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<b>let </b><font color="Maroon" title="c3">S1</font>, <font color="Maroon" title="c4">S2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pcs_0.html#L1" title="PCS_0:struct.1">TolStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="pcs_0.html#K8" title="PCS_0:func.8">[^</a><span class="default"><font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font></span><a href="pcs_0.html#K8" title="PCS_0:func.8">^]</a></span> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b2">y</font> iff ( <font color="Olive" title="b1">x</font> <a href="pcs_0.html#K10" title="PCS_0:func.10">^`1</a>  <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b2">y</font> <a href="pcs_0.html#K10" title="PCS_0:func.10">^`1</a>  or <font color="Olive" title="b1">x</font> <a href="pcs_0.html#K11" title="PCS_0:func.11">^`2</a>  <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Olive" title="b2">y</font> <a href="pcs_0.html#K11" title="PCS_0:func.11">^`2</a>  ) )</span><br/><b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="pcs_0.html#K8" title="PCS_0:func.8">[^</a><span class="default"><font color="Maroon" title="c3">S1</font>,<font color="Maroon" title="c4">S2</font></span><a href="pcs_0.html#K8" title="PCS_0:func.8">^]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c6">y</font> iff ( <font color="Maroon" title="c5">x</font> <a href="pcs_0.html#K10" title="PCS_0:func.10">^`1</a>  <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c6">y</font> <a href="pcs_0.html#K10" title="PCS_0:func.10">^`1</a>  or <font color="Maroon" title="c5">x</font> <a href="pcs_0.html#K11" title="PCS_0:func.11">^`2</a>  <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c6">y</font> <a href="pcs_0.html#K11" title="PCS_0:func.11">^`2</a>  ) )</span><br/>



<a NAME="E1:75"/><i><font color="Green" title="E27">A1</font></i>: 
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S1</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">S2</font> &amp; <font color="Maroon" title="c5">x</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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E2:75"/>
 ex <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S1</font> &amp; <font color="Olive" title="b2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">S2</font> &amp; <font color="Maroon" title="c6">y</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="Olive" title="b1">c</font>,<font color="Olive" title="b2">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E3:75"/><b>then </b>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="pcs_0.html#K10" title="PCS_0:func.10">^`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="pcs_0.html#K11" title="PCS_0:func.11">^`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="pcs_0.html#K10" title="PCS_0:func.10">^`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="pcs_0.html#K11" title="PCS_0:func.11">^`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 
<b>by </b><i><a class="txt" href="pcs_0.html#E1:75"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="mcart_1.html#T8" title="MCART_1:th.8">MCART_1:8</a></i>;<br/>
<b>hence </b><a NAME="E4:75"/>
( <font color="Maroon" title="c5">x</font> <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c6">y</font> iff ( <font color="Maroon" title="c5">x</font> <a href="pcs_0.html#K10" title="PCS_0:func.10">^`1</a>  <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c6">y</font> <a href="pcs_0.html#K10" title="PCS_0:func.10">^`1</a>  or <font color="Maroon" title="c5">x</font> <a href="pcs_0.html#K11" title="PCS_0:func.11">^`2</a>  <a href="pcs_0.html#R1" title="PCS_0:pred.1">(--)</a> <font color="Maroon" title="c6">y</font> <a href="pcs_0.html#K11" title="PCS_0:func.11">^`2</a>  ) )
 <b>by </b><i><a class="ref" href="pcs_0.html#T6" target="_self" title="PCS_0:th.6">Th6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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