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<b>set </b><font color="Maroon" title="c2">V</font> =  <a href="vectmetr.html#K4" title="VECTMETR:func.4">RLMSpace</a> <font color="Maroon" title="c1">n</font>;<br/>
<b>let </b><font color="Maroon" title="c3">G1</font>, <font color="Maroon" title="c4">G2</font> be  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">G1</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="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">G1</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> ) &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G2</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="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c4">G2</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> ) implies <font color="Maroon" title="c3">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:30_1_2"/><i><font color="Green" title="E12">A12</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">G1</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>
 <b>and </b><br/><a NAME="E2:30_1_2"/><i><font color="Green" title="E13">A13</font></i>: 
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="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">G1</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>
 <b>and </b><br/><a NAME="E3:30_1_2"/><i><font color="Green" title="E14">A14</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G2</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>
 <b>and </b><br/><a NAME="E4:30_1_2"/><i><font color="Green" title="E15">A15</font></i>: 
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="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c4">G2</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">g</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">G1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">G1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c4">G2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><br/><b>consider </b><font color="Maroon" title="c7">f1</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_2_1"/><i><font color="Green" title="E16">A16</font></i>: 
<font color="Maroon" title="c7">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font>
 <b>and </b><br/><a NAME="E3:30_1_2_1"/>
<font color="Maroon" title="c7">f1</font> is <a href="vectmetr.html#V3" title="VECTMETR:attr.3">isometric</a>
 <b>by </b><i><a class="txt" href="vectmetr.html#E1:30_1_2"><i><font color="Green" title="E12">A12</font></i></a>, <a class="ref" href="vectmetr.html#D4" target="_self" title="VECTMETR:def.4">Def4</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">g1</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_2_1"/><i><font color="Green" title="E17">A17</font></i>: 
<font color="Maroon" title="c8">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font>
 <b>and </b><br/><a NAME="E6:30_1_2_1"/>
<font color="Maroon" title="c8">g1</font> is <a href="vectmetr.html#V3" title="VECTMETR:attr.3">isometric</a>
 <b>by </b><i><a class="txt" href="vectmetr.html#E1:30_1_2"><i><font color="Green" title="E12">A12</font></i></a>, <a class="ref" href="vectmetr.html#D4" target="_self" title="VECTMETR:def.4">Def4</a></i>;<br/><b>thus </b>the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">G1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font> = 
<font color="Maroon" title="c7">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">g1</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E1:30_1_2"><i><font color="Green" title="E12">A12</font></i></a>, <a class="txt" href="vectmetr.html#E2:30_1_2"><i><font color="Green" title="E13">A13</font></i></a>, <a class="txt" href="vectmetr.html#E2:30_1_2_1"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="vectmetr.html#E5:30_1_2_1"><i><font color="Green" title="E17">A17</font></i></a></i>
<br/>.= 
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c4">G2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E1:30_1_2"><i><font color="Green" title="E12">A12</font></i></a>, <a class="txt" href="vectmetr.html#E4:30_1_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="vectmetr.html#E2:30_1_2_1"><i><font color="Green" title="E16">A16</font></i></a>, <a class="txt" href="vectmetr.html#E5:30_1_2_1"><i><font color="Green" title="E17">A17</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:30_1_2"/>
<font color="Maroon" title="c3">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G2</font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E1:30_1_2"><i><font color="Green" title="E12">A12</font></i></a>, <a class="txt" href="vectmetr.html#E3:30_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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