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<span class="kw">let </span><font color="Maroon" title="c5">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">n</font> is  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  implies <font color="Maroon" title="c5">n</font> is  <a href="number08.html#V2" title="NUMBER08:attr.2">having_at_least_three_different_prime_divisors</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:137_1_1"/><span class="lab"><font color="Green" title="E103">A1</font></span>: 
<font color="Maroon" title="c5">n</font> is  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">n</font> is  <a href="number08.html#V2" title="NUMBER08:attr.2">having_at_least_three_different_prime_divisors</a> </span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c2">P2</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="number08.html#D8" target="_self" onmouseover="rs('number08/D8')" onmouseout="rh()">NUMBER08:def 8</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">q2</font>, <font color="Olive" title="b2">q3</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c2">P2</font>,<font color="Olive" title="b1">q2</font>,<font color="Olive" title="b2">q3</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_distinct</a>  &amp; <font color="Maroon" title="c2">P2</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Olive" title="b1">q2</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Olive" title="b2">q3</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">n</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c3">P3</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">q3</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Maroon" title="c2">P2</font>,<font color="Maroon" title="c3">P3</font>,<font color="Olive" title="b1">q3</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_distinct</a>  &amp; <font color="Maroon" title="c2">P2</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c3">P3</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Olive" title="b1">q3</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">n</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c4">P5</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">P2</font>,<font color="Maroon" title="c3">P3</font>,<font color="Maroon" title="c4">P5</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_distinct</a>  &amp; <font color="Maroon" title="c2">P2</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c3">P3</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c4">P5</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">n</font> )</span><br/>

<span class="kw">thus </span><a NAME="E2:137_1_1"/>
( <font color="Maroon" title="c2">P2</font>,<font color="Maroon" title="c3">P3</font>,<font color="Maroon" title="c4">P5</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_distinct</a>  &amp; <font color="Maroon" title="c2">P2</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c3">P3</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c4">P5</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:137_1_1"><span class="lab"><font color="Green" title="E103">A1</font></span></a>, <a class="ref" href="nat_d.html#T6" onmouseover="rs('nat_d/T6')" onmouseout="rh()">NAT_D:6</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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