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<span class="kw">let </span><font color="Maroon" title="c1">b1</font>, <font color="Maroon" title="c2">b2</font>, <font color="Maroon" title="c3">a</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b2</font> iff  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">b2</font> )</span><br/>





<span class="kw">thus </span><a NAME="E1:6"/>
( <font color="Maroon" title="c1">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b2</font> implies  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">b2</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">b2</font> implies <font color="Maroon" title="c1">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b2</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:6_1"/>
<font color="Maroon" title="c1">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">b2</font></span><br/>

<a NAME="E2:6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E5">A1</font></span>: 
<font color="Maroon" title="c1">b1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">b1</font> or <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">b2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:6_1"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">b1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">b2</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">c</font> = <font color="Maroon" title="c4">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a></span>;<br/>
<a NAME="E5:6_1"/>
<font color="Maroon" title="c5">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">b1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="calcul_2.html#T1" target="_self" title="CALCUL_2:th.1">Th1</a></span>;<br/>
<a NAME="E6:6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A3</font></span>: 
<font color="Maroon" title="c5">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E7:6_1"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="calcul_2.html#T1" target="_self" title="CALCUL_2:th.1">Th1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:6_1"/>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:6_1"><span class="lab"><font color="Green" title="E7">A3</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/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:6"/><span class="lab"><font color="Green" title="E5">A4</font></span>: 
 <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">b1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">b2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:6_2"/>
<font color="Maroon" title="c1">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b2</font></span><br/><a NAME="E2:6_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">b1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_2.html#K2" title="CALCUL_2:func.2">seq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="calcul_2.html#T3" target="_self" title="CALCUL_2:th.3">Th3</a></span>;<br/><a NAME="E3:6_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">b1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6"><span class="lab"><font color="Green" title="E5">A4</font></span></a>, <a class="ref" href="calcul_2.html#T1" target="_self" title="CALCUL_2:th.1">Th1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:6_2"/>
<font color="Maroon" title="c1">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</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>
<span class="kw">hence </span><a NAME="E4:6"/>
<font color="Maroon" title="c1">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b2</font>
 ; <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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