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<b>let </b><font color="Maroon">it1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="vectsp_1.html#V3">strict</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> , <font color="Maroon">it2</font> be  non <a href="struct_0.html#V3">empty</a> <a href="vectsp_1.html#V3">strict</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:187_1_2"/><i><font color="Green">E194</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">it1</font> iff <font color="Olive">x</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> )
 <b>and </b><br/><a NAME="E2:187_1_2"/><i><font color="Green">E195</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it1</font><br/> for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> holds <br/><font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="polynom1.html#K22" target="_self">+</a> <font color="Olive">q</font>
 <b>and </b><br/><a NAME="E3:187_1_2"/><i><font color="Green">E196</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it1</font><br/> for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> holds <br/><font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Olive">q</font>
 <b>and </b><br/><a NAME="E4:187_1_2"/><i><font color="Green">E199</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">it1</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26" target="_self">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E5:187_1_2"/><i><font color="Green">E201</font></i>: 
 <a href="vectsp_1.html#K2">1_</a> <font color="Maroon">it1</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K27" target="_self">1_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E6:187_1_2"/><i><font color="Green">E202</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">it2</font> iff <font color="Olive">x</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> )
 <b>and </b><br/><a NAME="E7:187_1_2"/><i><font color="Green">E204</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it2</font><br/> for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> holds <br/><font color="Olive">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="polynom1.html#K22" target="_self">+</a> <font color="Olive">q</font>
 <b>and </b><br/><a NAME="E8:187_1_2"/><i><font color="Green">E205</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it2</font><br/> for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> holds <br/><font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Olive">q</font>
 <b>and </b><br/><a NAME="E9:187_1_2"/><i><font color="Green">E206</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">it2</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K26" target="_self">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E10:187_1_2"/><i><font color="Green">E207</font></i>: 
 <a href="vectsp_1.html#K2">1_</a> <font color="Maroon">it2</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K27" target="_self">1_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 ;<br/>

<div><i><font color="Green">E208</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="187_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:187_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">it1</font>
 ;<br/><a NAME="E2:187_1_2_1_1"/><b>then </b>
<font color="Maroon">x</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T30" target="_self">Th30</a></i>;<br/><b>hence </b><a NAME="E3:187_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">it2</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:187_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">it2</font>
 ;<br/><a NAME="E3:187_1_2_1"/><b>then </b>
<font color="Maroon">x</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:187_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">it1</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T30" target="_self">Th30</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:187_1_2"/><b>then </b><i><font color="Green">E211</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">it1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">it2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="187_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it1</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it1</font>;<br/><b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font>, <font color="Maroon">b1</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it1</font> ;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">a</font>, <font color="Maroon">q</font> = <font color="Maroon">b</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> <b>by </b><i><a class="ref" href="polynom1.html#T30" target="_self">Th30</a></i>;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it2</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">a1'</font> = <font color="Maroon">a'</font>, <font color="Maroon">b1'</font> = <font color="Maroon">b'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it2</font> ;<br/><b>thus </b>the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">it1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> = 
<font color="Maroon">a1</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">b1</font>

<br/>.= 
<font color="Maroon">p</font> <a href="polynom1.html#K22" target="_self">+</a> <font color="Maroon">q</font>
<b>by </b><i><a class="ref" href="polynom1.html#T31" target="_self">Th31</a></i>
<br/>.= 
<font color="Maroon">a1'</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">b1'</font>
<b>by </b><i/>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">it2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>

;<br/></div><b>end;</b></div>
<a NAME="E14:187_1_2"/><b>then </b><i><font color="Green">E212</font></i>: 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">it1</font> <a href="funct_2.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">it2</font>
 
<b>by </b><i>, <a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>
<div><i><font color="Green">E213</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="187_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it1</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it1</font>;<br/><b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font>, <font color="Maroon">b1</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it1</font> ;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">a</font>, <font color="Maroon">q</font> = <font color="Maroon">b</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> <b>by </b><i><a class="ref" href="polynom1.html#T30" target="_self">Th30</a></i>;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it2</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">a1'</font> = <font color="Maroon">a'</font>, <font color="Maroon">b1'</font> = <font color="Maroon">b'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">it2</font> ;<br/><b>thus </b>the <a href="group_1.html#U1">mult</a> of <font color="Maroon">it1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> = 
<font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b1</font>

<br/>.= 
<font color="Maroon">p</font> <a href="polynom1.html#K28" target="_self">*'</a> <font color="Maroon">q</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a1'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b1'</font>
<b>by </b><i/>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">it2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>

;<br/></div><b>end;</b></div>
<a NAME="E16:187_1_2"/><i><font color="Green">E214</font></i>: 
the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">it1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">it2</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T32" target="_self">Th32</a>, </i>;<br/>
<a NAME="E17:187_1_2"/>
the <a href="vectsp_1.html#U1">unity</a> of <font color="Maroon">it1</font> <a href="hidden.html#R1">=</a> the <a href="vectsp_1.html#U1">unity</a> of <font color="Maroon">it2</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E18:187_1_2"/>
<font color="Maroon">it1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">it2</font>
 <b>by </b><i>, , , , <a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


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