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<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <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">$2,$1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c1">I</font>, <font color="Maroon" title="c2">J</font>, <font color="Maroon" title="c3">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">F</font>, <font color="Olive" title="b2">G</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c3">D</font><br/> for <font color="Olive" title="b3">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c3">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 <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">D</font><br/> for <font color="Olive" title="b5">X</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> <font color="Maroon" title="c1">I</font><br/> for <font color="Olive" title="b6">Y</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> <font color="Maroon" title="c2">J</font>  st <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>  &amp; ( <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Olive" title="b6">Y</font>,<font color="Olive" title="b5">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <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="b2">G</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  holds <br/><font color="Olive" title="b1">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Olive" title="b5">X</font>,<font color="Olive" title="b6">Y</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Olive" title="b6">Y</font>,<font color="Olive" title="b5">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Olive" title="b4">g</font>,<font color="Olive" title="b3">f</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">F</font>, <font color="Maroon" title="c5">G</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c3">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">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c3">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 <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">D</font><br/> for <font color="Olive" title="b3">X</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> <font color="Maroon" title="c1">I</font><br/> for <font color="Olive" title="b4">Y</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> <font color="Maroon" title="c2">J</font>  st <font color="Maroon" title="c4">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; ( <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Olive" title="b4">Y</font>,<font color="Olive" title="b3">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c4">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  ) &amp; <font color="Maroon" title="c5">G</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  holds <br/><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Olive" title="b3">X</font>,<font color="Olive" title="b4">Y</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Olive" title="b4">Y</font>,<font color="Olive" title="b3">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c3">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 <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">D</font><br/> for <font color="Olive" title="b2">X</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> <font color="Maroon" title="c1">I</font><br/> for <font color="Olive" title="b3">Y</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> <font color="Maroon" title="c2">J</font>  st <font color="Maroon" title="c4">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; ( <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Olive" title="b3">Y</font>,<font color="Olive" title="b2">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c4">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  ) &amp; <font color="Maroon" title="c5">G</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  holds <br/><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Olive" title="b2">X</font>,<font color="Olive" title="b3">Y</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Olive" title="b1">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Olive" title="b3">Y</font>,<font color="Olive" title="b2">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Olive" title="b1">g</font>,<font color="Maroon" title="c6">f</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">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">X</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> <font color="Maroon" title="c1">I</font><br/> for <font color="Olive" title="b2">Y</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> <font color="Maroon" title="c2">J</font>  st <font color="Maroon" title="c4">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; ( <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Olive" title="b2">Y</font>,<font color="Olive" title="b1">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c4">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  ) &amp; <font color="Maroon" title="c5">G</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  holds <br/><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Olive" title="b2">Y</font>,<font color="Olive" title="b1">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c8">X</font> be    <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="c1">I</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">Y</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> <font color="Maroon" title="c2">J</font>  st <font color="Maroon" title="c4">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; ( <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Olive" title="b1">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c4">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  ) &amp; <font color="Maroon" title="c5">G</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  holds <br/><font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c8">X</font>,<font color="Olive" title="b1">Y</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Olive" title="b1">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c9">Y</font> be    <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="c2">J</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="c4">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; ( <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c4">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  ) &amp; <font color="Maroon" title="c5">G</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  implies <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c8">X</font>,<font color="Maroon" title="c9">Y</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> )</span><br/>











<span class="kw">assume </span><a NAME="E1:35"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
( <font color="Maroon" title="c4">F</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Maroon" title="c4">F</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; ( <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c4">F</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</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="c5">G</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  or <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c8">X</font>,<font color="Maroon" title="c9">Y</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c10">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E3:35"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">h</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"><font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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="c2">J</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon" title="c10">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</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>,<font color="Olive" title="b2">y</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_3.html#S2" title="FUNCT_3:sch.2">FUNCT_3:sch 2</a>();<br/></span>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">z1</font>, <font color="Maroon" title="c12">z2</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="c11">z1</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="c10">h</font> &amp; <font color="Maroon" title="c12">z2</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="c10">h</font> &amp; <font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">z2</font> implies <font color="Maroon" title="c11">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z2</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35_1"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
<font color="Maroon" title="c11">z1</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="c10">h</font>
 <span class="kw">and </span><br/><a NAME="E2:35_1"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
<font color="Maroon" title="c12">z2</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="c10">h</font>
 <span class="kw">and </span><br/><a NAME="E3:35_1"/><span class="lab"><font color="Green" title="E18">A5</font></span>: 
<font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">z2</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="c11">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z2</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c13">x2</font>, <font color="Maroon" title="c14">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:35_1"/><span class="lab"><font color="Green" title="E19">A6</font></span>: 
<font color="Maroon" title="c12">z2</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="c13">x2</font>,<font color="Maroon" title="c14">y2</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="#E3:35"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E2:35_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></span>;<br/><a NAME="E6:35_1"/>
( <font color="Maroon" title="c13">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">J</font> &amp; <font color="Maroon" title="c14">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E2:35_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E5:35_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E7:35_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A7</font></span>: 
<font color="Maroon" title="c10">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">x2</font>,<font color="Maroon" title="c14">y2</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="c14">y2</font>,<font color="Maroon" title="c13">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="#E3:35"><span class="lab"><font color="Green" title="E15">A2</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">x1</font>, <font color="Maroon" title="c16">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:35_1"/><span class="lab"><font color="Green" title="E21">A8</font></span>: 
<font color="Maroon" title="c11">z1</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="c15">x1</font>,<font color="Maroon" title="c16">y1</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="#E3:35"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E1:35_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></span>;<br/><a NAME="E10:35_1"/>
( <font color="Maroon" title="c15">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">J</font> &amp; <font color="Maroon" title="c16">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E1:35_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E9:35_1"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E11:35_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A9</font></span>: 
<font color="Maroon" title="c10">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">x1</font>,<font color="Maroon" title="c16">y1</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">y1</font>,<font color="Maroon" title="c15">x1</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="#E3:35"><span class="lab"><font color="Green" title="E15">A2</font></span></a></span>;<br/><a NAME="E12:35_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E9:35_1"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="txt" href="#E5:35_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E7:35_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><span class="kw">hence </span><a NAME="E13:35_1"/>
<font color="Maroon" title="c11">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E9:35_1"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="txt" href="#E5:35_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E11:35_1"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="txt" href="#E7:35_1"><span class="lab"><font color="Green" title="E20">A7</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="E5:35"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A10</font></span>: 
<font color="Maroon" title="c10">h</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="E6:35"/><span class="lab"><font color="Green" title="E17">A11</font></span>: 
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> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <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"><font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">J</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">x</font> <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"><font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> implies <font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <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"><font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">J</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_2"/><span class="lab"><font color="Green" title="E18">A12</font></span>: 
<font color="Maroon" title="c11">x</font> <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"><font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <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"><font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">J</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">y</font>, <font color="Maroon" title="c13">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:35_2"/><span class="lab"><font color="Green" title="E19">A13</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</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="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></span>;<br/>
<a NAME="E4:35_2"/><span class="lab"><font color="Green" title="E20">A14</font></span>: 
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">J</font> &amp; <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_2"><span class="lab"><font color="Green" title="E18">A12</font></span></a>, <a class="txt" href="#E3:35_2"><span class="lab"><font color="Green" title="E19">A13</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E5:35_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</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="c13">z</font>,<font color="Maroon" title="c12">y</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="#E3:35"><span class="lab"><font color="Green" title="E15">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:35_2"/>
<font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <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"><font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">J</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="#E3:35_2"><span class="lab"><font color="Green" title="E19">A13</font></span></a>, <a class="txt" href="#E4:35_2"><span class="lab"><font color="Green" title="E20">A14</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</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="E7:35"/><span class="lab"><font color="Green" title="E18">A15</font></span>: 
<font color="Maroon" title="c5">G</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</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="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c8">X</font>,<font color="Maroon" title="c9">Y</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span></span><br/>

<a NAME="E8:35"/><span class="lab"><font color="Green" title="E19">A16</font></span>: 
<font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">h</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c11">G</font> = <font color="Maroon" title="c5">G</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"><font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c3">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c3">D</font> ;<br/>
<a NAME="E2:35_3"/><span class="lab"><font color="Green" title="E20">A17</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> <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"><font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c1">I</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/>
<a NAME="E3:35_3"/><span class="lab"><font color="Green" title="E21">A18</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <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"><font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">J</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/>
<a NAME="E4:35_3"/><span class="lab"><font color="Green" title="E22">A19</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">h</font> &amp; <font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> iff ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">h</font> &amp; <font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:35_3_1"/>
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> implies ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">h</font> &amp; <font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</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="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">h</font> &amp; <font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> implies <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:35_3_1_1"/><span class="lab"><font color="Green" title="E23">A20</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</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="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">h</font> &amp; <font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">y</font>, <font color="Maroon" title="c14">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:35_3_1_1"/><span class="lab"><font color="Green" title="E24">A21</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c14">z</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="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></span>;<br/>
<a NAME="E4:35_3_1_1"/><span class="lab"><font color="Green" title="E25">A22</font></span>: 
( <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">J</font> &amp; <font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_3"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E1:35_3_1_1"><span class="lab"><font color="Green" title="E23">A20</font></span></a>, <a class="txt" href="#E3:35_3_1_1"><span class="lab"><font color="Green" title="E24">A21</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E5:35_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c14">z</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="c14">z</font>,<font color="Maroon" title="c13">y</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="#E3:35"><span class="lab"><font color="Green" title="E15">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:35_3_1_1"/>
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">h</font> &amp; <font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E3:35_3"><span class="lab"><font color="Green" title="E21">A18</font></span></a>, <a class="txt" href="#E3:35_3_1_1"><span class="lab"><font color="Green" title="E24">A21</font></span></a>, <a class="txt" href="#E4:35_3_1_1"><span class="lab"><font color="Green" title="E25">A22</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</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><span class="kw">that </span><br/><a NAME="E2:35_3_1"/><span class="lab"><font color="Green" title="E23">A23</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">h</font>
 <span class="kw">and </span><br/><a NAME="E3:35_3_1"/>
<font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</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="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E4:35_3_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E2:35_3_1"><span class="lab"><font color="Green" title="E23">A23</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</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="E5:35_3"/>
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>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_3_2"/><span class="lab"><font color="Green" title="E23">A24</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">y</font>, <font color="Maroon" title="c14">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:35_3_2"/><span class="lab"><font color="Green" title="E24">A25</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c14">z</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="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">z</font> = <font color="Maroon" title="c14">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_3"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E1:35_3_2"><span class="lab"><font color="Green" title="E23">A24</font></span></a>, <a class="txt" href="#E3:35_3_2"><span class="lab"><font color="Green" title="E24">A25</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">y</font> = <font color="Maroon" title="c13">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">J</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_3"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="txt" href="#E1:35_3_2"><span class="lab"><font color="Green" title="E23">A24</font></span></a>, <a class="txt" href="#E3:35_3_2"><span class="lab"><font color="Green" title="E24">A25</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E6:35_3_2"/><span class="lab"><font color="Green" title="E25">A26</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">z</font>,<font color="Maroon" title="c16">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_3"><span class="lab"><font color="Green" title="E21">A18</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E7:35_3_2"/><span class="lab"><font color="Green" title="E26">A27</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">y</font>,<font color="Maroon" title="c15">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_3"><span class="lab"><font color="Green" title="E20">A17</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c16">y</font>,<font color="Maroon" title="c15">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_3_2"><span class="lab"><font color="Green" title="E24">A25</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c15">z</font>,<font color="Maroon" title="c16">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E15">A2</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c11">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">z</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_3_2"><span class="lab"><font color="Green" title="E25">A26</font></span></a>, <a class="ref" href="finseqop.html#T82" title="FINSEQOP:th.82">FINSEQOP:82</a></span>
<br/>.= 
<font color="Maroon" title="c11">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">y</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="ref" href="binop_1.html#D2" title="BINOP_1:def.2">BINOP_1:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">y</font>,<font color="Maroon" title="c15">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:35_3_2"><span class="lab"><font color="Green" title="E26">A27</font></span></a>, <a class="ref" href="finseqop.html#T82" title="FINSEQOP:th.82">FINSEQOP:82</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35_3_2"><span class="lab"><font color="Green" title="E24">A25</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">hence </span><a NAME="E6:35_3"/>
<font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_3"><span class="lab"><font color="Green" title="E22">A19</font></span></a>, <a class="ref" href="funct_1.html#T20" title="FUNCT_1:th.20">FUNCT_1:20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:35"/><span class="lab"><font color="Green" title="E20">A28</font></span>: 
( <font color="Maroon" title="c8">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">I</font> &amp; <font color="Maroon" title="c9">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">J</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/>
<a NAME="E10:35"/>
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c8">X</font>,<font color="Maroon" title="c9">Y</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span> iff <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">h</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c8">X</font>,<font color="Maroon" title="c9">Y</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span> iff <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">h</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span> )</span><br/>

<span class="kw">thus </span><a NAME="E1:35_4"/>
( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c8">X</font>,<font color="Maroon" title="c9">Y</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span> implies <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">h</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">h</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span> implies <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c8">X</font>,<font color="Maroon" title="c9">Y</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:35_4_1"/><span class="lab"><font color="Green" title="E21">A29</font></span>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c8">X</font>,<font color="Maroon" title="c9">Y</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">h</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">y1</font>, <font color="Maroon" title="c13">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:35_4_1"/><span class="lab"><font color="Green" title="E22">A30</font></span>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y1</font>,<font color="Maroon" title="c13">x1</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="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></span>;<br/>
<a NAME="E4:35_4_1"/><span class="lab"><font color="Green" title="E23">A31</font></span>: 
( <font color="Maroon" title="c12">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> &amp; <font color="Maroon" title="c13">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_4_1"><span class="lab"><font color="Green" title="E21">A29</font></span></a>, <a class="txt" href="#E3:35_4_1"><span class="lab"><font color="Green" title="E22">A30</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E5:35_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A32</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c12">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c12">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">h</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35"><span class="lab"><font color="Green" title="E20">A28</font></span></a>, <a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E6:35_4_1"/>
<font color="Maroon" title="c10">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c12">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35"><span class="lab"><font color="Green" title="E20">A28</font></span></a>, <a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E3:35_4_1"><span class="lab"><font color="Green" title="E22">A30</font></span></a>, <a class="txt" href="#E4:35_4_1"><span class="lab"><font color="Green" title="E23">A31</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:35_4_1"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">h</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_4_1"><span class="lab"><font color="Green" title="E24">A32</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="E2:35_4"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">h</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c8">X</font>,<font color="Maroon" title="c9">Y</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:35_4"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">h</font>
 <span class="kw">and </span><br/><a NAME="E5:35_4"/><span class="lab"><font color="Green" title="E21">A33</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E6:35_4"/><span class="lab"><font color="Green" title="E22">A34</font></span>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</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="c13">x1</font>, <font color="Maroon" title="c14">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:35_4"/><span class="lab"><font color="Green" title="E23">A35</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c14">y1</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="#E5:35_4"><span class="lab"><font color="Green" title="E21">A33</font></span></a>, <a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></span>;<br/>
<a NAME="E9:35_4"/><span class="lab"><font color="Green" title="E24">A36</font></span>: 
( <font color="Maroon" title="c13">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font> &amp; <font color="Maroon" title="c14">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_4"><span class="lab"><font color="Green" title="E21">A33</font></span></a>, <a class="txt" href="#E8:35_4"><span class="lab"><font color="Green" title="E23">A35</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E10:35_4"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c14">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:35_4"><span class="lab"><font color="Green" title="E22">A34</font></span></a>, <a class="txt" href="#E8:35_4"><span class="lab"><font color="Green" title="E23">A35</font></span></a></span>;<br/>
<a NAME="E11:35_4"/><span class="kw">then </span>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">y1</font>,<font color="Maroon" title="c13">x1</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="#E9:35"><span class="lab"><font color="Green" title="E20">A28</font></span></a>, <a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E9:35_4"><span class="lab"><font color="Green" title="E24">A36</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:35_4"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c8">X</font>,<font color="Maroon" title="c9">Y</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35_4"><span class="lab"><font color="Green" title="E24">A36</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</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="E11:35"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A37</font></span>: 
<font color="Maroon" title="c10">h</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c8">X</font>,<font color="Maroon" title="c9">Y</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">h</font> = <font color="Maroon" title="c10">h</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"><font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">J</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="#E3:35"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E6:35"><span class="lab"><font color="Green" title="E17">A11</font></span></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>;<br/>
<font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c8">X</font>,<font color="Maroon" title="c9">Y</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> = 
<font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">h</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E5:35"><span class="lab"><font color="Green" title="E16">A10</font></span></a>, <a class="txt" href="#E11:35"><span class="lab"><font color="Green" title="E21">A37</font></span></a>, <a class="ref" href="setwop_2.html#T8" title="SETWOP_2:th.8">SETWOP_2:8</a></span>
<br/>.= 
<font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:35"><span class="lab"><font color="Green" title="E19">A16</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E14:35"/>
<font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c8">X</font>,<font color="Maroon" title="c9">Y</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</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>
