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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   <a href="gtarski1.html#V2" title="GTARSKI1:attr.2">satisfying_CongruenceSymmetry</a>   <a href="gtarski1.html#V3" title="GTARSKI1:attr.3">satisfying_CongruenceEquivalenceRelation</a>   <a href="gtarski1.html#V4" title="GTARSKI1:attr.4">satisfying_CongruenceIdentity</a>   <a href="gtarski1.html#L1" title="GTARSKI1:struct.1">TarskiGeometryStruct</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">d</font> being   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font>  st ( <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font> or <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Olive" title="b4">b</font>,<font color="Olive" title="b3">a</font> or <font color="Olive" title="b2">q</font>,<font color="Olive" title="b1">p</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font> or <font color="Olive" title="b2">q</font>,<font color="Olive" title="b1">p</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Olive" title="b4">b</font>,<font color="Olive" title="b3">a</font> ) &amp; ( <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Olive" title="b5">c</font>,<font color="Olive" title="b6">d</font> or <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Olive" title="b6">d</font>,<font color="Olive" title="b5">c</font> or <font color="Olive" title="b2">q</font>,<font color="Olive" title="b1">p</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Olive" title="b5">c</font>,<font color="Olive" title="b6">d</font> or <font color="Olive" title="b2">q</font>,<font color="Olive" title="b1">p</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Olive" title="b6">d</font>,<font color="Olive" title="b5">c</font> ) holds <br/>( <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Olive" title="b6">d</font>,<font color="Olive" title="b5">c</font> &amp; <font color="Olive" title="b4">b</font>,<font color="Olive" title="b3">a</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Olive" title="b5">c</font>,<font color="Olive" title="b6">d</font> &amp; <font color="Olive" title="b4">b</font>,<font color="Olive" title="b3">a</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Olive" title="b6">d</font>,<font color="Olive" title="b5">c</font> &amp; <font color="Olive" title="b5">c</font>,<font color="Olive" title="b6">d</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font> &amp; <font color="Olive" title="b6">d</font>,<font color="Olive" title="b5">c</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font> &amp; <font color="Olive" title="b5">c</font>,<font color="Olive" title="b6">d</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Olive" title="b4">b</font>,<font color="Olive" title="b3">a</font> &amp; <font color="Olive" title="b6">d</font>,<font color="Olive" title="b5">c</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Olive" title="b4">b</font>,<font color="Olive" title="b3">a</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c7">d</font> be   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> or <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font> or <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c2">p</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> or <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c2">p</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font> ) &amp; ( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font> or <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c6">c</font> or <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c2">p</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font> or <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c2">p</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c6">c</font> ) implies ( <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c6">c</font> &amp; <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font> &amp; <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c6">c</font> &amp; <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c6">c</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c6">c</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:5"/>
( ( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> or <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font> or <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c2">p</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> or <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c2">p</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font> ) &amp; ( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font> or <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c6">c</font> or <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c2">p</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font> or <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c2">p</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c6">c</font> ) )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c6">c</font> &amp; <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font> &amp; <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c6">c</font> &amp; <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c6">c</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c6">c</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font> )</span><br/>

<a NAME="E2:5"/><span class="kw">then </span>
( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski4.html#T1" target="_self" onmouseover="rs('gtarski4/T1')" onmouseout="rh()">Prelim01</a></span>;<br/>
<a NAME="E3:5"/><span class="kw">then </span>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski1.html#D6" onmouseover="rs('gtarski1/D6')" onmouseout="rh()">GTARSKI1:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:5"/>
( <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c6">c</font> &amp; <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font> &amp; <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c6">c</font> &amp; <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c6">c</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c6">c</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski4.html#T1" target="_self" onmouseover="rs('gtarski4/T1')" onmouseout="rh()">Prelim01</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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