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<span class="kw">let </span><font color="Maroon" title="c1">A1</font>, <font color="Maroon" title="c2">A2</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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">Y1</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A1</font><br/> for <font color="Olive" title="b2">Y2</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">A2</font><br/> for <font color="Olive" title="b3">f</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</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">A1</font>,<font color="Maroon" title="c1">A1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">A1</font><br/> for <font color="Olive" title="b4">g</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">A2</font>,<font color="Maroon" title="c2">A2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c2">A2</font><br/> for <font color="Olive" title="b5">F</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">Y1</font>,<font color="Olive" title="b1">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<font color="Olive" title="b1">Y1</font>  st <font color="Olive" title="b5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Olive" title="b1">Y1</font> holds <br/>for <font color="Olive" title="b6">G</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b2">Y2</font>,<font color="Olive" title="b2">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<font color="Olive" title="b2">Y2</font>  st <font color="Olive" title="b6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">g</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Olive" title="b2">Y2</font> holds <br/><span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Olive" title="b5">F</font>,<font color="Olive" title="b6">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Olive" title="b3">f</font>,<font color="Olive" title="b4">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">Y1</font>,<font color="Olive" title="b2">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">Y1</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A1</font>; <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">Y2</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">A2</font><br/> for <font color="Olive" title="b2">f</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</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">A1</font>,<font color="Maroon" title="c1">A1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">A1</font><br/> for <font color="Olive" title="b3">g</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">A2</font>,<font color="Maroon" title="c2">A2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c2">A2</font><br/> for <font color="Olive" title="b4">F</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c3">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<font color="Maroon" title="c3">Y1</font>  st <font color="Olive" title="b4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">Y1</font> holds <br/>for <font color="Olive" title="b5">G</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">Y2</font>,<font color="Olive" title="b1">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<font color="Olive" title="b1">Y2</font>  st <font color="Olive" title="b5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Olive" title="b1">Y2</font> holds <br/><span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Olive" title="b4">F</font>,<font color="Olive" title="b5">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Olive" title="b2">f</font>,<font color="Olive" title="b3">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Olive" title="b1">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">Y2</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">A2</font>; <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">f</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</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">A1</font>,<font color="Maroon" title="c1">A1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">A1</font><br/> for <font color="Olive" title="b2">g</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">A2</font>,<font color="Maroon" title="c2">A2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c2">A2</font><br/> for <font color="Olive" title="b3">F</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c3">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<font color="Maroon" title="c3">Y1</font>  st <font color="Olive" title="b3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">Y1</font> holds <br/>for <font color="Olive" title="b4">G</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c4">Y2</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<font color="Maroon" title="c4">Y2</font>  st <font color="Olive" title="b4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c4">Y2</font> holds <br/><span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Olive" title="b3">F</font>,<font color="Olive" title="b4">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</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">A1</font>,<font color="Maroon" title="c1">A1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">A1</font>; <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">g</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">A2</font>,<font color="Maroon" title="c2">A2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c2">A2</font><br/> for <font color="Olive" title="b2">F</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c3">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<font color="Maroon" title="c3">Y1</font>  st <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">Y1</font> holds <br/>for <font color="Olive" title="b3">G</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c4">Y2</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<font color="Maroon" title="c4">Y2</font>  st <font color="Olive" title="b3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c4">Y2</font> holds <br/><span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Olive" title="b2">F</font>,<font color="Olive" title="b3">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Olive" title="b1">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">g</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">A2</font>,<font color="Maroon" title="c2">A2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c2">A2</font>; <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">F</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c3">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<font color="Maroon" title="c3">Y1</font>  st <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">Y1</font> holds <br/>for <font color="Olive" title="b2">G</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c4">Y2</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<font color="Maroon" title="c4">Y2</font>  st <font color="Olive" title="b2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c4">Y2</font> holds <br/><span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">F</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c3">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<font color="Maroon" title="c3">Y1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">Y1</font> implies for <font color="Olive" title="b1">G</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c4">Y2</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<font color="Maroon" title="c4">Y2</font>  st <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c4">Y2</font> holds <br/><span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Olive" title="b1">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> )</span><br/>













<span class="kw">assume </span><a NAME="E1:4"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">Y1</font>
 ; <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">G</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c4">Y2</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<font color="Maroon" title="c4">Y2</font>  st <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c4">Y2</font> holds <br/><span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Olive" title="b1">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span><br/>

<a NAME="E2:4"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:4"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">G</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c4">Y2</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<font color="Maroon" title="c4">Y2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c4">Y2</font> implies <span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E3:4"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
<font color="Maroon" title="c8">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c4">Y2</font>
 ; <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="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c9">X</font> =  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span>;<br/>
<a NAME="E4:4"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:4"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></span>;<br/>
<a NAME="E5:4"/><span class="lab"><font color="Green" title="E7">A5</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:4_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">x</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"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> or <font color="Maroon" title="c10">x</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"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:4_1_1"/>
<font color="Maroon" title="c10">x</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"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_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"> <font color="Maroon" title="c10">x</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"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">x11</font>, <font color="Maroon" title="c12">x21</font>, <font color="Maroon" title="c13">x12</font>, <font color="Maroon" title="c14">x22</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:4_1_1"/><span class="lab"><font color="Green" title="E8">A6</font></span>: 
<font color="Maroon" title="c10">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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x11</font>,<font color="Maroon" title="c13">x12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x21</font>,<font color="Maroon" title="c14">x22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:4_1_1"/><span class="lab"><font color="Green" title="E9">A7</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x11</font>,<font color="Maroon" title="c12">x21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">F</font>
 <span class="kw">and </span><br/><a NAME="E5:4_1_1"/><span class="lab"><font color="Green" title="E10">A8</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x12</font>,<font color="Maroon" title="c14">x22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D3" title="FUNCT_4:def.3">FUNCT_4:def 3</a></span>;<br/>
<a NAME="E6:4_1_1"/><span class="lab"><font color="Green" title="E11">A9</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c4">Y2</font>,<font color="Maroon" title="c4">Y2</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="#E3:4"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="ref" href="relat_1.html#T87" title="RELAT_1:th.87">RELAT_1:87</a></span>;<br/>
<a NAME="E7:4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A10</font></span>: 
<font color="Maroon" title="c13">x12</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:4_1_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E8:4_1_1"/><span class="lab"><font color="Green" title="E13">A11</font></span>: 
<font color="Maroon" title="c14">x22</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:4_1_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="txt" href="#E6:4_1_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E9:4_1_1"/><span class="lab"><font color="Green" title="E14">A12</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c3">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="#E1:4"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="ref" href="relat_1.html#T87" title="RELAT_1:th.87">RELAT_1:87</a></span>;<br/>
<a NAME="E10:4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">x21</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:4_1_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E11:4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A13</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x21</font>,<font color="Maroon" title="c14">x22</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="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</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="#E8:4_1_1"><span class="lab"><font color="Green" title="E13">A11</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E12:4_1_1"/>
<font color="Maroon" title="c11">x11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:4_1_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E9:4_1_1"><span class="lab"><font color="Green" title="E14">A12</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E13:4_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x11</font>,<font color="Maroon" title="c13">x12</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="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</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="#E7:4_1_1"><span class="lab"><font color="Green" title="E12">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="E14:4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A14</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></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="#E3:4_1_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E11:4_1_1"><span class="lab"><font color="Green" title="E15">A13</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E15:4_1_1"/>
<font color="Maroon" title="c10">x</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"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:4"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E4:4"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="txt" href="#E3:4_1_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E4:4_1_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E5:4_1_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="ref" href="funct_4.html#D3" title="FUNCT_4:def.3">FUNCT_4:def 3</a></span>;<br/>
<a NAME="E16:4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></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="#E14:4_1_1"><span class="lab"><font color="Green" title="E16">A14</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:4_1_1"/>
<font color="Maroon" title="c10">x</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"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</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><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c10">x</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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">x</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"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> or <font color="Maroon" title="c10">x</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"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> )</span><br/>

<a NAME="E2:4_1"/><span class="lab"><font color="Green" title="E8">A15</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c3">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="#E1:4"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:4_1"/>
<font color="Maroon" title="c10">x</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"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x</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"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span></span><br/>

<a NAME="E4:4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A16</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/>
<a NAME="E5:4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A17</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E6:4_1"/><span class="lab"><font color="Green" title="E11">A18</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c4">Y2</font>,<font color="Maroon" title="c4">Y2</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="#E3:4"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/>
<a NAME="E7:4_1"/>
<font color="Maroon" title="c10">x</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"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:4_1"><span class="lab"><font color="Green" title="E9">A16</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">x11</font>, <font color="Maroon" title="c12">x21</font>, <font color="Maroon" title="c13">x12</font>, <font color="Maroon" title="c14">x22</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:4_1"/><span class="lab"><font color="Green" title="E12">A19</font></span>: 
<font color="Maroon" title="c10">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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x11</font>,<font color="Maroon" title="c13">x12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x21</font>,<font color="Maroon" title="c14">x22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E10:4_1"/><span class="lab"><font color="Green" title="E13">A20</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x11</font>,<font color="Maroon" title="c12">x21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">and </span><br/><a NAME="E11:4_1"/><span class="lab"><font color="Green" title="E14">A21</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x12</font>,<font color="Maroon" title="c14">x22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D3" title="FUNCT_4:def.3">FUNCT_4:def 3</a></span>;<br/>
<a NAME="E12:4_1"/><span class="lab"><font color="Green" title="E15">A22</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x21</font>,<font color="Maroon" title="c14">x22</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="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</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="#E9:4_1"><span class="lab"><font color="Green" title="E12">A19</font></span></a>, <a class="txt" href="#E5:4_1"><span class="lab"><font color="Green" title="E10">A17</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E13:4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A23</font></span>: 
<font color="Maroon" title="c14">x22</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y2</font>
 
<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="E14:4_1"/><span class="lab"><font color="Green" title="E17">A24</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x11</font>,<font color="Maroon" title="c13">x12</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="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</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="#E9:4_1"><span class="lab"><font color="Green" title="E12">A19</font></span></a>, <a class="txt" href="#E5:4_1"><span class="lab"><font color="Green" title="E10">A17</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E15:4_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">x12</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y2</font>
 
<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="E16:4_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x12</font>,<font color="Maroon" title="c14">x22</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="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c4">Y2</font>,<font color="Maroon" title="c4">Y2</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="#E13:4_1"><span class="lab"><font color="Green" title="E16">A23</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E17:4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A25</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x12</font>,<font color="Maroon" title="c14">x22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:4_1"><span class="lab"><font color="Green" title="E14">A21</font></span></a>, <a class="txt" href="#E6:4_1"><span class="lab"><font color="Green" title="E11">A18</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E18:4_1"/><span class="lab"><font color="Green" title="E19">A26</font></span>: 
<font color="Maroon" title="c12">x21</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:4_1"><span class="lab"><font color="Green" title="E15">A22</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E19:4_1"/>
<font color="Maroon" title="c11">x11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:4_1"><span class="lab"><font color="Green" title="E17">A24</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E20:4_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x11</font>,<font color="Maroon" title="c12">x21</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="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c3">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="#E18:4_1"><span class="lab"><font color="Green" title="E19">A26</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E21:4_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x11</font>,<font color="Maroon" title="c12">x21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:4_1"><span class="lab"><font color="Green" title="E13">A20</font></span></a>, <a class="txt" href="#E2:4_1"><span class="lab"><font color="Green" title="E8">A15</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:4_1"/>
<font color="Maroon" title="c10">x</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"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:4_1"><span class="lab"><font color="Green" title="E12">A19</font></span></a>, <a class="txt" href="#E17:4_1"><span class="lab"><font color="Green" title="E18">A25</font></span></a>, <a class="ref" href="funct_4.html#D3" title="FUNCT_4:def.3">FUNCT_4:def 3</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><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E8">A27</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">x</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"> ( <font color="Maroon" title="c10">x</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"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> implies <span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:4_2"/><span class="lab"><font color="Green" title="E9">A28</font></span>: 
<font color="Maroon" title="c10">x</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"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_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="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">x11</font>, <font color="Maroon" title="c12">x21</font>, <font color="Maroon" title="c13">x12</font>, <font color="Maroon" title="c14">x22</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:4_2"/><span class="lab"><font color="Green" title="E10">A29</font></span>: 
<font color="Maroon" title="c10">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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x11</font>,<font color="Maroon" title="c13">x12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x21</font>,<font color="Maroon" title="c14">x22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:4_2"/><span class="lab"><font color="Green" title="E11">A30</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x11</font>,<font color="Maroon" title="c12">x21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">F</font>
 <span class="kw">and </span><br/><a NAME="E5:4_2"/><span class="lab"><font color="Green" title="E12">A31</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x12</font>,<font color="Maroon" title="c14">x22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D3" title="FUNCT_4:def.3">FUNCT_4:def 3</a></span>;<br/><span class="kw">thus </span><span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> = 
<span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x11</font>,<font color="Maroon" title="c13">x12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x21</font>,<font color="Maroon" title="c14">x22</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="#E3:4_2"><span class="lab"><font color="Green" title="E10">A29</font></span></a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">x11</font>,<font color="Maroon" title="c12">x21</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">x12</font>,<font color="Maroon" title="c14">x22</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:4_2"><span class="lab"><font color="Green" title="E11">A30</font></span></a>, <a class="txt" href="#E5:4_2"><span class="lab"><font color="Green" title="E12">A31</font></span></a>, <a class="ref" href="funct_4.html#D3" title="FUNCT_4:def.3">FUNCT_4:def 3</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x11</font>,<font color="Maroon" title="c12">x21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x12</font>,<font color="Maroon" title="c14">x22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:4"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E4:4_2"><span class="lab"><font color="Green" title="E11">A30</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">x11</font>,<font color="Maroon" title="c12">x21</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">x12</font>,<font color="Maroon" title="c14">x22</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:4"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E5:4_2"><span class="lab"><font color="Green" title="E12">A31</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x11</font>,<font color="Maroon" title="c13">x12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x21</font>,<font color="Maroon" title="c14">x22</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="#E2:4"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E4:4"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="txt" href="#E4:4_2"><span class="lab"><font color="Green" title="E11">A30</font></span></a>, <a class="txt" href="#E5:4_2"><span class="lab"><font color="Green" title="E12">A31</font></span></a>, <a class="ref" href="funct_4.html#D3" title="FUNCT_4:def.3">FUNCT_4:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:4"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E1:4_2"><span class="lab"><font color="Green" title="E9">A28</font></span></a>, <a class="txt" href="#E3:4_2"><span class="lab"><font color="Green" title="E10">A29</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</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><span class="kw">end;</span></div>
<a NAME="E7:4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A32</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>  st <font color="Olive" title="b1">x</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"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> holds <br/><span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
;<br/>
<a NAME="E8:4"/><span class="lab"><font color="Green" title="E10">A33</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:4_3"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:4_3"/><span class="lab"><font color="Green" title="E11">A34</font></span>: 
<font color="Maroon" title="c11">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"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:4_3"/><span class="lab"><font color="Green" title="E12">A35</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E5:4_3"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:4"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E6:4"><span class="lab"><font color="Green" title="E8">A27</font></span></a>, <a class="txt" href="#E3:4_3"><span class="lab"><font color="Green" title="E11">A34</font></span></a>, <a class="txt" href="#E4:4_3"><span class="lab"><font color="Green" title="E12">A35</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:4_3"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:4"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E3:4_3"><span class="lab"><font color="Green" title="E11">A34</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</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><span class="kw">end;</span></div>
<a NAME="E9:4"/><span class="lab"><font color="Green" title="E11">A36</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D18" title="RELAT_1:def.18">RELAT_1:def 18</a></span>;<br/>
<a NAME="E10:4"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</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="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<a NAME="E11:4"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</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="#E8:4"><span class="lab"><font color="Green" title="E10">A33</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E12:4"/><span class="kw">then </span>
<span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</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="#E9:4"><span class="lab"><font color="Green" title="E11">A36</font></span></a>, <a class="txt" href="#E5:4"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:4"/>
<span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</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="#E5:4"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E7:4"><span class="lab"><font color="Green" title="E9">A32</font></span></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</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/>


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