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<b>defpred </b><font color="Maroon">S<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> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b2">g</font> );<br/>
<a NAME="E1:30_1_1"/><i><font color="Green" title="E12">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="vectmetr.html#K2" title="VECTMETR:func.2">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span>  ex <font color="Olive" title="b3">z</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="vectmetr.html#K2" title="VECTMETR:func.2">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="vectmetr.html#K2" title="VECTMETR:func.2">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon" title="c4">x1</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="vectmetr.html#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E2:30_1_1_1"/><i><font color="Green" title="E13">A2</font></i>: 
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font>
 <b>and </b><br/><a NAME="E3:30_1_1_1"/><i><font color="Green" title="E14">A3</font></i>: 
<font color="Maroon" title="c4">x1</font> is <a href="vectmetr.html#V3" title="VECTMETR:attr.3">isometric</a>
 <b>by </b><i><a class="ref" href="vectmetr.html#D4" target="_self" title="VECTMETR:def.4">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">y1</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="vectmetr.html#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="vectmetr.html#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E5:30_1_1_1"/><i><font color="Green" title="E15">A4</font></i>: 
<font color="Maroon" title="c5">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font>
 <b>and </b><br/><a NAME="E6:30_1_1_1"/><i><font color="Green" title="E16">A5</font></i>: 
<font color="Maroon" title="c5">y1</font> is <a href="vectmetr.html#V3" title="VECTMETR:attr.3">isometric</a>
 <b>by </b><i><a class="ref" href="vectmetr.html#D4" target="_self" title="VECTMETR:def.4">Def4</a></i>;<br/>
<a NAME="E7:30_1_1_1"/>
<font color="Maroon" title="c4">x1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">y1</font> is <a href="vectmetr.html#V3" title="VECTMETR:attr.3">isometric</a>
 
<b>by </b><i><a class="txt" href="vectmetr.html#E3:30_1_1_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="vectmetr.html#E6:30_1_1_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="vectmetr.html#T4" target="_self" title="VECTMETR:th.4">Th4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">z</font> = <font color="Maroon" title="c4">x1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">y1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="vectmetr.html#K2" title="VECTMETR:func.2">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="ref" href="vectmetr.html#D4" target="_self" title="VECTMETR:def.4">Def4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">z</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c4">x1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c5">y1</font>
;<br/>

<b>thus </b><a NAME="E9:30_1_1_1"/>
( <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c5">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">y1</font> )
 <b>by </b><i><a class="txt" href="vectmetr.html#E2:30_1_1_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="vectmetr.html#E5:30_1_1_1"><i><font color="Green" title="E15">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c2">o</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of  <a href="vectmetr.html#K2" title="VECTMETR:func.2">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E3:30_1_1"/><i><font color="Green" title="E13">A6</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="vectmetr.html#K2" title="VECTMETR:func.2">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Maroon" title="c2">o</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="vectmetr.html#E1:30_1_1"><i><font color="Green" title="E12">A1</font></i></a>);<br/></i>
<b>take </b><font color="Maroon" title="c3">G</font> =  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="vectmetr.html#K2" title="VECTMETR:func.2">ISOM</a> <span class="p2">(<span class="default"><a href="vectmetr.html#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">o</font> #);<br/>

<a NAME="E4:30_1_1"/>
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectmetr.html#K2" title="VECTMETR:func.2">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectmetr.html#K2" title="VECTMETR:func.2">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span> holds <br/>the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c3">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b2">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:30_1_1_2"/><i><font color="Green" title="E14">A7</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectmetr.html#K2" title="VECTMETR:func.2">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>and </b><br/><a NAME="E2:30_1_1_2"/><i><font color="Green" title="E15">A8</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectmetr.html#K2" title="VECTMETR:func.2">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon" title="c6">f1</font>, <font color="Maroon" title="c7">g1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:30_1_1_2"/><i><font color="Green" title="E16">A9</font></i>: 
<font color="Maroon" title="c6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font>
 <b>and </b><br/><a NAME="E5:30_1_1_2"/><i><font color="Green" title="E17">A10</font></i>: 
<font color="Maroon" title="c7">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font>
 <b>and </b><br/><a NAME="E6:30_1_1_2"/><i><font color="Green" title="E18">A11</font></i>: 
<font color="Maroon" title="c2">o</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">g1</font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E3:30_1_1"><i><font color="Green" title="E13">A6</font></i></a>, <a class="txt" href="vectmetr.html#E1:30_1_1_2"><i><font color="Green" title="E14">A7</font></i></a>, <a class="txt" href="vectmetr.html#E2:30_1_1_2"><i><font color="Green" title="E15">A8</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:30_1_1_2"/>
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c3">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">g</font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E4:30_1_1_2"><i><font color="Green" title="E16">A9</font></i></a>, <a class="txt" href="vectmetr.html#E5:30_1_1_2"><i><font color="Green" title="E17">A10</font></i></a>, <a class="txt" href="vectmetr.html#E6:30_1_1_2"><i><font color="Green" title="E18">A11</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:30_1_1"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectmetr.html#K2" title="VECTMETR:func.2">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp; ( for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectmetr.html#K2" title="VECTMETR:func.2">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="vectmetr.html#K2" title="VECTMETR:func.2">ISOM</a> <span class="p1">(<span class="default"><a href="vectmetr.html#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font></span>)</span> holds <br/>the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c3">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b2">g</font> ) )
 ;<br/>


</div>
