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<b>let </b><font color="Maroon" title="c1">a</font> be   <a href="prvect_1.html#NM1" title="PRVECT_1:NM.1">Domain-Sequence</a>;<br/><b>let </b><font color="Maroon" title="c2">b</font> be    <a href="prvect_1.html#M1" title="PRVECT_1:mode.1">BinOps</a> of <font color="Maroon" title="c1">a</font>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">a</font>;<br/>





<b>assume </b><a NAME="E1:46"/><i><font color="Green" title="E30">A1</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">a</font> holds  <font color="Maroon" title="c3">f</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Olive" title="b1">i</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c2">b</font> <a href="prvect_1.html#K10" title="PRVECT_1:func.10">.</a> <font color="Olive" title="b1">i</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c1">a</font>;<br/><a NAME="E1:46_1"/><i><font color="Green" title="E31">A2</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span> <a href="prvect_1.html#K13" title="PRVECT_1:func.13">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">a</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span> <a href="prvect_1.html#K13" title="PRVECT_1:func.13">.</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">a</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">a</font> )
 <b>by </b><i><a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:46_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">a</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c6">i</font> = <font color="Maroon" title="c5">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">a</font> ;<br/><a NAME="E3:46_1_1"/>
( <span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span> <a href="prvect_1.html#K13" title="PRVECT_1:func.13">.</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prvect_1.html#K10" title="PRVECT_1:func.10">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c6">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c6">i</font></span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c6">i</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c2">b</font> <a href="prvect_1.html#K10" title="PRVECT_1:func.10">.</a> <font color="Maroon" title="c6">i</font> )
 <b>by </b><i><a class="txt" href="prvect_1.html#E1:46"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="prvect_1.html#D10" target="_self" title="PRVECT_1:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E4:46_1_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span> <a href="prvect_1.html#K13" title="PRVECT_1:func.13">.</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T11" title="BINOP_1:th.11">BINOP_1:11</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:46_1"/>
<span class="p1"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span> <a href="prvect_1.html#K13" title="PRVECT_1:func.13">.</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="prvect_1.html#E1:46_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:46_1_2"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">a</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c6">i</font> = <font color="Maroon" title="c5">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">a</font> ;<br/><a NAME="E3:46_1_2"/>
( <span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span> <a href="prvect_1.html#K13" title="PRVECT_1:func.13">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">f</font></span>)</span> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="prvect_1.html#K10" title="PRVECT_1:func.10">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c6">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c6">i</font></span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="prvect_1.html#K9" title="PRVECT_1:func.9">.</a> <font color="Maroon" title="c6">i</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c2">b</font> <a href="prvect_1.html#K10" title="PRVECT_1:func.10">.</a> <font color="Maroon" title="c6">i</font> )
 <b>by </b><i><a class="txt" href="prvect_1.html#E1:46"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="prvect_1.html#D10" target="_self" title="PRVECT_1:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E4:46_1_2"/>
<span class="p1">(<span class="default"><span class="p2"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span> <a href="prvect_1.html#K13" title="PRVECT_1:func.13">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T11" title="BINOP_1:th.11">BINOP_1:11</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:46_1"/>
<span class="p1"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span> <a href="prvect_1.html#K13" title="PRVECT_1:func.13">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="prvect_1.html#E1:46_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:46"/>
<font color="Maroon" title="c3">f</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <span class="p1"><a href="prvect_1.html#K14" title="PRVECT_1:func.14">[:</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="prvect_1.html#K14" title="PRVECT_1:func.14">:]</a></span>
 <b>by </b><i><a class="ref" href="binop_1.html#T11" title="BINOP_1:th.11">BINOP_1:11</a></i>;<br/>


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