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<span class="kw">thus </span><a NAME="E1:15_3_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">-</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="relat_1.html#D19" onmouseover="rs('relat_1/D19')" onmouseout="rh()">RELAT_1:def 19</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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="15_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">-</a> <font color="Maroon" title="c3">q</font></span>)</span> or <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:15_3_1_1"/><span class="lab"><font color="Green" title="E3">A5</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">-</a> <font color="Maroon" title="c3">q</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"> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:15_3_1_1"/><span class="lab"><font color="Green" title="E4">A6</font></span>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">-</a> <font color="Maroon" title="c3">q</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">-</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_3_1_1"><span class="lab"><font color="Green" title="E3">A5</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_3_1_1"><span class="lab"><font color="Green" title="E4">A6</font></span></a></span>;<br/>
<a NAME="E5:15_3_1_1"/>
<font color="Maroon" title="c2">p</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">-</a> <font color="Maroon" title="c3">q</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">p</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">+</a> <span class="p1">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c3">q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D7" onmouseover="rs('polynom1/D7')" onmouseout="rh()">POLYNOM1:def 7</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom1.html#K7" title="POLYNOM1:func.7">-</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c6">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="vfunct_1.html#K5" title="VFUNCT_1:func.5">-</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#T15" onmouseover="rs('polynom1/T15')" onmouseout="rh()">POLYNOM1:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#T17" onmouseover="rs('polynom1/T17')" onmouseout="rh()">POLYNOM1:17</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:15_3_1_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_3_1_1"><span class="lab"><font color="Green" title="E4">A6</font></span></a>, <a class="ref" href="int_1.html#D2" onmouseover="rs('int_1/D2')" onmouseout="rh()">INT_1:def 2</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>


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