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<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><br/> for <font color="Olive" title="b2">H</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  st ( for <font color="Olive" title="b3">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">H</font> holds <br/>( <font color="Olive" title="b3">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> &amp;  ex <font color="Olive" title="b4">X1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b5">Y1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> st <font color="Olive" title="b3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b4">X1</font>,<font color="Olive" title="b5">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> ) ) holds <br/><span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p5">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b2">H</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p5">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b2">H</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">H</font> being  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  st ( for <font color="Olive" title="b2">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">H</font> holds <br/>( <font color="Olive" title="b2">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> &amp;  ex <font color="Olive" title="b3">X1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b4">Y1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> st <font color="Olive" title="b2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b3">X1</font>,<font color="Olive" title="b4">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> ) ) holds <br/><span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p5">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b1">H</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p5">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b1">H</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">H</font> be   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">H</font> holds <br/>( <font color="Olive" title="b1">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> &amp;  ex <font color="Olive" title="b2">X1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b3">Y1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> st <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b2">X1</font>,<font color="Olive" title="b3">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> ) ) implies <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p5">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p5">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:14"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">H</font> holds <br/>( <font color="Olive" title="b1">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> &amp;  ex <font color="Olive" title="b2">X1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ex <font color="Olive" title="b3">Y1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> st <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b2">X1</font>,<font color="Olive" title="b3">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p5">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p5">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">u</font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p5">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p5">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">u</font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p5">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p5">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )</span><br/>



<span class="kw">assume </span><a NAME="E2:14"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p5">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p5">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c4">H</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font></span><br/>

<a NAME="E3:14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p3">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c4">H</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E4:14"/>
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p3">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c4">H</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:14"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">and </span><br/><a NAME="E7:14"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c4">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:14"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:14"/><span class="lab"><font color="Green" title="E12">A6</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">H</font>
 <span class="kw">and </span><br/><a NAME="E11:14"/><span class="lab"><font color="Green" title="E13">A7</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:14"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">X1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c10">Y1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font><span class="kw"> such that </span><br/><a NAME="E13:14"/><span class="lab"><font color="Green" title="E14">A8</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c9">X1</font>,<font color="Maroon" title="c10">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E10:14"><span class="lab"><font color="Green" title="E12">A6</font></span></a></span>;<br/>
<a NAME="E14:14"/><span class="lab"><font color="Green" title="E15">A9</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:14"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="txt" href="#E11:14"><span class="lab"><font color="Green" title="E13">A7</font></span></a>, <a class="ref" href="funct_3.html#T9" title="FUNCT_3:th.9">FUNCT_3:9</a></span>;<br/>
<a NAME="E15:14"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:14"><span class="lab"><font color="Green" title="E12">A6</font></span></a></span>;<br/>
<a NAME="E16:14"/><span class="kw">then </span>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D6" title="FUNCT_3:def.6">FUNCT_3:def 6</a></span>;<br/>
<a NAME="E17:14"/><span class="kw">then </span>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D1" title="FUNCT_3:def.1">FUNCT_3:def 1</a></span>;<br/>
<a NAME="E18:14"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c4">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:14"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/>
<a NAME="E19:14"/><span class="kw">then </span>
<font color="Maroon" title="c10">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_3.html#K1" title="FUNCT_3:func.1">.:</a> <span class="p2">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c4">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:14"><span class="lab"><font color="Green" title="E14">A8</font></span></a>, <a class="txt" href="#E14:14"><span class="lab"><font color="Green" title="E15">A9</font></span></a>, <a class="ref" href="borsuk_1.html#T13" target="_self" title="BORSUK_1:th.13">Th13</a></span>;<br/>
<a NAME="E20:14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A10</font></span>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></span>;<br/>
<a NAME="E21:14"/>
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:14"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="txt" href="#E11:14"><span class="lab"><font color="Green" title="E13">A7</font></span></a>, <a class="txt" href="#E13:14"><span class="lab"><font color="Green" title="E14">A8</font></span></a>, <a class="txt" href="#E14:14"><span class="lab"><font color="Green" title="E15">A9</font></span></a>, <a class="ref" href="borsuk_1.html#T13" target="_self" title="BORSUK_1:th.13">Th13</a></span>;<br/>
<a NAME="E22:14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A11</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:14"><span class="lab"><font color="Green" title="E14">A8</font></span></a>, <a class="txt" href="#E20:14"><span class="lab"><font color="Green" title="E16">A10</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E23:14"/>
<font color="Maroon" title="c8">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E10:14"><span class="lab"><font color="Green" title="E12">A6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:14"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">v</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:14"><span class="lab"><font color="Green" title="E17">A11</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
