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<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="binop_1.html#NM2">BinOp</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:3_1_1"/><i><font color="Green">E3</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K7">+</a> <font color="Olive">b<sub>2</sub></font> ) &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="binop_1.html#V2">associative</a> )
 <b>by </b><i><a class="txt" href="real_ns1.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
;<br/>

<b>thus </b><a NAME="E3:3_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K7">+</a> <font color="Olive">b<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="real_ns1.html#E2:3_1_1"><i><font color="Green">E3</font></i></a></i>;<br/>


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