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<span class="kw">set </span><font color="Maroon" title="c5">D</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Olive" title="b1">lamb</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Inv</font>)</span>)</span> where <font color="Olive" title="b1">lamb</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Lamb</font> : <font color="Olive" title="b1">lamb</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">Lamb</font> </span> } </span> ;<br/>
<span class="kw">take </span><font color="Maroon" title="c6">U</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Olive" title="b1">lamb</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Inv</font>)</span>)</span> where <font color="Olive" title="b1">lamb</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Lamb</font> : <font color="Olive" title="b1">lamb</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">Lamb</font> </span> } </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">  for <font color="Olive" title="b1">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">U</font> iff  ex <font color="Olive" title="b2">lamb</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">lamb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Lamb</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Olive" title="b2">lamb</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Inv</font>) ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">U</font> iff  ex <font color="Olive" title="b1">lamb</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">lamb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Lamb</font> &amp; <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Olive" title="b1">lamb</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Inv</font>) ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:3_1_1"/>
( <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">U</font> implies  ex <font color="Olive" title="b1">lamb</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">lamb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Lamb</font> &amp; <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Olive" title="b1">lamb</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Inv</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">lamb</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">lamb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Lamb</font> &amp; <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Olive" title="b1">lamb</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Inv</font>) ) implies <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">U</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="3_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:3_1_1_1"/>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">U</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">lamb</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">lamb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Lamb</font> &amp; <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Olive" title="b1">lamb</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Inv</font>) )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">d</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:3_1_1_1"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
( <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c8">d</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Olive" title="b1">lamb</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Inv</font>)</span>)</span> where <font color="Olive" title="b1">lamb</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Lamb</font> : <font color="Olive" title="b1">lamb</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">Lamb</font> </span> } </span>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<a NAME="E4:3_1_1_1"/>
 ex <font color="Olive" title="b1">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Lamb</font> st <br/>( <font color="Maroon" title="c8">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Olive" title="b1">l</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Inv</font>) &amp; <font color="Olive" title="b1">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">Lamb</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_1_1_1"><span class="lab"><font color="Green" title="E3">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:3_1_1_1"/>
 ex <font color="Olive" title="b1">lamb</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">lamb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Lamb</font> &amp; <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Olive" title="b1">lamb</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Inv</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_1_1_1"><span class="lab"><font color="Green" title="E3">A1</font></span></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/>


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

<span class="kw">given </span><font color="Maroon" title="c8">lamb</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><a NAME="E3:3_1_1"/><span class="lab"><font color="Green" title="E3">A2</font></span>: 
( <font color="Maroon" title="c8">lamb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Lamb</font> &amp; <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c8">lamb</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Inv</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="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">U</font></span><br/>

<a NAME="E4:3_1_1"/>
 <a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Maroon" title="c8">lamb</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Inv</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="surreali.html#K2" title="SURREALI:func.2">divs</a> (<font color="Olive" title="b1">lamb</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Inv</font>)</span>)</span> where <font color="Olive" title="b1">lamb</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Lamb</font> : <font color="Olive" title="b1">lamb</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">Lamb</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_1_1"><span class="lab"><font color="Green" title="E3">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:3_1_1"/>
<font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_1_1"><span class="lab"><font color="Green" title="E3">A2</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</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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