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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:15_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><b>assume </b><a NAME="E2:15_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>(  not 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K24">*</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> or  not <font color="Olive">b<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> or  not <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><a NAME="E3:15_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> or  not 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K24">*</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> or  not <font color="Olive">b<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> )
 ;<br/><b>hence </b><a NAME="E4:15_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="int_2.html#V1">prime</a>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:15_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="nat_4.html#E14"><i><font color="Green">Lemma14</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:15"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> iff ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="binop_2.html#K24">*</a> <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="int_2.html#V1">prime</a> holds <br/> not <font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Olive">b<sub>1</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="nat_4.html#E14"><i><font color="Green">Lemma14</font></i></a></i>;<br/>


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