</font>,<font color="Maroon" title="c10">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:24_3"><span class="lab"><font color="Green" title="E29">A73</font></span></a>, <a class="txt" href="#E9:24_3"><span class="lab"><font color="Green" title="E30">A74</font></span></a>, <a class="txt" href="#E18:24_3"><span class="lab"><font color="Green" title="E36">A88</font></span></a>, <a class="ref" href="arrow.html#T5" target="_self" title="ARROW:th.5">Th5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:24_3"/>
<font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c10">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E15:24_3"><span class="lab"><font color="Green" title="E35">A86</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">set </span><font color="Maroon" title="c4">b</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">nb</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>, <font color="Maroon" title="c6">pI</font>, <font color="Maroon" title="c7">pII</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>)<span class="kw"> such that </span><br/><a NAME="E8:24"/><span class="lab"><font color="Green" title="E24">A94</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">nb</font> holds <br/><font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>
 <span class="kw">and </span><br/><a NAME="E9:24"/><span class="lab"><font color="Green" title="E25">A95</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>, the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>]
 <span class="kw">and </span><br/><a NAME="E10:24"/><span class="lab"><font color="Green" title="E26">A96</font></span>: 
( (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> holds <br/> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">pI</font>,<font color="Olive" title="b1">a</font> ) &amp; (  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">pII</font>, the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ) )
 <span class="kw">and </span><br/><a NAME="E11:24"/><span class="lab"><font color="Green" title="E27">A97</font></span>: 
 for <font color="Olive" title="b1">p</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>)<br/>  for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> &amp; <font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">nb</font>,<font color="Olive" title="b3">c</font> holds <br/><font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E23">A67</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">nb</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>)<br/>  for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">nb</font>,<font color="Olive" title="b3">b</font> holds <br/><font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">b</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">p</font> be    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>); <a class="txt" 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>, <font color="Olive" title="b2">b</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">nb</font>,<font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font>,<font color="Olive" title="b2">b</font></span><br/><span class="kw">let </span><font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c10">a9</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">nb</font>,<font color="Maroon" title="c10">a9</font> implies <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c10">a9</font> )</span><br/>





<span class="kw">assume </span><a NAME="E12:24"/><span class="lab"><font color="Green" title="E28">A98</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c8">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">nb</font>,<font color="Maroon" title="c10">a9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c10">a9</font></span><br/>

<a NAME="E13:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A99</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_4"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:24_4"/>
( ( <font color="Maroon" title="c9">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c10">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ) or <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c10">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_4_1"><span class="kw">suppose </span></a><a NAME="E1:24_4_1"/>
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c10">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c10">a9</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:24_4_1"/>
<font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c10">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:24"><span class="lab"><font color="Green" title="E27">A97</font></span></a>, <a class="txt" href="#E12:24"><span class="lab"><font color="Green" title="E28">A98</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_4_2"><span class="kw">suppose </span></a><a NAME="E1:24_4_2"/><span class="lab"><font color="Green" title="E30">A100</font></span>: 
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c10">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c10">a9</font></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c11">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E3:24_4_2"/><span class="lab"><font color="Green" title="E31">A101</font></span>: 
( <font color="Maroon" title="c11">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">a</font> &amp; <font color="Maroon" title="c11">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">a9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="ref" href="arrow.html#T2" target="_self" title="ARROW:th.2">Th2</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">nc</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>, <font color="Maroon" title="c13">pI9</font>, <font color="Maroon" title="c14">pII9</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>)<span class="kw"> such that </span><br/><a NAME="E5:24_4_2"/>
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">nc</font> holds <br/><font color="Maroon" title="c13">pI9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">pII9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>
 <span class="kw">and </span><br/><a NAME="E6:24_4_2"/>
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">pI9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c11">c</font>]
 <span class="kw">and </span><br/><a NAME="E7:24_4_2"/>
 for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c</font> holds <br/><font color="Maroon" title="c11">c</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">pI9</font>,<font color="Olive" title="b1">a</font>
 <span class="kw">and </span><br/><a NAME="E8:24_4_2"/>
 for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c</font> holds <br/><font color="Olive" title="b1">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">pII9</font>,<font color="Maroon" title="c11">c</font>
 <span class="kw">and </span><br/><a NAME="E9:24_4_2"/><span class="lab"><font color="Green" title="E32">A102</font></span>: 
 for <font color="Olive" title="b1">p</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">N</font>,<span class="p1">(<span class="default"><a href="arrow.html#K2" title="ARROW:func.2">LinPreorders</a> <font color="Maroon" title="c1">A</font></span>)</span>)<br/>  for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">a9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c</font> &amp; <font color="Olive" title="b3">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c</font> &amp; <font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">nc</font>,<font color="Olive" title="b3">a9</font> holds <br/><font color="Olive" title="b2">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E23">A67</font></span></a></span>;<br/><a NAME="E10:24_4_2"/>
<font color="Maroon" title="c12">nc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">nb</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_4_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_4_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:24_4_2_1_1"/>
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c10">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_4_2"><span class="lab"><font color="Green" title="E30">A100</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_4_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:24_4_2_1_1_1"/><span class="lab"><font color="Green" title="E33">A103</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">nc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">nb</font></span><br/><div class="add"><span class="kw">assume </span><a NAME="E2:24_4_2_1_1_1"/><span class="lab"><font color="Green" title="E34">A104</font></span>: 
<font color="Maroon" title="c12">nc</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">nb</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:24_4_2_1_1_1"/>
(  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">nc</font>,<font color="Maroon" title="c10">a9</font> or <font color="Maroon" title="c10">a9</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">nc</font>, the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24"><span class="lab"><font color="Green" title="E25">A95</font></span></a>, <a class="txt" href="#E13:24"><span class="lab"><font color="Green" title="E29">A99</font></span></a>, <a class="txt" href="#E1:24_4_2_1_1_1"><span class="lab"><font color="Green" title="E33">A103</font></span></a></span>;<br/><a NAME="E4:24_4_2_1_1_1"/><span class="kw">then </span>
( (  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">nc</font>,<font color="Maroon" title="c10">a9</font> &amp; <font color="Maroon" title="c10">a9</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">pII</font>, the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ) or ( <font color="Maroon" title="c10">a9</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">nc</font>, the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> &amp;  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">pI</font>,<font color="Maroon" title="c10">a9</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24"><span class="lab"><font color="Green" title="E24">A94</font></span></a>, <a class="txt" href="#E10:24"><span class="lab"><font color="Green" title="E26">A96</font></span></a>, <a class="txt" href="#E13:24"><span class="lab"><font color="Green" title="E29">A99</font></span></a>, <a class="txt" href="#E1:24_4_2_1_1_1"><span class="lab"><font color="Green" title="E33">A103</font></span></a>, <a class="txt" href="#E2:24_4_2_1_1_1"><span class="lab"><font color="Green" title="E34">A104</font></span></a></span>;<br/><a NAME="E5:24_4_2_1_1_1"/><span class="kw">then </span>
( (  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">nc</font>,<font color="Maroon" title="c10">a9</font> &amp; <font color="Maroon" title="c10">a9</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">pII</font>, the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ) or ( <font color="Maroon" title="c10">a9</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">nc</font>, the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> &amp;  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">pI</font>,<font color="Maroon" title="c10">a9</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a></span>;<br/><span class="kw">hence </span><a NAME="E6:24_4_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_4_2"><span class="lab"><font color="Green" title="E31">A101</font></span></a>, <a class="txt" href="#E9:24_4_2"><span class="lab"><font color="Green" title="E32">A102</font></span></a>, <a class="txt" href="#E1:24_4_2_1_1_1"><span class="lab"><font color="Green" title="E33">A103</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_4_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:24_4_2_1_1_2"/><span class="lab"><font color="Green" title="E33">A105</font></span>: 
<font color="Maroon" title="c10">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">nc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">nb</font></span><br/><div class="add"><span class="kw">assume </span><a NAME="E2:24_4_2_1_1_2"/><span class="lab"><font color="Green" title="E34">A106</font></span>: 
<font color="Maroon" title="c12">nc</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">nb</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:24_4_2_1_1_2"/>
(  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">nc</font>,<font color="Maroon" title="c9">a</font> or <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">nc</font>, the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24"><span class="lab"><font color="Green" title="E25">A95</font></span></a>, <a class="txt" href="#E13:24"><span class="lab"><font color="Green" title="E29">A99</font></span></a>, <a class="txt" href="#E1:24_4_2_1_1_2"><span class="lab"><font color="Green" title="E33">A105</font></span></a></span>;<br/><a NAME="E4:24_4_2_1_1_2"/><span class="kw">then </span>
( (  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">nc</font>,<font color="Maroon" title="c9">a</font> &amp; <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">pII</font>, the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ) or ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">nc</font>, the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> &amp;  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">pI</font>,<font color="Maroon" title="c9">a</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24"><span class="lab"><font color="Green" title="E24">A94</font></span></a>, <a class="txt" href="#E10:24"><span class="lab"><font color="Green" title="E26">A96</font></span></a>, <a class="txt" href="#E13:24"><span class="lab"><font color="Green" title="E29">A99</font></span></a>, <a class="txt" href="#E1:24_4_2_1_1_2"><span class="lab"><font color="Green" title="E33">A105</font></span></a>, <a class="txt" href="#E2:24_4_2_1_1_2"><span class="lab"><font color="Green" title="E34">A106</font></span></a></span>;<br/><a NAME="E5:24_4_2_1_1_2"/><span class="kw">then </span>
( (  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c7">pII</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">nc</font>,<font color="Maroon" title="c9">a</font> &amp; <font color="Maroon" title="c9">a</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">pII</font>, the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> ) or ( <font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c6">pI</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">nc</font>, the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> &amp;  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> <a href="arrow.html#R1" title="ARROW:pred.1">&lt;=_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">pI</font>,<font color="Maroon" title="c9">a</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arrow.html#T4" target="_self" title="ARROW:th.4">Th4</a></span>;<br/><span class="kw">hence </span><a NAME="E6:24_4_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_4_2"><span class="lab"><font color="Green" title="E31">A101</font></span></a>, <a class="txt" href="#E9:24_4_2"><span class="lab"><font color="Green" title="E32">A102</font></span></a>, <a class="txt" href="#E1:24_4_2_1_1_2"><span class="lab"><font color="Green" title="E33">A105</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E11:24_4_2"/>
<font color="Maroon" title="c9">a</font> <a href="arrow.html#NR3" title="ARROW:NR.3">&lt;_</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c10">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:24"><span class="lab"><font color="Green" title="E28">A98</font></span></a>, <a class="txt" href="#E3:24_4_2"><span class="lab"><font color="Green" title="E31">A101</font></span></a>, <a class="txt" href="#E9:24_4_2"><span class="lab"><font color="Green" title="E32">A102</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
