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<div><span class="lab"><font color="Green" title="E45">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:55_1_1_1"/>
<font color="Maroon" title="c4">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  )</span><br/><a NAME="E2:55_1_1_1"/><span class="kw">then </span>
(  <a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>, <a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font> <a href="arytm_3.html#R1" title="ARYTM_3:pred.1">are_relative_prime</a>  &amp;  <a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T23" target="_self" title="ARYTM_3:th.23">Th23</a>, <a class="ref" href="arytm_3.html#T24" target="_self" title="ARYTM_3:th.24">Th24</a></span>;<br/><span class="kw">hence </span><a NAME="E3:55_1_1_1"/>
(  <a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_3.html#T39" target="_self" title="ARYTM_3:th.39">Th39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E2:55_1_1"/>
 <a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E3:55_1_1"/>
( ( <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  ) &amp; (  <a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  <a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  ) &amp; ( not <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; not  <a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_3.html#K4" title="ARYTM_3:func.4">RED</a> <font color="Maroon" title="c4">j</font>,<font color="Maroon" title="c3">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1_1"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="arytm_3.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="txt" href="arytm_3.html#E48"><span class="lab"><font color="Green" title="E33">Lm5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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