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<span class="kw">let </span><font color="Maroon" title="c1">D</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> holds   <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>,<font color="Olive" title="b2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Olive" title="b2">p2</font>,<font color="Olive" title="b2">p2</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>,<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c3">p2</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span><br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_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="E1:112_1"/>
( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> or <font color="Maroon" title="c2">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_1_1"><span class="kw">suppose </span></a><a NAME="E1:112_1_1"/>
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>,<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c3">p2</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:112_1_1"/>
 <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>,<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c3">p2</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_6.html#T101" target="_self" title="FINSEQ_6:th.101">Th101</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_1_2"><span class="kw">suppose </span></a><a NAME="E1:112_1_2"/><span class="lab"><font color="Green" title="E92">A1</font></span>: 
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>,<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c3">p2</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span><br/><div class="add"><a NAME="E2:112_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="finseq_6.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/><a NAME="E3:112_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><span class="kw">hence </span> <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>,<font color="Maroon" title="c3">p2</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="finseq_5.html#K2" title="FINSEQ_5:func.2">:-</a> <font color="Maroon" title="c3">p2</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p2</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_6.html#D2" target="_self" title="FINSEQ_6:def.2">Def2</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="finseq_5.html#K1" title="FINSEQ_5:func.1">-:</a> <font color="Maroon" title="c3">p2</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_2"><span class="lab"><font color="Green" title="E92">A1</font></span></a>, <a class="ref" href="finseq_6.html#T57" target="_self" title="FINSEQ_6:th.57">Th57</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_2"><span class="lab"><font color="Green" title="E92">A1</font></span></a>, <a class="ref" href="finseq_6.html#T53" target="_self" title="FINSEQ_6:th.53">Th53</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">p2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_6.html#T50" target="_self" title="FINSEQ_6:th.50">Th50</a></span>
<br/>.= 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c3">p2</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D9" title="FINSEQ_1:def.9">FINSEQ_1:def 9</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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