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<span class="kw">let </span><font color="Maroon" title="c1">f</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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">O</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="waybel_0.html#V16" title="WAYBEL_0:attr.16">connected</a>  <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a><br/>  for <font color="Olive" title="b2">T</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="exchsort.html#NM3" title="EXCHSORT:NM.3">array</a> of <font color="Olive" title="b1">O</font><br/>  for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b2">T</font>  st <font color="Olive" title="b3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">q</font> &amp; <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Olive" title="b2">T</font>,<font color="Olive" title="b3">p</font>,<font color="Olive" title="b4">q</font>) <a href="exchsort.html#K7" title="EXCHSORT:func.7">incl</a>  holds <br/> for <font color="Olive" title="b5">x1</font>, <font color="Olive" title="b6">x2</font>, <font color="Olive" title="b7">y1</font>, <font color="Olive" title="b8">y2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b2">T</font>  st <font color="Olive" title="b5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b7">y1</font> &amp; <font color="Olive" title="b6">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b8">y2</font> &amp; <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b5">x1</font>,<font color="Olive" title="b7">y1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b6">x2</font>,<font color="Olive" title="b8">y2</font>) holds <br/>( <font color="Olive" title="b5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">x2</font> &amp; <font color="Olive" title="b7">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">y2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">O</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="waybel_0.html#V16" title="WAYBEL_0:attr.16">connected</a>  <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</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">T</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="exchsort.html#NM3" title="EXCHSORT:NM.3">array</a> of <font color="Maroon" title="c2">O</font><br/>  for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">T</font>  st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">q</font> &amp; <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Olive" title="b1">T</font>,<font color="Olive" title="b2">p</font>,<font color="Olive" title="b3">q</font>) <a href="exchsort.html#K7" title="EXCHSORT:func.7">incl</a>  holds <br/> for <font color="Olive" title="b4">x1</font>, <font color="Olive" title="b5">x2</font>, <font color="Olive" title="b6">y1</font>, <font color="Olive" title="b7">y2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">T</font>  st <font color="Olive" title="b4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b6">y1</font> &amp; <font color="Olive" title="b5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b7">y2</font> &amp; <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b4">x1</font>,<font color="Olive" title="b6">y1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b5">x2</font>,<font color="Olive" title="b7">y2</font>) holds <br/>( <font color="Olive" title="b4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">x2</font> &amp; <font color="Olive" title="b6">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">y2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">T</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="exchsort.html#NM3" title="EXCHSORT:NM.3">array</a> of <font color="Maroon" title="c2">O</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>, <font color="Olive" title="b2">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">T</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">q</font> &amp; <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon" title="c3">T</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font>) <a href="exchsort.html#K7" title="EXCHSORT:func.7">incl</a>  holds <br/> for <font color="Olive" title="b3">x1</font>, <font color="Olive" title="b4">x2</font>, <font color="Olive" title="b5">y1</font>, <font color="Olive" title="b6">y2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">T</font>  st <font color="Olive" title="b3">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">y1</font> &amp; <font color="Olive" title="b4">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b6">y2</font> &amp; <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b3">x1</font>,<font color="Olive" title="b5">y1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b4">x2</font>,<font color="Olive" title="b6">y2</font>) holds <br/>( <font color="Olive" title="b3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">x2</font> &amp; <font color="Olive" title="b5">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">y2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">q</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">T</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">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>) <a href="exchsort.html#K7" title="EXCHSORT:func.7">incl</a>  implies  for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">y1</font>, <font color="Olive" title="b4">y2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">T</font>  st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">y1</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">y2</font> &amp; <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">x1</font>,<font color="Olive" title="b3">y1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b4">y2</font>) holds <br/>( <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font> &amp; <font color="Olive" title="b3">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">y2</font> ) )</span><br/>









<span class="kw">assume </span><a NAME="E1:112"/><span class="lab"><font color="Green" title="E64">Z0</font></span>: 
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>) <a href="exchsort.html#K7" title="EXCHSORT:func.7">incl</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">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">y1</font>, <font color="Olive" title="b4">y2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">T</font>  st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">y1</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">y2</font> &amp; <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">x1</font>,<font color="Olive" title="b3">y1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">x2</font>,<font color="Olive" title="b4">y2</font>) holds <br/>( <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font> &amp; <font color="Olive" title="b3">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">y2</font> )</span><br/>

<a NAME="E2:112"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">Z1</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q</font>
 
;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">x2</font>, <font color="Maroon" title="c8">y1</font>, <font color="Maroon" title="c9">y2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">T</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">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">y1</font> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">y2</font> &amp; <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c8">y1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) implies ( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:112"/><span class="lab"><font color="Green" title="E66">Z2</font></span>: 
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">y1</font> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">and </span><br/><a NAME="E4:112"/><span class="lab"><font color="Green" title="E67">Z4</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c8">y1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c10">X</font> =  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">T</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">i</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">T</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">s</font> =  <a href="exchsort.html#K5" title="EXCHSORT:func.5">Swap</a> (<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">T</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c13">h</font> = <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><span class="p2">(<span class="default"><a href="exchsort.html#K5" title="EXCHSORT:func.5">Swap</a> (<span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">T</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="exchsort.html#K5" title="EXCHSORT:func.5">Swap</a> (<span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">T</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>)</span>)</span></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">Y</font> = <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">p</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">Z1</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">Z2</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">g</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></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="c6">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c6">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q</font> ) or ( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">q</font> ) or ( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> ) or ( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">x1</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> ) or ( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> ) or ( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> ) or ( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">y1</font> ) or ( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">y1</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E66">Z2</font></span></a>, <a class="ref" href="exchsort.html#T2" target="_self" title="EXCHSORT:th.2">Eight</a></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"/><span class="lab"><font color="Green" title="E68">C1</font></span>: 
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c6">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">D1</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c8">y1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c8">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T67" target="_self" title="EXCHSORT:th.67">Case1</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_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="E1:112_1_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>8</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E66">Z2</font></span></a>, <a class="ref" href="exchsort.html#T2" target="_self" title="EXCHSORT:th.2">Eight</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:112_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T67" target="_self" title="EXCHSORT:th.67">Case1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_1_1_1"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_1"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:112_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c5">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T68" target="_self" title="EXCHSORT:th.68">Case23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_1_1_2"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_1"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_1"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:112_1_1_1_3"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c4">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T68" target="_self" title="EXCHSORT:th.68">Case23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_1_1_3"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_1"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_1"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:112_1_1_1_4"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E66">Z2</font></span></a>, <a class="ref" href="exchsort.html#T69" target="_self" title="EXCHSORT:th.69">Case47</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_1_1_4"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_1"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_1_1_5"><span class="kw">suppose </span></a><a NAME="E1:112_1_1_1_5"/>
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T66" target="_self" title="EXCHSORT:th.66">Case5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_1_1_5"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_1"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_1_1_6"><span class="kw">suppose </span></a><a NAME="E1:112_1_1_1_6"/>
<font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_1_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T70" target="_self" title="EXCHSORT:th.70">Case68</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_1_1_6"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_1"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_1"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_1_1_7"><span class="kw">suppose </span></a><a NAME="E1:112_1_1_1_7"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_1_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E66">Z2</font></span></a>, <a class="ref" href="exchsort.html#T69" target="_self" title="EXCHSORT:th.69">Case47</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_1_1_7"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_1"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_1_1_8"><span class="kw">suppose </span></a><a NAME="E1:112_1_1_1_8"/>
<font color="Maroon">S<sub>8</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_1_1_8"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T70" target="_self" title="EXCHSORT:th.70">Case68</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_1_1_8"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_1"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_1"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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><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="E68">C1</font></span>: 
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">q</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">D1</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c8">y1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c8">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E66">Z2</font></span></a>, <a class="ref" href="exchsort.html#T69" target="_self" title="EXCHSORT:th.69">Case47</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_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="E1:112_1_2_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>8</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E66">Z2</font></span></a>, <a class="ref" href="exchsort.html#T2" target="_self" title="EXCHSORT:th.2">Eight</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:112_1_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T67" target="_self" title="EXCHSORT:th.67">Case1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_2_1_1"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_2"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:112_1_2_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_2_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c5">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T68" target="_self" title="EXCHSORT:th.68">Case23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_2_1_2"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_2"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_2"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:112_1_2_1_3"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_2_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c4">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T68" target="_self" title="EXCHSORT:th.68">Case23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_2_1_3"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_2"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_2"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_2_1_4"><span class="kw">suppose </span></a><a NAME="E1:112_1_2_1_4"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_2_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E66">Z2</font></span></a>, <a class="ref" href="exchsort.html#T69" target="_self" title="EXCHSORT:th.69">Case47</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_2_1_4"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_2"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_2_1_5"><span class="kw">suppose </span></a><a NAME="E1:112_1_2_1_5"/>
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_2_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T66" target="_self" title="EXCHSORT:th.66">Case5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_2_1_5"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_2"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_2_1_6"><span class="kw">suppose </span></a><a NAME="E1:112_1_2_1_6"/>
<font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_2_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T70" target="_self" title="EXCHSORT:th.70">Case68</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_2_1_6"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:112"><span class="lab"><font color="Green" title="E65">Z1</font></span></a>, <a class="txt" href="#E1:112_1_2"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_2"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_2_1_7"><span class="kw">suppose </span></a><a NAME="E1:112_1_2_1_7"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_2_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E66">Z2</font></span></a>, <a class="ref" href="exchsort.html#T69" target="_self" title="EXCHSORT:th.69">Case47</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_2_1_7"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_2"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_2_1_8"><span class="kw">suppose </span></a><a NAME="E1:112_1_2_1_8"/><span class="lab"><font color="Green" title="E70">C2</font></span>: 
<font color="Maroon">S<sub>8</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_2_1_8"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T70" target="_self" title="EXCHSORT:th.70">Case68</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_2_1_8"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_2"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E1:112_1_2_1_8"><span class="lab"><font color="Green" title="E70">C2</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_2"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_1_3"><span class="kw">suppose </span></a><a NAME="E1:112_1_3"/><span class="lab"><font color="Green" title="E68">C1</font></span>: 
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">D1</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c8">y1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c8">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T66" target="_self" title="EXCHSORT:th.66">Case5</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_1_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="E1:112_1_3_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>8</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E66">Z2</font></span></a>, <a class="ref" href="exchsort.html#T2" target="_self" title="EXCHSORT:th.2">Eight</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:112_1_3_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T67" target="_self" title="EXCHSORT:th.67">Case1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_3_1_1"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_3"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:112_1_3_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_3_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c5">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T68" target="_self" title="EXCHSORT:th.68">Case23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_3_1_2"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_3"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_3"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_3_1_3"><span class="kw">suppose </span></a><a NAME="E1:112_1_3_1_3"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_3_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c4">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T68" target="_self" title="EXCHSORT:th.68">Case23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_3_1_3"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:112"><span class="lab"><font color="Green" title="E65">Z1</font></span></a>, <a class="txt" href="#E1:112_1_3"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_3"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_3_1_4"><span class="kw">suppose </span></a><a NAME="E1:112_1_3_1_4"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_3_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E66">Z2</font></span></a>, <a class="ref" href="exchsort.html#T69" target="_self" title="EXCHSORT:th.69">Case47</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_3_1_4"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_3"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_3_1_5"><span class="kw">suppose </span></a><a NAME="E1:112_1_3_1_5"/>
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_3_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T66" target="_self" title="EXCHSORT:th.66">Case5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_3_1_5"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_3"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_3_1_6"><span class="kw">suppose </span></a><a NAME="E1:112_1_3_1_6"/>
<font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_3_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T70" target="_self" title="EXCHSORT:th.70">Case68</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_3_1_6"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:112"><span class="lab"><font color="Green" title="E65">Z1</font></span></a>, <a class="txt" href="#E1:112_1_3"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_3"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_3_1_7"><span class="kw">suppose </span></a><a NAME="E1:112_1_3_1_7"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_3_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E66">Z2</font></span></a>, <a class="ref" href="exchsort.html#T69" target="_self" title="EXCHSORT:th.69">Case47</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_3_1_7"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_3"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_3_1_8"><span class="kw">suppose </span></a><a NAME="E1:112_1_3_1_8"/>
<font color="Maroon">S<sub>8</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_3_1_8"/><span class="kw">then </span>
( <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c9">y2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T70" target="_self" title="EXCHSORT:th.70">Case68</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_3_1_8"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_3"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_3"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_1_4"><span class="kw">suppose </span></a><a NAME="E1:112_1_4"/><span class="lab"><font color="Green" title="E68">C1</font></span>: 
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">x1</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">D1</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c8">y1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c8">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E66">Z2</font></span></a>, <a class="ref" href="exchsort.html#T69" target="_self" title="EXCHSORT:th.69">Case47</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_1_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="E1:112_1_4_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>8</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E66">Z2</font></span></a>, <a class="ref" href="exchsort.html#T2" target="_self" title="EXCHSORT:th.2">Eight</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_1_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:112_1_4_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T67" target="_self" title="EXCHSORT:th.67">Case1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_4_1_1"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_4"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:112_1_4_1_2"/><span class="lab"><font color="Green" title="E70">C2</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c5">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T68" target="_self" title="EXCHSORT:th.68">Case23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_4_1_2"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_4_1_2"><span class="lab"><font color="Green" title="E70">C2</font></span></a>, <a class="txt" href="#E1:112_1_4"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_4"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_4_1_3"><span class="kw">suppose </span></a><a NAME="E1:112_1_4_1_3"/><span class="lab"><font color="Green" title="E70">C2</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_4_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c4">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T68" target="_self" title="EXCHSORT:th.68">Case23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_4_1_3"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_4_1_3"><span class="lab"><font color="Green" title="E70">C2</font></span></a>, <a class="txt" href="#E1:112_1_4"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_4"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_4_1_4"><span class="kw">suppose </span></a><a NAME="E1:112_1_4_1_4"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_4_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E66">Z2</font></span></a>, <a class="ref" href="exchsort.html#T69" target="_self" title="EXCHSORT:th.69">Case47</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_4_1_4"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_4"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_4_1_5"><span class="kw">suppose </span></a><a NAME="E1:112_1_4_1_5"/>
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_4_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T66" target="_self" title="EXCHSORT:th.66">Case5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_4_1_5"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_4"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_4_1_6"><span class="kw">suppose </span></a><a NAME="E1:112_1_4_1_6"/>
<font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_4_1_6"/><span class="kw">then </span>
( <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c9">y2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T70" target="_self" title="EXCHSORT:th.70">Case68</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_4_1_6"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_4"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_4"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_4_1_7"><span class="kw">suppose </span></a><a NAME="E1:112_1_4_1_7"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_4_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E66">Z2</font></span></a>, <a class="ref" href="exchsort.html#T69" target="_self" title="EXCHSORT:th.69">Case47</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_4_1_7"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_4"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_4_1_8"><span class="kw">suppose </span></a><a NAME="E1:112_1_4_1_8"/>
<font color="Maroon">S<sub>8</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_4_1_8"/><span class="kw">then </span>
( <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c9">y2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T70" target="_self" title="EXCHSORT:th.70">Case68</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_4_1_8"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_4"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_4"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_1_5"><span class="kw">suppose </span></a><a NAME="E1:112_1_5"/><span class="lab"><font color="Green" title="E68">C1</font></span>: 
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">D1</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c8">y1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c5">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T68" target="_self" title="EXCHSORT:th.68">Case23</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_1_5_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_5_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>8</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E66">Z2</font></span></a>, <a class="ref" href="exchsort.html#T2" target="_self" title="EXCHSORT:th.2">Eight</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_1_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:112_1_5_1_1"/><span class="lab"><font color="Green" title="E70">C2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_5_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T67" target="_self" title="EXCHSORT:th.67">Case1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_5_1_1"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_5_1_1"><span class="lab"><font color="Green" title="E70">C2</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_5"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:112_1_5_1_2"/><span class="lab"><font color="Green" title="E70">C2</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_5_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c5">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T68" target="_self" title="EXCHSORT:th.68">Case23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_5_1_2"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_5_1_2"><span class="lab"><font color="Green" title="E70">C2</font></span></a>, <a class="txt" href="#E1:112_1_5"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_5"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_5_1_3"><span class="kw">suppose </span></a><a NAME="E1:112_1_5_1_3"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_5_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c4">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T68" target="_self" title="EXCHSORT:th.68">Case23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_5_1_3"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:112"><span class="lab"><font color="Green" title="E65">Z1</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_5"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_5_1_4"><span class="kw">suppose </span></a><a NAME="E1:112_1_5_1_4"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_5_1_4"/><span class="kw">then </span>
( <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">x1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_5"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E66">Z2</font></span></a>, <a class="ref" href="exchsort.html#T69" target="_self" title="EXCHSORT:th.69">Case47</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_5_1_4"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_5"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_5_1_5"><span class="kw">suppose </span></a><a NAME="E1:112_1_5_1_5"/>
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_5_1_5"/><span class="kw">then </span>
( <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">x1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_5"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T66" target="_self" title="EXCHSORT:th.66">Case5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_5_1_5"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_5"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_5_1_6"><span class="kw">suppose </span></a><a NAME="E1:112_1_5_1_6"/>
<font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_5_1_6"/><span class="kw">then </span>
( <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c9">y2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T70" target="_self" title="EXCHSORT:th.70">Case68</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_5_1_6"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_5"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_5_1_7"><span class="kw">suppose </span></a><a NAME="E1:112_1_5_1_7"/><span class="lab"><font color="Green" title="E70">C2</font></span>: 
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_5_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E66">Z2</font></span></a>, <a class="ref" href="exchsort.html#T69" target="_self" title="EXCHSORT:th.69">Case47</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_5_1_7"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_5"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E1:112_1_5_1_7"><span class="lab"><font color="Green" title="E70">C2</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_5"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_5_1_8"><span class="kw">suppose </span></a><a NAME="E1:112_1_5_1_8"/>
<font color="Maroon">S<sub>8</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_5_1_8"/><span class="kw">then </span>
( <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c6">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_5"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T70" target="_self" title="EXCHSORT:th.70">Case68</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_5_1_8"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_5"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_1_6"><span class="kw">suppose </span></a><a NAME="E1:112_1_6"/><span class="lab"><font color="Green" title="E68">C1</font></span>: 
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">D1</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c8">y1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c4">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T68" target="_self" title="EXCHSORT:th.68">Case23</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_1_6_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_6_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>8</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E66">Z2</font></span></a>, <a class="ref" href="exchsort.html#T2" target="_self" title="EXCHSORT:th.2">Eight</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_1_6_1_1"><span class="kw">suppose </span></a><a NAME="E1:112_1_6_1_1"/><span class="lab"><font color="Green" title="E70">C2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_6_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T67" target="_self" title="EXCHSORT:th.67">Case1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_6_1_1"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_6_1_1"><span class="lab"><font color="Green" title="E70">C2</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_6"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_6_1_2"><span class="kw">suppose </span></a><a NAME="E1:112_1_6_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_6_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c5">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T68" target="_self" title="EXCHSORT:th.68">Case23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_6_1_2"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:112"><span class="lab"><font color="Green" title="E65">Z1</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_6"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_6_1_3"><span class="kw">suppose </span></a><a NAME="E1:112_1_6_1_3"/><span class="lab"><font color="Green" title="E70">C2</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_6_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c4">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T68" target="_self" title="EXCHSORT:th.68">Case23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_6_1_3"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_6_1_3"><span class="lab"><font color="Green" title="E70">C2</font></span></a>, <a class="txt" href="#E1:112_1_6"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_6"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_6_1_4"><span class="kw">suppose </span></a><a NAME="E1:112_1_6_1_4"/><span class="lab"><font color="Green" title="E70">C2</font></span>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_6_1_4"/><span class="kw">then </span>
( <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c6">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_6"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E66">Z2</font></span></a>, <a class="ref" href="exchsort.html#T69" target="_self" title="EXCHSORT:th.69">Case47</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_6_1_4"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_6_1_4"><span class="lab"><font color="Green" title="E70">C2</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_6"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_6_1_5"><span class="kw">suppose </span></a><a NAME="E1:112_1_6_1_5"/><span class="lab"><font color="Green" title="E70">C2</font></span>: 
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_6_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T66" target="_self" title="EXCHSORT:th.66">Case5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_6_1_5"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_6_1_5"><span class="lab"><font color="Green" title="E70">C2</font></span></a>, <a class="txt" href="#E2:112"><span class="lab"><font color="Green" title="E65">Z1</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_6"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_6_1_6"><span class="kw">suppose </span></a><a NAME="E1:112_1_6_1_6"/>
<font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_6_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T70" target="_self" title="EXCHSORT:th.70">Case68</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_6_1_6"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E1:112_1_6"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_6"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_6_1_7"><span class="kw">suppose </span></a><a NAME="E1:112_1_6_1_7"/><span class="lab"><font color="Green" title="E70">C2</font></span>: 
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_6_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E66">Z2</font></span></a>, <a class="ref" href="exchsort.html#T69" target="_self" title="EXCHSORT:th.69">Case47</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_6_1_7"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:112"><span class="lab"><font color="Green" title="E65">Z1</font></span></a>, <a class="txt" href="#E1:112_1_6_1_7"><span class="lab"><font color="Green" title="E70">C2</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_6"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_6_1_8"><span class="kw">suppose </span></a><a NAME="E1:112_1_6_1_8"/>
<font color="Maroon">S<sub>8</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_6_1_8"/><span class="kw">then </span>
( <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c6">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_6"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T70" target="_self" title="EXCHSORT:th.70">Case68</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_6_1_8"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_6"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_1_7"><span class="kw">suppose </span></a><a NAME="E1:112_1_7"/><span class="lab"><font color="Green" title="E68">C1</font></span>: 
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">y1</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">D1</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c8">y1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c8">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T70" target="_self" title="EXCHSORT:th.70">Case68</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_1_7_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_7_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>8</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E66">Z2</font></span></a>, <a class="ref" href="exchsort.html#T2" target="_self" title="EXCHSORT:th.2">Eight</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_1_7_1_1"><span class="kw">suppose </span></a><a NAME="E1:112_1_7_1_1"/><span class="lab"><font color="Green" title="E70">C2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_7_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T67" target="_self" title="EXCHSORT:th.67">Case1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_7_1_1"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_7_1_1"><span class="lab"><font color="Green" title="E70">C2</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_7"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_7_1_2"><span class="kw">suppose </span></a><a NAME="E1:112_1_7_1_2"/><span class="lab"><font color="Green" title="E70">C2</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_7_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c5">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T68" target="_self" title="EXCHSORT:th.68">Case23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_7_1_2"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E1:112_1_7_1_2"><span class="lab"><font color="Green" title="E70">C2</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_7"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_7_1_3"><span class="kw">suppose </span></a><a NAME="E1:112_1_7_1_3"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_7_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c4">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T68" target="_self" title="EXCHSORT:th.68">Case23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_7_1_3"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E1:112_1_7"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_7"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_7_1_4"><span class="kw">suppose </span></a><a NAME="E1:112_1_7_1_4"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_7_1_4"/><span class="kw">then </span>
( <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E66">Z2</font></span></a>, <a class="ref" href="exchsort.html#T69" target="_self" title="EXCHSORT:th.69">Case47</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_7_1_4"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_7"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_7_1_5"><span class="kw">suppose </span></a><a NAME="E1:112_1_7_1_5"/><span class="lab"><font color="Green" title="E70">C2</font></span>: 
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_7_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T66" target="_self" title="EXCHSORT:th.66">Case5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_7_1_5"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:112"><span class="lab"><font color="Green" title="E65">Z1</font></span></a>, <a class="txt" href="#E1:112_1_7_1_5"><span class="lab"><font color="Green" title="E70">C2</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_7"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_7_1_6"><span class="kw">suppose </span></a><a NAME="E1:112_1_7_1_6"/><span class="lab"><font color="Green" title="E70">C2</font></span>: 
<font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_7_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T70" target="_self" title="EXCHSORT:th.70">Case68</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_7_1_6"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_7_1_6"><span class="lab"><font color="Green" title="E70">C2</font></span></a>, <a class="txt" href="#E1:112_1_7"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_7"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_7_1_7"><span class="kw">suppose </span></a><a NAME="E1:112_1_7_1_7"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_7_1_7"/><span class="kw">then </span>
( <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E66">Z2</font></span></a>, <a class="ref" href="exchsort.html#T69" target="_self" title="EXCHSORT:th.69">Case47</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_7_1_7"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_7"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_7_1_8"><span class="kw">suppose </span></a><a NAME="E1:112_1_7_1_8"/>
<font color="Maroon">S<sub>8</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_7_1_8"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T70" target="_self" title="EXCHSORT:th.70">Case68</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_7_1_8"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:112"><span class="lab"><font color="Green" title="E65">Z1</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_7"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_1_8"><span class="kw">suppose </span></a><a NAME="E1:112_1_8"/><span class="lab"><font color="Green" title="E68">C1</font></span>: 
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">y1</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">D1</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c8">y1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c8">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T70" target="_self" title="EXCHSORT:th.70">Case68</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_1_8_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_8_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] or <font color="Maroon">S<sub>8</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>] )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E66">Z2</font></span></a>, <a class="ref" href="exchsort.html#T2" target="_self" title="EXCHSORT:th.2">Eight</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_1_8_1_1"><span class="kw">suppose </span></a><a NAME="E1:112_1_8_1_1"/><span class="lab"><font color="Green" title="E70">C2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_8_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T67" target="_self" title="EXCHSORT:th.67">Case1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_8_1_1"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_8_1_1"><span class="lab"><font color="Green" title="E70">C2</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_8"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_8_1_2"><span class="kw">suppose </span></a><a NAME="E1:112_1_8_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_8_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c5">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">x2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T68" target="_self" title="EXCHSORT:th.68">Case23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_8_1_2"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_8"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_8_1_3"><span class="kw">suppose </span></a><a NAME="E1:112_1_8_1_3"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_8_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c4">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">x2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T68" target="_self" title="EXCHSORT:th.68">Case23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_8_1_3"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_8"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_8_1_4"><span class="kw">suppose </span></a><a NAME="E1:112_1_8_1_4"/><span class="lab"><font color="Green" title="E70">C2</font></span>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_8_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E66">Z2</font></span></a>, <a class="ref" href="exchsort.html#T69" target="_self" title="EXCHSORT:th.69">Case47</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_8_1_4"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_8"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E1:112_1_8_1_4"><span class="lab"><font color="Green" title="E70">C2</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_8"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_8_1_5"><span class="kw">suppose </span></a><a NAME="E1:112_1_8_1_5"/>
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_8_1_5"/><span class="kw">then </span>
( <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_8"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T66" target="_self" title="EXCHSORT:th.66">Case5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_8_1_5"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_8"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_8_1_6"><span class="kw">suppose </span></a><a NAME="E1:112_1_8_1_6"/>
<font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_8_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T70" target="_self" title="EXCHSORT:th.70">Case68</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_8_1_6"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:112"><span class="lab"><font color="Green" title="E65">Z1</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_8"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_8_1_7"><span class="kw">suppose </span></a><a NAME="E1:112_1_8_1_7"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_8_1_7"/><span class="kw">then </span>
( <font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c9">y2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">y1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_8"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E66">Z2</font></span></a>, <a class="ref" href="exchsort.html#T69" target="_self" title="EXCHSORT:th.69">Case47</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_8_1_7"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_8"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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="112_1_8_1_8"><span class="kw">suppose </span></a><a NAME="E1:112_1_8_1_8"/><span class="lab"><font color="Green" title="E70">C2</font></span>: 
<font color="Maroon">S<sub>8</sub></font>[<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/><div class="add"><a NAME="E2:112_1_8_1_8"/><span class="kw">then </span>
<font color="Maroon" title="c1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x2</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E64">Z0</font></span></a>, <a class="ref" href="exchsort.html#T70" target="_self" title="EXCHSORT:th.70">Case68</a></span>;<br/><span class="kw">hence </span><a NAME="E3:112_1_8_1_8"/>
( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1_8_1_8"><span class="lab"><font color="Green" title="E70">C2</font></span></a>, <a class="txt" href="#E1:112_1_8"><span class="lab"><font color="Green" title="E68">C1</font></span></a>, <a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E67">Z4</font></span></a>, <a class="txt" href="#E2:112_1_8"><span class="lab"><font color="Green" title="E69">D1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</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>

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