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<span class="kw">let </span><font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c2">D</font>, <font color="Maroon" title="c3">E</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">B</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">C</font><br/>  for <font color="Olive" title="b2">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">D</font><br/>  for <font color="Olive" title="b3">H</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c3">E</font><br/>  for <font color="Olive" title="b4">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c3">E</font>  st <font color="Olive" title="b3">H</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Olive" title="b3">H</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; ( <font color="Olive" title="b1">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Olive" title="b3">H</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  ) &amp; <font color="Olive" title="b2">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  holds <br/><font color="Olive" title="b3">H</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Olive" title="b1">B</font></span>)</span>,<font color="Olive" title="b4">h</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">H</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Olive" title="b1">B</font>,<span class="p1">(<span class="default"><font color="Olive" title="b4">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">f</font></span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c4">B</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">C</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">C</font>,<font color="Maroon" title="c2">D</font><br/>  for <font color="Olive" title="b2">H</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c3">E</font><br/>  for <font color="Olive" title="b3">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c3">E</font>  st <font color="Olive" title="b2">H</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Olive" title="b2">H</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; ( <font color="Maroon" title="c4">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Olive" title="b2">H</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  ) &amp; <font color="Olive" title="b1">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  holds <br/><font color="Olive" title="b2">H</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c4">B</font></span>)</span>,<font color="Olive" title="b3">h</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">H</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c4">B</font>,<span class="p1">(<span class="default"><font color="Olive" title="b3">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">f</font></span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c5">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">D</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">H</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c3">E</font><br/>  for <font color="Olive" title="b2">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c3">E</font>  st <font color="Olive" title="b1">H</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Olive" title="b1">H</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; ( <font color="Maroon" title="c4">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Olive" title="b1">H</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  ) &amp; <font color="Maroon" title="c5">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  holds <br/><font color="Olive" title="b1">H</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c4">B</font></span>)</span>,<font color="Olive" title="b2">h</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">H</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c4">B</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">f</font></span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c6">H</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c3">E</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c3">E</font>  st <font color="Maroon" title="c6">H</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Maroon" title="c6">H</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; ( <font color="Maroon" title="c4">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c6">H</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  ) &amp; <font color="Maroon" title="c5">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  holds <br/><font color="Maroon" title="c6">H</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c4">B</font></span>)</span>,<font color="Olive" title="b1">h</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">H</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c4">B</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">f</font></span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c7">h</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c3">E</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">H</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Maroon" title="c6">H</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; ( <font color="Maroon" title="c4">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c6">H</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  ) &amp; <font color="Maroon" title="c5">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  implies <font color="Maroon" title="c6">H</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c4">B</font></span>)</span>,<font color="Maroon" title="c7">h</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">H</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c4">B</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">f</font></span>)</span>) )</span><br/>













<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:6"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
( <font color="Maroon" title="c6">H</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Maroon" title="c6">H</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; ( <font color="Maroon" title="c4">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c6">H</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  ) )
 <span class="kw">and </span><br/><a NAME="E2:6"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
<font color="Maroon" title="c5">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">H</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c4">B</font></span>)</span>,<font color="Maroon" title="c7">h</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">H</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c4">B</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">f</font></span>)</span>)</span><br/>

<span class="kw">set </span><font color="Maroon" title="c8">s</font> = <font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">B</font>;<br/>
<a NAME="E3:6"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
(  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c4">B</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">B</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T83" onmouseover="rs('relat_1/T83')" onmouseout="rh()">RELAT_1:83</a>, <a class="ref" href="relat_1.html#T115" onmouseover="rs('relat_1/T115')" onmouseout="rh()">RELAT_1:115</a></span>;<br/>
<a NAME="E4:6"/>
<font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finsub_1.html#D5" onmouseover="rs('finsub_1/D5')" onmouseout="rh()">FINSUB_1:def 5</a></span>;<br/>
<a NAME="E5:6"/><span class="kw">then </span>
<font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E6:6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A4</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T62" onmouseover="rs('relat_1/T62')" onmouseout="rh()">RELAT_1:62</a></span>;<br/>
<a NAME="E7:6"/>
<font color="Maroon" title="c5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">B</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="txt" href="#E2:6"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="funct_1.html#T52" onmouseover="rs('funct_1/T52')" onmouseout="rh()">FUNCT_1:52</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:6"/>
<font color="Maroon" title="c6">H</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c4">B</font></span>)</span>,<font color="Maroon" title="c7">h</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">H</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c4">B</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">f</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E6:6"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="txt" href="#E3:6"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="setwop_2.html#T5" target="_self" onmouseover="rs('setwop_2/T5')" onmouseout="rh()">Th5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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