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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_1_1_1"/>
( <font color="Maroon">r</font> is <a href="xxreal_0.html#V2">positive</a> or  not <font color="Maroon">r</font> is <a href="xxreal_0.html#V2">positive</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_1_1"><b>suppose </b></a><a NAME="E1:55_1_1_1_1"/>
<font color="Maroon">r</font> is <a href="xxreal_0.html#V2">positive</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as  <a href="xxreal_0.html#V2">positive</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/><a NAME="E3:55_1_1_1_1"/>
 <a href="toprealb.html#K7" target="_self">Tcircle</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> is <a href="borsuk_2.html#V1">arcwise_connected</a>
 ;<br/><b>hence </b><a NAME="E4:55_1_1_1_1"/>
 <a href="toprealb.html#K7" target="_self">Tcircle</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> is <a href="borsuk_2.html#V1">arcwise_connected</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_1_2"><b>suppose </b></a><a NAME="E1:55_1_1_1_2"/>
not <font color="Maroon">r</font> is <a href="xxreal_0.html#V2">positive</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as  non <a href="xxreal_0.html#V2">positive</a> non <a href="xxreal_0.html#V3">negative</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/><a NAME="E3:55_1_1_1_2"/>
 <a href="toprealb.html#K7" target="_self">Tcircle</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> is <a href="realset2.html#V3">trivial</a>
 ;<br/><b>hence </b><a NAME="E4:55_1_1_1_2"/>
 <a href="toprealb.html#K7" target="_self">Tcircle</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> is <a href="borsuk_2.html#V1">arcwise_connected</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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