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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">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">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">a</font>,<font color="Olive" title="b2">b</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> holds <br/>( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Olive" title="b4">d</font>,<font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> &amp; <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Olive" title="b4">d</font>,<font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b4">d</font>,<font color="Olive" title="b3">c</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b4">d</font>,<font color="Olive" title="b3">c</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">c1</font>, <font color="Maroon" title="c3">c2</font>, <font color="Maroon" title="c4">c3</font>, <font color="Maroon" title="c5">c4</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">c1</font>,<font color="Maroon" title="c3">c2</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c5">c4</font> implies ( <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c4">c3</font> &amp; <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c2">c1</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c5">c4</font> &amp; <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c2">c1</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c4">c3</font> &amp; <font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c5">c4</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> &amp; <font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c4">c3</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> &amp; <font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c5">c4</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c2">c1</font> &amp; <font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c4">c3</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c2">c1</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:4"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c5">c4</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">c1</font>,<font color="Maroon" title="c3">c2</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c4">c3</font> &amp; <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c2">c1</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c5">c4</font> &amp; <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c2">c1</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c4">c3</font> &amp; <font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c5">c4</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> &amp; <font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c4">c3</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> &amp; <font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c5">c4</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c2">c1</font> &amp; <font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c4">c3</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c2">c1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:4"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
(  not <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c4">c3</font> or  not <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c2">c1</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c5">c4</font> or  not <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c2">c1</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c4">c3</font> or  not <font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c5">c4</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> or  not <font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c4">c3</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> or  not <font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c5">c4</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c2">c1</font> or  not <font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c4">c3</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c2">c1</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"> contradiction</span><br/>

<a NAME="E3:4"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c2">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski1.html#D5" onmouseover="rs('gtarski1/D5')" onmouseout="rh()">GTARSKI1:def 5</a></span>;<br/>
<a NAME="E4:4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A4</font></span>: 
<font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c5">c4</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c3">c2</font>,<font color="Maroon" title="c2">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:4"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="gtarski1.html#D6" onmouseover="rs('gtarski1/D6')" onmouseout="rh()">GTARSKI1:def 6</a></span>;<br/>
<a NAME="E5:4"/>
<font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski3.html#T1" onmouseover="rs('gtarski3/T1')" onmouseout="rh()">GTARSKI3:1</a></span>;<br/>
<a NAME="E6:4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A5</font></span>: 
<font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c5">c4</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:4"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="gtarski1.html#D6" onmouseover="rs('gtarski1/D6')" onmouseout="rh()">GTARSKI1:def 6</a></span>;<br/>
<a NAME="E7:4"/>
<font color="Maroon" title="c4">c3</font>,<font color="Maroon" title="c5">c4</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Maroon" title="c5">c4</font>,<font color="Maroon" title="c4">c3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gtarski1.html#D5" onmouseover="rs('gtarski1/D5')" onmouseout="rh()">GTARSKI1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:4"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E2:4"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E3:4"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E4:4"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E6:4"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="ref" href="gtarski1.html#D6" onmouseover="rs('gtarski1/D6')" onmouseout="rh()">GTARSKI1:def 6</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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