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<b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="vectsp_1.html#V8">left_unital</a> <a href="vectsp_1.html#V9">Field-like</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/>

<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>, <a href="numbers.html#K5">NAT</a> ;<br/>
<a NAME="E2:37"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="group_1.html#K10">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>4</sub></font> &amp; ( <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> ) )
 
<b>by </b><i><a class="txt" href="int_3.html#E24"><i><font color="Green">Lemma21</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:37"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="int_3.html#V1" target="_self">Euclidian</a>
 <b>by </b><i><a class="ref" href="int_3.html#D9" target="_self">Def9</a></i>;<br/>


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