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<span class="kw">let </span><font color="Maroon" title="c1">D</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">X</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">Y</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st not <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> holds <br/>for <font color="Olive" title="b4">F</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b4">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Olive" title="b4">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Olive" title="b4">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  holds <br/>for <font color="Olive" title="b5">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font></span>)</span>,<font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b6">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Olive" title="b2">Y</font></span>)</span>,<font color="Maroon" title="c1">D</font>  st ( for <font color="Olive" title="b7">H</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font><br/> for <font color="Olive" title="b8">SF</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Olive" title="b2">Y</font></span>)</span>  st <font color="Olive" title="b8">SF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b9">h</font> where <font color="Olive" title="b9">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Olive" title="b2">Y</font> : <font color="Olive" title="b9">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">H</font> </span>}</span>  holds <br/><font color="Olive" title="b4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Olive" title="b8">SF</font>,<font color="Olive" title="b6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">H</font> ) holds <br/><font color="Olive" title="b4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font></span>)</span></span>)</span>,<font color="Olive" title="b5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Olive" title="b2">Y</font></span>)</span></span>)</span>,<font color="Olive" title="b6">g</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</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">Y</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st not <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/>for <font color="Olive" title="b3">F</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b3">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Olive" title="b3">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Olive" title="b3">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  holds <br/>for <font color="Olive" title="b4">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Olive" title="b1">Y</font></span>)</span>,<font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b5">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Olive" title="b1">Y</font></span>)</span>,<font color="Maroon" title="c1">D</font>  st ( for <font color="Olive" title="b6">H</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X</font>,<font color="Olive" title="b1">Y</font><br/> for <font color="Olive" title="b7">SF</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Olive" title="b1">Y</font></span>)</span>  st <font color="Olive" title="b7">SF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b8">h</font> where <font color="Olive" title="b8">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Olive" title="b1">Y</font> : <font color="Olive" title="b8">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">H</font> </span>}</span>  holds <br/><font color="Olive" title="b3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Olive" title="b7">SF</font>,<font color="Olive" title="b5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">H</font> ) holds <br/><font color="Olive" title="b3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Olive" title="b1">Y</font></span>)</span></span>)</span>,<font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Olive" title="b1">Y</font></span>)</span></span>)</span>,<font color="Olive" title="b5">g</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">Y</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</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">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/>for <font color="Olive" title="b2">F</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b2">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Olive" title="b2">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Olive" title="b2">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  holds <br/>for <font color="Olive" title="b3">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span>,<font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b4">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font></span>)</span>,<font color="Maroon" title="c1">D</font>  st ( for <font color="Olive" title="b5">H</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font><br/> for <font color="Olive" title="b6">SF</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font></span>)</span>  st <font color="Olive" title="b6">SF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b7">h</font> where <font color="Olive" title="b7">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font> : <font color="Olive" title="b7">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">H</font> </span>}</span>  holds <br/><font color="Olive" title="b2">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Olive" title="b6">SF</font>,<font color="Olive" title="b4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">H</font> ) holds <br/><font color="Olive" title="b2">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>,<font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>,<font color="Olive" title="b4">g</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">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"> ( not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> implies for <font color="Olive" title="b1">F</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b1">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Olive" title="b1">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Olive" title="b1">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  holds <br/>for <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span>,<font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b3">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font></span>)</span>,<font color="Maroon" title="c1">D</font>  st ( for <font color="Olive" title="b4">H</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font><br/> for <font color="Olive" title="b5">SF</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font></span>)</span>  st <font color="Olive" title="b5">SF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b6">h</font> where <font color="Olive" title="b6">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font> : <font color="Olive" title="b6">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">H</font> </span>}</span>  holds <br/><font color="Olive" title="b1">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Olive" title="b5">SF</font>,<font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">H</font> ) holds <br/><font color="Olive" title="b1">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>,<font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>,<font color="Olive" title="b3">g</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:80"/><span class="lab"><font color="Green" title="E68">A1</font></span>: 
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</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="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b1">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Olive" title="b1">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Olive" title="b1">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  holds <br/>for <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span>,<font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b3">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font></span>)</span>,<font color="Maroon" title="c1">D</font>  st ( for <font color="Olive" title="b4">H</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font><br/> for <font color="Olive" title="b5">SF</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font></span>)</span>  st <font color="Olive" title="b5">SF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b6">h</font> where <font color="Olive" title="b6">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font> : <font color="Olive" title="b6">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">H</font> </span>}</span>  holds <br/><font color="Olive" title="b1">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Olive" title="b5">SF</font>,<font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">H</font> ) holds <br/><font color="Olive" title="b1">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>,<font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>,<font color="Olive" title="b3">g</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">Xx</font> = <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">FXY</font> =  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">FXxY</font> =  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">B</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span>,<font color="Maroon" title="c3">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:80"/><span class="lab"><font color="Green" title="E69">A2</font></span>: 
<font color="Maroon" title="c8">B</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 <span class="kw">and </span><br/><a NAME="E4:80"/><span class="lab"><font color="Green" title="E70">A3</font></span>: 
for <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font><br/> for <font color="Olive" title="b2">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font>  st <font color="Olive" title="b2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> holds <br/><font color="Maroon" title="c8">B</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="ref" href="matrix11.html#T57" target="_self" title="MATRIX11:th.57">Th57</a></span>;<br/>
<a NAME="E5:80"/><span class="lab"><font color="Green" title="E71">A4</font></span>: 
 <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fraenkel.html#T16" title="FRAENKEL:th.16">FRAENKEL:16</a></span>;<br/>
<a NAME="E6:80"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span>,<font color="Maroon" title="c3">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">domB</font> =  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">B</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span>,<font color="Maroon" title="c3">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:80"><span class="lab"><font color="Green" title="E71">A4</font></span></a>, <a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></span>;<br/>
<a NAME="E8:80"/>
 <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fraenkel.html#T16" title="FRAENKEL:th.16">FRAENKEL:16</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">FXY9</font> =  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></span>;<br/>
<a NAME="E10:80"/><span class="lab"><font color="Green" title="E72">A5</font></span>: 
 <a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fraenkel.html#T16" title="FRAENKEL:th.16">FRAENKEL:16</a>, <a class="ref" href="matrix_2.html#D17" title="MATRIX_2:def.17">MATRIX_2:def 17</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">Y9</font> = <font color="Maroon" title="c3">Y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c3">Y</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c12">F</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</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="c12">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Maroon" title="c12">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Maroon" title="c12">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  implies for <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span>,<font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b2">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font></span>)</span>,<font color="Maroon" title="c1">D</font>  st ( for <font color="Olive" title="b3">H</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font><br/> for <font color="Olive" title="b4">SF</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font></span>)</span>  st <font color="Olive" title="b4">SF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b5">h</font> where <font color="Olive" title="b5">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font> : <font color="Olive" title="b5">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">H</font> </span>}</span>  holds <br/><font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Olive" title="b4">SF</font>,<font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">H</font> ) holds <br/><font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>,<font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>,<font color="Olive" title="b2">g</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E12:80"/><span class="lab"><font color="Green" title="E73">A6</font></span>: 
<font color="Maroon" title="c12">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a> 
 <span class="kw">and </span><br/><a NAME="E13:80"/><span class="lab"><font color="Green" title="E74">A7</font></span>: 
<font color="Maroon" title="c12">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> 
 <span class="kw">and </span><br/><a NAME="E14:80"/><span class="lab"><font color="Green" title="E75">A8</font></span>: 
<font color="Maroon" title="c12">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</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">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span>,<font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b2">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font></span>)</span>,<font color="Maroon" title="c1">D</font>  st ( for <font color="Olive" title="b3">H</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font><br/> for <font color="Olive" title="b4">SF</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font></span>)</span>  st <font color="Olive" title="b4">SF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b5">h</font> where <font color="Olive" title="b5">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font> : <font color="Olive" title="b5">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">H</font> </span>}</span>  holds <br/><font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Olive" title="b4">SF</font>,<font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">H</font> ) holds <br/><font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>,<font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>,<font color="Olive" title="b2">g</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c13">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span>,<font color="Maroon" title="c1">D</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="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font></span>)</span>,<font color="Maroon" title="c1">D</font>  st ( for <font color="Olive" title="b2">H</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font><br/> for <font color="Olive" title="b3">SF</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font></span>)</span>  st <font color="Olive" title="b3">SF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b4">h</font> where <font color="Olive" title="b4">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font> : <font color="Olive" title="b4">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">H</font> </span>}</span>  holds <br/><font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Olive" title="b3">SF</font>,<font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">H</font> ) holds <br/><font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>,<font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>,<font color="Olive" title="b1">g</font></span><br/><span class="kw">let </span><font color="Maroon" title="c14">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font></span>)</span>,<font color="Maroon" title="c1">D</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">H</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font><br/> for <font color="Olive" title="b2">SF</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font></span>)</span>  st <font color="Olive" title="b2">SF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">h</font> where <font color="Olive" title="b3">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font> : <font color="Olive" title="b3">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">H</font> </span>}</span>  holds <br/><font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Olive" title="b2">SF</font>,<font color="Maroon" title="c14">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font> ) implies <font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>,<font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>,<font color="Maroon" title="c14">g</font> )</span><br/>



<span class="kw">assume </span><a NAME="E15:80"/><span class="lab"><font color="Green" title="E76">A9</font></span>: 
for <font color="Olive" title="b1">H</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font><br/> for <font color="Olive" title="b2">SF</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font></span>)</span>  st <font color="Olive" title="b2">SF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">h</font> where <font color="Olive" title="b3">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font> : <font color="Olive" title="b3">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">H</font> </span>}</span>  holds <br/><font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Olive" title="b2">SF</font>,<font color="Maroon" title="c14">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>,<font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>,<font color="Maroon" title="c14">g</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c15">gB</font> = <font color="Maroon" title="c14">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">B</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span>,<font color="Maroon" title="c3">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">D</font> ;<br/>
<a NAME="E17:80"/>
for <font color="Olive" title="b1">z</font> being   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font> holds  <font color="Maroon" title="c13">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c11">Y9</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c15">gB</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Olive" title="b1">z</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">z</font> be    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</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="c13">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c11">Y9</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c15">gB</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c16">z</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c17">Z</font> = <font color="Maroon" title="c16">z</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">SF</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font> : <font color="Olive" title="b1">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Z</font> </span>}</span> ;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">z</font>,$1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">q</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E3:80_1"/><span class="lab"><font color="Green" title="E77">A10</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c19">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> holds <br/><font color="Maroon" title="c19">q</font> <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> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></span>
<a NAME="E4:80_1"/><span class="lab"><font color="Green" title="E78">A11</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/>
<a NAME="E5:80_1"/><span class="kw">then </span>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">Z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span>,<font color="Maroon" title="c3">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">ZY</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">Z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span>,<font color="Maroon" title="c3">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></span>;<br/>
<a NAME="E7:80_1"/>
for <font color="Olive" title="b1">x9</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">ZY</font> iff <font color="Olive" title="b1">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">q</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c3">Y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">x9</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="c21">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">ZY</font> iff <font color="Maroon" title="c21">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">q</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c3">Y</font> )</span><br/>

<span class="kw">thus </span><a NAME="E1:80_1_1"/>
( <font color="Maroon" title="c21">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">ZY</font> implies <font color="Maroon" title="c21">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">q</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c3">Y</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="c21">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">q</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c3">Y</font> implies <font color="Maroon" title="c21">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">ZY</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:80_1_1_1"/>
<font color="Maroon" title="c21">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">ZY</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="c21">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">q</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c3">Y</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">z9</font>, <font color="Maroon" title="c23">y9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:80_1_1_1"/><span class="lab"><font color="Green" title="E79">A12</font></span>: 
<font color="Maroon" title="c22">z9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">Z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E4:80_1_1_1"/><span class="lab"><font color="Green" title="E80">A13</font></span>: 
<font color="Maroon" title="c23">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">and </span><br/><a NAME="E5:80_1_1_1"/><span class="lab"><font color="Green" title="E81">A14</font></span>: 
<font color="Maroon" title="c21">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">z9</font>,<font color="Maroon" title="c23">y9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E6:80_1_1_1"/><span class="lab"><font color="Green" title="E82">A15</font></span>: 
<font color="Maroon" title="c16">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">z9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:80_1_1_1"><span class="lab"><font color="Green" title="E79">A12</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E7:80_1_1_1"/>
<font color="Maroon" title="c19">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">z</font>,<font color="Maroon" title="c23">y9</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="#E3:80_1"><span class="lab"><font color="Green" title="E77">A10</font></span></a>, <a class="txt" href="#E4:80_1_1_1"><span class="lab"><font color="Green" title="E80">A13</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:80_1_1_1"/>
<font color="Maroon" title="c21">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">q</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:80_1"><span class="lab"><font color="Green" title="E77">A10</font></span></a>, <a class="txt" href="#E4:80_1_1_1"><span class="lab"><font color="Green" title="E80">A13</font></span></a>, <a class="txt" href="#E5:80_1_1_1"><span class="lab"><font color="Green" title="E81">A14</font></span></a>, <a class="txt" href="#E6:80_1_1_1"><span class="lab"><font color="Green" title="E82">A15</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</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">assume </span><a NAME="E2:80_1_1"/>
<font color="Maroon" title="c21">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">q</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c3">Y</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="c21">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">ZY</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">y9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:80_1_1"/><span class="lab"><font color="Green" title="E79">A16</font></span>: 
<font color="Maroon" title="c22">y9</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> <font color="Maroon" title="c19">q</font>
 <span class="kw">and </span><br/><a NAME="E5:80_1_1"/><span class="lab"><font color="Green" title="E80">A17</font></span>: 
<font color="Maroon" title="c22">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">and </span><br/><a NAME="E6:80_1_1"/><span class="lab"><font color="Green" title="E81">A18</font></span>: 
<font color="Maroon" title="c21">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/>
<a NAME="E7:80_1_1"/>
<font color="Maroon" title="c21">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">z</font>,<font color="Maroon" title="c22">y9</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="#E3:80_1"><span class="lab"><font color="Green" title="E77">A10</font></span></a>, <a class="txt" href="#E4:80_1_1"><span class="lab"><font color="Green" title="E79">A16</font></span></a>, <a class="txt" href="#E6:80_1_1"><span class="lab"><font color="Green" title="E81">A18</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:80_1_1"/>
<font color="Maroon" title="c21">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">ZY</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:80_1_1"><span class="lab"><font color="Green" title="E80">A17</font></span></a>, <a class="ref" href="zfmisc_1.html#T128" title="ZFMISC_1:th.128">ZFMISC_1:128</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="E8:80_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A19</font></span>: 
<font color="Maroon" title="c19">q</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">ZY</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></span>;<br/>
<a NAME="E9:80_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A20</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c19">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">ZY</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:80_1"><span class="lab"><font color="Green" title="E77">A10</font></span></a>, <a class="ref" href="relat_1.html#T146" title="RELAT_1:th.146">RELAT_1:146</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c21">x1</font>, <font color="Maroon" title="c22">x2</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="c21">x1</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> <font color="Maroon" title="c19">q</font> &amp; <font color="Maroon" title="c22">x2</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> <font color="Maroon" title="c19">q</font> &amp; <font color="Maroon" title="c19">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x2</font> implies <font color="Maroon" title="c21">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">x2</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:80_1_2"/><span class="lab"><font color="Green" title="E81">A21</font></span>: 
<font color="Maroon" title="c21">x1</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> <font color="Maroon" title="c19">q</font>
 <span class="kw">and </span><br/><a NAME="E2:80_1_2"/><span class="lab"><font color="Green" title="E82">A22</font></span>: 
<font color="Maroon" title="c22">x2</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> <font color="Maroon" title="c19">q</font>
 <span class="kw">and </span><br/><a NAME="E3:80_1_2"/><span class="lab"><font color="Green" title="E83">A23</font></span>: 
<font color="Maroon" title="c19">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x2</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="c21">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">x2</font></span><br/><a NAME="E4:80_1_2"/><span class="lab"><font color="Green" title="E84">A24</font></span>: 
<font color="Maroon" title="c19">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">z</font>,<font color="Maroon" title="c22">x2</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="#E3:80_1"><span class="lab"><font color="Green" title="E77">A10</font></span></a>, <a class="txt" href="#E2:80_1_2"><span class="lab"><font color="Green" title="E82">A22</font></span></a></span>;<br/><a NAME="E5:80_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">z</font>,<font color="Maroon" title="c21">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:80_1"><span class="lab"><font color="Green" title="E77">A10</font></span></a>, <a class="txt" href="#E1:80_1_2"><span class="lab"><font color="Green" title="E81">A21</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:80_1_2"/>
<font color="Maroon" title="c21">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:80_1_2"><span class="lab"><font color="Green" title="E83">A23</font></span></a>, <a class="txt" href="#E4:80_1_2"><span class="lab"><font color="Green" title="E84">A24</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</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="E11:80_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A25</font></span>: 
<font color="Maroon" title="c19">q</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></span>;<br/>
<a NAME="E12:80_1"/>
<font color="Maroon" title="c20">ZY</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span>,<font color="Maroon" title="c3">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80_1"><span class="lab"><font color="Green" title="E78">A11</font></span></a>, <a class="ref" href="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">q</font> = <font color="Maroon" title="c19">q</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">Y</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span>,<font color="Maroon" title="c3">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:80_1"><span class="lab"><font color="Green" title="E77">A10</font></span></a>, <a class="txt" href="#E9:80_1"><span class="lab"><font color="Green" title="E80">A20</font></span></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">gBq</font> = <font color="Maroon" title="c15">gB</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c21">q</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c1">D</font> ;<br/>
<a NAME="E15:80_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">gB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span>,<font color="Maroon" title="c3">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">C</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E17:80_1"/><span class="lab"><font color="Green" title="E82">A26</font></span>: 
<span class="p1">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c15">gB</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c16">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">C</font>
 <span class="kw">and </span><br/><a NAME="E18:80_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c23">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">and </span><br/><a NAME="E19:80_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c23">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">gB</font>
 <span class="kw">and </span><br/><a NAME="E20:80_1"/><span class="lab"><font color="Green" title="E83">A27</font></span>: 
for <font color="Olive" title="b1">y9</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> holds <br/><font color="Maroon" title="c23">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">gB</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c16">z</font>,<font color="Olive" title="b1">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_5.html#T36" title="FUNCT_5:th.36">FUNCT_5:36</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c24">C</font> = <font color="Maroon" title="c23">C</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c1">D</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:80_1"><span class="lab"><font color="Green" title="E82">A26</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c25">x9</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="c25">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> implies <font color="Maroon" title="c24">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">gBq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x9</font> )</span><br/><span class="kw">assume </span><a NAME="E1:80_1_3"/><span class="lab"><font color="Green" title="E84">A28</font></span>: 
<font color="Maroon" title="c25">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</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="c24">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">gBq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x9</font></span><br/><a NAME="E2:80_1_3"/><span class="lab"><font color="Green" title="E85">A29</font></span>: 
<font color="Maroon" title="c21">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">z</font>,<font color="Maroon" title="c25">x9</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="#E3:80_1"><span class="lab"><font color="Green" title="E77">A10</font></span></a>, <a class="txt" href="#E1:80_1_3"><span class="lab"><font color="Green" title="E84">A28</font></span></a></span>;<br/><a NAME="E3:80_1_3"/>
<font color="Maroon" title="c24">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">gB</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c16">z</font>,<font color="Maroon" title="c25">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:80_1"><span class="lab"><font color="Green" title="E83">A27</font></span></a>, <a class="txt" href="#E1:80_1_3"><span class="lab"><font color="Green" title="E84">A28</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:80_1_3"/>
<font color="Maroon" title="c24">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">gBq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80_1_3"><span class="lab"><font color="Green" title="E84">A28</font></span></a>, <a class="txt" href="#E2:80_1_3"><span class="lab"><font color="Green" title="E85">A29</font></span></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</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="E23:80_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A30</font></span>: 
<font color="Maroon" title="c24">C</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c22">gBq</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></span>;<br/>
<a NAME="E24:80_1"/><span class="lab"><font color="Green" title="E85">A31</font></span>: 
<font color="Maroon" title="c8">B</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c20">ZY</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font> : <font color="Olive" title="b1">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Z</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c25">b</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="c25">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c20">ZY</font> or <font color="Maroon" title="c25">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font> : <font color="Olive" title="b1">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Z</font> </span>}</span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:80_1_4"/>
<font color="Maroon" title="c25">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c20">ZY</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="c25">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font> : <font color="Olive" title="b1">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Z</font> </span>}</span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">zy</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:80_1_4"/>
<font color="Maroon" title="c26">zy</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> <font color="Maroon" title="c8">B</font>
 <span class="kw">and </span><br/><a NAME="E4:80_1_4"/><span class="lab"><font color="Green" title="E86">A32</font></span>: 
<font color="Maroon" title="c26">zy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">ZY</font>
 <span class="kw">and </span><br/><a NAME="E5:80_1_4"/><span class="lab"><font color="Green" title="E87">A33</font></span>: 
<font color="Maroon" title="c25">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">zy</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c27">z9</font>, <font color="Maroon" title="c28">y9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:80_1_4"/><span class="lab"><font color="Green" title="E88">A34</font></span>: 
<font color="Maroon" title="c27">z9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">Z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E8:80_1_4"/><span class="lab"><font color="Green" title="E89">A35</font></span>: 
<font color="Maroon" title="c28">y9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">and </span><br/><a NAME="E9:80_1_4"/><span class="lab"><font color="Green" title="E90">A36</font></span>: 
<font color="Maroon" title="c26">zy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c27">z9</font>,<font color="Maroon" title="c28">y9</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="#E4:80_1_4"><span class="lab"><font color="Green" title="E86">A32</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E10:80_1_4"/>
<font color="Maroon" title="c3">Y</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c28">y9</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:80_1_4"><span class="lab"><font color="Green" title="E89">A35</font></span></a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c29">F1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font><span class="kw"> such that </span><br/><a NAME="E12:80_1_4"/><span class="lab"><font color="Green" title="E91">A37</font></span>: 
<font color="Maroon" title="c29">F1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Z</font>
 <span class="kw">and </span><br/><a NAME="E13:80_1_4"/><span class="lab"><font color="Green" title="E92">A38</font></span>: 
<font color="Maroon" title="c29">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="ref" href="stirl2_1.html#T67" title="STIRL2_1:th.67">STIRL2_1:67</a></span>;<br/>
<a NAME="E14:80_1_4"/>
<font color="Maroon" title="c27">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:80_1_4"><span class="lab"><font color="Green" title="E88">A34</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E15:80_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c8">B</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c27">z9</font>,<font color="Maroon" title="c28">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">F1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80"><span class="lab"><font color="Green" title="E70">A3</font></span></a>, <a class="txt" href="#E12:80_1_4"><span class="lab"><font color="Green" title="E91">A37</font></span></a>, <a class="txt" href="#E13:80_1_4"><span class="lab"><font color="Green" title="E92">A38</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:80_1_4"/>
<font color="Maroon" title="c25">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font> : <font color="Olive" title="b1">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Z</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:80_1_4"><span class="lab"><font color="Green" title="E87">A33</font></span></a>, <a class="txt" href="#E9:80_1_4"><span class="lab"><font color="Green" title="E90">A36</font></span></a>, <a class="txt" href="#E12:80_1_4"><span class="lab"><font color="Green" title="E91">A37</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E25:80_1"/><span class="lab"><font color="Green" title="E86">A39</font></span>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font> : <font color="Olive" title="b1">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Z</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c20">ZY</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_5"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:80_1_5"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E2:80_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A40</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c25">sf</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="c25">sf</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font> : <font color="Olive" title="b1">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Z</font> </span>}</span>  or <font color="Maroon" title="c25">sf</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c20">ZY</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:80_1_5"/>
<font color="Maroon" title="c25">sf</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font> : <font color="Olive" title="b1">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Z</font> </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="c25">sf</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c20">ZY</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font><span class="kw"> such that </span><br/><a NAME="E5:80_1_5"/><span class="lab"><font color="Green" title="E88">A41</font></span>: 
<font color="Maroon" title="c26">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">sf</font>
 <span class="kw">and </span><br/><a NAME="E6:80_1_5"/><span class="lab"><font color="Green" title="E89">A42</font></span>: 
<font color="Maroon" title="c26">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Z</font>
 ;<br/>
<a NAME="E7:80_1_5"/><span class="lab"><font color="Green" title="E90">A43</font></span>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span>,<font color="Maroon" title="c3">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_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> <font color="Maroon" title="c8">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E8:80_1_5"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c26">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E9:80_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A44</font></span>: 
<font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">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> <font color="Maroon" title="c26">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:80_1_5"><span class="lab"><font color="Green" title="E87">A40</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E10:80_1_5"/><span class="lab"><font color="Green" title="E92">A45</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c26">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font>
 
<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="E11:80_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A46</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">z</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span>,<font color="Maroon" title="c3">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:80_1_5"><span class="lab"><font color="Green" title="E91">A44</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E12:80_1_5"/>
<font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c17">Z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E13:80_1_5"/><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="c16">z</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">ZY</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:80_1_5"><span class="lab"><font color="Green" title="E91">A44</font></span></a>, <a class="txt" href="#E10:80_1_5"><span class="lab"><font color="Green" title="E92">A45</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E14:80_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c8">B</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c16">z</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c26">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c20">ZY</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:80_1_5"><span class="lab"><font color="Green" title="E93">A46</font></span></a>, <a class="txt" href="#E7:80_1_5"><span class="lab"><font color="Green" title="E90">A43</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:80_1_5"/>
<font color="Maroon" title="c25">sf</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c20">ZY</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80"><span class="lab"><font color="Green" title="E70">A3</font></span></a>, <a class="txt" href="#E5:80_1_5"><span class="lab"><font color="Green" title="E88">A41</font></span></a>, <a class="txt" href="#E6:80_1_5"><span class="lab"><font color="Green" title="E89">A42</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c25">SF</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">h</font> where <font color="Olive" title="b1">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font> : <font color="Olive" title="b1">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Z</font> </span>}</span>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:80_1"><span class="lab"><font color="Green" title="E85">A31</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E27:80_1"/>
<font color="Maroon" title="c8">B</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c20">ZY</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">SF</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:80_1"><span class="lab"><font color="Green" title="E85">A31</font></span></a>, <a class="txt" href="#E25:80_1"><span class="lab"><font color="Green" title="E86">A39</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E28:80_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A47</font></span>: 
<font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c20">ZY</font></span>)</span>,<font color="Maroon" title="c14">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:80"><span class="lab"><font color="Green" title="E76">A9</font></span></a></span>;<br/>
<a NAME="E29:80_1"/>
<font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c20">ZY</font></span>)</span>,<font color="Maroon" title="c14">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c20">ZY</font>,<font color="Maroon" title="c15">gB</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:80"><span class="lab"><font color="Green" title="E74">A7</font></span></a>, <a class="txt" href="#E14:80"><span class="lab"><font color="Green" title="E75">A8</font></span></a>, <a class="txt" href="#E3:80"><span class="lab"><font color="Green" title="E69">A2</font></span></a>, <a class="ref" href="setwop_2.html#T8" title="SETWOP_2:th.8">SETWOP_2:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E30:80_1"/>
<font color="Maroon" title="c13">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c11">Y9</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2" title="CAT_2:func.2">curry</a> <font color="Maroon" title="c15">gB</font></span>)</span> <a href="cat_2.html#K1" title="CAT_2:func.1">.</a> <font color="Maroon" title="c16">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:80"><span class="lab"><font color="Green" title="E74">A7</font></span></a>, <a class="txt" href="#E14:80"><span class="lab"><font color="Green" title="E75">A8</font></span></a>, <a class="txt" href="#E28:80_1"><span class="lab"><font color="Green" title="E87">A47</font></span></a>, <a class="txt" href="#E11:80_1"><span class="lab"><font color="Green" title="E81">A25</font></span></a>, <a class="txt" href="#E8:80_1"><span class="lab"><font color="Green" title="E79">A19</font></span></a>, <a class="txt" href="#E17:80_1"><span class="lab"><font color="Green" title="E82">A26</font></span></a>, <a class="txt" href="#E23:80_1"><span class="lab"><font color="Green" title="E84">A30</font></span></a>, <a class="ref" href="setwop_2.html#T8" title="SETWOP_2:th.8">SETWOP_2:8</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="E18:80"/><span class="kw">then </span>
<font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c10">FXY9</font>,<font color="Maroon" title="c11">Y9</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<font color="Maroon" title="c15">gB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c10">FXY9</font>,<font color="Maroon" title="c13">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:80"><span class="lab"><font color="Green" title="E73">A6</font></span></a>, <a class="txt" href="#E13:80"><span class="lab"><font color="Green" title="E74">A7</font></span></a>, <a class="txt" href="#E14:80"><span class="lab"><font color="Green" title="E75">A8</font></span></a>, <a class="ref" href="matrix_3.html#T32" title="MATRIX_3:th.32">MATRIX_3:32</a></span>;<br/>
<a NAME="E19:80"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A48</font></span>: 
<font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c9">domB</font>,<font color="Maroon" title="c15">gB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c10">FXY9</font>,<font color="Maroon" title="c13">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E20:80"/><span class="lab"><font color="Green" title="E78">A49</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:80"><span class="lab"><font color="Green" title="E69">A2</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E21:80"/>
<font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c9">domB</font></span>)</span>,<font color="Maroon" title="c14">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c9">domB</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">B</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:80"><span class="lab"><font color="Green" title="E74">A7</font></span></a>, <a class="txt" href="#E14:80"><span class="lab"><font color="Green" title="E75">A8</font></span></a>, <a class="txt" href="#E3:80"><span class="lab"><font color="Green" title="E69">A2</font></span></a>, <a class="ref" href="setwop_2.html#T8" title="SETWOP_2:th.8">SETWOP_2:8</a></span>;<br/>
<a NAME="E22:80"/><span class="kw">then </span>
<font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>,<font color="Maroon" title="c14">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c9">domB</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">B</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:80"><span class="lab"><font color="Green" title="E78">A49</font></span></a>, <a class="txt" href="#E10:80"><span class="lab"><font color="Green" title="E72">A5</font></span></a>, <a class="ref" href="relat_1.html#T146" title="RELAT_1:th.146">RELAT_1:146</a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:80"/>
<font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>,<font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<font color="Maroon" title="c3">Y</font></span>)</span></span>)</span>,<font color="Maroon" title="c14">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:80"><span class="lab"><font color="Green" title="E77">A48</font></span></a>, <a class="ref" href="matrix_2.html#D17" title="MATRIX_2:def.17">MATRIX_2:def 17</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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