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<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="coh_sp.html#K20" title="COH_SP:func.20">FuncsT</a> <font color="Maroon" title="c2">X</font> iff  ex <font color="Olive" title="b1">T1</font>, <font color="Olive" title="b2">T2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K17" title="COH_SP:func.17">TOL</a> <font color="Maroon" title="c2">X</font> st <br/>( ( <font color="Olive" title="b2">T2</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Olive" title="b1">T1</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; <font color="Maroon" title="c1">x</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">T1</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">T2</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> </span>)</span> ) )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">F</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">T</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b2">TT</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> </span>)</span></span>)</span> where <font color="Olive" title="b1">T</font>, <font color="Olive" title="b2">TT</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K17" title="COH_SP:func.17">TOL</a> <font color="Maroon" title="c2">X</font> : verum </span>}</span> ;<br/>
<span class="kw">thus </span><a NAME="E1:63"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="coh_sp.html#K20" title="COH_SP:func.20">FuncsT</a> <font color="Maroon" title="c2">X</font> implies  ex <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K17" title="COH_SP:func.17">TOL</a> <font color="Maroon" title="c2">X</font> st <br/>( ( <font color="Olive" title="b2">B</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Olive" title="b1">A</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; <font color="Maroon" title="c1">x</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">B</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> </span>)</span> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">T1</font>, <font color="Olive" title="b2">T2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K17" title="COH_SP:func.17">TOL</a> <font color="Maroon" title="c2">X</font> st <br/>( ( <font color="Olive" title="b2">T2</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Olive" title="b1">T1</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; <font color="Maroon" title="c1">x</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">T1</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">T2</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> </span>)</span> ) implies <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="coh_sp.html#K20" title="COH_SP:func.20">FuncsT</a> <font color="Maroon" title="c2">X</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:63_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="coh_sp.html#K20" title="COH_SP:func.20">FuncsT</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K17" title="COH_SP:func.17">TOL</a> <font color="Maroon" title="c2">X</font> st <br/>( ( <font color="Olive" title="b2">B</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Olive" title="b1">A</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; <font color="Maroon" title="c1">x</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">B</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> </span>)</span> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">C</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:63_1"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E4:63_1"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
<font color="Maroon" title="c4">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">T</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b2">TT</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> </span>)</span></span>)</span> where <font color="Olive" title="b1">T</font>, <font color="Olive" title="b2">TT</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K17" title="COH_SP:func.17">TOL</a> <font color="Maroon" title="c2">X</font> : verum </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">A</font>, <font color="Maroon" title="c6">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K17" title="COH_SP:func.17">TOL</a> <font color="Maroon" title="c2">X</font><span class="kw"> such that </span><br/><a NAME="E6:63_1"/><span class="lab"><font color="Green" title="E37">A3</font></span>: 
<font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_1"><span class="lab"><font color="Green" title="E36">A2</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">A</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">B</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K17" title="COH_SP:func.17">TOL</a> <font color="Maroon" title="c2">X</font> st <br/>( ( <font color="Olive" title="b1">B</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c5">A</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; <font color="Maroon" title="c1">x</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">B</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> </span>)</span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c6">B</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="c6">B</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c5">A</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; <font color="Maroon" title="c1">x</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> </span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E7:63_1"/>
( ( <font color="Maroon" title="c6">B</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c5">A</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; <font color="Maroon" title="c1">x</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> </span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E6:63_1"><span class="lab"><font color="Green" title="E37">A3</font></span></a>, <a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</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">given </span><font color="Maroon" title="c4">A</font>, <font color="Maroon" title="c5">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K17" title="COH_SP:func.17">TOL</a> <font color="Maroon" title="c2">X</font><span class="kw"> such that </span><a NAME="E3:63"/><span class="lab"><font color="Green" title="E35">A4</font></span>: 
( ( <font color="Maroon" title="c5">B</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c4">A</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; <font color="Maroon" title="c1">x</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> </span>)</span> )
 ; <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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="coh_sp.html#K20" title="COH_SP:func.20">FuncsT</a> <font color="Maroon" title="c2">X</font></span><br/>

<a NAME="E4:63"/><span class="lab"><font color="Green" title="E36">A5</font></span>: 
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">T</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b2">TT</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> </span>)</span></span>)</span> where <font color="Olive" title="b1">T</font>, <font color="Olive" title="b2">TT</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K17" title="COH_SP:func.17">TOL</a> <font color="Maroon" title="c2">X</font> : verum </span>}</span> 
 
;<br/>
<a NAME="E5:63"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:63"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="coh_sp.html#K20" title="COH_SP:func.20">FuncsT</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</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>
