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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>

<b>set </b><font color="Maroon" title="c2">cT</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:97_1"/><i><font color="Green" title="E60">A1</font></i>: 
for <font color="Olive" title="b1">f</font> being  <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">f</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> holds <br/><font color="Olive" title="b1">f</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 ;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c1">T</font>;<br/><b>assume </b><a NAME="E2:97_1"/><i><font color="Green" title="E61">A2</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a>
 ;<br/><b>set </b><font color="Maroon" title="c4">fcT</font> = <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>;<br/><a NAME="E3:97_1"/>
<font color="Maroon" title="c3">f</font> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:97_1"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="pscomp_1.html#E2:97_1"><i><font color="Green" title="E61">A2</font></i></a></i>;<br/><a NAME="E4:97_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>
 <b>by </b><i><a class="ref" href="seq_2.html#D5" title="SEQ_2:def.5">SEQ_2:def 5</a></i>;<br/><a NAME="E5:97_1"/><b>then </b><i><font color="Green" title="E62">A3</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 <b>by </b><i><a class="ref" href="pscomp_1.html#D12" target="_self" title="PSCOMP_1:def.12">Def12</a></i>;<br/><b>assume </b><a NAME="E6:97_1"/>
not <font color="Maroon" title="c3">f</font> is <a href="pscomp_1.html#V7" title="PSCOMP_1:attr.7">with_max</a>
 ;<br/><a NAME="E7:97_1"/><b>then </b><i><font color="Green" title="E63">A4</font></i>: 
not <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> is <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">with_max</a>
 <b>by </b><i><a class="ref" href="pscomp_1.html#D14" target="_self" title="PSCOMP_1:def.14">Def14</a></i>;<br/><a NAME="E8:97_1"/><b>then </b><i><font color="Green" title="E64">A5</font></i>: 
not  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E5:97_1"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="pscomp_1.html#D3" target="_self" title="PSCOMP_1:def.3">Def3</a></i>;<br/><b>set </b><font color="Maroon" title="c5">b</font> =  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c6">bf</font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">-</a> <font color="Maroon" title="c3">f</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c7">g</font> =  <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">Inv</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c8">f'</font> = <font color="Maroon" title="c3">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/><b>reconsider </b><font color="Maroon" title="c9">bf'</font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">-</a> <font color="Maroon" title="c3">f</font></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/><div><i><font color="Green" title="E65">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_1_1"/>
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c10">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:97_1_1"/><i><font color="Green" title="E66">A7</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:97_1_1"/><i><font color="Green" title="E67">A8</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">x</font> = <font color="Maroon" title="c10">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="pscomp_1.html#E3:97_1_1"><i><font color="Green" title="E66">A7</font></i></a></i>;<br/><font color="Maroon" title="c9">bf'</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">x</font> = 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">-</a> <font color="Maroon" title="c8">f'</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">x</font></span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D17" target="_self" title="PSCOMP_1:def.17">Def17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D16" target="_self" title="PSCOMP_1:def.16">Def16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">x</font></span>)</span>

;<br/><b>hence </b><a NAME="E7:97_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pscomp_1.html#E8:97_1"><i><font color="Green" title="E64">A5</font></i></a>, <a class="txt" href="pscomp_1.html#E4:97_1_1"><i><font color="Green" title="E67">A8</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/></div><b>end;</b></div><b>set </b><font color="Maroon" title="c10">gcT</font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>;<br/><a NAME="E12:97_1"/>
 <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">-</a> <font color="Maroon" title="c3">f</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E2:97_1"><i><font color="Green" title="E61">A2</font></i></a>, <a class="ref" href="pscomp_1.html#T55" target="_self" title="PSCOMP_1:th.55">Th55</a></i>;<br/><a NAME="E13:97_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">-</a> <font color="Maroon" title="c3">f</font></span>)</span> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a>
 <b>by </b><i><a class="ref" href="pscomp_1.html#T56" target="_self" title="PSCOMP_1:th.56">Th56</a></i>;<br/><a NAME="E14:97_1"/><b>then </b><i><font color="Green" title="E66">A9</font></i>: 
 <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">Inv</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E11:97_1"><i><font color="Green" title="E65">A6</font></i></a>, <a class="ref" href="pscomp_1.html#T57" target="_self" title="PSCOMP_1:th.57">Th57</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_2"><b>now </b></a><div class="add"><a NAME="E1:97_1_2"/>
 <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">Inv</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> is <a href="seq_2.html#V3" title="SEQ_2:attr.3">bounded</a>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:97_1"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="pscomp_1.html#E14:97_1"><i><font color="Green" title="E66">A9</font></i></a></i>;<br/><a NAME="E2:97_1_2"/><b>then </b>
 <a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">Inv</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>
 <b>by </b><i><a class="ref" href="seq_2.html#D5" title="SEQ_2:def.5">SEQ_2:def 5</a></i>;<br/><a NAME="E3:97_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 <b>by </b><i><a class="ref" href="pscomp_1.html#D12" target="_self" title="PSCOMP_1:def.12">Def12</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">p</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E5:97_1_2"/><i><font color="Green" title="E67">A10</font></i>: 
for <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p</font>
 <b>by </b><i><a class="ref" href="seq_4.html#D1" title="SEQ_4:def.1">SEQ_4:def 1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:97_1_2_1"/>
( <font color="Maroon" title="c11">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 or <font color="Maroon" title="c11">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_1_2_1_1"><b>suppose </b></a><a NAME="E1:97_1_2_1_1"/><i><font color="Green" title="E68">A11</font></i>: 
<font color="Maroon" title="c11">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c12">1'</font> = 1 as  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/><b>take </b><font color="Maroon" title="c13">1'</font> = <font color="Maroon" title="c12">1'</font>;<br/><b>thus </b><a NAME="E3:97_1_2_1_1"/>
<font color="Maroon" title="c13">1'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><b>let </b><font color="Maroon" title="c14">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E4:97_1_2_1_1"/>
<font color="Maroon" title="c14">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 ;<br/><b>hence </b><a NAME="E5:97_1_2_1_1"/>
<font color="Maroon" title="c14">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">1'</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:97_1_2_1_1"><i><font color="Green" title="E68">A11</font></i></a>, <a class="txt" href="pscomp_1.html#E5:97_1_2"><i><font color="Green" title="E67">A10</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_1_2_1_2"><b>suppose </b></a><a NAME="E1:97_1_2_1_2"/><i><font color="Green" title="E68">A12</font></i>: 
<font color="Maroon" title="c11">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c12">p</font> = <font color="Maroon" title="c11">p</font>;<br/><b>thus </b><a NAME="E2:97_1_2_1_2"/>
<font color="Maroon" title="c12">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:97_1_2_1_2"><i><font color="Green" title="E68">A12</font></i></a></i>;<br/><b>thus </b><a NAME="E3:97_1_2_1_2"/>
for <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E5:97_1_2"><i><font color="Green" title="E67">A10</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c11">p</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E17:97_1"/><i><font color="Green" title="E67">A13</font></i>: 
<font color="Maroon" title="c11">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>and </b><br/><a NAME="E18:97_1"/><i><font color="Green" title="E68">A14</font></i>: 
for <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p</font>
 ;<br/><a NAME="E19:97_1"/>
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c11">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="pscomp_1.html#E17:97_1"><i><font color="Green" title="E67">A13</font></i></a>, <a class="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E21:97_1"/><i><font color="Green" title="E69">A15</font></i>: 
<font color="Maroon" title="c12">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <b>and </b><br/><a NAME="E22:97_1"/><i><font color="Green" title="E70">A16</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E5:97_1"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E24:97_1"/><i><font color="Green" title="E71">A17</font></i>: 
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )
 <b>and </b><br/><a NAME="E25:97_1"/><i><font color="Green" title="E72">A18</font></i>: 
<font color="Maroon" title="c12">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">x</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E21:97_1"><i><font color="Green" title="E69">A15</font></i></a>, <a class="ref" href="funct_2.html#T115" title="FUNCT_2:th.115">FUNCT_2:115</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">x</font> = <font color="Maroon" title="c13">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="pscomp_1.html#E24:97_1"><i><font color="Green" title="E71">A17</font></i></a></i>;<br/><a NAME="E27:97_1"/>
( <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span> )
 <b>by </b><i><a class="txt" href="pscomp_1.html#E5:97_1"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="pscomp_1.html#E7:97_1"><i><font color="Green" title="E63">A4</font></i></a>, <a class="txt" href="pscomp_1.html#E21:97_1"><i><font color="Green" title="E69">A15</font></i></a>, <a class="txt" href="pscomp_1.html#E25:97_1"><i><font color="Green" title="E72">A18</font></i></a>, <a class="ref" href="pscomp_1.html#D3" target="_self" title="PSCOMP_1:def.3">Def3</a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/><a NAME="E28:97_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E29:97_1"/><b>then </b><i><font color="Green" title="E73">A19</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><i><font color="Green" title="E74">A20</font></i>: <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">x</font> = 
1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">bf'</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">x</font></span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D18" target="_self" title="PSCOMP_1:def.18">Def18</a></i>
<br/>.= 
1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">-</a> <font color="Maroon" title="c8">f'</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D17" target="_self" title="PSCOMP_1:def.17">Def17</a></i>
<br/>.= 
1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D16" target="_self" title="PSCOMP_1:def.16">Def16</a></i>
<br/>.= 
1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span>

;<br/><a NAME="E31:97_1"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">Inv</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">+</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">-</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/><a NAME="E32:97_1"/><b>then </b>
1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E18:97_1"><i><font color="Green" title="E68">A14</font></i></a>, <a class="txt" href="pscomp_1.html#E30:97_1"><i><font color="Green" title="E74">A20</font></i></a></i>;<br/><a NAME="E33:97_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E29:97_1"><i><font color="Green" title="E73">A19</font></i></a>, <a class="ref" href="real_2.html#T178" title="REAL_2:th.178">REAL_2:178</a></i>;<br/><a NAME="E34:97_1"/><b>then </b>
1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c11">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E17:97_1"><i><font color="Green" title="E67">A13</font></i></a>, <a class="ref" href="real_2.html#T177" title="REAL_2:th.177">REAL_2:177</a></i>;<br/><a NAME="E35:97_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c11">p</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><b>hence </b><a NAME="E36:97_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pscomp_1.html#E22:97_1"><i><font color="Green" title="E70">A16</font></i></a>, <a class="txt" href="pscomp_1.html#E25:97_1"><i><font color="Green" title="E72">A18</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:97"/><i><font color="Green" title="E60">A21</font></i>: 
for <font color="Olive" title="b1">f</font> being  <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">f</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a> holds <br/><font color="Olive" title="b1">f</font> is <a href="pscomp_1.html#V7" title="PSCOMP_1:attr.7">with_max</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">f</font> be   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c1">T</font>;<br/>

<b>assume </b><a NAME="E3:97"/><i><font color="Green" title="E61">A22</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a>
 ;<br/>

<a NAME="E4:97"/><b>then </b>
<font color="Maroon" title="c3">f</font> is <a href="pscomp_1.html#V7" title="PSCOMP_1:attr.7">with_max</a>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E2:97"><i><font color="Green" title="E60">A21</font></i></a></i>;<br/>
<a NAME="E5:97"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> is <a href="pscomp_1.html#V1" title="PSCOMP_1:attr.1">with_max</a>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D14" target="_self" title="PSCOMP_1:def.14">Def14</a></i>;<br/>
<a NAME="E6:97"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D3" target="_self" title="PSCOMP_1:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E7:97"/>
<font color="Maroon" title="c3">f</font> is <a href="seq_2.html#V1" title="SEQ_2:attr.1">bounded_above</a>
 <b>by </b><i><a class="ref" href="pscomp_1.html#D12" target="_self" title="PSCOMP_1:def.12">Def12</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_2.html#D5" title="SEQ_2:def.5">SEQ_2:def 5</a><br/>

<a NAME="E8:97"/>
<font color="Maroon" title="c3">f</font> is <a href="pscomp_1.html#V8" title="PSCOMP_1:attr.8">with_min</a>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E2:97"><i><font color="Green" title="E60">A21</font></i></a>, <a class="txt" href="pscomp_1.html#E3:97"><i><font color="Green" title="E61">A22</font></i></a>, <a class="ref" href="pscomp_1.html#T60" target="_self" title="PSCOMP_1:th.60">Th60</a></i>;<br/>
<a NAME="E9:97"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> is <a href="pscomp_1.html#V2" title="PSCOMP_1:attr.2">with_min</a>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D15" target="_self" title="PSCOMP_1:def.15">Def15</a></i>;<br/>
<a NAME="E10:97"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D4" target="_self" title="PSCOMP_1:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E11:97"/>
<font color="Maroon" title="c3">f</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a>
 <b>by </b><i><a class="ref" href="pscomp_1.html#D11" target="_self" title="PSCOMP_1:def.11">Def11</a></i>;<br/>


</div>
