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<span class="kw">let </span><font color="Maroon" title="c1">U1</font> be   <a href="unialg_1.html#NM1" title="UNIALG_1:NM.1">Universal_Algebra</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">A</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">U1</font><br/>  for <font color="Olive" title="b2">o</font> being   <a href="unialg_2.html#NM3" title="UNIALG_2:NM.3">operation</a> of <font color="Maroon" title="c1">U1</font>  st <font color="Olive" title="b1">A</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Olive" title="b2">o</font> holds <br/> <a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">o</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <font color="Olive" title="b2">o</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">U1</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">o</font> being   <a href="unialg_2.html#NM3" title="UNIALG_2:NM.3">operation</a> of <font color="Maroon" title="c1">U1</font>  st <font color="Maroon" title="c2">A</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Olive" title="b1">o</font> holds <br/> <a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">o</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <font color="Olive" title="b1">o</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">o</font> be   <a href="unialg_2.html#NM3" title="UNIALG_2:NM.3">operation</a> of <font color="Maroon" title="c1">U1</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="c2">A</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c3">o</font> implies  <a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">o</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c3">o</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:12"/>
<font color="Maroon" title="c2">A</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c3">o</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">  <a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">o</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c3">o</font></span><br/>

<a NAME="E2:12"/><span class="kw">then </span>
<font color="Maroon" title="c3">o</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c3">o</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D5" target="_self" onmouseover="rs('unialg_2/D5')" onmouseout="rh()">Def5</a></span>;<br/>
<a NAME="E3:12"/><span class="kw">then </span>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">o</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">o</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c3">o</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E4">A1</font></span>:  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">o</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c2">A</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c3">o</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c3">o</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c2">A</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#T22" onmouseover="rs('margrel1/T22')" onmouseout="rh()">MARGREL1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c3">o</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c2">A</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#T21" onmouseover="rs('margrel1/T21')" onmouseout="rh()">MARGREL1:21</a></span>
;<br/>
<a NAME="E5:12"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">o</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">o</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#T22" onmouseover="rs('margrel1/T22')" onmouseout="rh()">MARGREL1:22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:12"/>
 <a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">o</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c3">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="finseq_2.html#T110" onmouseover="rs('finseq_2/T110')" onmouseout="rh()">FINSEQ_2:110</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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