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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="rfunct_3.html#M2">Element</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="comput_1.html#V5" target="_self">with_the_same_arity</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="comput_1.html#K5" target="_self">HFuncs</a> <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:82"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="funct_6.html#K6">:&gt;</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="funct_6.html#K6">:&gt;</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="funct_6.html#K6">:&gt;</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K4" target="_self">arity</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="comput_1.html#T45" target="_self">Th45</a>, <a class="ref" href="relat_1.html#T44">RELAT_1:44</a></i>;<br/>
<b>hence </b><a NAME="E2:82"/><i><font color="Green">E49</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="funct_6.html#K6">:&gt;</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="comput_1.html#K4" target="_self">arity</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>

<a NAME="E3:82"/>
<span class="p1">(<span class="default"><a href="comput_1.html#K4" target="_self">arity</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i><a class="ref" href="msualg_1.html#T12">MSUALG_1:12</a></i>;<br/>
<a NAME="E4:82"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="funct_6.html#K6">:&gt;</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i><a class="txt" href="comput_1.html#E2:82"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:82"/>
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="funct_6.html#K6">:&gt;</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T45">RELAT_1:45</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<b>hence </b><a NAME="E6:82"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="funct_6.html#K6">:&gt;</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>

<a NAME="E7:82"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="funct_6.html#K6">:&gt;</a></span> is   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</a> ,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="comput_1.html#E4:82"><i><font color="Green">E50</font></i></a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E8:82"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="funct_6.html#K6">:&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="rfunct_3.html#K4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="partfun1.html#T119">PARTFUN1:119</a></i>;<br/>
<a NAME="E9:82"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="funct_6.html#K6">:&gt;</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="rfunct_3.html#M2">Element</a> of  <a href="rfunct_3.html#K4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="comput_1.html#V4" target="_self">homogeneous</a> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E10:82"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="funct_6.html#K6">:&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="comput_1.html#K5" target="_self">HFuncs</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


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